Essentials of Fuzzy Soft Multisets: Theory and Applications 9811927596, 9789811927591

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Essentials of Fuzzy Soft Multisets

Anjan Mukherjee · Ajoy Kanti Das

Essentials of Fuzzy Soft Multisets Theory and Applications

Anjan Mukherjee Department of Mathematics Tripura University Agartala, Tripura, India

Ajoy Kanti Das Bir Bikram Memorial College Agartala, Tripura, India

ISBN 978-981-19-2759-1 ISBN 978-981-19-2760-7 (eBook) https://doi.org/10.1007/978-981-19-2760-7 Mathematics Subject Classification: 03E72, 03E75, 03E02, 03E70, 03B52, 06D72, 54A40 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

1

Introduction to Fuzzy Sets, Soft Sets, Multisets and Their Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fuzzy Sets and Their Generalizations . . . . . . . . . . . . . . . . . . . . . . . 1.3 Soft Sets and Their Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Multisets and Their Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Useful Algorithms for Solving Decision-Making Problems . . . . . 1.5.1 Roy-Maji Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Feng’s Algorithm Using Choice Values . . . . . . . . . . . . . . 1.5.3 Jiang’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Salleh-Alkhazaleh Algorithm . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 9 13 13 13 14 14 15

2

Fuzzy Soft Multiset Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 A study on FSMSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Restricted Union and Restricted Intersection . . . . . . . . . . . . . . . . . 2.3 Extended Union and Extended Intersection . . . . . . . . . . . . . . . . . . . 2.4 AND Operator and OR Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Absolute and Null FSMSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 21 29 38 42

3

Relation on Fuzzy Soft Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fuzzy Soft Multi Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 De Morgan Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Associative Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Distributive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Inverse of Fuzzy Soft Multi Relation . . . . . . . . . . . . . . . . 3.4.2 Various Types of Fuzzy Soft Multi Relations . . . . . . . . . .

45 45 47 47 47 48 49

4

Topology on Fuzzy Soft Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Some Results on Absolute FSMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fuzzy Soft Multi Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fuzzy Soft Multi Basis and Sub Basis . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 61

v

vi

Contents

4.4 4.5

Neighbourhoods and Neighbourhood Systems . . . . . . . . . . . . . . . . Fuzzy Soft Multi Subspace Topology . . . . . . . . . . . . . . . . . . . . . . . .

62 68

5

Fuzzy Soft Multi Points and Their Sequences . . . . . . . . . . . . . . . . . . . . 5.1 Fuzzy Soft Multi Points and Their Properties . . . . . . . . . . . . . . . . . 5.2 Sequence of Fuzzy Soft Multi Points . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Convergence Sequence of Fuzzy Soft Multi Points . . . . . . . . . . . . 5.4 Cluster Fuzzy Soft Multi Point and Its Properties . . . . . . . . . . . . .

71 71 75 78 79

6

Fuzzy Soft Multi Compactness and Separation Axioms . . . . . . . . . . . 6.1 Fuzzy Soft Multi Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fuzzy Soft Multi Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . .

81 81 83

7

Fuzzy Soft Multiset Based Applications . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 An Adjustable Approach Based on Feng’s Algorithm . . . . . . . . . . 7.2 Algorithm 2 (Feng’s Algorithm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Algorithm 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Advantages of Our Algorithm (Algorithm 5) are as Follows . . . . 7.5 Modified Feng’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Algorithm 6 (Modified Feng’s Algorithm) . . . . . . . . . . . . . . . . . . . 7.7 Algorithm 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Application of FSMS Theory in Information Systems . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 87 88 92 92 93 94 97 99

8

Generalization of Fuzzy Soft Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Intuitionistic Fuzzy Soft Multi Set . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Results on IFSMSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 A Study on IFSMSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 102 106

9

Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Intuitionistic Fuzzy Soft Multi Relations . . . . . . . . . . . . . . . . . . . . . 9.2 Various Types of IFSMRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Inverse of IFSMRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Weakly-Reflexive and w-reflexive IFSMRs . . . . . . . . . . . . . . . . . . . 9.5 IFSM-Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Parameterized Topological Space Induced by an IFSM-Topological Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 IFSM-Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 IFSM-Points and Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Sequences of IFSMSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Application of Intuitionistic Fuzzy Soft Multisets . . . . . . . . . . . . . . . . . 10.1 An Adjustable Approach Based on IFSMSs . . . . . . . . . . . . . . . . . . 10.2 Algorithm 3 (Jiang’s Algorithm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Algorithm 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 114 116 121 124 128 129 131 134 139 139 140 140 147

Contents

10.5 Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Application of IFSMS Theory in Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

148 149 149 151

About the Authors

Anjan Mukherjee is former Pro Vice-Chancellor and former Professor at the Department of Mathematics, Tripura University, India. He has completed his Bachelor and Master degrees in mathematics from the University of Calcutta and obtained his Ph.D. from Tripura University. Professor Mukherjee has more than 35 years of experience in research and teaching. He has published more than 170 research papers in different national and international journals and conference proceedings and has delivered several invited talks. Professor Mukherjee is on the editorial board of the Universal Journal of Computational Mathematics and is associated with the Fuzzy and Rough Sets Association. He has presented his work at the University of Texas (USA), City College of New York (USA), AMC 5th Asian Mathematical Conference (Malaysia) and several universities in Bangladesh, Turkey and many other countries. Ajoy Kanti Das is Assistant Professor at the Department of Mathematics, Bir Bikram Memorial College, Maharaj Bir Bikram University, Agartala, Tripura. He has completed his Bachelor and Master degrees in mathematics from Tripura University and completed his Ph.D. from the Department of Mathematics, Tripura University. A gold medalist in both Bachelor and Master degrees for his excellent performance in mathematics, his current research interest is in fuzzy set theory, soft set and soft computing. With more than 10 years of experience in research and teaching, Dr. Das has published more than 40 research papers in different national and international journals and conference proceedings. He participated and presented his papers at more than 20 national and international conferences. He was also invited as a speaker at international conferences in several countries. He was awarded Junior Research Fellowship from the CSIR, UGC and DST to conduct his research work.

ix

Chapter 1

Introduction to Fuzzy Sets, Soft Sets, Multisets and Their Generalizations

1.1 Introduction Uncertainty, ambiguity and the representation of imperfect knowledge have been an issue in a variety of domains, including artificial intelligence, network and communication, signal processing, machine learning, computer science, information technology, medical science, economics and engineering. The problem is defined in a variety of ways. If a predicate has borderline cases, it is considered vague. Because there appears to be no specific height at which someone becomes tall, the predicate “is tall” is ambiguous. Alternatively, a predicate is sometimes described as vague if its application is ambiguous in some instances, such as when competent speakers of the language cannot agree on whether the predicate applies. The debate over whether a hotdog counts as a sandwich suggests that the term “sandwich” is ambiguous. The Sorites paradox is a common example of ambiguity. A conventional version of this conundrum includes a 2000-man succession of progressively taller men, beginning with a paradigm instance of a short man at one extreme and ending with a paradigm example of a tall man at the other extreme. • Base step: Man 1 is short. • Induction step: If man n is short, then man n + 1 is short. • Conclusion: Man 2000 is short. The Sorites paradox is a common example of ambiguity. A conventional version of this conundrum includes a 2000-man succession of progressively taller men, beginning with a paradigm instance of a short man at one extreme and ending with a paradigm example of a tall man at the other extreme. The No Sharp Boundaries thesis concerning vague predicates is based on this intuition, and it plays an important role in vagueness theories. The problem with vagueness is that it is difficult to articulate its specific type of indeterminacy.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_1

1

2

1 Introduction to Fuzzy Sets, Soft Sets, Multisets and Their Generalizations

In today’s world, there is a lot of ambiguity, inaccuracy, and vagueness. In reality, many of the situations we dealt with were ambiguous rather than specific. Classical approaches are not often effective when dealing with such a wide range of uncertain data, because numerous sorts of uncertainties are involved in these problems. There are numerous mathematical tools available to deal with ambiguity. Fuzzy set theory [1], intuitionistic fuzzy set theory [2], multiset theory [3], soft set theory [4] and soft multiset theory [5, 6] are a few examples. Information sciences, intelligent systems, machine learning, cybernetics, the smoothness of functions, game theory, operations research, measurement theory, probability theory and other fields have all used these theories. Since Zadeh [1] established the fuzzy set, several new methodologies and ideas for dealing with imprecision and uncertainty have been proposed. Fuzzy sets are used in a variety of fields, including databases, pattern recognition, neural systems, fuzzy modelling, medicine, economics and multicriteria decision-making (see [7–10]). Molodtsov [4] pioneered soft set theory as a universal mathematical method for modelling ambiguous concepts in 1999. The description of objects in soft set theory is not constrained, so researchers can choose the form of parameters they need, substantially simplifying the decision-making process and making it more efficient in the absence of partial knowledge. Although various mathematical tools for modelling uncertainties exist, such as probability theory, fuzzy set theory, rough set theory, interval valued fuzzy set theory and so on, each of these techniques has its own set of challenges. Furthermore, all of these strategies lack parameterization of the instruments, which means they can’t be used to solve problems, especially in areas like economics, the environment and social issues. Soft set theory is unusual, in that, it is unaffected by the aforementioned challenges. Maji et al. [11] went on to give some new definitions of soft sets as well as a detailed discussion of their use in decision-making situations. Ali et al. [12] proposed some new algebraic operations for soft sets based on the analysis of many operations on soft sets introduced in [13] and established that certain De Morgan’s law holds in soft set theory with respect to these new definitions. Maji et al. [14] defined fuzzy soft sets by combining soft and fuzzy sets, which have a lot of potential for solving decision-making problems. As a generalization of Molodtsov’s soft set, Alhazaymeh and others [5, 6, 15–17] provided the definition of a soft multi set and its basic operations such as complement, union and intersection, among others. Mukherjee [18, 19] recently investigated the topic of soft multi topological spaces in depth. They also looked into the concepts of soft multiset relative complement, soft multipoint, soft multi-open set, soft multi-closed set, soft multi basis, soft multi sub basis, neighbourhood and neighbourhood systems and soft multiset interior and closure. A parameterized family of topological spaces can be obtained from a soft multi topological space. Alkhazaleh and Salleh [20] developed the concept of fuzzy soft multiset theory in 2012 and used the Roy-Maji Algorithm [21] to apply it to decision-making situations. Maji et al. [13] were the first to propose using soft sets to solve decision-making problems, and in 2007, they published an application on fuzzy soft sets-based decision-making problems [21]. The Roy-Maji algorithm [21] was incorrect, according to Kong et al. [22], and they developed a corrected algorithm.

1.2 Fuzzy Sets and Their Generalizations

3

The validity of the Roy-Maji algorithm [21] was investigated by Feng et al. [23], who noted that the Roy-Maji algorithm [21] has several drawbacks. They also proposed leveraging thresholds and choice values to develop an adaptable method to fuzzy soft set-based decision-making challenges. Then, using level soft sets of intuitionistic fuzzy soft sets, Jiang et al. [24] summarize the customizable approach to deal with fuzzy soft sets-based decision-making and propose a movable way to deal with intuitionistic fuzzy soft sets-based decision-making. For our future research, we provide the basic notions of fuzzy sets, intuitionistic fuzzy set theory, soft sets, multisets and their generalization. We discuss various practical techniques for simulating real-world decision-making difficulties. We also provide all of the necessary concepts for understanding and defining the many concepts that will be discussed in this book.

1.2 Fuzzy Sets and Their Generalizations Fuzzy sets are useful mathematical tools for simulating various kinds of uncertainty. Zadeh [1] proposed the fuzzy set concept in 1965 as an extension of the classical notion of an ordinary (crisp) set. The membership of items in a set is evaluated in binary terms in classical set theory; according to a bivalent condition, an element either belongs to or does not belong to a set. Fuzzy set theory, on the other hand, allows for a gradual assessment of the membership of elements in a set described by a membership function with a value in the real unit interval [0, 1]. This idea has been intensively investigated by both mathematicians and computer scientists since its inception. Fuzzy logic, fuzzy neural networks, fuzzy automata, fuzzy control systems and other applications of fuzzy set theory have emerged over time. Definition 1.2.1 Let U [1] be a nonempty set. Then a fuzzy set Γ on U is a set having the form Γ = {(x, μΓ (x)) : x ∈ U}, where the function μΓ : U → [0, 1] is called the membership function and μΓ (x) represents the degree of membership of each element x ∈ U. If μΓ (x) = 1, ∀x ∈ U, then Γ becomes an ordinary (crisp) set. We denote the class of all fuzzy sets on U by FS(U). Definition 1.2.2 [1] L et Γ, Ψ ∈ FS(U). Then • Union of Γ and Ψ denoted by Γ

Γ

.

.

Ψ, is a fuzzy set defined by

Ψ = {(x, m(μΓ (x), μΨ (x))) : x ∈ U}

4

1 Introduction to Fuzzy Sets, Soft Sets, Multisets and Their Generalizations

• Intersection of Γ and Ψ, denoted by Γ

Γ

.

.

Ψ, is a fuzzy set defined by

Ψ = {(x, m(μΓ (x), μΨ (x))) : x ∈ U}

• Complement of Γ, denoted by Γ c , is a fuzzy set defined by

Γ c = {(x, 1 − μΓ (x)) : x ∈ U} • Γ is said to be a fuzzy subset of Ψ, denoted by Γ ⊆ Ψ if

μΓ (x) ≤ μΨ (x), ∀x ∈ U. Zadeh [1] defined the concept of a fuzzy relation on a set, and several authors have expanded on it. Definition 1.2.3 Let X, Y [25] be two universal sets, then σ = {((x, y), μσ (x, y)) : (x, y) ∈ X × Y } is called a fuzzy relation on X × Y, where X × Y denotes the Cartesian product of two crisp sets X and Y and μσ : X × Y → [0, 1] is a mapping. If a fuzzy relation σ on X×X satisfies the following conditions, we call it a “fuzzy equivalence relation” or “fuzzy similarity relation”: (i) Reflexive relation: μσ (x, x) = 1, ∀x ∈ X (ii) Symmetric relation: μσ (x, y) = μσ (y, x), ∀(x, y) ∈ X × X (iii) Transitive relation: . . μσ (x, z) ≥ m min[μσ (x, y), μσ (y, z)] , ∀(x, y), (y, z), (x, z) ∈ X × X. y

Following Zadeh’s [26] exposition of the fuzzy set notion, various researchers focused on the generalization of the fuzzy set concept. As a result, Atanassov [2] published an intuitionistic fuzzy set in 1986. Definition 1.2.4 Let U [2] be a nonempty set. Then an intuitionistic fuzzy set (briefly, IFS) Γ is an object having the form Γ = {x, μΓ (x), vΓ (x) : x ∈ U}, where the functions μΓ : U → [0, 1] and vΓ : U → [0, 1] are called membership function and non-membership function, respectively. μΓ (x) and vΓ (x) represent the degree of membership and the degree of non-membership, respectively, of each element x ∈ U and 0 ≤ μΓ (x) + vΓ (x) ≤ 1, for each x ∈ U. We denote the class of all IFSs on U by IFS(U).

1.3 Soft Sets and Their Generalizations

5

Definition 1.2.5 Let Γ, Ψ ∈ IFS(U ) [2]. Then • Union of Γ and Ψ, denoted by Γ Γ

.

.

Ψ, is an IFS defined by

Ψ = {x, m(μΓ (x), μΨ (x)), m(vΓ (x), vΨ (x)) : x ∈ U}

• Intersection of Γ and Ψ, denoted by Γ Γ

.

.

Ψ, is an IFS defined by

Ψ = {x, m(μΓ (x), μΨ (x)), m(vΓ (x), vΨ (x)) : x ∈ U}

• Complement of Γ, denoted by Γ c , is an IFS defined by Γ c = {x, vΓ (x), μΓ (x) : x ∈ U} • Γ is said to be an intuitionistic fuzzy subset (briefly, IF-subset) of Ψ, denoted by Γ ⊆ Ψ if μΓ (x) ≤ μΨ (x) and vΓ (x) ≥ vΨ (x), ∀x ∈ U.

1.3 Soft Sets and Their Generalizations Soft set theory research has accelerated because it may be easily applied to a variety of fields, including artificial intelligence, computer science, medical science, machine learning, information technology, economics, the environment and engineering. The practical application of soft computing techniques, particularly soft computing techniques for handling decision-making problems, has become increasingly important. Molodstov [4] pioneered the use of soft sets as a useful mathematical tool for analysing vagueness, uncertainty and undefined things in 1999. Definition 1.3.1 Let U [4] be a universe set and E be a set of parameters. Let P(U ) denote the power set of U and A ⊆ E. Then the pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P(U). In other words, the soft set is not a kind of set, but a parameterized family of subsets of U . For e ∈ A, F(e) ⊆ U may be considered as the set of e-approximate elements of the soft set (F, A). Example 1.3.2 Suppose [4] that there are six houses in the universe set given by U = {h1 , h2 , h3 , h4 , h5 , h6 } and A = {e1 , e2 , e3 , e4 , e5 } is the set of parameters, where e1 stands for the parameter “expensive”, e2 stands for the parameter “beautiful”, e3 stands for the parameter “wooden”, e4 stands for the parameter “cheap” and e5 stands for the parameter “in green surroundings”. In this case, to define soft set means to point out expensive houses, beautiful houses and so on. The soft set (F, A) may describe the “attractiveness of the houses”

6

1 Introduction to Fuzzy Sets, Soft Sets, Multisets and Their Generalizations

that Mr. X is going to buy. Suppose that F(e1 ) = {h2 , h4 }, F(e2 ) = {h1 , h3 }, F(e3 ) = {h3 , h4 , h5 }, F(e4 ) = {h1 , h3 , h5 }, F(e5 ) = {h1 }. Then the soft set (F, A) is a parameterized family {F(ei ) : i = 1, 2, 3, 4, 5} of subsets of U and gives us a collection of approximate descriptions of an object. Definition 1.3.3 For [4] two soft sets (F, A) and (G, B) over a common universe U, we say that (F, A) is a soft subset of (G, B) if (i) A ⊆ B (ii) F(e) ⊆ G(e), ∀e ∈ A. We denote this by (F, A) ⊆ (G, B). Definition 1.3.4 The [4] extended union of two soft sets (F, A) and (G, B) over a common universe U is the soft set (H, C), where C = A ∪ B and ∀e ∈ C. ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ F(e) ∪ G(e), if e ∈ A ∩ B We write Definition 1.3.5 The [4] extended intersection of two soft sets (F, A) and (G, B) over a common universe U is the soft set (H, C), where C = A ∪ B and ∀e ∈ C. ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ F(e) ∩ G(e), if e ∈ A ∩ B ˜ E (G, B) = (H, C). We write(F, A)∩ The concept of fuzzy soft sets was introduced by Maji et al. [14], which can be thought of as a fuzzy generalization of (crisp) soft sets. Definition 1.3.6 Let U [14] be a universe set, E be a set of parameters and A ⊆ E. Then the pair (F, A) is called a fuzzy soft set (FSS) over U , where F is a mapping given by F : A → F S(U ). It is easy to see that every (classical) soft set may be considered as a FSS. For e ∈ A, F(e) is a fuzzy subset of U and is called the fuzzy value set of the parameter ‘e’. Let us denote μF(e) (x) by the membership degree that object ‘x’ holds parameter ‘e’, where e ∈ A and x ∈ U . Then F(e) can be written as a fuzzy set, such that F(e) =

{( ) } x, μF(e) (x) : x ∈ U

Definition 1.3.7 The [14] union of two FSSs (F, A) and (G, B) over a common universe U is a FSS (H, C), where C = A ∪ B and ∀e ∈ C, x ∈ U. ⎧ if e ∈ A − B ⎨ μ F(e) (x), μ H (e) (x) = μG(e) (x), ) if e ∈ B − A ⎩ ( m μ F(e) (x), μG(e) (x) , if e ∈ A ∩ B

1.3 Soft Sets and Their Generalizations

7

We write (F, A) ∪(G, B) = (H, C). Definition 1.3.8 The [14] intersection of two FSSs (F, A) and (G, B) over a common universe U is a FSS (H, C), where C = A ∪ B and ∀e ∈ C, x ∈ U. ⎧ if e ∈ A − B ⎨ μ F(e) (x), μ H (e) (x) = μG(e) (x), ) if e ∈ B − A ⎩ ( m μ F(e) (x), μG(e) (x) , if e ∈ A ∩ B We write (F, A) ∩ (G, B) = (H, C). Definition 1.3.9 Let . = (F, A) [23] be a FSS over U , where A ⊆ E and E is a set of parameters. For t ∈ [0, 1], {the t-level soft set of }. is a crisp soft set L(. ; t) = (Ft , A) defined by Ft (e) = u ∈ U : μ F(e) (u) ≥ t , ∀e ∈ A. Definition 1.3.10 Let . = (F, A) [23] be a FSS over U, where A ⊆ E and E is the parameter set. Let λ : A → [0, 1] be a fuzzy set in A, which is called a threshold fuzzy set. The level soft set of the FSS . with { respect to the fuzzy set λ}is a crisp soft set L(. ; λ) = (Fλ , A) defined by Fλ (e) = u ∈ U : μ F(e) (u) ≥ λ(e) , ∀e ∈ A. Definition 1.3.11 Let . = (F, A) [23] be a FSS over a finite universe U, where A ⊆ E and E is the parameter set. The mid-threshold . of the FSS . define a fuzzy set mid A → [0, 1] by ∀e ∈ A, mid σ (e) = |U1 | u∈U μ F(e) (u) and the level soft set of . with respect to the mid-threshold fuzzy set mid . , namely L(. ; mid. ) is called the mid-level soft set of . . The mid-level decision rule will mean using the mid-threshold and considering the mid-level soft set in FSS-based decision-making. Definition 1.3.12 Let . = (F, A) [23] be a FSS over a finite universe U , where A ⊆ E and E is the parameter set. The max-threshold of the FSS . defines a fuzzy set maxσ : A → [0, 1] by ∀e ∈ A, maxσ (e) = maxu∈U μ F(e) (u) and the level soft set of . with respect to the max-threshold fuzzy set max max . , namely L(. ; max. ) is called the top-level soft set of . . The top-level decision rule will mean using the max-threshold and considering the top-level soft set in FSS-based decision-making. By introducing the concept of IFSs into the theory of soft sets, Maji et al. [14] proposed the concept of intuitionistic fuzzy soft sets as follows: Definition 1.3.13 Let U [14] be a universe set, E be a set of parameters and A ⊆ E. Then the pair (F, A) is called an intuitionistic fuzzy soft set (briefly, IFSS) over U , where F is a mapping given by F : A → IFS(U). For e ∈ A, F(e) is an intuitionistic fuzzy subset of U and is called the intuitionistic fuzzy value set of the parameter ‘e’. Let us denote μ F(e) (X) by the membership degree that object ‘x’ holds parameter ‘e’ and ν F(e) (X) by the membership degree that object ‘x’ doesn’t hold parameter ‘e’, where e ∈ A and x ∈ U. Then F(e) can be written as an IFS, such that

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1 Introduction to Fuzzy Sets, Soft Sets, Multisets and Their Generalizations

F(e) =

{(

) } X, μ F(e) (X), ν F(e) (X) : X ∈ U .

If ∀x ∈ U, νF(e) (x) = 1 − μ F(e) (x), then F(e) will generate to be a standard fuzzy set and then (F, A) will be generated to be a traditional fuzzy soft set. Definition 1.3.14 The [14] union of two IFSSs (F, A) and (G, B) over a common universe U is an IFSS (H, C), where C = A ∪ B and ∀e ∈ C, x ∈ U. ⎧ ⎨ μ F(e) (x), μ H (e) (x) = μG(e) (x), ) ⎩ ( m μ F(e) (x), μG(e) (x) , ⎧ ⎨ v F(e) (x), v H (e) (x) = vG(e) (x), ) ⎩ ( m v F(e) (x), vG(e) (x) ,

if e ∈ A − B if e ∈ B − A if e ∈ A ∩ B if e ∈ A − B if e ∈ B − A if e ∈ A ∩ B

We write (F, A) ∪ (G, B) = (H, C). Definition 1.3.15 The [14] intersection of two IFSSs (F, A) and (G, B) over a common universe U is an IFSS (H, C), where C = A ∪ B and ∀e ∈ C, x ∈ U. ⎧ ⎨ μ F(e) (x), μ H (e) (x) = μG(e) (x), ) ⎩ ( m μ F(e) (x), μG(e) (x) , ⎧ ⎨ v F(e) (x), v H (e) (x) = vG(e) (x), ) ⎩ ( m v F(e) (x), vG(e) (x) ,

if e ∈ A − B if e ∈ B − A if e ∈ A ∩ B if e ∈ A − B if e ∈ B − A if e ∈ A ∩ B

We write (F, A) ∩ (G, B) = (H, C). Definition 1.3.16 Let . = (F, A) [27] be an IFSS over a finite universe U, where A ⊆ E and E is the parameter soft set of . is ( set. For) s, t ∈ [0, 1], the (s, t)-level { s, t) = F , A defined by F = u ∈ U : μF(e) (u) ≥ s a crisp soft set L(., (e) (s,t) (s,t) )} and vF(e) (u) ≤ t , ∀e ∈ A. Definition 1.3.17 Let . = (F, A) [27] be an IFSS over a finite universe U, where A ⊆ E and E is the parameter set. Let λ : A → [0, 1]×[0, 1] be an IFS in A, which is called a threshold IFS. The level soft set of{ . with respect to λ is a crisp soft set L(. ; λ)} = (Fλ , A) defined by Fλ (e) = u ∈ U : μF(e) (u) ≥ μλ (e) and vF(e) (u) ≤ vλ (e) , ∀e ∈ A. According to the definition, there are four different types of special-level soft sets: Mid-level soft set L(. ;mid), Top–Bottom-level soft set L (. ; toptop) and Bottom-bottom-level soft set L(. ; bottombottom). Definition 1.3.18 Let . = (F, A) [27] be an IFSS over a finite universe U, where A ⊆ E and E is the parameter set. The mid-threshold or mid-level soft set of . is denoted by L(. ; mid) and defined as mid. : A → [0, 1]×[0, 1] by ∀e ∈ A,

1.4 Multisets and Their Generalizations

μmidw (e) =

9

1 . 1 . μ F(e) (u) and vmidw (e) = v F(e) (u). |U | u∈U |U | u∈U

Definition 1.3.19 Let . = (F, A) [27] be an IFSS over a finite universe U, where A ⊆ E and E is the parameter set. The top–bottom-threshold of . denoted by L(. ; topbottom) and defined as topbottom : A → [0, 1]×[0, 1] by ∀e ∈ A, μtopbottomw (e) = maxu∈U μF(e) (u) and vtopbottom (e) = minu∈U vF(e) (u). Definition 1.3.20 Let . = (F, A) [27] be an IFSS over a finite universe U, where A ⊆ E and E is the parameter set. The top-top-threshold of the IFSS . denoted by L( . ; toptop) and defined as toptop a . : A → [0, 1]×[0, 1] by ∀e ∈ A, μtoptopw (e) = maxu∈U μF(e) (u) and vtoptopw (e) = maxu∈U vF(e) (u). Definition 1.3.21 Let . = (F, A) [27] be an IFSS over a finite universe U, where A ⊆ E and E is the parameter set. The bottom-bottom-threshold of . define denoted by L( . ; bottombottom) and defined as bottombottom: A → [0, 1]×[0, 1] by ∀e ∈ . A, μbottombottomω (e) = minu∈U μF(e) (u) and

1.4 Multisets and Their Generalizations A set is a well-defined collection of distinct objects in classical set theory. The mathematical structure known as multiset is formed when repeated occurrences of any object are allowed in a set. A multiset is distinguished from a set in that each element has a multiplicity, which is a natural number that does not necessarily represent how many times it is a member of the multiset. A multiset is a collection of objects in which there is a lot of repetition of elements. Yager [19] created an elementary algebra of multisets and presented a formal definition for the multiset. Definition 1.4.1 Amultiset M [19] drawn from the set X is represented by a function Count M or C M defined as C M : X → N where N represents the set of non-negative integers. The word “multiset” is often shortened to “mset”. Definition 1.4.2 Let M1 and M2 [19] be two msets drawn from a set X . An mset M1 is a submset of M2 ( M1 ⊆ M2 ) if C M1 (x) ≤ C M2 (x) for all x ∈ X . Definition 1.4.3 The [19] union of two msets M1 and M2 drawn from a set X is a {mset M denoted} by M = M1 ∪ M2 such that for all x ∈ X, C M (x) = Max C M1 (x), C M2 (x) . Definition 1.4.4 The [19] intersection of two msets M1 and M2 drawn from a set X is{ a mset M denoted } by M = M1 ∩ M2 such that for all x ∈ X, C M (x) = Min C M1 (x), C M2 (x) . Definition 1.4.5 A [19] submset N of M is a whole submset of M with each element in N having full multiplicity as in M. i.e., C N (x) = C M (x) for every x in N.

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1 Introduction to Fuzzy Sets, Soft Sets, Multisets and Their Generalizations

Notation 1.4.6 Let mset from X with x appearing n times in M. It is { M [19] be a } kn k1 k2 denoted by M = x1 , x2 , . . . ., xn where M is an mset with X 1 appearing k1 times, X 2 appearing k2 times, and so on. Definition 1.4.7 Let [X ]m [19] denotes the set of all msets whose elements are in X such that no element in the mset occurs more than m times. For any mset M ∈ [X ]m , the support set of M denoted by M ∗ is a subset of X and M ∗ = {x ∈ X : C M (x) > 0}, i.e., M ∗ is an ordinary set. M ∗ is also called the root set of M. As a generalization of Molodtsov’s soft set, Alhazaymeh and others [5, 6, 15, 16, 28] introduced the definition of a soft multi set and investigated its basic operations such as complement, union and intersection. Definition{ 1.4.8 Let {U } i : i ∈ I } [6] be a collection of universes such that . ∩i∈I Ui = φ and let EUi : i ∈ I be a collection of sets of.parameters. Let U = i∈I P(Ui ), where P(Ui ) denotes the power set of Ui , E = i∈I EUi and A ⊆ E. A pair (F, A) is called a soft multiset over U, where F is a mapping given by F : A → U. Definition 1.4.9 A [6] soft multiset (F, A) over U is called a null soft multiset ˜ if for all a ∈ A, F(a) = φ. denoted by φ, Definition 1.4.10 A [6] soft multiset (F, A) over U is called an absolute soft multiset ˜ if for all a ∈ A, F(a) = U . denoted by A, ( ) Definition 1.4.11 For [6] any soft multiset (F, A), a pair eUi , j , FeUi , j is called a Ui -soft multipart, ∀eUi , j ∈ ak and FeUi , j ⊆ F(A) is an approximate value set, where ak ∈ A, k = {1, 2, 3, .., n}, i ∈ {1, 2, 3, .., m} and j ∈ {1, 2, 3, .., r}. Definition 1.4.12 For [6] two soft multisets (F, A) and (G, B) over U, (F, A) is called a soft multisubset of (G, B) if 1. A ⊆ B and ( ) ( ) 2. ∀eUi , j ∈ ak , eUi , j , FeUi , j ⊆ eUi , j , G eUi , j , Where ak ∈ A, k = {1, 2, 3, . . . , n}, i ∈ {1, 2, 3, . . . , m} and j ∈ {1, 2, 3, .., r}. This relation is denoted by (F, A) ⊆ (G, B). Definition 1.4.13 Two [6] soft multisets (F, A) and (G, B) over U are said to be equal if (F, A) is a soft multi subset of (G, B) and (G, B) is a soft multi subset of (F, A). Definition 1.4.14 Union [6] of two soft multisets (F, A) and (G, B) over U denoted by (F, A) ∼ (G, B) is the soft multiset (H, C), where C = A ∪ B and ∀e ∈ C, ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ F(e) ∪ G(e), if e ∈ A ∩ B

1.4 Multisets and Their Generalizations

11

Definition 1.4.15 Intersection [6] of two soft multisets (F, A) and (G, B) over U ˜ denoted by (F, A) ∩(G, B) is the soft multiset (H, C), where C = A ∪ B and ∀e ∈ C, ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ F(e) ∩ G(e), if e ∈ A ∩ B .m Definition 1.4.16 Let E = i=1 EUi [6] where .mEUi is a set of parameters.{The NOT by ¬E is defined by ¬E = set of E denoted i=1 ¬E Ui where ¬E Ui = ¬eUi , j = } not eUi , j , ∀i, j . Definition 1.4.17 The [6] complement of a soft multiset (F, A) over U is denoted by (F, A)c and defined by (F, A)c = (Fc , ¬A), where Fc : ¬A → U is a mapping given by Fc (α) = U − F(¬α), ∀α ∈ ¬A. Alkhazaleh and Salleh [20] developed the notion of fuzzy soft multiset theory in 2012 and investigated how these sets may be used in decision-making. Definition 1.4.18 Let { {Ui : i ∈ }I } [20] be a collection of universes, such that EUi : i ∈ I be a collection of sets of parameters. . ∩ U = φ and let Let U = i∈I i . F S(U where F S(U denotes the set of all fuzzy subsets of U , E = ), ) i i i i∈I i∈I E Ui and A ⊆ E. A pair (F, A) is called a FSMS (briefly, FSMS) over U, where F is a mapping given by F : A → U. Definition 1.4.19 For [20] ( ) any FSMS (F, A), where A ⊆ E and E is a set of parameters. A pair eUi , j , FeUi , j is called a Ui -FSMS-part of (F, A) over U, ∀eUi , j ∈ ak and FeUi , j ⊆ F(A) is an approximate value set, where ak ∈ A, k ∈ {1, 2, 3, .., n}, i ∈ {1, 2, 3, .., m} and j ∈ {1, 2, 3, .., r}. Definition 1.4.20 A [20] FSMS (F, A) over U is called fuzzy soft multi subset of FSMS (G, B) if (a) A ⊆ B and ( ) ( ) (b) ∀eUi , j ∈ ak , eUi , j , FeUi , j is a fuzzy subset of eUi , j , G eUi , j where ak ∈ A, k ∈ {1, 2, 3, .., n}, i ∈ {1, 2, 3, .., m} and j ∈ {1, 2, 3, .., r}. This relation is denoted by (F, A) . (G, B). Definition 1.4.21 Two [20] FSMSs (F, A) and (G, B) over U are called equal if (F, A) is a fuzzy soft multi subset of (G, B) and (G, B) is a fuzzy soft multi subset of (F, A). Definition 1.4.22 A [20] FSMS (F, A) over U is called a null FSMS, denoted by (F, A)φ , if all the FSMS parts of (F, A) equals φ. (F, A) over U is called an absolute FSMS, denoted Definition 1.4.23 ( A [20] FSMS ) ˙ by (F, A)U , if eUi , j , FeUi , j = Ui , ∀i.

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1 Introduction to Fuzzy Sets, Soft Sets, Multisets and Their Generalizations

Definition 1.4.24 Union [20] of two FSMSs (F, A) and (G, B) over U is a FSMS (H,D) and written as (F, A) ∼ (G, B) = (H, D), where D = A ∪ B and ∀e ∈ D, ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ ∪(F(e), G(e)), if e ∈ A ∩ B where

.

) ( (F(e), G(e)) = s FeUi , j , G eUi , j , ∀i ∈ {1, 2, 3, .., m} with s as an s-norm.

Proposition 1.4.25 If [20] (F, A), (G, B) and (H, D) are three FSMSs over U, then (a) (b) (c) (d)

( ) ˜ ˜ D) = ((F, A) ∼ (G, B))∪(H, D), (F, A) ∼ (G, B)∪(H, (F, A) ∼ (F, A) = (F, A), (F, A) ∼ (G, A)φ = (F, A), (F, A) ∼ (G, A)U = (G, A)U .

Definition 1.4.26 Intersection [20] of two FSMSs (F, A) and (G, B) over U is a FSMS (H, D) where D = A ∩ B and ∀e ∈ D, ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ ∩(F(e), G(e)), if e ∈ A ∩ B ( ) . where (F(e), G(e)) = t FeUi , j , G eUi , j , ∀i ∈ {1, 2, 3, . . . , m} with t as a t-norm and is written as (F, A) ∼ (G, B) = (H, C) Proposition 1.4.27 If [20] (F, A), (G, B) and (H, D) are three FSMSs over U, then (a) (b) (c) (d)

( ) ( ) ˜ (G, B)∩(H, ˜ ˜ ˜ D) = (F, A)∩(G, B) ∩(H, D), (F, A)∩ ˜ A) = (F, A), (F, A)∩(F, ˜ A)φ = (G, A)φ , (F, A)∩(G, ˜ A)U = (F, A). (F, A)∩(G,

Definition 1.4.28 The [20] complement of FSMS (F, A) over U is denoted by (F, A)c and is defined by (F, A)c = (Fc , A), where Fc : A → U is a mapping given by Fc (α) = c(F(α)), ∀α ∈ A, where c is fuzzy complement. Proposition 1.4.29 For [20] a FSMS (F, A) over U, (a) ((F, A)c )c = (F, A) (b) (F, A)cφ = (F, A)U (c) (F, A)Uc = (F, A)φ

1.5 Useful Algorithms for Solving Decision-Making Problems

13

1.5 Useful Algorithms for Solving Decision-Making Problems For our future research, we provide some techniques for solving decision-making difficulties.

1.5.1 Roy-Maji Algorithm To solve fuzzy soft set-based decision-making problems, Roy and Maji [21] employed the following approach. Algorithm 1 (Roy-Maji algorithm) Step 1. Input the fuzzy soft sets (F, A), (G, B) and (H, C). Step 2. Input the parameter set P as observed by the observer. Step 3. Compute the corresponding resultant fuzzy soft set (S, P) from the fuzzy soft sets (F, A), (G, B) and (H , C) and place it in tabular form. Step 4. Construct the comparison table of the fuzzy soft set (S, P) and compute ri and ti for oi , for all i. Step 5. Compute the score of oi , for all i. Step 6. The decision is Sk if Sk = maxi Si . Step 7. If k has more than one value, then any one of ok may be chosen.

1.5.2 Feng’s Algorithm Using Choice Values The following are the details of Feng’s Algorithm [23] for solving a fuzzy soft set decision-making problem: Algorithm 2 (Feng’s Algorithm) Step 1. Input the fuzzy soft set . = (F, A). Step 2. Input a threshold fuzzy set λ : A → [0, 1] (or select a threshold value t ∈ [0, 1] or select mid-level decision criterion or select top-level decision criterion) for solving the decision-making problems. Step 3. Obtain the level soft set L(. ; λ) of . with respect to the threshold fuzzy set λ (or L(. ; t) or L(. ; mid) or L(. ; max)). Step 4. Present the level soft set L(. ; λ) (or L(. ; t); or L(. ; mid); or L(. ; max)) as in tabular form and also, obtain the choice value Si of u i ∈ U, ∀i. Step 5. The final optimal decision to be select uk if Sk = maxi Si . Step 6. If k has more than one value, then any one of U k may be chosen.

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1 Introduction to Fuzzy Sets, Soft Sets, Multisets and Their Generalizations

1.5.3 Jiang’s Algorithm Jiang et al. [24] used level soft sets to implement the following adjustable approach to IFSS-based decision-making. Algorithm 3 (Jiang’s algorithm) Step 1. Input the (resultant) IFSS. = (F, A). Step 2. Input a threshold IFS λ : A → [0, 1]×[0, 1] (or give a threshold value (s, t) ∈ [0, 1]×[0, 1]; or choose the mid-level decision rule; or choose the top– bottom-level decision rule, or choose the top-top-level decision rule or choose the bottom-bottom-level decision rule) for decision-making. Step 3. Compute the LS-set L(. ; λ) of . with respect to the threshold IFS λ (or the (s,t)-LS-set L(. ; s, t); or the mid-LS-set L(. ; mid); or the top–bottom-LS-set L(. ; topbottom); or the top-top-LS-set L(. ; toptop), or the bottom-bottom-LS-set L(. ; bottombottom)). Step 4. Present the LS-set L(. ; λ) (or L(. ; s, t); or L(. ; mid); or L(. ; topbottom); or L(. ; toptop); or L(. ; bottombottom)) in tabular form and compute the choice value Si of u i ∈ U, ∀i. Step 5. The optimal decision is to select uk if Sk = maxi Si . Step 6. If k has more than one value then any one of u k may be chosen.

1.5.4 Salleh-Alkhazaleh Algorithm Salleh and Alkhazaleh [20] give thefollowing algorithm to solve FSMSs based decision-making problems. Algorithm 4 (Salleh-Alkhazaleh algorithm) Step 1. Input the FSMS(H,C) which is introduced by making any operations between (F,A) and (G,B). Step 2. Apply the Roy-Maji algorithm to the first FSMSpart in (H,C) to get the decision Sk1 . Step 3. Redefine the FSMS(H,C) by keeping all values in each row where Sk1 is maximum and replacing the values in the other rows by zero, to get (H,C)1 . Step 4. Apply the Roy-Maji algorithm to the second FSMSpart in (H,C)1 to get the decision Sk2 . Step 5. Redefine the FSMS(H,C)1 by keeping the first and second parts and applying the method in step (c) to the third part. Step 6. Apply the Roy-Maji algorithm to the third FSMSpart in (H,C)2 to get the decision Sk3 . Step 7. The decision is (Sk1, Sk2, Sk3 ).

References

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Chapter 2

Fuzzy Soft Multiset Theory

We have investigated existing basic conceptions and results on fuzzy soft multisets in this chapter (FSMSs). Our research has yielded some new findings. In this paper, we will define some new operations in FSMS theory, such as restricted union, restricted intersection, extended union, extended intersection, AND operator and OR operator, and show that the associative laws, distributive laws and De Morgan’s type of results apply to these newly defined operations in our way. These qualities can be applied to real-world situations such as decision-making, inventory control and so on.

2.1 A study on FSMSs In this section, we describe some additional FSMS operations and show that De Morgan’s type of results holds in FSMS theory with respect to these newly specified operations. Definition { 2.1.1 Let} {Ui : i ∈ I } be a collection of universes, such that .∩i∈I Ui = φ and let EUi : i ∈ I be a collection of sets of parameters. Let U = i∈I F S(Ui ), . where F S(Ui ) denotes the set of all fuzzy subsets of Ui , E = i∈I EUi and A ⊆ E. A pair (F, A) is called a FSMS over U, where F is a mapping given by F : A → U and defined by ∀e ∈ A (. F(e) =

u μ F(e) (u)

. : u ∈ Ui

) :i ∈I .

For illustration, we consider the following examples. Example 2.1.2 Let us consider three universes U1 = {h 1 , h 2 , h 3 , h 4 }, U2 = {c1 , c2 , c3 } and U3 = {v1 , v2 } be the sets of “houses,” “cars” and “hotels”, respectively. Suppose Mr. X has a budget to buy a house, a car and rent a venue to hold a wedding celebration. Let us consider a FSMS (F, A) which describes “houses,” © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_2

17

18

2 Fuzzy Soft Multiset Theory

“cars” and “hotels” that Mr. X is considering for accommodation purchase, transportation purchase { } and a venue to hold a wedding celebration, respectively. Let EU1 , EU2 , EU3 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎫ ⎪ ⎪ eU1 ,1 = modern, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = wooden, e ⎨ ⎬ U1 ,2 EU1 = eU1 ,3 = in green surroundings, , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ eU1 ,5 = in good repair, ⎧ ⎫ ⎪ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU2 ,2 = new model, ⎬ E U2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition, ⎧ ⎫ ⎪ ⎪ eU3 ,1 = expensive, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = cheap, e ⎨ U3 ,2 ⎬ EU3 = eU3,3 = in Kuala Lumpur, . ⎪ ⎪ ⎪ ⎪ ⎪ eU3 ,4 = majestic, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U3,5 = beautiful, Let U =

.3 i=1

F S(Ui ), E =

.3 i=1

EUi and A ⊆ E, such that

) )} { ( ( A = a1 = eU1 ,1 , eU2 ,1 , eU3 ,1 , a2 = eU11 ,2 , eU2 ,2 , eU3 ,2 . Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of choice parameters. Suppose that . ) (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , F(a1 ) = 0.2 0.4 0.8 0.5 0.8 0.5 0.4 0.8 0.7 . { ) (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , . F(a2 ) = 0.7 0.7 1 0.8 0.8 0.6 0.3 0.5 0.4 Then FSMS (F, A) can be written as . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , (F, A) = a1 , 0.2 0.4 0.8 0.5 0.8 0.5 0.4 0.8 0.7 . { ) . (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , } , (a2 , 0.7 0.7 1 0.8 0.8 0.6 0.3 0.5 0.4

2.1 A study on FSMSs

19

FSMS as in tabular form. Example 2.1.3 Let us consider three universes U1 = {h 1 , h 2 , h 3 , h 4 , h 5 }, U2 = {c1 , c2 , c3 , c4 } and U3 = {v1 , v2 , v3 } be the sets of “houses,” “cars” and “hotels”, respectively. Suppose Mr. X has a budget to buy a house, a car and rent a venue to hold a wedding celebration. Let us consider a FSMS (F, A) which describes “houses,” “cars” and “hotels” that Mr. X is considering for accommodation purchase, transportation purchase and a venue to hold a wedding celebration, respectively. Let { } EU1 , EU2 , EU3 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎫ ⎪ ⎪ eU1 ,1 = modern, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,2 = wooden, ⎨ ⎬ EU1 = eU1 ,3 = in green surroundings, , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ eU1 ,5 = in good repair, ⎧ ⎫ ⎪ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU2 ,2 = new model, ⎬ , E U2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition, ⎧ ⎫ ⎪ ⎪ eU3 ,1 = expensive, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU3 ,2 = cheap, ⎬ EU3 = eU3,3 = in Kuala Lumpur, . ⎪ ⎪ ⎪ ⎪ ⎪ eU3 ,4 = majestic, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ = beautiful, U3,5 Let U =

.3 i=1

F S(Ui ), E =

.3 i=1

EUi and A ⊆ E, such that

) ) ) { ( ( ( A = a1 = eU1 ,1 , eU2 ,1 , eU3 ,1 , a2 = eU1 ,4 , eU2 ,2 , eU3 ,3 , a3 = eU1 ,2 , eU2 ,3 , eU3 ,1 , ) ) )} ( ( ( a4 = eU1 ,3 , eU2 ,3 , eU3 ,5 , a5 = eU1 ,5 , eU2 ,4 , eU3 ,4 , a6 = eU1 ,1 , eU2 ,5 , eU3 ,2 . Assume Mr. X wants to select things from a set of given objects based on a set of choice parameters. Then, as shown in Table 2.1, FSMS (F, A) can be expressed. Definition 2.1.4 For any FSMS (F, A), where A ⊆ E and E are a set of parameters, a Ui − FSMS part of (F, A) over U, is denoted by (F, A)i and defined as ∀e ∈ A . F(e)i =

u μ F(e) (u)

. : u ∈ Ui .

For illustration, we consider the following examples.

20

2 Fuzzy Soft Multiset Theory

Table 2.1 The tabular representation of the FSMS (F, A) Ui h1 U1

U2

a1

a2

a3

a4

a5

a6

0.8

0.6

0.5

0.4

0.7

0.8

h2

0.4

0.5

0.7

0.5

0.5

0.4

h3

0.9

0

1

0.1

1

0.9

h4

0.4

0.4

0.4

0

0.4

0.4

h5

0.1

1

0.8

0.8

1

0.1

c1

0.8

0.9

0.6

0.6

0

0.7

c2

1

0.6

0.1

0.1

0.1

0.1

c3

0.8

1

0

0

0.7

0.9

c4

1

0

1

1

1

0.1

v1

0.8

0.6

0.8

1

1

0.1

v2

0.7

0.6

0.7

0.7

0.8

0.9

v3

0.6

0

0.6

0

0

0.1

Example 2.1.5 If we consider the FSMS (F, A) as in Example 2.1.2, then, we have . . .. .. h1 h2 h3 h4 h1 h2 h3 h4 , , , , , , , , (F, A)1 = 0.2 0.4 0.8 0.5 0.7 0.7 1 0.8 {{ c c2 c3 } { c1 c2 c3 }} 1 , , , , , , (F, A)2 = 0.5 } 0.4 0.8 }}0.6 0.3 { {{ 0.8 v1 v2 v1 v2 , , , . (F, A)3 = 0.8 0.7 0.5 0.4 Example 2.1.6 If we consider the FSMS (F, A) as in Table 2.1, then the Ui -FSMS part, i 1, 1, 2, 3 of (F, A) can be represented as follows (Tables 2.2, 2.3 and 2.4). Definition 2.1.7 A FSMS (F, A) over U is called a fuzzy soft multi subset of a FSMS (G, B) and denoted by (F, A) ⊆ (G, B). If (a) A ⊆ B and (b) ∀e ∈ A, F(e) ⊆ G(e) ⇐⇒ μ F(e) (u) ≤ μG(e) (u), ∀u ∈ Ui , i ∈ I

Table 2.2 The tabular representation of the U1 —FSMS part of (F, A) U1

a1

a2

a3

a4

a5

a6

h1

0.8

0.6

0.5

0.4

0.7

0.8

h2

0.4

0.5

0.7

0.5

0.5

0.4

h3

0.9

0

1

0.1

1

0.9

h4

0.4

0.4

0.4

0

0.4

0.4

h5

0.1

1

0.8

0.8

1

0.1

2.2 Restricted Union and Restricted Intersection

21

Table 2.3 The tabular representation of the U2 − FSMS part of (F, A) U2

a1

a2

a3

a4

a5

a6

c1

0.8

0.9

0.6

0.6

0

0.7

c2

1

0.6

0.1

0.1

0.1

0.1

c3

0.8

1

0

0

0.7

0.9

c4

1

0

1

1

1

0.1

Table 2.4 The tabular representation of the U3 —FSMS part of (F, A) U3

a1

a2

a3

a4

a5

a6

v1

0.8

0.6

0.8

1

1

0.1

v2

0.7

0.6

0.7

0.7

0.8

0.9

v3

0.6

0

0.6

0

0

0.1

Definition 2.1.8 The complement of a FSMS (F, A) over U is denoted by (F, A)c c c and is defined ) ∀e ∈ A ({by (F, A) = (F ,}A), where u c F (e) = 1−μ F(e) (u) : u ∈ Ui : i ∈ I .

2.2 Restricted Union and Restricted Intersection Definition 2.2.1 The restricted union of two FSMSs (F, A) and (G, B) over U is a B and ∀e ∈ C FSMS (H, C), where C = A ∩ ({ } ) U u : i ∈ I and written H (e) = (F(e), G(e)) = : u ∈ U i max{μ F(e) (u),μG(e) (u)} (F, A)∼ R (G, B) = (H, C) Definition 2.2.2 The restricted intersection of two FSMSs (F, A) and (G, B) over U is a FSMS (H, D), where D ({ = A ∩ B and ∀e ∈ D, } ) . u : i ∈ I and is written as H (e) = (F(e), G(e)) = m {μ (u),μ : u ∈ U i F(e) G(e) (u)} . ∼ (F, A) R (G, B) = (H, D). Example 2.2.3{Let us consider there are two universes U1 = {h 1 , h 2 , h 3 }, U2 = } {c1 , c2 } and let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎫ ⎪ ⎪ eU1,1 = modern, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,2 = wooden, ⎨ ⎬ EU1 = eU1 ,3 = in green surroundings, , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ eU1 ,5 = in good repair,

22

2 Fuzzy Soft Multiset Theory

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

E U2

eU2 ,1 = beautiful, eUU2 ,2 = new model, = eU2 ,3 = sporty, ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎩ eU2 ,5 = in good condition,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.2 .2 Let U = i=1 F S(Ui ), E = i=1 E and ) ) )} { ( ( Ui ( A = {e1 = (eU1 ,1 , eU2 ,1 ), e2 = (eU1 ,2 , eU2 ,2 ), e3 = (eU1 ,3 , eU2 ,3 )}, B = e1 = eU1 ,1 , eU2 ,1 , e2 = eU1 ,2 , eU2 ,2 , e4 = eU1 ,3 , eU2 ,4 . Suppose that . )) .( (. h 1 h 2 h 3 { c1 c2 } , , , , , (F, A) = e1 , 0.2 0.4 0.8 0.8 0.5 ( (. . { )) c1 c2 } h1 h2 h3 e2 , , , , , , 0.7 0.7 1 0.8 0.6 ( (. . { )). c1 c2 } h1 h2 h3 e3 , , , , , , 0.9 0.7 1 0.3 0.5 . { )) .( (. c1 c2 } h1 h2 h3 , , , , , (G, B) = e1 , 0.3 0.3 0.7 0.7 0.6 ( (. . { )) c1 c2 } h1 h2 h3 e2 , , , , , , 0.5 0.8 1 0.4 0.6 . { ) . (. c1 c2 } h1 h2 h3 , , , , } , (e4 , 0.8 0.9 1 0.5 0.2 Then we have A)U˜ R( (G,({B) }) } { c1 c2 }))} {( ({ h 1 h 2 h 3 } { c1 (F, c2 h1 h2 h3 , , e2 , 0.7 , , 0.4 , 0.8 , 0.8 , 0.6 , 0.8 , 1 , 0.8 , = e1 , 0.3 {( ({ h 1 h 2 h 3 } { c1 c2 })) ( ({ h 10.6 h 2 h 3 } { c1 c2 })} ˜ . (ii) (F, A)∩ R (G, B) = e1 , 0.2 , 0.3 , 0.7 , 0.7 , 0.5 , e2 , 0.5 , 0.7 , 1 , 0.4 , 0.6 (i)

Proposition 2.2.4 (Associative Laws) Let (F, A), (G, B) and (H, C) are three FSMSs over U, then we have the following properties: ( ) ( ) ˜ R (H, C) = (F, A)∩˜ R (G, B) ∩˜ R (H, C) [i] (F, A)∩˜ R (G, B)∩ ( ) ( ) ˜ R (H, C) = (F, A)∪˜ R (G, B) U˜ R (H, C) [ii] (F, A)∪˜ R (G, B)∪

2.2 Restricted Union and Restricted Intersection

23

Proof [i] Assume that (G, B)∩˜ R (H, ({ C) = (I, D), where D = B }∩ C and )∀e ∈ D, u I (e) = G(e) ∩ H (e) = m {μ (u),μ : u ∈ Ui : i ∈ I . G(e) ( ) H (e) (u)} ˜ R (G, B)∩˜ R (H, C) = (F, A)∩˜ R (I, D), we suppose that Since (F, A)∩ ˜ (F, A)∩ R (I, D) = (K ,({ M) where M = A} ∩ D =) A ∩ B ∩ C and ∀e ∈ M, K (e) = F(e) ∩ I (e) = μ K (e)u (u) : u ∈ Ui : i ∈ I , where { } μ K (e) (u) =m μ F(e) (u), μ I (e) (u) { { }} =m μ F(e) (u), m μG(e) (u), μ H (e) (u) { } =m μ F(e) (u), μG(e) (u), μ H (e) (u) . ˜ R (G, B) = (J, N ), where N = A ∩ B and ∀e ∈ N , Suppose that (F, A)∩ } ) ({ u : i ∈ I . J (e) = F(e) ∩ G(e) = m {μ (u),μ : u ∈ U i F(e) G(e) (u)} ( ) ˜ R (G, B) ∩ ˜ R (H, C) = (J, N )∩˜ R (H, C), we suppose that Since (F, A)∩ ˜ (J, N )∩ R (H, C) = (O, N ∩ C), where. ({ ∀e ∈ N ∩ }C = ) A ∩ B ∩ C, O(e) = J (e) ∩ H (e) = u : u ∈ Ui : i ∈ I , μ O(e) (u) where { } μ O(e) (u) =m μ J (e) (u), μ H (e) (u) { { } } =m m μ F(e) (u), μG(e) (u) , μ H (e) (u) { } =m μ F(e) (u), μG(e) (u), μ H (e) (u) =μ K (e) (u). Consequently, K( and O are the same ) operators. ( ) ˜ R (H, C). Thus, (F, A)∩˜ R (G, B)∩˜ R (H, C) = (F, A)∩˜ R (G, B) ∩ [ii] Assume that (G, B)∼ ˜ R (H,({ C) = (I, D), where D = B }∩ C and)∀e ∈ D, u I (e) = G(e) ∪ H (e) = m {μ (u),μ : u ∈ Ui : i ∈ I . (u)} G(e)

(

H (e)

)

Since (F, A)U˜ R (G, B)U˜ R (H, C) = (F, A)U˜ R (I, D), we suppose that (F, A)U˜ R (I, D) = (K , M), where M = A ∩ D = A ∩ B ∩ C and ∀e ∈ M, where (. K (e) = F(e) ∪ I (e) =

u μ K (e) (u)

. : u ∈ Ui

{ } μ K (e) (u) =m μ F(e) (u), μ I (e) (u)

) :i ∈I

24

2 Fuzzy Soft Multiset Theory

{ { }} =m μ F(e) (u), m μG(e) (u), μ H (e) (u) { } =m μ F(e) (u), μG(e) (u), μ H (e) (u) . Suppose that (F, A)U˜ R (G, B) = (J, N ), where N = A ∩ B and ∀e ∈ N , (. J (e) = F(e) ∪ G(e) =

u { } : u ∈ Ui m μ F(e) (u), μG(e) (u)

.

) :i ∈I .

( ) Since (F, A)U˜ R (G, B) U˜ R (H, C) = (J, N )U˜ R (H, C), we suppose that (J, N )∪˜ R (H, C) = (O, N ∩ C) where ∀e ∈ N ∩ C = A ∩ B ∩ C, (. O(e) = J (e) ∪ H (e) =

u μ O(e) (u)

. : u ∈ Ui

) :i ∈I ,

where { } μ O(e) (u) =m μ J (e) (u), μ H (e) (u) { { }} =m μ F(e) (u), m μG(e) (u), μ H (e) (u) { } =m μ F(e) (u), μG(e) (u), μ H (e) (u) =μK(e) (u). Consequently, K(and O are the same ) operators. ( ) Thus, (F, A)U˜ R (G, B)U˜ R (H, C) = (F, A)U˜ R (G, B) ∼ R (H, C). Proposition 2.2.5 (Distributive Laws) Let (F, A), (G, B) and (H, C) are three FSMSs over U, then we have the following properties: ) ( ) ( ) ( ˜ R (F, A)U˜ R (H, C) (1) (F, A)∪˜ R (G, B)∩˜ R (H, C) = (F, A)U˜ R (G, B) Ω ( ) ( ) ( ) ˜ R (H, C) (2) (F, A)∩˜ R (G, B)∪˜ R (H, C) = (F, A)∩˜ R (G, B) ∪˜ R (F, A)∩ Proof (1) Assume that (G, B)∩˜ R (H, C) = (I, D), where D = B ∩ C and ∀e ∈ D, I (e) } ) ({ = G(e) ∩ H (e) u : i ∈ I . = m {μ (u),μ : u ∈ U i G(e) H (e) (u)} ( ) ˜ Since (F, A)U R (G, B)∩˜ R (H, C) = (F, A)U˜ R (I, D), we suppose that (F, A)U˜ R (I, D) = (K , M) where M = A ∩ D = A ∩ B ∩ C and ∀e ∈ M . ) (. u : u ∈ Ui : i ∈ I , K (e) = F(e) ∪ I (e) = μ K (e) (u) where

2.2 Restricted Union and Restricted Intersection

25

{ } μ K (e) (u) = m μ F(e){(u), μ I (e) (u) }} { (u), m μ (u), μ (u) = m μ F(e) G(e) H (e) } { }} { { = m m μ F(e) (u), μG(e) (u) , m μ F(e) (u), μ H (e) (u) . Suppose that (F, A)∼ R (G, B) = (J, N ), where N = A ∩ B and ∀e ∈ N, (. J (e) = F(e) ∪ G(e) =

u { } : u ∈ Ui m μ F(e) (u), μG(e) (u)

.

) :i ∈I .

Again, let (F, A)∼ R (H, C) = (S, T ), where T = A ∩ C and ∀e ∈ T , (. S(e) = F(e) ∪ H (e) =

u { } : u ∈ Ui m μ F(e) (u), μ H (e) (u)

.

) :i ∈I .

( ) ( ) ˜ R (S, T ), Since (F, A)U˜ R (G, B) ∩˜ R (F, A)U˜ R (H, C) = (J, N )∩ ˜ R (S, T ) = (O, N ∩ T ), where ∀e ∈ N ∩ T = we suppose that (J, N )Ω A ∩ B ∩ C, . ) (. u : u ∈ Ui : i ∈ I , O(e) = J (e) ∩ S(e) = μ O(e) (u) where { } μ O(e) (u) =m μ J (e) (u), μ S(e) (u) { { } { }} =m m μ F(e) (u), μG(e) (u) , m μ F(e) (u), μ H (e) (u) =μ K (e) (u). Consequently, K and O are( the same ) ˜ R (G, B)∩˜ R (H, C) = operators. Thus, (F, A) U ) ( ) ( ˜ R (F, A)U˜ R (H, C) . (F, A)U˜ R (G, B) ∩ (2) Assume that (G, B)U˜ R (H, C) = (I, D), where D = B ∩ C and ∀e ∈ D, I (e) = G(e) ∪ H (e) } ) ({ u = m {μ (u),μ : u ∈ Ui : i ∈ I . (u)} G(e)

H (e)

( ) ˜ R (G, B)U˜ R (H, C) = (F, A)∩ ˜ R (I, D), we suppose that Since (F, A)∩ (F, A)∩˜ R (I, D) = (K , M) where M = A ∩ D = A ∩ B ∩ C and ∀e ∈ M, (. K (e) = F(e) ∩ I (e) =

u μ K (e) (u)

. : u ∈ Ui

) :i ∈I ,

26

2 Fuzzy Soft Multiset Theory

where { } μ K (e) (u) =m μ F(e) (u), μ I (e) (u) { { }} =m μ F(e) (u), m μG(e) (u), μ H (e) (u) { { } { }} =m m μ F(e) (u), μG(e) (u) , m μ F(e) (u), μ H (e) (u) . ˜ R (G, B) = (J, N ), where N = A ∩ B and ∀e ∈ N , Suppose that (F, A)∩ (. J (e) = F(e) ∩ G(e) =

u { } : u ∈ Ui m μ F(e) (u), μG(e) (u)

.

) :i ∈I .

Again, let (F, A)∩˜ R (H, C) = (S, T ), where T = A ∩ C and ∀e ∈ T , (. S(e) = F(e) ∩ H (e) =

u { } : u ∈ Ui m μ F(e) (u), μ H (e) (u)

.

) :i ∈I .

( ) ( ) ˜ R (F, A)∩˜ R (H, C) = ( J, N )U˜ R (S, T ). Since (F, A)∩˜ R (G, B) ∪ We suppose that (J, N )∪˜ R (S, T ) = (O, N ∩ T ) where ∀e ∈ N ∩ T = A ∩ B ∩ C, } ) ({ O(e) = J (e) ∪ S(e) = μ O(e)u (u) : u ∈ Ui : i ∈ I { } { μ S(e) (u) }} { { μ O(e) (u) = m μ}J (e) (u), = m m μ F(e) (u), μG(e) (u) , m μ F(e) (u), μ H (e) (u) = μ K (e) (u). Consequently, K( and O are the same ) operators. ( ) ( ) ˜ R (H, C) ˜ R (G, B) ∪˜ R (F, A)∩ Thus, (F, A)∩˜ R (G, B)∪˜ R (H, C) = (F, A)∩ Proposition 2.2.6 For two FSMSs (F, A) and (G, B) over U, we have ( )c [i] (F, A)U˜ R (G, B) ⊆ (F, A)c U˜ R (G, B)c ) ( ˜ R (G, B)c ⊆ ˜ (F, A)∩ ˜ R (G, B) c [ii] (F, A)c ∩ Proof ˜ R (G, B) = (H, C), where C = A ∩ B and ∀e ∈ C, [i] Let (F, A)∪ . H (e) = (F(e), G(e)) =

(.

u { } : u ∈ Ui m μ F(e) (u), μG(e) (u)

.

) :i ∈I .

( )c Thus, (F, A)U˜ R (G, B) = (H, C)c = (H c , C), where C = A ∩ B and ∀e ∈ C,

2.2 Restricted Union and Restricted Intersection

27

. H c (e) =( (F(e), G(e)))c (. . ) u { } : u ∈ Ui : i ∈ I = 1 − m μ F(e) (u), μG(e) (u) .. . ) u { } : u ∈ Ui : i ∈ I . = m 1 − μ F(e) (u), 1 − μG(e) (u) Again, (F, A)c U˜ R (G, B)c = (F c , A)U˜ R (G c , B) = (K , D). Where D = A ∩ B and ∀e ∈ D, .( ) F c (e), G c (e) K (e) = (. . ) u { } : u ∈ Ui : i ∈ I . = m 1 − μ F(e) (u), 1 − μG(e) (u) c We see(that C = D and ∀e )c ∈ C, H)c(e) ⊆ K (e). ˜ Thus, (F, A)U˜ R (G, B) ⊆F, A U˜ R (G, B)c .

˜ R (G, B) = (H, C), where C = A ∩ B and ∀e ∈ C, [ii] Let (F, A)∩ . H (e) = (F(e), G(e)) =

(.

u { } : u ∈ Ui m μ F(e) (u), μG(e) (u)

.

) :i ∈I .

( ) ˜ R (G, B) c = (H, C)c = (H c , C), where C = A ∩ B and Thus, (F, A)∩ ∀e ∈ C, } ) ({ u : i ∈ I H c (e) = (∩(F(e), G(e)))c = : u ∈ U i {{ 1−m {μ F(e) (u),μG(e) (u)} } ) u = : u ∈ Ui : i ∈ I . m {1−μ (u),1−μ (u)} F(e)

G(e)

˜ R (G c , B) = (K , D). Where D = ˜ R (G, B)c = (F c , A)∩ Again, (F, A)c ∩ A ∩ B and ∀e ∈ D, } ) ({ u : i ∈ I . K (e) = ∩(F c (e), G c (e)) = : u ∈ U i m {1−μ F(e) (u),1−μG(e) (u)} We see that C = D and ∀e ∈ C, K (e) ⊆ H c (e) ( )c Thus, (F, A)c ∩˜ R (G, B)c ⊆ (F, A)∩˜ R (G, B) . Proposition 2.2.7 (De Morgan Laws) For two FSMSs (F, A) and ( , B) over U, we have ( )c [i] (F, A)∪˜ R (G, B) = (F, A)c ∩˜ R (G, B)c ( )c [ii] (F, A)∩˜ R (G, B) = (F, A)c ∪˜ R (G, B)c Proof

28

2 Fuzzy Soft Multiset Theory

˜ R (G, B) = (H, C), where C = A ∩ B and ∀e ∈ C, [i] Let (F, A)∪ . H (e) = (F(e), G(e)) =

(.

u { } : u ∈ Ui m μ F(e) (u), μG(e) (u)

.

) :i ∈I .

Thus, ((F, A)∼ R (G, B))c = (H, C)c = (H c , C), where C = A ∩ B and ∀e ∈ C, } ) ({ u : i ∈ I H c (e) = : u ∈ U i {{ 1−m {μ F(e) (u),μG(e) (u)} } ) u = : u ∈ Ui : i ∈ I . m {1−μ (u),1−μ (u)} F(e)

G(e)

˜ R (G c , B) = (K , D), where D = ˜ R (G, B)c = (F c , A)∩ Again, (F, A)c ∩ A ∩ B and ∀e ∈ D, } ) ({ u : i ∈ I . K (e) = ∩(F c (e), G c (e)) = m {1−μ (u),1−μ : u ∈ U i F(e) G(e) (u)} c We see that C = D and ∀e ∈ C, H (e) = K (e). Hence proved. ˜ R (G, B) = (H, C), where C = A ∩ B and ∀e ∈ C, [ii] Let (F, A)∩ . H (e) = (F(e), G(e)) =

(.

u { } : u ∈ Ui m μ F(e) (u), μG(e) (u)

.

) :i ∈I .

)c ( Thus, (F, A)∩˜ R (G, B) = (H, C)c = (H c , C), where C = A ∩ B and ∀e ∈ C, } ) ({ u : i ∈ I H c (e) = : u ∈ U i ({ 1−m {μ F(e) (u),μG(e) (u)} } ) u = : i ∈ I . : u ∈ U i m {1−μ (u),1−μ (u)} F(e)

G(e)

˜ R (G, B)c = (F c , A)U˜ R (G c , B) = (K , D), where D = Again, (F, A)c ∪ A ∩ B and ∀e ∈ D, } ) ({ U u : i ∈ I . K (e) = (F c (e), G c (e)) = m {1−μ (u),1−μ : u ∈ U i F(e) G(e) (u)} c We see that C = D and ∀e ∈ C, H (e) = K (e). Hence proved.

2.3 Extended Union and Extended Intersection

29

2.3 Extended Union and Extended Intersection Definition 2.3.1 The extended union of two FSMSs (F, A) and (G, B) over U is a FSMS (H, D), where D = A ∪ B and ∀e ∈ D, ⎧ if e ∈ A-B, ⎨ F(e), H (e) = G(e), if e ∈ B − A, ⎩ ∪(F(e), G(e)), if e ∈ A ∩ B, } ) ({ u : i ∈ I where U (F(e), G(e)) = and is written as : u ∈ U i max{μ F(e) (u),μG(e) (u)} ˜ (F, A)U E (G, B) = (H, D). Definition 2.3.2 The extended intersection of two FSMSs (F, A) and (G, B) over U is a FSMS (H, D), where D = A ∪ B and ∀e ∈ D, ⎧ if e ∈ A − B, ⎨ F(e), H (e) = G(e), if e ∈ B − A, ⎩ ∩(F(e), G(e)), if e ∈ A ∩ B, } ) ({ u : i ∈ I where ∩(F(e), G(e)) = and is written as : u ∈ U i min{μ F(e) (u),μG(e) (u)} ~ there are two universes (F, A)∩ E (G, B) = (H, D) Example 2.3.3 {Let us consider } U1 = {h 1 , h 2 , h 3 }, U2 = {c1 , c2 } and let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

E U1

eU1 ,1 = modern, eU1 ,2 = wooden, = eU1 ,3 = in green surroundings, ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎩ eU1 ,5 = in good repair,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

E U2

Let U =

.2

⎫ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ eU2 ,2 = new model, ⎬ , = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = black, e U2 ,4 ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition ,

i=1 F S(Ui ),

E=

.2

i=1 E Ui

and

) ) )} { ( ( ( A = {e1 = (eU1 ,1 , eU2 ,1 ), e2 = (eU1 ,2 , eU2 ,2 ), e3 = (eU1 ,3 , eU2 ,3 )}, B = e1 = eU1 ,1 , eU2 ,1 , e2 = eU1 ,2 , eU2 ,2 , e4 = eU1 ,3 , eU2 ,4 .

30

2 Fuzzy Soft Multiset Theory

Suppose that } { c1 c2 })) ( ({ h 1 h 2 h 3 } { c1 c2 })) h2 h3 , 0.8 , 0.6 , , 0.4 , 0.8 , 0.7 1 ( , ({0.8h 1, 0.5h 2 h,3 }e2{, c1 0.7c2, })}} , } { , 0.7})) , 1 ( , 0.3 })) ({,h0.5 {( {{ h 1 h 2 h 3e3}, { c0.9 c2 h2 h3 c1 c2 1 1 , , e , 0.4 , , 0.3 , 0.7 , , , , , 0.6 (G, B) = e1 , 0.3 2 0.8 1 ( ({0.7h 1 0.6h 2 h 3 } { c1 0.5c2 })}} , e4 , 0.8 , 0.9 , 1 , 0.5 , 0.2 (F, A) =

{(

e1 ,

{{ h 1

0.2

Then we have {( ({ h 1 h 2 h 3 } { c1 c2 }) (i). (F, A) U˜ E((G,({ B) = e1 , } 0.3 { c,1 0.4c,2 0.8 })) , 0.8 , 0.6 , h1 h2 h3 })), 1( , ({0.8h,1 0.6h 2 h, 3 } { c1 c2 }))} ( ({ h 1 h 2 h 3 }e2{, c1 0.7c,2 0.8 , , , , })) , e3 , 0.9 , 0.7 , 1 , 0.3 , 0.5 {( , e({ 4, 0.8 0.9 }1 { 0.5 0.2 ˜ E (G, B) = e1 , h 1 , h 2 , h 3 , c1 , c2 , (ii). (F, A)∩ 0.7 0.7 0.5 ( ({ h 1 h 2 h 3 } {0.2c1 0.3c2 })) , , , , , , e 0.4 0.6 })) 1( ({ } { c1 c2 }))} ( {{ h 1 h 2 h 3 }2 { c1 0.5 c2 0.7 h1 h2 h3 . , 0.9 , 1 , 0.5 , 0.2 e3 , 0.9 , 0.7 , 1 , 0.3 , 0.5 , e4 , 0.8 Proposition 2.3.4 (Associative Laws) Let (F, A), (G, B) and (H, C) are three FSMSs over U, then we have the following properties: ( ) ( ) ˜ E (G, B)U˜ E (H, C) = (F, A)∪ ˜ E (G, B) ∪ ˜ E (H, C) [i] (F, A)∪ ) ( ) ( ˜ E (G, B)∩ ˜ E (G, B) ∩ ˜ E (H, C) ˜ E (H, C) = (F, A)∩ [ii] (F, A)∩ Proof ˜ E (H, C) = (I, D), where D = B ∪ C and ∀e ∈ D, [i] Suppose that (G, B)∪ ⎧ if e ∈ B − C, ⎨ G(e), I (e) = H (e), if e ∈ C − B, ⎩ ∪(G(e), H (e)), if e ∈ B ∩ C. ( ) Since(F, A)U˜ E (G, B)U˜ E (H, C) = (F, A)U˜ E (I, D), we suppose that (F, A)U˜ E (I, D) = ( J, M), where M = A ∪ D = A ∪ B ∪ C and∀e ∈ M, ⎧ G(e), ⎪ ⎪ ⎪ ⎪ ⎪ H (e), ⎪ ⎪ ⎪ ⎪ F(e), ⎨U J (e) = (G(e), H (e)), U ⎪ ⎪ ⎪ (F(e), H (e)), ⎪ ⎪ U ⎪ ⎪ (G(e), F(e)), ⎪ ⎪ ⎩ ∪(F(e), G(e), H (e)),

if e if e if e if e if e if e if e

∈ B − C − A, ∈ C − B − A, ∈ A − B − C, ∈ B ∩ C − A, ∈ A ∩ C − B, ∈ A ∩ B − C, ∈ A ∩ B ∩ C.

˜ E (G, B) = (K , S), where S = A ∪ B and ∀e ∈ S, Assume that (F, A)∪

2.3 Extended Union and Extended Intersection

31

⎧ if e ∈ A − B, ⎨ F(e), K (e) = G(e), if e ∈ B − A, ⎩ ∪(F(e), G(e)), if e ∈ A ∩ B. ( ) Since (F, A)U˜ E (G, B) U˜ E (H, C) = (K , S)U˜ E (H, C), we suppose that (K , S)U˜ E (H, C) = (L , T ) where T = S ∪ C = A ∪ B ∪ C and ∀e ∈ T , ⎧ G(e), ⎪ ⎪ ⎪ ⎪ ⎪ H (e), ⎪ ⎪ ⎪ ⎪ ⎨ F(e), L(e) = ∪(G(e), H (e)), ⎪ ⎪ ⎪ ∪(F(e), H (e)), ⎪ ⎪ U ⎪ ⎪ (G(e), F(e)), ⎪ ⎪ ⎩ ∪(F(e), G(e), H (e)),

if e ∈ B − C − A, if e ∈ C − B − A, if e ∈ A − B − C, if e ∈ B ∩ C − A, if e ∈ A ∩ C − B, if e ∈ A ∩ B − C, if e ∈ A ∩ B ∩ C.

Therefore, it is clear that M = T and ∀e ∈ M, ( J (e) = L(e), )that is J and L are the same operators. Thus, (F, A)∪˜ E (G, B)∪˜ E (H, C) = ) ( ˜ E (G, B) U˜ E (H, C). (F, A)∪ ˜ E (H, C) = (I, D), where D = B ∪ C and ∀e ∈ D, [ii] Suppose that (G, B)Ω ⎧ if e ∈ B − C, ⎨ G(e), I (e) = H (e), if e ∈ C − B, ⎩ ∩(G(e), H (e)), if e ∈ B ∩ C. ( ) ˜ E (H, C) = (F, A)∩ ˜ E (I, D), we suppose that Since, (F, A)∩˜ E (G, B)∩ (F, A)∩˜ E (I, D) = (J, M) where M = A ∪ D = A ∪ B ∪ C and ∀e ∈ M, ⎧ G(e), ⎪ ⎪ ⎪ ⎪ ⎪ H (e), ⎪ ⎪ ⎪ ⎪ ⎨ F(e), J (e) = ∩(G(e), H (e)), ⎪ ⎪ ⎪ ∩(F(e), H (e)), ⎪ ⎪ ⎪ ⎪ ⎪ ∩(G(e), F(e)), ⎪ ⎩ ∩(F(e), G(e), H (e)),

if e ∈ B − C − A, if e ∈ C − B − A, if e ∈ A − B − C, if e ∈ B ∩ C − A, if e ∈ A ∩ C − B, if e ∈ A ∩ B − C, if e ∈ A ∩ B ∩ C.

˜ E (G, B) = (K , S), where S = A ∪ B and ∀e ∈ S, Assume that (F, A)∩ ⎧ if e ∈ A − B, ⎨ F(e), K (e) = G(e), if e ∈ B − A, ⎩ ∩(F(e), G(e)), if e ∈ A ∩ B.

32

2 Fuzzy Soft Multiset Theory

( ) ˜ E (G, B) ∩ ˜ E (H, C) = (K , S)∩˜ E (H, C), we suppose that Since (F, A)∩ (K , S)∩˜ E (H, C) = (L , T ) where T = S ∪ C = A ∪ B ∪ C and ∀e ∈ T , ⎧ G(e), ⎪ ⎪ ⎪ ⎪ ⎪ H (e), ⎪ ⎪ ⎪ ⎪ ⎨ F(e), L(e) = ∩(G(e), H (e)), ⎪ ⎪ ⎪ ∩(F(e), H (e)), ⎪ ⎪ ⎪ ⎪ ∩(G(e), F(e)), ⎪ ⎪ ⎩ ∩(F(e), G(e), H (e)),

if e ∈ B − C − A, if e ∈ C − B − A, if e ∈ A − B − C, if e ∈ B ∩ C − A, if e ∈ A ∩ C − B, if e ∈ A ∩ B − C, if e ∈ A ∩ B ∩ C.

Therefore, it is clear that M = T and ∀e ∈ M, ( J (e) = L(e), )that is J and L are the same operators. Thus, (F, A)∩˜ E (G, B)∩˜ E (H, C) = ( ) ˜ E (G, B) ∩ ˜ E (H, C). (F, A)∩ Proposition 2.3.5 (Distributive Laws) Let (F, A), (G, B) and (H, C) are three FSMSs over U, then we have the following properties: ( ) ( ) ˜ E (G, B) ∼ (1) (F, A)∩˜ E ((G, B)∼ ˜ E ((F, A)∩˜ E (H, C) ) ˜ E (H, C)) = ((F, A)∩ ) ( ) ˜ E (F, A)U˜ E (H, C) (2) (F, A)U˜ E (G, B)∩˜ E (H, C) = (F, A)U˜ E (G, B) ∩ Proof (1) Suppose that (G, B)U˜ E (H, C) = (I, D), where D = B ∪ C and ∀e ∈ D, ⎧ if e ∈ B − C, ⎨ G(e), I (e) = H (e), if e ∈ C − B, ⎩ U (G(e), H (e)), if e ∈ B ∩ C. ( ) Since (F, A)∩˜ E (G, B)U˜ E (H, C) = (F, A)∩˜ E (I, D), we suppose that (F, A)∩˜ E (I, D) = (J, M) where M = A ∪ D = A ∪ B ∪ C and ∀e ∈ M, ⎧ G(e), ⎪ ⎪ ⎪ ⎪ ⎪ H (e), ⎪ ⎪ ⎪ ⎪ F(e), ⎨U J (e) = (G(e), H (e)), ⎪ ⎪ ⎪ ∩(F(e), H (e)), ⎪ ⎪ ⎪ ⎪ ∩(G(e), F(e)), ⎪ ⎪ ⎩ ∩(F(e), (G(e) ∪ H (e))),

if e ∈ B − C − A, if e ∈ C − B − A, if e ∈ A − B − C, if e ∈ B ∩ C − A, if e ∈ A ∩ C − B, if e ∈ A ∩ B − C, if e ∈ A ∩ B ∩ C.

˜ E (G, B) = (K , S), where S = A ∪ B and ∀e ∈ S, Assume that (F, A)∩

2.3 Extended Union and Extended Intersection

33

⎧ if e ∈ A − B, ⎨ F(e), K (e) = G(e), if e ∈ B − A, ⎩ ∩(F(e), G(e)), if e ∈ A ∩ B, and (F, A)∩˜ E (H, C) = (N , T ), where T = A ∪ C and ∀e ∈ T , ⎧ if e ∈ A − C, ⎨ F(e), N (e) = H (e), if e ∈ C − A, ⎩ ∩(F(e), H (e)), if e ∈ A ∩ C. ) ( ) ( ˜ E (G, B) U˜ E (F, A)∩˜ E (H, C) = (K , S)U˜ E (N , T ), we Since (F, A)∩ suppose that (K , S)U˜ E (N , T ) = (O, P) where P = S ∪ T = (A ∪ B) ∪ ( A ∪ C) = A ∪ B ∪ C and ∀e ∈ P, ⎧ G(e), ⎪ ⎪ ⎪ ⎪ ⎪ H (e), ⎪ ⎪ ⎪ ⎪ ⎨ F(e), O(e) = ∪(G(e), H (e)), ⎪ ⎪ ⎪ ∩(F(e), H (e)), ⎪ ⎪ ⎪ ⎪ ∩(G(e), F(e)), ⎪ ⎪ ⎩ ∩(F(e), ∪(G(e), H (e))),

if e ∈ B − C − A, if e ∈ C − B − A, if e ∈ A − B − C, if e ∈ B ∩ C − A, if e ∈ A ∩ C − B, if e ∈ A ∩ B − C, if e ∈ A ∩ B ∩ C.

Therefore, it is clear that M = P and ∀e ∈ M, (J (e) = O(e), that ) is “J” and “O” are the same operators. Thus, (F, A)∩˜ E (G, B)∪˜ E (H, C) = ) ( ) ( ˜ E (G, B) U˜ E (F, A)∩˜ E (H, C) . (F, A)∩ (2) Suppose that (G, B)∩˜ E (H, C) = (I, D), where D = B ∪ C and ∀e ∈ D, ⎧ if e ∈ B − C, ⎨ G(e), I (e) = H(e), if e ∈ C − B, ⎩ ∩(G(e), H (e)), if e ∈ B ∩ C. ( ) ˜ E (H, C) = (F, A)U˜ E (I, D), we suppose that Since (F, A)U˜ E (G, B)∩ (F, A)U˜ E (I, D) = (J, M) where M = A ∪ D = A ∪ B ∪ C and ∀e ∈ M, ⎧ ⎪ ⎪ G(e), ⎪ ⎪ ⎪ H(e), ⎪ ⎪ ⎪ ⎪ ⎨ F(e), J (e) = ∩(G(e), H (e)), ⎪ ⎪ ⎪ ∪(F(e), H (e)), ⎪ ⎪ ⎪ ⎪ ∪(G(e), F(e)), ⎪ ⎪ ⎩ ∪(F(e), ∩(G(e), H (e))),

if e ∈ B − C − A, if e ∈ C − B − A, if e ∈ A − B − C, if e ∈ B ∩ C − A, if e ∈ A ∩ C − B, if e ∈ A ∩ B − C, if e ∈ A ∩ B ∩ C.

34

2 Fuzzy Soft Multiset Theory

˜ E (G, B) = (K , S), where S = A ∪ B and ∀e ∈ S, Assume that (F, A)∪ ⎧ if e ∈ A − B, ⎨ F(e), K (e) = G(e), if e ∈ B − A, ⎩ ∪(F(e), G(e)), if e ∈ A ∩ B and (F, A)U˜ E (H, C) = (N , T ), where T = A ∪ C and ∀e ∈ T , ⎧ if e ∈ A − C, ⎨ F(e), N (e) = H(e), if e ∈ C − A, ⎩ ∪(F(e), H (e)), if e ∈ A ∩ C. ( ) ( ) ˜ E (N , T ), we ˜ E (F, A)U˜ E (H, C) = (K , S)∩ Since (F, A)U˜ E (G, B) ∩ suppose that (K , S)∩˜ E (N , T ) = (O, P) where P = S ∪ T = (A ∪ B) ∪ ( A ∪ C) = A ∪ B ∪ C and ∀e ∈ P, ⎧ G(e), ⎪ ⎪ ⎪ ⎪ ⎪ H(e), ⎪ ⎪ ⎪ ⎪ F(e), ⎨ O(e) = ∩(G(e), H (e)), ⎪ ⎪ ⎪ ∪(F(e), H (e)), ⎪ ⎪ ⎪ ⎪ ∪(G(e), F(e)), ⎪ ⎪ ⎩ ∪(F(e), ∩(G(e), H (e))),

if e ∈ B − C − A, if e ∈ C − B − A, if e ∈ A − B − C, if e ∈ B ∩ C − A, if e ∈ A ∩ C − B, if e ∈ A ∩ B − C, if e ∈ A ∩ B ∩ C.

Therefore, it is clear that M = P and ∀e ∈ M, J (e) = O(e), that is “ J “ and “O” are the same operators. ) ( ) ( ) ( ˜ E (H, C) = (F, A)U˜ E (G, B) ∩ ˜ E (F, A)U˜ E (H, C) . Thus (F, A)U˜ E (G, B)∩ Proposition 2.3.6 For two FSMSs (F, A) and ( , B) overU , then we have [i] ((F, A)∼ E (G, B))c ⊆((F, A)c ∼ E (G, B))c c ∼ ∼ [ii] (F, A)c ∩ E (G, B)c ⊆ (F, A)∩ E (G, B) Proof [i] Let (F, A)U˜ E (G, B) = (H, C), where C = A ∪ B and ∀e ∈ C, ⎧ if e ∈ A − B, ⎨ F(e), H (e) = G(e), if e ∈ B − A, ⎩ ∪(F(e), G(e)), if e ∈ A ∩ B, where

U

(F(e), G(e)) =

({

u max{μ F(e) (u),μG(e) (u)}

} ) : u ∈ Ui : i ∈ I .

2.3 Extended Union and Extended Intersection

35

( )c Thus, (F, A)∪˜ E (G, B) = (H, C)c = (H c , C), where C = A ∪ B and ∀e ∈ C, ⎧ c if e ∈ A − B, ⎨ F (e), H c (e) = G c (e), if e ∈ B − A, ⎩ (∪(F(e), G(e)))c , if e ∈ A ∩ B, } ) ({ U u : i ∈ I . where ( (F(e), G(e)))c = 1−m {μ (u),μ : u ∈ U i F(e) G(e) (u) ({ }} ) u = m {1−μ (u),1−μ (u)} : u ∈ Ui : i ∈ I . F(e) G(e) ˜ E (G c , B) = (K , D). Where D = ˜ E (G, B)c = (F c , A)∪ Again, (F, A)c ∪ A ∪ B and ∀e ∈ D, ⎧ c if e ∈ A − B, ⎨ F (e), K (e) = G c (e), if e ∈ B − A ⎩ ∪(F c (e), G c (e)), if e ∈ A ∩ B, } ) ({ U u : i ∈ I . where (F c (e), G c (e)) = m {1−μ (u),1−μ : u ∈ U i F(e) G(e) (u)} c We see that C = D and ∀e ∈ C, H (e) ⊆ K (e). ( )c ˜ Thus (F, A)U˜ E (G, B) ⊆(F, A)c U˜ E (G, B)c . ˜ E (G, B) = (H, C), where C = A ∪ B and ∀e ∈ C, [ii] Let (F, A)∩ ⎧ if e ∈ A − B, ⎨ F(e), H (e) = G(e), if e ∈ B − A, ⎩ ∩(F(e), G(e)), if e ∈ A ∩ B, } ) ({ u : i ∈ I where ∩(F(e), G(e)) = min{μ (u),μ : u ∈ U i F(e) G(e) (u)} ( )c ˜ E (G, B) = (H, C)c = (H c , C), where C = A ∪ B and Thus, (F, A)Ω ∀e ∈ C, ⎧ c if e ∈ A − B, ⎨ F (e), H c (e) = G c (e), if e ∈ B − A, ⎩ (∩(F(e), G(e)))c , if e ∈ A ∩ B, } ) ({ where (∩(F(e), G(e)))c = 1−min{μ u(u),μ (u)} : u ∈ Ui : i ∈ I . F(e) G(e) } ) ({ u : i ∈ I . = m {1−μ (u),1−μ : u ∈ U i F(e) G(e) (u)} c˜ c c c Again, (F, A) ∩ E (G, B) = (F , A)∩˜ E (G , B) = (K , D). Where D = A ∪ B and ∀e ∈ D,

36

2 Fuzzy Soft Multiset Theory

⎧ c if e ∈ A − B, ⎨ F (e), K (e) = G c (e), if e ∈ B − A, ⎩ ∩(F c (e), G c (e)), if e ∈ A ∩ B, } ) ({ u : i ∈ I where ∩(F c (e), G c (e)) = . We : u ∈ U i min{1−μ F(e) (u),1−μG(e) (u)} see that C = D and ∀e ∈ C, K( (e) ⊆ H c (e). ) ˜ (F, A)∩ ˜ E (G, B) c . Thus, (F, A)c ∩˜ E (G, B)c ⊆ Proposition 2.3.7 For two FSMSs (F, A) and (G, B) over U, we have ( )c ˜ R (G, B) ⊆(F, ˜ [i] (F, A)Ω A)c U˜ E (G, B)c ( ) ˜ E (G, B) c ˜ R (G, B)c ⊆ (F, A)∪ [ii] (F, A)c ∩ Proof ˜ R (G, B) = (H, C), where C = A ∩ B and ∀e ∈ C, [i] Let (F, A)∩ . H (e) = (F(e), G(e)) =

(.

u { } : u ∈ Ui m μ F(e) (u), μG(e) (u)

.

) :i ∈I

)c ( Thus, (F, A)∩˜ R (G, B) = (H, C)c = (H c , C), where C = A ∩ B and ∀e ∈ C, } ) ({ u : i ∈ I . H c (e) = : u ∈ U i ({ 1−m {μ F(e) (u),μG(e) (u)} } ) u = : u ∈ Ui : i ∈ I . m {1−μ (u),1−μ (u)} F(e)

G(e)

Again, (F, A)c U˜ E (G, B)c = (F c , A)U˜ E (G c , B) = (K , D), where D = A ∪ B and ∀e ∈ D, ⎧ c if e ∈ A − B, ⎨ F (e), K (e) = G c (e), if e ∈ B − A, ⎩ ∪(F c (e), G c (e)), if e ∈ A ∩ B, } ) ({ u : i ∈ I . where U (F c (e), G c (e)) = m {1−μ (u),1−μ : u ∈ U i F(e) G(e) (u)} We see( that C ⊆ D and )∀e ∈ C, H c (e) = K (e). ˜ R (G, B) c ⊆(F, ˜ Thus, (F, A)∩ A)c U˜ E (G, B)c . [ii] Let (F, A)c ∩˜ R (G, B)c = (F c , A)∩˜ R (G c , B) = (H, C), where C = A ∩ B and ∀e ∈ C, } ) ({ . u : i ∈ I . H (e) = (F c (e), G c (e)) = m {1−μ (u),1−μ : u ∈ U i F(e) G(e) (u) ({ }} ) u = : u ∈ Ui : i ∈ I . 1−m {μ (u),μ (u)} F(e)

G(e)

2.3 Extended Union and Extended Intersection

37

Again, let (F, A)U˜ E (G, B) = (K , D). Where D = A ∪ B and ∀e ∈ D, ⎧ if e ∈ A − B, ⎨ F(e), K (e) = G(e), if e ∈ B − A, ⎩ ∪(F(e), G(e)), if e ∈ A ∩ B, } ) ({ u : i ∈ I where U (F(e), G(e)) = Thus, : u ∈ U i max{μ F(e) (u),μG(e) (u)} ( )c (F, A)U˜ E (G, B) = (K , D)c = (K c , D), where D = A ∪ B and ∀e ∈ D, K c (e) = ⎧ c c ⎪ ⎨ FU(e), G (e), − B, ( (F(e), G(e)))c , if e ∈ A ({ ⎪ ⎩ where (U(F(e), G(e)))c =

⎫ ⎪ ⎬

⎞

⎟ } ⎪ : i ∈ I ⎠. u : u ∈ Ui ⎭ 1−m {μ F(e) (u),μG(e) (u)} We see that C ⊆ D and ∀e ∈( C, H (e) = K c (e). )c Thus, (F, A)c ∩˜ R (G, B)c ⊆ (F, A)U˜ E (G, B) . Proposition 2.3.8 (De Morgan Laws) For two FSMSs (F, A) and (G, B) over U, we have ( )c [i] (F, A)U˜ E (G, B) = (F, A)c ∩˜ E (G, B)c ( )c ˜ E (G, B)c [ii] (F, A)∩˜ E (G, B) = (F, A)c ∪ Proof [i] Let (F, A)∪˜ E (G, B) = ⎧ (H, C), where C = A ∪ B and ∀e ∈ C, if e ∈ A − B, ⎨ F(e), H (e) = G(e), if e ∈ B − A, ⎩ ∪(F(e), G(e)), if e ∈ A ∩ B, } ({

) u : i ∈ I . : u ∈ U where U (F(e), G(e)) = m {μ (u),μ i F(e) G(e) (u)} ( )c Thus, (F, A)U˜ E (G, B) = (H, C)c = (H c , C), where C = A ∪ B and ∀e ∈ C, ⎧ c if e ∈ A − B, ⎨ F (e), H c (e) = G c (e), if e ∈ B − A, ⎩ (U (F(e), G(e)))c , if e ∈ A ∩ B, } ) ({ U u : i ∈ I where ( (F(e), G(e)))c = 1−m {μ (u),μ : u ∈ U i F(e) G(e) ({ }(u)} ) u = : i ∈ I . : u ∈ U i m {1−μ F(e) (u),1−μG(e) (u)} c˜ c c c ˜ E (G , B) = (K , D), where D = Again, (F, A) ∩ E (G, B) = (F , A)∩ A ∪ B and ∀e ∈ D,

38

2 Fuzzy Soft Multiset Theory

⎧ c if e ∈ A − B, ⎨ F (e), K (e) = G c (e), if e ∈ B − A, ⎩ ∩(F c (e), G c (e)), if e ∈ A ∩ B, } ) ({ u : i ∈ I where ∩(F c (e), G c (e)) = min{1−μ (u),1−μ . : u ∈ U i F(e) G(e) (u)} We see that C = D and ∀e ∈ C, H c (e) = K (e). Thus, the result. [ii]. Let (F, A)∩˜ E (G, B) = (H, C), where C = A ∪ B and ∀e ∈ C, ⎧ if e ∈ A − B, ⎨ F(e), H (e) = G(e), if e ∈ B − A, ⎩ ∩(F(e), G(e)), if e ∈ A ∩ B, } ) ({ u : i ∈ I . where ∩(F(e), G(e)) = min{μ (u),μ : u ∈ U i F(e) G(e) (u)} ( )c c c ˜ Thus, (F, A)∩ E (G, B) = (H, C) = (H , C), where C = A ∪ B and ∀e ∈ C, ⎧ c G c (e), ⎪ ⎨ F (e), c if e ∈ A − B, H c (e) = (∩(F(e), G(e))) , ({ ⎪ 1 ⎩ where (∩(F(e), G(e)))c = : u, A, A ∩ B, 1−m {μ F(e) }(u),μG(e) (u) )} ({ u = : u ∈ Ui : i ∈ I . m {1−μ (u),1−μ (u)} F(e)

G(e)

Again, (F, A)c U˜ E (G, B)c = (F c , A)U˜ E (G c , B) = (K , D), where D = A ∪ B and ∀e ∈ D, ⎧ c if e ∈ A − B, ⎨ F (e), c K (e) = G (e), if e ∈ B − A, ⎩ ∪(F c (e), G c (e)), if e ∈ A ∩ B, } ) ({ u : i ∈ I where ∪(F c (e), G c (e)) = max{1−μ (u),1−μ . : u ∈ U i F(e) G(e) (u)} c We see that C = D and ∀e ∈ C, H (e) = K (e). Hence proved.

2.4 AND Operator and OR Operator Definition 2.4.1 If (F, A) and (G, B) be two FSMSs over U, then “(F, A) AND (G, B) “ is a FSMS denoted by (F, A) ∧ (G, B) and is defined by(F, A) ∧ (G, B) = (H, A × B), where H is mapping given by H: A × B → U and (. ∀(a, b) ∈ A × B, H (a, b) =

u { } : u ∈ Ui m μ F(a) (u), μG(b) (u)

.

) :i ∈I .

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

2.4 AND Operator and OR Operator

39

Definition 2.4.2 If (F, A) and (G, B) be two FSMSs over U, then “(F, A) OR (G, B)” is a FSMS denoted by (F, A) ∨ (G, B) and is defined by (F, A) ∨ (G, B) = (K , A × B), where K is mapping given by K : A × B → U and (. ∀(a, b) ∈ A × B, K (a, b) =

u { } : u ∈ Ui m μ F(a) (u), μG(b) (u)

.

) :i ∈I .

Example 2.4.3 {Let us consider there are two universes U1 = {h 1 , h 2 , h 3 }, U2 = } {c1 , c2 } and let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎫ ⎪ ⎪ eU1 ,1 = modern, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,2 = wooden, ⎨ ⎬ EU1 = eU1 ,3 = in green surroundings, , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ eU1 ,5 = in good repair, ⎧ ⎫ ⎪ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU2 ,2 = new model, ⎬ . E U2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition, Let U =

.2

i=1 F S(Ui ),

E=

.2

i=1 E Ui

and

{ ( ( ( ) ) )} A = {e1 = (eU1 ,1 , eU2 ,1 ), e2 = (eU1 ,2 , eU2 ,2 ), e3 = (eU1 ,3 , eU2 ,3 )}, B = e1 = eU1 ,1 , eU2 ,1 , e2 = eU1 ,2 , eU2 ,2 , e4 = eU1 ,3 , eU2 ,4 . Suppose that {( ({ h 1 h 2 h 3 } { c1 c2 }) ( ({ h 1 h 2 h 3 } { c1 c2 }) e1 , 0.2 , 0.4 , 0.8 , , 0.8 , 0.6 , 0.7 1 ( , ({0.8h 1, 0.5h 2 ,h,3 }e2{, c1 0.7c2, })} , , , , , , e 3 0.3 0.5 }) 1 ( ({ } { c1 c2 })) {( ({ h 1 h 2 h 3 } { c10.9 c20.7 h1 h2 h3 , , }e2{, 0.5 , 0.4 , 0.6 , , 0.3 , 0.7( , ({ , , 0.8 , (G, B) = { e1 , 0.3 0.7 0.6 1 })} c1 c2 h1 h2 h3 e4 , 0.8 , 0.9 , 1 , 0.5 , 0.2 . (F, A) =

Then, we have A) ∧((G, B) ({ }) } { c1 c2 })) ({ h 1 h 2 h 3 } { c1 (F, {( c2 h1 h2 h3 , 0.4 , 0.8 } , { 0.4 , 0.5 })) , = ((e1 , e1 ), ({ 0.2 , 0.3 , 0.7 }, { 0.7 , 0.5 })), , ( (e1 , e2 ), ({ 0.2 h1 h2 h3 c1 c2 h1 h2 h3 c1 c2 , , , , , , , , e , , , , e (e (e ), ), 1 4 2 1 0.4 0.8 } { 0.5 0.2 })) ( 0.3 0.7} { 0.7 0.6})) ({0.2 ({ 0.3 ( c1 c2 h1 h2 h3 c1 c2 h1 h2 h3 , e , , , , e , , , , , , , (e ), ), (e 2 2 2 4 0.7 1 } {0.4 0.6 })) ( 0.7 1 } {0.5 0.2 })) ({ 0.5 ({0.7 ( h1 h2 h3 c1 c2 h1 h2 h3 c1 c2 , (e}3 , e{2 ), 0.5 , , 0.3(, 0.7 , 0.3 , 0.7 , 1 , 0.3 , 0.5 (e3 , e1 ), 0.3 }))} ({, h0.5 h h c c , (e3 , e4 ), 0.81 , 0.72 , 13 , 0.31 , 0.22

40

2 Fuzzy Soft Multiset Theory

})∨((G, B) {{ h 1 h 2 h 3 } { c1 c2 }) ({ h 1 h 2 h 3 } { c1(F,c2A) {( = ( (e1 , e1 ),({ 0.3 , 0.4 , 0.8} ,{ 0.8 , 0.6})) , ( (e1 , e2 ), ({ 0.5 , 0.8 , 1 } , { 0.8 , 0.6})), c1 c2 h1 h2 h3 c1 c2 h1 h2 h3 , 0.9 , 1 }, { 0.8 , 0.5 , 0.7 , 1 }, { 0.8 , 0.6 })), ((e2 , e1 ), ({ 0.7 })), ((e1 , e4 ), ({ 0.8 c1 c2 h1 h2 h3 c1 c2 h1 h2 h3 , , , , , e , , , , e , , , (e (e ), ), 2 4 2 2 0.8 1 } { 0.8 0.6 })) ( 0.9 1 } { 0.8 0.6 })) ({ 0.8 ({ 0.7 ( c1 c2 h1 h2 h3 c1 c2 h1 h2 h3 , , , , 0.7(, 1 , 0.7 , , e , , , (e ), (e3 , e1 ), 0.9 3} {2 0.9}))} 0.8 1 0.4 0.6 ({ 0.6 c1 c2 h1 h2 h3 . , 0.9 , 1 , 0.5 , 0.5 (e3 , e4 ), 0.9 Proposition 2.4.4 (Associative Laws) Let (F, A), (G, B) and (H, C) are three FSMSs over U, then we have the following properties: [i] (F, A) ∧ ((G, B) ∧ (H, C)) = ((F, A) ∧ (G, B)) ∧ (H, C) [ii] (F, A) ∨ ((G, B) ∨ (H, C)) = ((F, A) ∨ (G, B)) ∨ (H, C) Proof [i] Assume that (G,({ B) ∧ (H, C) = (I, B × C), where } ∀(b,)c) ∈ B × C, I (b, c) = u G(b) ∩ H (c) = min{μ (u),μ :i ∈I . : u ∈ U i G(b) H (c) (u)} Since (F, A) ∧ ((G, B) ∧ (H, C)) = (F, A) ∧ (I, B × C), we suppose that (F, A) ∧ (I, B × C) = (K , A × (B × C)) where ∀(a, b, c) ∈ A × (B × C) = A × B × C, . ) (. u : u ∈ Ui : i ∈ I , K (a, b, c) =F(a) ∩ I (b, c) = μ K (a,b,c) (u) { } where μ K (a,b,c) (u) = m μ F(e) (u), μ I (b,c) (u) { { }} =m μ F(e) (u), m μG(b) (u), μ H (c) (u) { } =m μ F(e) (u), μG(b) (u), μ H (c) (u) . We take (a, b) ∈ A × B. Suppose that (F, A) ∧ (G, B) = (J, A × B), where ∈ A ×} B, J (a, = F(a) ∩ G(b) = ({ ∀(a, b) ) b) u : u ∈ Ui : i ∈ I . m {μ F(a) (u),μG(b) (u)} Since ((F, A) ∧ (G, B)) ∧ (H, C) = ( J, A × B) ∧ (H, C), we suppose that (J, A × B) ∧ (H, C) = (O, (A × B) × C) where ∀(a, b, c) ∈ (A × B) × C = A × B × C, . ) (. u : u ∈ Ui : i ∈ I , O(a, b, c) = J (a, b) ∩ H (c) = μ O(a,b,c) (u) where { } μ O(a,b,c) {(u) { = m μ J (a,b) (u), μ}H (c) (u) } = { m m μ F(e) (u), μG(b) (u) }, μ H (c) (u) = m μ F(e) (u), μG(b) (u), μ H (c) (u) = μ K (a,b,c) (u).

2.4 AND Operator and OR Operator

41

Consequently, K and O are the same operators. Thus (F, A) ∧ ((G, B) ∧ (H, C)) = ((F, A) ∧ (G, B)) ∧ (H, C). [ii] Assume that (G, B) ∨ (H, C) = (I, B × C), ∀(b, c) ∈ ) B × C, u I (b, c) = G(b) ∪ H (c) = ( max{μ (u),μ (u)} : u ∈ Ui } : i ∈ I . G(b) H (c) Since (F, A) ∨ ((G, B) ∨ (H, C)) = (F, A) ∨ (I, B × C), we suppose that (F, A) ∨ (I, B × C) = (K , A × (B × C)) where )= ({ ∀(a, b, c) ∈ A ×} (B × C) u A × B × C, K (a, b, c) = F(a) ∪ I (b, c) = μ K (a,b,c) (u) : u ∈ Ui : i ∈ I , where { } μ K (a,b,c) (u) =m μ F(e) (u), μ I (b,c) (u) { { }} =m μ F(e) (u), m μG(b) (u), μ H (c) (u) { } =m μ F(e) (u), μG(b) (u), μ H (c) (u) . We take(a, b) ∈ A × B. Suppose that (F, A) ∨ (G, B) = (J, ({ A × B), where∀(a, b) ∈ } A ×) B,J (a, b) = F(a) ∪ G(b) = u : u ∈ Ui : i ∈ I . max{μ F(a) (u),μG(b) (u)} Since ((F, A) ∨ (G, B)) ∨ (H, C) = ( J, A × B) ∨ (H, C), we suppose that (J, A × B)∨(H, C) = (O, ( A × B) × C), where ({ ∀(a, b, c) ∈ (A}× B)×C) = u A × B × C O(a, b, c) = J (a, b) ∪ H (c) = μ O(a,b,c) : u ∈ Ui : i ∈ I , (u) where { } μ O(a,b,c) {(u) { = m μ J (a,b) (u), μ}H (c) (u) } = { m m μ F(e) (u), μG(b) (u) ,}μ H (c) (u) = m μ F(e) (u), μG(b) (u), μ H (c) (u) = μ K (a,b.c) (u). Consequently, K and O are the same operators. Thus (F, A) ∨ ((G, B) ∨ (H, C)) = ((F, A) ∨ (G, B)) ∨ (H, C). Proposition 2.4.5 For two FSMSs (F, A) and (G, B) over U, we have the following [i] ((F, A) ∧ (G, B))c = (F, A)c ∨ (G, B)c [ii] ((F, A) ∨ (G, B))c = (F, A)c ∧ (G, B)c Proof [i] Let (F, A) ∧ (G, B) = (H, A × B), where ∀a ∈ A and ∀b ∈ B, H (a, b) =

.

(. (F(a), G(b)) =

u { } : u ∈ Ui m μ F(a) (u), μG(b) (u)

.

) :i ∈I

Thus ((F, A) ∧ (G, B))c = (H, A × B)c = (H c , A × B), where ∀(a, b) ∈ A × B,

42

2 Fuzzy Soft Multiset Theory

H c (a, b) = =

}

({

)

u : u ∈ Ui : i ∈ I } ) ({ 1−m {μ F(a) (u),μG(b) (u)} u : i ∈ I : u ∈ U i m {1−μ F(a) (u),1−μG(b) (u)}

c Again, let (F, A)c ∨ (G, B)c = (F c , A) ∨ (G = (K , A × U , B) c where ∀(a, b) ∈ A × B, K (a, b) = (a), G c (b)) = B), (F } ) ({ u : u ∈ Ui : i ∈ I . max{1−μ (u),1−μ (u)} F(a)

G(b)

Thus it follows that ((F, A) ∧ (G, B))c = (F, A)c ∨ (G, B)c . [ii] Let (F, A) ∨ (G, B) ∀b ∈ B, H (a, b) = ({= (H, A × B), where ∀a ∈} A and ) U u (F(a), G(b)) = max{μ (u),μ (u)} : u ∈ Ui : i ∈ I . F(a)

G(b)

c c c Thus, ((F, A) ∨ (G, ({ B)) = (H, A × B) = (H ,}A × B),)where ∀(a, b) ∈ u c A × B, H (a, b) = 1−max{μ (u),μ (u)} : u ∈ Ui : i ∈ I . F(a) G(b) } ) ({ u : i ∈ I = . : u ∈ U i (min{1−μ F(a) (u),1−μG(b) (u)} c c c Again, let (F, A) ∧ (G, B) = (F , A) ∧ (G c , B) = (K , A × c c B), where ∀(a, b) ∈ A × }B, K (a, ({ ) b) = ∩(F (a), G (b)) = u : u ∈ Ui : i ∈ I . min{1−μ (u),1−μ (u)} F(a)

G(b)

Thus, it follows that ((F, A) ∨ (G, B))c = (F, A)c ∧ (G, B)c .

2.5 Absolute and Null FSMSs Here, we define absolute FSMS and null FSMS in a different approach and study their fundamental properties. i∈I Ui = φ and let { Let {Ui :}i ∈ I } be a collection of universes, such that ∩. EUi : i ∈ I be a collection of sets of parameters. Let.U = i∈I F S(Ui ), where F S(Ui ) denotes the set of all fuzzy subsets of Ui , E = i∈I EUi and A ⊆ E Definition 2.5.1 A FSMS (F, A) over U is called an absolute FSMS, denoted by (F, A)U , if ∀e ∈ A, μ F(e) (u) = 1, ∀u ∈ Ui , i ∈ I . Definition 2.5.2 A null FSMS Ω A over U, is a FSMS in which ∀e ∈ A, μ F(e) (u) = 0, ∀u ∈ Ui , i ∈ I . We consider an absolute FSMS (F, A) over U and FSMS A ( F, A) denote the family of all fuzzy soft multi subsets of (F, A) in which all the parameter set A are the same. Proposition 2.5.3 If (F, A) and (G, A) are two FSMSs in FS M S (F, A), then we have the following: ˜ ˜ (i) ((F, A)∪(G, A))c = (F, A)c ∩(G, A)c c c ˜ (ii) ((F, A)∩(G, A)) = (F, A) ∼ (G, A)c Definition 2.5.4 Let I be an arbitrary index set and {(Fi , A) : i ∈ I } be a subfamily of F S M S A (F, A).

2.5 Absolute and Null FSMSs

43

[i] The union of these FSMSs is the FSMSs (H, A), where H (e) = ∪i∈I Fi (e), for each e ∈ A. We write U˜ i∈I (Fi , A) = (H, A). [ii] The intersection of these FSMSs is the FSMSs (M, A), where M(e) = ∩i∈I Fi (e), for each e ∈ A. ˜ i∈I (Fi , A) = (M, A). We write Ω Proposition 2.5.5 If (F, A) is a FSMS in F S M S A ( F, A), then. (i) (ii) (iii) (iv)

˜ A = Ω A , (F, A)∩Ω ˜ (F, A)∩(F, A) = (F, A), (F, A) ∼ Ω A = (F, A), (F, A) ∼ (F, A) = (F, A).

Chapter 3

Relation on Fuzzy Soft Multisets

In this segment, we define the product of two fuzzy soft multisets (FSMSs) and introduce the concept of FSMS relations. It is necessary to distinguish between the concepts of null relation and absolute relation. Also, we introduce the inverse of the fuzzy soft multi relation and analyse certain key aspects of the aforesaid idea as well as some fundamental properties of these notions.

3.1 Fuzzy Soft Multi Relations Definition 3.1.1 The product (F, A) × (G, B) of two FSMSs (F, A) and (G, B) over U is a FSMS (H, A × B) where H is mapping given by H : A × B → U and ∀(a, b) ∈ A × B, (. H (a, b) = ∩(F(a), G(b)) =

u { } : u ∈ Ui m μ F(a) (u), μG(b) (u)

.

) :i ∈I .

Example 3.1.2 { Give us}a chance to consider two universes U1 = {h 1 , h 2 , h 3 }, U2 = {c1 , c2 }. Let EU1 , EU2 be a collection of sets }of decision } related to { { parameters the above universes, where EU1 = eU1 ,1 , eU1 ,2 , EU2 = eU2 ,1 , eU2 ,2 . Let U = 2 . 2 .i=1 F S(Ui ), E = EUi and A, B ⊆ E, such that (Tables 3.1, 3.2 and 3.3) i=1

) ) )} { ( ( ( a1 = ) eU1 ,1 ,(eU2 ,1 , a2 = )} eU1 ,1 , eU2 ,2 , a3 = eU1 ,2 , eU2 ,2 , B { A (= b1 = eU1 ,2 , eU2 ,1 , b2 = eU1 ,2 , eU2 ,2 .

=

Definition 3.1.3 Let (F, A) be a FSMS over U. Then a fuzzy soft multi relation R on (F, A) will be a fuzzy soft multi subset of the product set (F, A) × (F, A) and is characterized as a couple (R, A × A), where R is mapping given by R : A×A → U. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_3

45

46

3 Relation on Fuzzy Soft Multisets

Table 3.1 FSMS (F, A) (F, A)

a1

a2

a3

h1

0.7

0.5

0.3

h2

0.3

0.1

0.7

h3

0.7

0.3

0.5

c1

0.7

0.3

0.6

c2

0.4

0.4

0.4

Table 3.2 FSMS (G, B) (G, B)

b1

b2

h1

0.5

0.3

h2

0.7

0.5

h3

0.3

0.5

c1

0.4

0.7

c2

0.7

0.4

Table 3.3 The product set (H, A × B) (a1 , b1 )

(a1 , b2 )

(a2 , b1 )

(a2 , b2 )

(a3 , b1 )

(a3 , b2 )

h1

0.5

0.3

0.5

0.3

0.3

0.3

h2

0.3

0.3

0.1

0.1

0.7

0.5

h3

0.3

0.5

0.3

0.3

0.3

0.5

c1

0.4

0.7

0.3

0.3

0.6

0.6

c2

0.4

0.4

0.4

0.4

0.4

0.4

The collection of all fuzzy soft multi relations R on (F, A) over U is indicated by F S M RU (F, A). Example 3.1.4 Consider the FSMS (F, A) given in Table 3.1, then a fuzzy soft multi relation R on (F, A) is given in Table 3.4. Definition 3.1.5 Let R1 , R2 ∈ F S M RU (F, A), then we define for (a, b) ∈ A × A: Table 3.4 Fuzzy soft multi relation R R

(a1 , a1 )

(a1 , a2 )

( a1 , a3 )

(a2 , a1 )

(a2 , a2 )

(a2 , a3 )

(a3 , a1 )

(a3 , a2 )

(a3 , a3 )

h1

0.6

0.4

0.3

0.5

0.4

0.3

0.3

0.3

0.3

h2

0.2

0.1

0.3

0.1

0.1

0.1

0.3

0.1

0.6

h3

0.5

0.3

0.5

0.3

0.3

0.3

0.5

0.3

0.5

c1

0.7

0.3

0.6

0.3

0.3

0.3

0.2

0.3

0.6

c2

0.4

0.4

0.4

0.2

0.4

0.4

0.2

0.4

0.2

3.4 Distributive Laws

47

(i) R1 ≤ R2 if and only if R1 (a, b) ⊆ R2 (a, b), for (a, b) ∈ A × A. (ii) R1 ∨ R2 as (R1 ∨ R2 )(a, b) = R1 (a, b) ∪ R2 (a,b), where ∪ denotes the fuzzy union. (iii) R1 ∧ R2 as (R1 ∧ R2 )(a, b) = R1 (a, b) ∩ R2 (a, b), where ∩ denotes the fuzzy intersection. (iv) R1C as R1C (a, b) = C[R1 (a, b)], where C denotes the fuzzy complement. Result 3.1.6 For three fuzzy soft multi relations R1 , R2 , R3 ∈ FF M RU (F, A), the following properties hold:

3.2 De Morgan Laws (a) (R1 ∨ R2 )C = RC1 ∧ RC2 . (b) (R1 ∧ R2 )C = RC1 ∨ RC2 .

3.3 Associative Laws (a) R1 ∨ (R2 ∨ R3 ) = (R1 ∨ R2 ) ∨ R3 . (b) R1 ∧ (R2 ∧ R1 ) = (R1 ∧ R2 ) ∧ R3 .

3.4 Distributive Laws (a) R1 ∧ (R2 ∨ R3 ) = (R1 ∧ R2 ) ∨ (R1 ∧ R3 ). (b) R1 ∨ (R2 ∧ R3 ) = (R1 ∨ R2 ) ∧ (R1 ∨ R3 ). Definition 3.1.7 A null fuzzy soft multi relation R. ∈ F S M RU (F, A) is defined as R. = (R. , A × A). , where (R. , A × A). is the null FSMS and an absolute fuzzy soft multi relation RU ∈ F S M RU (F, A) is defined as RU = (RU , A × A)U , where (RU , A × A)U is the absolute FSMS. Remark 3.1.8 For any fuzzy soft multi relation R ∈ F S M RU (F, A), we have (i) (ii) (iii) (iv)

R∨ R∧ R∨ R∧

R. R. RU RU

= R. = R. . = RU . = R.

48

3 Relation on Fuzzy Soft Multisets

3.4.1 Inverse of Fuzzy Soft Multi Relation In this segment, we introduce the inverse of the fuzzy soft multi relation and analyse certain key aspects of the aforesaid idea as well as some fundamental properties of these notions. Definition 3.2.1 Let R ∈ FSMRU (F, A) be a fuzzy soft multi relation on (F, A). Then R −1 is defined as ∀a, b ∈ A, R −1 (a, b) = R(b, a), i.e. μ R −1 (a,b) (u) = μ R(b,a) (u), ∀u ∈ Ui , ∀i ∈ I. Example 3.2.2 If we consider the fuzzy soft multi relation R as in Table 3.4, then R −1 is given as in Table 3.5. Proposition 3.2.3 If R is a fuzzy soft multi relation on (F, A), then R −1 is also a fuzzy soft multi relation on (F, A). Proof R −1 (a, b) = R(b, a) ⊆ F(b) ∩ F(a) = F(a) ∩ F(b)∀a, b ∈ A, i.e. R −1 (a, b) ⊆ F(a) ∩ F(b), ∀a, b ∈ A, implies R −1 is a fuzzy soft multi subset of the product set (F, A) × (F, A) and hence R −1 is a fuzzy soft multi relation on (F, A). Proposition 3.2.4 If R1 and R2 be two fuzzy soft multi relations on (F, A), then ( )−1 (i) R1−1 = R1 . (ii) R1 ⊆ R2 ⇒ R1−1 ⊆ R2−1 . Proof Let us consider R1 and R2 be two fuzzy soft multi relations on (F, A), then ∀a, b ∈ A, )−1 )−1 ( ( [i]. R1−1 (a, b) = R1−1 (b, a) = R1 (a, b). Hence R1−1 = R1 [ii]. R1 (a, b) ⊆ R2 (a, b) ⇒ R1−1 (b, a) ⊆ R2−1 (b, a) ⇒ R1−1 ⊆ R2−1 Proposition 3.2.5 If R1 and R2 be two fuzzy soft multi relations on (F, A), then (a) (R1 ∨ R2 )−1 = R1−1 ∨ R2−1 . (b) (R1 ∧ R2 )−1 = R1−1 ∧ R2−1 Table 3.5 R−1 R−1

(a1 , a1 )

(a1 , a2 )

( a1 , a3 )

(a2 , a1 )

(a2 , a2 )

(a2 , a3 )

(a3 , a1 )

(a3 , a2 )

(a3 , a3 )

h1

0.6

0.5

0.3

0.4

0.4

0.3

0.3

0.3

0.3

h2

0.2

0.1

0.3

0.1

0.1

0.1

0.3

0.1

0.6

h3

0.5

0.3

0.5

0.3

0.3

0.3

0.5

0.3

0.5

c1

0.7

0.3

0.2

0.3

0.3

0.3

0.6

0.3

0.6

c2

0.4

0.2

0.2

0.4

0.4

0.4

0.4

0.4

0.2

3.4 Distributive Laws

49

Table 3.6 Fuzzy soft multiset (F, A)

(F, A)

a

b

c

h1

1

1

1

h2

1

1

1

h3

1

1

1

c1

1

1

1

c2

1

1

1

Proof Let us consider R1 and R2 be two fuzzy soft multi relations on (F, A),∀a, b ∈ A, (R1 ∨ R2 )(−1 (a, b)= )(R1 ∨ R2 )(b, a) = R1 (b, a) ∨ R2 (b, a) = R1−1 (a, b) ∨ R2−1 (a, b) = R1−1 ∨ R21 (a, b) then . Hence (R1 ∨ R2 )−1 = R1−1 ∨ R2−1 . Similarly, we can be proved the other.

3.4.2 Various Types of Fuzzy Soft Multi Relations Definition 3.3.1 A fuzzy soft multi relation R ∈ F S M RU (F, A) is known as reflexive if μ R(a,a) (u) = 1, ∀u ∈ Ui , ∀i ∈ I and ∀a ∈ A. Example 3.3.2 { Give us}a chance to consider two universes U1 = {h 1 , h 2 , h 3 }, U2 = {c1 , c2 }. Let EU1 , EU2 be a collection parameters related { of sets of decision { } } to the above universes, where EU1 = eU1 ,1 , eU1 ,2 , eU1 ,3 , EU2 = eU2 ,1 , eU2 ,2 . Let 2 2 . . U= F S(Ui ), E = EUi and A ⊆ E, such that i=1

i=1

) ( ) ( )} { ( A = a = eU1 ,1 , eU2 ,1 , b = eU1 ,1 , eU2 ,2 , c = eU1 ,2 , eU2 ,1 . Then the fuzzy soft multi relation (R, A × A) on (F, A) as in Table 3.6, is a reflexive fuzzy soft multi relation on (F, A) (Table 3.7). Proposition 3.3.3 R is reflexive on (F, A) if and only if R −1 is reflexive. Proof Let us consider R is reflexive relation on (F, A). Then ∀(a, a) ∈ A × A.

Table 3.7 Reflexive fuzzy soft multi relation R R

(a, a)

(a, b)

(a, c)

(b, a)

(b, b)

(b, c)

(c, a)

(c, b)

(c, c)

h1

1

0.3

0.1

0.1

1

0.4

0.4

1

1

h2

1

0.5

0.7

0.6

1

0.1

0.1

0

1

h3

1

0.8

0.9

1

1

0.3

0.3

0.9

1

c1

1

0.7

0.8

1

1

0.3

0.3

0.8

1

c2

1

0.1

0.5

1

1

0.4

0.4

0

1

50

3 Relation on Fuzzy Soft Multisets

Table 3.8 Symmetric fuzzy soft multi relation R R

(a1 , a1 ) (a1 , a2 ) (a1 , a3 ) (a2 , a1 ) (a2 , a2 )

(a2 , a3 )

(a3 , a1 ) (a3 , a2 ) (a3 , a3 )

h1

0.6

0.3

0.3

0.4

0.3

0.4

0.4

0.3

0.3

h2

0.2

0.1

0.3

0.1

0.1

0.1

0.3

0.1

0.6

h3

0.5

0.3

0.5

0.3

0.3

0.3

0.5

0.3

0.5

c1

0.7

0.3

0.6

0.3

0.3

0.3

0.6

0.3

0.6

c2

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.2

μ R −1 (a,a) (u) = μ R(a,a) (u) = 1, ∀u ∈ Ui , ∀i ∈ I. Thus R −1 is reflexive relation on (F, A). Conversely, let R −1 is reflexive relation on (F, A). Then ∀(a, a) ∈ A × A, μ R(a,a) (u) = μ R −1 (a,a) (u) = 1, ∀u ∈ Ui , ∀i ∈ I. Thus, R is reflexive relation on (F, A). Definition 3.3.4 A fuzzy soft multi relation R ∈ FSMRU (F, A) is said to be symmetric if μ R(a,b) (u) = μ R(b,a) (u), ∀u ∈ Ui , ∀i ∈ I and ∀a ∈ A. Example 3.3.5 Consider the FSMS (F, A) as in Table 3.1. Then the relation R be characterized as in Table 3.8, is a symmetric fuzzy soft multi relation on (F, A). Proposition 3.3.6 R is symmetric fuzzy soft multi relation on (F, A) if and only if R −1 is symmetric. Proof Let us consider R is symmetric fuzzy soft multi relation on (F, A). Then ∀(a, b) ∈ A × A μ R −1 (a,b) (u) = μ R(b,a) (u) = μ R(a,b) (u) = μ R −1 (b,a) (u), ∀u ∈ Ui , ∀i ∈ I. i.e. R −1 (a, b) = R(b, a) = R(a, b) = R −1 (b, a). Thus, R −1 is symmetric fuzzy soft multi relation on (F, A). Conversely, let R −1 is symmetric fuzzy soft multi relation on (F, A). Then ∀(a, b) ∈ A × A, μ R(a,b) (u) = μ R −1 (b,a) (u) = μ R −1 (a,b) (u) = μ R(b,a) (u), ∀u ∈ Ui , ∀i ∈ I. i.e R(a, b) = R −1 (b, a) = R −1 (a, b) = R(b, a). Thus, R is symmetric fuzzy soft multi relation on (F, A). Proposition 3.3.7 R is symmetric fuzzy soft multi relation on (F, A) if and only if R −1 = R.

3.4 Distributive Laws

51

Proof Let us consider R is a symmetric fuzzy soft multi relation on (F, A). Then ∀(a, b) ∈ A × A μ R −1 (a,b) (u) = μ R(b,a) (u) = μ R(a,b) (u), ∀u ∈ Ui , ∀i ∈ I. i.e. R −1 (a, b) = R(b, a) = R(a, b). Thus, R −1 = R. Conversely, let R −1 = R. Then ∀(a, b) ∈ A × A, μ R(a,b) (u) = μ R −1 (a,b) (u) = μ R(b,a) (u), ∀u ∈ Ui , ∀i ∈ I. i.e. R(a, b) = R −1 (a, b) = R(b, a). Thus, R is symmetric fuzzy soft multi relation on (F, A). Definition 3.3.8 Let R1 , R2 ∈ FSMRU (F, A) be two fuzzy soft multi relations on (F, A). Then the composition of R1 and R2 is denoted by R1 oR2 and defined by R1 oR2 = (R1 oR2 , A × A), where R1 oR2 : A× A → U is defined as ∀u ∈ Ui , ∀i ∈ I and ∀a, b, c ∈ A. { ( )} μ R1 OR2 (a,b) (u) = max m μ R1 (a,c) (u), μ R1 (c,b) (u) c

Proposition 3.3.9 If R1 and R2 be two fuzzy soft multi relations on (F, A), then (R1 oR2 )−1 = R2−1 oR1−1 . Proof Let us consider R1 and R2 be two fuzzy soft multi relations on (F, A), then ∀a, b ∈ A, { ( )} −1 −1 m μ μ R2−1 OR−1 (u) = max (u), μ (u) (a,b) R (a,c) R (c,b) 1 2 1 c { ( )} = max m . . . μ R2 (c,a) (u), μ R1 (b,c) (u) c )} { ( = max m = μ R1 (b,c) (u), μ R2 (c,a) (u) c

= μ R1 O R2 (b,a) (u) = μ(R1 OR2 )−1 (a,b) (u), ∀u ∈ Ui , ∀i ∈ I. Hence, (R1 oR2 )−1 = R2−1 oR1−1 . Proposition 3.3.10 If R is symmetric fuzzy soft multi relation on (F, A) then RoR is symmetric on (F, A). Proof Let us consider R is symmetric fuzzy soft multi relation on (F, A). Then R −1 = R. Now (RoR)−1 = R −1 oR −1 = RoR. Hence, RoR is symmetric on (F, A). Proposition 3.3.11 If R1 and R2 be two symmetric fuzzy soft multi relations on (F, A), then R1 oR2 is symmetric on (F, A) if and only if R1 oR2 = R2 oR1 .

52

3 Relation on Fuzzy Soft Multisets

Table 3.9 Transitive fuzzy soft multi relation R on (F, A) R

(a, a)

(a, b)

(a, c)

(b, a)

(b, b)

(b, c)

(c, a)

(c, b)

(c, c)

h1

0.5

0.4

0.4

0.3

0.4

0.3

0.3

0.4

0.5

h2

0.3

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.3

h3

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.3

c1

0.4

0.1

0.1

0.1

0.3

0.1

0.1

0.1

0.4

c2

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

Proof R1 and R2 be two symmetric fuzzy soft multi relations on (F, A), i.e. R1−1 = R1 and R2−1 = R2 . Now (R1 oR2 )−1 = R2−1 oR1−1 . R1 oR2 is symmetric on (F, A) implies R1 oR2 = (R1 oR2 )−1 = R2−1 oR1−1 = R2 oR1 . Conversely, (R1 oR2 )−1 = R2−1 oR1−1 = R2 oR1 = R1 oR2 . So R1 oR2 is symmetric on (F, A). Definition 3.3.12 A fuzzy soft multi relation R ∈ F S M RU (F, A) is said to be transitive if and only if RoR ⊆ R. Example 3.3.13 Consider the FSMS (F, A) as in Table 3.1. Then the relation R be characterized as in Table 3.9, is a transitive fuzzy soft multi relation on (F, A). Proposition 3.3.14 R is transitive relation on (F, A) if and only if R −1 is transitive relations on (F, A). Proof Let us consider R be transitive relation on (F, A), then ∀a, b ∈ A, μ R −1 (a,b) (u) = μ R(b,a) (u) ≥ μ R O R(b,a) (u) { ( )} = max m μ R(b,c) (u), μ R(c,a) (u) c { ( )} = max m μ R(c,a) (u), μ R(b,c) (u) c { ( )} = max m μ R −1 (a,c) (u), μ R −1 (c,b) (u) c

= μ R −1 R −1 (a,b) (u), ∀u ∈ Ui , ∀i ∈ I Hence, R −1 is transitive relations on (F, A).

3.4 Distributive Laws

53

Conversely, let R −1 is transitive relation on (F, A). Then ∀(a, a) ∈ A × A, μ R(a,b) (u) = μ R −1 (b,a) (u) ≥ μ R −1 R −1 (b,a) (u) { ( )} = max m μ R −1 (b,c) (u), μ R −1 (c,a) (u) c { ( )} = max m μ R −1 (c,a) (u), μ R −1 (b,c) (u) c )} { ( = max m μ R(a,c) (u), μ R(c,b) (u) c

= μ R O R(a,b) (u), ∀u ∈ Ui , ∀i ∈ I. Hence, R is transitive relation on (F, A).

Chapter 4

Topology on Fuzzy Soft Multisets

The notion of fuzzy soft multi topologies, as well as their basic features, are discussed in this chapter. The concepts of fuzzy soft multi-open sets, fuzzy soft multi-closed sets, fuzzy soft multi basis, fuzzy soft multi sub basis, neighbourhoods and neighbourhood systems, the interior and closure of a fuzzy soft multiset (FSMS) and their basic features will be introduced and examined. A parameterized family of fuzzy topological spaces is obtained from a fuzzy soft multi topological space. The ideas of fuzzy soft multi topological sub-spaces are also introduced, and some basic features of these concepts are investigated.

4.1 Some Results on Absolute FSMS Let { {Ui : i ∈} I } be a collection of universes, such that ∩i∈I .Ui = φ and let = EUi : i ∈ I be a collection of sets of parameters. Let U i∈I F S(Ui ), where . F S(Ui ) denotes the set of all fuzzy subsets of Ui , E = i∈I EUi and A ⊆ E. We consider an absolute FSMS (F, A) over U and F S M S A (F, A) denote the family of all fuzzy soft multi subsets of (F, A) in which all the parameter set A are the same. Throughout this chapter, (F, A) refers to an initial universal FSMS with fixed parameter set A. Definition 4.1.1 A null FSMS . A over (F, A), is a FSMS in which all the FSMS parts equals φ. Proposition 4.1.2 If (F, A) and (G, A) are two FSMSs in FSMS A ( F, A), then we have the following: ( )c ˜ ˜ A) = (F, A)c ∩(G, (i) (F, A)∪(G, A)c . ( )c c ˜ ˜ A) = (F, A) ∪(G, (ii) (F, A)∩(G, A)c . © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_4

55

56

4 Topology on Fuzzy Soft Multisets

Definition 4.1.3 Let I be an arbitrary index set and {(Fi , A) : i ∈ I } be a subfamily of F S M S A (F, A) [i]. The union of these FSMSs is the FSMSs (H, A), where H (e) = ∪i∈I Fi (e) for each e ∈ A. We write (F, A). [ii]. The intersection of these FSMSs is the FSMSs (M,A) where M(e) = ∩i∈I Fi (e) for each e ∈ A. We write F S M S A ( F, A). Proposition 4.1.4 If (F, A) and (G, A) are two FSMSs in F S M S A ( F, A), then (i) (ii) (iii) (iv)

˜ A = .A, (F, A)∩. ˜ A) = (F, A), (F, A)∩(F, (F, A) ∼ . A = (F, A), ˜ A) = (F, A). (F, A)∪(F,

4.2 Fuzzy Soft Multi Topological Spaces Definition 4.2.1 A subfamily τ of F S M S A (F, A) is called fuzzy soft multitopology on (F, A), if the following axioms are satisfied: [O1 ]. A , ( F, A) ∈ τ, [O2 ]{(Fi , A) : i ∈ I } ⊆ τ ⇒ U˜ i∈I (Fi , A) ∈ τ., [O3 ]I f (F, A), (G, A) ∈ τ, then (F, A) ∩ (G, A) ∈ τ. Then the pair ((F, A), τ ) is called fuzzy soft multi topological space. The members of τ are called fuzzy soft multi-open sets (or τ -open FSMSs or simply open sets) and the conditions [O1 ], [O2 ] and [O3 ] are called the axioms for fuzzy soft multi-open sets. Example 4.2.2 Let us consider there are three universes U1 , U { 2 and U3 . Let }U1 = {h 1 , h 2 , h 3 , h 4 }, , U2 = {c1 , c2 , c3 } and U3 = {v1 , v2 } . . . Let EU1 , EU2 , EU3 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎫ ⎪ ⎪ eU1,1 = modern, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU1 ,2 = quooden, ⎬ EU1 = eU1,3 = in green surroundings , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎪ ⎩e ⎭ = in good repair, U1,5 ⎧ ⎫ ⎪ ⎪ eU2 ,1 = geautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2,2 = new model, ⎨ ⎬ , E U2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = oblack, e U2 ,4 ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition,

4.2 Fuzzy Soft Multi Topological Spaces

E U3

Let U =

3 . i=1

⎧ ⎪ e ⎪ ⎪ U3 ,1 ⎪ ⎪ ⎨ eU3 ,2 = eU3 ,3 ⎪ ⎪ ⎪ ⎪ eU3 ,4 ⎪ ⎩e U3 ,5

57

= = = = =

F S(Ui ), E =

expensive cheap in Kuala Lumpur, majestic, obautiful, 3 .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

EUi , and

i=1

) )} { ( ( A = e1 = eU1 ,1 , eU2 ,1 , eU3 ,1 , e2 = eU1 ,1 , eU2 ,2 , eU3 ,1 . Suppose that . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , .A = e1 , 0 0 0 0 0 0 0 0 0 ( (. . . . { )). c1 C2 C3 v1 v2 } h1 h2 h3 h4 e2 , , , , , , , , , , 0 0 0 0 0 0 0 0 0 . . . ). . (. c1 C2 C3 { v1 v2 } h1 h2 h3 h4 , , , , , , , , , (F, A) = (e1 , 1 1 1 1 1 1 1 1 1 ( (. . . . )). c1 C2 C3 { v1 v2 } h1 h2 h3 h4 e2 , , , , , , , , , , 1 1 1 1 1 1 1 1 1 . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , e1 , (F1 , A) = 0.2 0.4 0.8 0.5 0.8 0.5 0.4 0.8 0.7 ( ( (. . { ). . c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 e2 , , , , , , , , , , 0.7 0.7 1 0.8 0.8 0.6 0.3 0.5 0.4 . { )) .( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , , e1 , (F2 , A) = 0.3 0.3 0.7 0.7 0.8 0.6 0.6 0.9 0.7 ( (. . { )). c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 e2 , , , , , , , , , , 0.8 0.9 1 0.8 0.8 0.8 0.5 0.7 0.6 (F3 , A) = (F1 , A) ∼ (F2 , A) .( (. . )) h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } = e1 , , , , , , , , , , 0.3 0.4 0.8 0.7 0.8 0.6 0.6 0.9 0.7 ( (. . { )). c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 e2 , , , , , , , , , 0.8 0.9 1 0.8 0.8 0.8 0.5 0.7 0.6 ˜ 2 , A) (F4 , A) = (F1 , A)∩(F .( (. . )) h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } = e1 , , , , , , , , , , 0.2 0.3 0.7 0.5 0.8 0.5 0.4 0.8 0.7 ( (. . { )). c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 e2 , , , , , , , , , 0.7 0.7 1 0.8 0.8 0.6 0.3 0.5 0.4

58

4 Topology on Fuzzy Soft Multisets

Then we observe that the sub family τ1 = {. A , (F, A), (F1 , A), (F2 , A), (F3 , A), (F4 , A)} of FS M S ( F, A) is a fuzzy soft multi topology on (F, A), since it satisfies the necessary three axioms [O1 ], [O2 ] and [O3 ] and ((F, A), τ1 ) is a fuzzy soft multi topological space. But the sub family τ2 = {. A , (F, A), (F1 , A), (F2 , A)} of F S M S A (F, A) is ˜ 2 , A) and the not a fuzzy soft multi topology on (F, A) since the union (F1 , A)∪(F ˜ 2 , A) does not belong to τ2 . intersection (F1 , A)∩(F Definition 4.2.3 As every fuzzy soft multi topology on (F, A) must contain the sets . A and (F, A), so the family I = {. A , (F, A)}, forms a fuzzy soft multi topology on (F, A). This topology is called indiscrete fuzzy soft multi topology and the pair ((F, A), I ) is called an indiscrete soft topological space. Definition 4.2.4 Let D denote the family of all fuzzy soft multi subsets of (F, A). Then we observe that D satisfies all the axioms for topology on (F, A). This topology is called discrete fuzzy soft multi topology and the pair ((F, A), D) is called a discrete soft topological space. Proposition 4.2.5 Let ((F, A), τ ) be a fuzzy soft multi topological space over (F, A). Then the collection τe = {F(e) : (F, A) ∈ τ } for each e ∈ A, defines a fuzzy topology on F(e). Proof [O1 ] Since . A , (F, A) ∈ τ implies that ϕ, F(e) ∈ τe , for each e ∈ E. [O2 ] Let {Fi (e) : i ∈ I } ⊆ τe , for some {(Fi , A) : i ∈ I } ⊆ τ . Since U˜ i∈I (Fi , A) ⊆ τ , so U˜ i∈I Fi (e) ⊆ τe , for each e ∈ E. ˜ A) ∈ [O3 ] Let F(e), G(e) ∈ τe , for some (F, A), (G, A) ∈ τ . Since (F, A)∩(G, ˜ τ , so F(e)∩G(e) ∈ τe , for each e ∈ E. Thus, τe defines a fuzzy topology on F(e) for each e ∈ E. Example 4.2.6 Let us consider the fuzzy soft multi topology τ1 = {. A , (F, A), (F1 , A), (F2 , A), (F3 , A), (F4 , A)} as in Example 4.2.6. Then it can be easily seen that τe1 = {ϕ, F(e1 ), F1 (e1 ), F2 (e1 ), F3 (e1 ), F4 (e1 )} and τe2 = {ϕ, F(e2 ), F1 (e2 ), F2 (e2 ), F3 (e2 ), F4 (e2 )} are fuzzy topologies on F(e1 ) and F(e2 ), respectively. Definition 4.2.7 Let ((F, A), τ1 ) and ((F, A), τ2 ) be two fuzzy soft multi topological spaces. If each (F, A) ∈ τ1 ⇒ (F, A) ∈ τ2 , then τ2 is called fuzzy soft multi finer (stronger) topology than τ1 and τ1 is called fuzzy soft multi coarser (or weaker) topology than τ2 . Two fuzzy soft multi topologies, one of which is finer than the other, are said to be comparable.

4.2 Fuzzy Soft Multi Topological Spaces

59

Example 4.2.8 The indiscrete fuzzy soft multi topology on (F, A) is the fuzzy soft multi coarsest (weakest) and discrete fuzzy soft multi topology on (F, A) is the fuzzy soft multi finest (strongest) of all topologies of (F, A). Any other fuzzy soft multi topology on (F, A) will be in between these two fuzzy soft multi topologies. Example 4.2.9 If we consider the fuzzy soft multi topologies τ1 as in Example 4.2.2 and τ3 = {. A , (F, A), (F1 , A)} on (F, A). Then τ1 is fuzzy soft multi finer topology than τ3 and τ3 is fuzzy soft multi coarser topology than τ1 . Theorem 4.2.10 Let {τi : i ∈ I } be an arbitrary collection of fuzzy soft multi topologies on (F, A). Then their intersection ∩i∈I τi is also a fuzzy soft multi topology on (F, A). ˜ E˜ ∈ τi , for each i ∈ I , hence φ, ˜ E˜ ∈ ∩i∈I τi . Proof [O1 ]. Since φ, {( ( ) } ) [O2 ]. Let F k , A : k ∈ K be an arbitrary (family)of FSMSs where F k , A ∈ ∩i∈I τi for each k ∈ K . Then for each i ∈ I, F k , A ∈ τi for k ∈ K and since ) ( for each i ∈ I, τi is a fuzzy soft multi topology, therefore, U˜ k∈K F k , A ∈ τi , for ( ) each i ∈ I . Hence, U˜ k∈K F k , A ∈ ∩i∈I τi [O3 ]. Let (F, A), (G, A) ∈ ∩i∈I τi , then (F, A), (G, A) ∈ τi , for each i ∈ I and since τi is a fuzzy soft multi topology for ˜ ˜ A) ∈ τi for each i ∈ I . Hence, (F, A)∩(G, A) ∈ each i ∈ I , therefore, (F, A)∩(G, ∩i∈I τi . Thus, ∩i∈I τi satisfies all the axioms of topology. Hence ∩i∈I τi forms a topology. But the union of topologies need not be a topology; we can show this with the following example. Example 4.2.11 The union of two fuzzy soft multi topologies may not be a fuzzy soft multi topology. If we consider Example 4.2.2, then the subfamilies τ3 = {. A , (F, A), (F1 , A)} and τ4 = {. A , (F, A), (F2 , A)} are the fuzzy soft multi topologies on (F, A). But their union τ3 ∪ τ4 = {. A , (F, A), (F1 , A), (F2 , A)} = τ2 which is not a fuzzy soft multi topology on (F, A). Definition 4.2.12 Let ((F, A), τ ) be a fuzzy soft multi topological space over (F, A). A fuzzy soft multi subset (F, A) of (F, A) is called fuzzy soft multi closed if its complement (F, A)c is a member of τ . Example 4.2.13 Let us consider Example 4.2.2, then the fuzzy soft multi-closed sets in ((F, A), τ1 ) are . )) h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , = e1 , 1 1 1 1 1 1 1 1 1 ( (. . { ) .. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 e2 , , , , , , , , , , 1 1 1 1 1 1 1 1 1 .( (. . { )) h1 h2 h3 h4 c1 c2 c3 } { v1 v2 } c , (F, A) = e1 , , , , , , , , , 0 0 0 0 0 0 0 0 0 .(

.cA

(.

60

4 Topology on Fuzzy Soft Multisets

( (. . )). h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } e2 , , , , , , , , , , 0 0 0 0 0 0 0 0 0 .( (. )) . h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } c , , , , , , , e1 , , (F1 , A) = 0.8 0.6 0.2 0.5 0.2 0.5 0.6 0.2 0.3 . { )). ( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , e2 , , , , , , , 0.3 0.3 0 0.2 0.2 0.4 0.7 0.5 0.6 . { )) .( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 c , , , , , , , , , e1 , (F2 , A) = 0.7 0.7 0.3 0.3 0.2 0.4 0.4 0.1 0.3 . { . . )). .( (. c1 c2 c3 } h 2 h 3 h 4 { c2 } h1 h2 h3 h4 c , , , , , , , , , , e2 , (F3 , A) = 0.2 0.1 0 0.2 0.2 0.2 0.5 0.6 0.2 0.3 0.4 . { ). )). ( (. } { } c1 c2 c3 c2 v 1 v2 c1 h 2 c2 h 3 h 4 , , , , , , , , , , (e2 , 0.2 0.4 0.1 0.4 0.2 0.2 0.2 0.5 0.3 0.3 0.4 .( (. . { )) h1 h2 h3 h4 c1 c2 c3 } { v1 v2 } c , e1 , , , , , , , , (F4 , A) = 0.8 0.7 0.3 0.5 0.2 0.5 0.6 0.2 0.3 . { )). ( . h1 h2 h3 h4 c1 c2 c3 } { v1 v2 } , , . , , , , , , e2 , 0.3 0.3 0 0.2 0.2 0.4 0.7 0.5 0.6 Definition 4.2.14 Let ((F, A), τ ) be a fuzzy soft multi topological space over (F, A). Then (1) . A and (F, A) are closed FSMSs over (F, A). (2) The intersection of an arbitrary collection of fuzzy soft multi-closed sets is a fuzzy soft multi-closed set over (F, A). (3) The union of any two fuzzy soft multi-closed sets is a fuzzy soft multi-closed set over (F, A). Proof c c (1) Since{(. A , (F, ) A) ∈ τ,}(. A ) and (F, A) are closed. k (2) Let F , A : k ∈ K be an arbitrary family of) fuzzy soft multi-closed ( sets in ((F, A), τ ) and let (F, A) = ∩k∈K F k , A . Now since (F, A)c = ( ( ( ( ))c )c )c ∩k∈K F k , A = ∪k∈K F k , A and F k , A ∈ τ , for each k ∈ K , so ( k )c ( k )c ∪k∈K F , A ∈ τ . Hence, F , A ∈ τ . Thus, (F, A) is a fuzzy soft multi-closed {( )set. } (3) Let F i , A |i = 1, 2, 3, . . . , n be a finite family ( i )of fuzzy soft multi-closed n F , A . Now since (G, A)c = sets in ((F, A), τ ) and let (G, A) = ∪i=1 ( n ( i ))c ( ( ( i )c ) ) c c n n ∪i=1 F , A F i , A and F i , A ∈ τ . So ∩i=1 F , A ∈ τ. = ∩i=1 Hence, (G, A)c ∈ τ . Thus, (G, A) is a fuzzy soft multi-closed set.

4.3 Fuzzy Soft Multi Basis and Sub Basis

61

4.3 Fuzzy Soft Multi Basis and Sub Basis Definition 4.3.1 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and B be a subfamily of τ . If every element of τ can be expressed as the arbitrary fuzzy soft multi union of some element of B, then B is called the fuzzy soft multi basis for the fuzzy soft multi topology τ . Definition 4.3.2 A collection S ⊆ τ is called a subbasis for the fuzzy soft multi topology τ if the set B(S) consisting of finite intersections of elements of S forms a basis for τ . Example 4.3.3 In Example 4.2.2, for the fuzzy soft multi topology τ1 , the subfamily B = {.A , (F, A), (F1 , A), (F2 , A), (F4 , A)} of FSMS A (F, A) is a basis and S = {. A , (F, A), (F1 , A), (F2 , A)} sub basis, since B(S) = {.A , ( F, A), (F1 , A), (F2 , A), (F4 , A)} is a basis for the fuzzy soft multi topology τ1 . Theorem 4.3.4 Let ((F, A), τ ) be fuzzy soft multi topological space on (F, A). A subfamily B of τ forms a base for the fuzzy soft multi topology τ on (F, A) if and ˜ only if (1). (F, A) = ∪{(G, A) : (G, A) ∈ B}. (2). For every (G 1 , A), (G 2 , A) ∈ B, (G1 , A)Q ∩(G2 , A) is the union of members of B. Proof Necessity: Let B be a base for a topology τ on (F, A). ˜ (1) Since (F, A) ∈ τ , we have (F, A) = ∪{(G, A) : (G, A) ∈ B}. (2) If (G 1 , A), (G 2 , A) ∈ B, then (G 1 , A), (G 2 , A) ∈ τ , since B subfamily of τ and since τ is a topology on, therefore (G 1 , A) ∼ (G 2 , A) ∈ τ and thus (G 1 , A) ∩ (G 2 , A) is the union of members of B. Sufficiency: Let B be a family with the given properties and let τ be the family of all unions of members of B. Now if we can prove that τ is a topology on (U, E), then it is obvious that B is a base for this topology. [O1 ].. A , ({ F, A) ∈ τ , since . A ∈ τ})is the union of empty sub-collection from B ˜ and (F, A) ∈ τ , by condition (1). (F, A) = (i.e. . A = ∪˜ (G, A) : (G, A) ∈ . A ⊆ ˜ ∪{(G, A) : (G,( A) ∈ )B}. ( ) [O2 ]. Let Fk , A ∈ τ for all k. By definition of τ , each Fk , A = ( k ) ( ) ˜ ˜ ∪{(G, A) : (G, A) ∈ B} is also A) : (G, A) ∈ B}. Hence, U˜ k F , A = U˜ k ∪{(G, B and so belongs to τ . Thus, τ satisfies the union of members of [O2 ].( ( ) ( ) ) ∈ τ . By definition of τ, F1 , A = [O3 ]. Let F1 , A , F2 , A ( 2 ) ˜ ˜ ∪{(G, A) : (H, A) ∈ B}. A) : (G, A) ∈ B} and F , A = ∪{(H, Hence (

) ( 2 ) ˜ F ,A F 1, A ∩

( ) ( ) ˜ ˜ ∪{(H, ˜ = ∪{(G, A) : (G, A) ∈ B} ∩ A) : (H, A) ∈ B}

62

4 Topology on Fuzzy Soft Multisets

{ } ˜ (G, A)∩(H, ˜ = ∪ A) : (G, A), (H, A) ∈ B . Condition (2) implies that ( 1 ) ( 2 ) ˜ F , A is expressible as the union of the member of B and hence is a F, A ∩ member of τ . The topology τ obtained as above forms a base is called the topology generated by the base B. Since the base, defined as above is a subfamily of τ , i.e. members of base are open, it is called an open base. Theorem 4.3.5 Let B is a basis for fuzzy soft multi topology τ . Denote Be = {F(e) : (F, A) ∈ B} and τe = {F(e) : (F, A) ∈ τ } for any e ∈ A. Then Be is a basis for the fuzzy soft multi topology τe . Proof Let e ∈ A. For any V ∈ τe , V = G(e) for some (G, A) { ∈ τ . Note that B is}a basis for τ . Then there exists B ' ⊆ B such that (G, A) = U˜ (S, A) : (S, A) ∈ B ' . } } { { So G(e) = U˜ S(e) : S(e) ∈ Be' , where Be' = F(e) : (F, A) ∈ B ' ⊆ Be . Thus, Be is the basis for the topology τe for any e ∈ A.

4.4 Neighbourhoods and Neighbourhood Systems Definition 4.4.1 Let τ be the fuzzy soft multi topology on (F, A). A FSMS (F, A) in F S M S A (F, A) is a neighbourhood of a FSMS (G, A) if and only if there exists ˜ an τ -open FSMS(H, A), i.e. (H, A) ∈ τ such that (G, A)⊆(H, A) . (F, A). Example 4.4.2 Let us consider the fuzzy soft multi topology τ1 as in Example 4.2.2, then the FSMS.

(F, A) =

{( ({ h 1 h 2 h 3 h 4 } { c1 c2 c3 } { v1 v2 })) e1 , ({ 0.3 , 0.6 , 0.8 , 0.5} ,{ 0.8 , 0.5 , 0.6} ,{ 0.8 , 0.7}) , h1 h2 h3 h4 c1 c2 c3 v1 v2 , 0.5 , 0.9 , 1 , 0.8 , 0.8 , 0.6 , 0.5 , 0.7 } , (e2 , 0.7

is a neighbourhood of the FSMS {( ({ h 1 h 2 h 3 h 4 } { c1 c2 c3 } { v1 v2 })) , 0.3 , 0.5 , 0.3} ,{ 0.4 , 0.5 , 0.2} ,{ 0.3 , 0.5}))}, (G, A) = ( e1 , ({ 0.1 c1 c2 c3 v1 v2 h1 h2 h3 h4 , 0.3 . , 0.4 , 1 , 0.5 , 0.5 , 0.4 , 0.3 , 0.4 e2 , 0.3 The reason is that there exists an τ1 -open FSMS {( ({ h 1 h 2 h 3 h 4 } { c1 c2 c3 } { v1 v2 })) , 0.8 , 0.5 , 0.8 , 0.4 , 0.8 , 0.7 , (F1 , A) = ( ({e1 , 0.2 } ,{0.5 } ,{0.4 }))} c1 c2 c3 v1 v2 h1 h2 h3 h4 ∈ τ1 , e2 , 0.7 , 0.7 , 1 , 0.8 , 0.8 , 0.6 , 0.3 , 0.5 , 0.4 ˜ ˜ 1 , A)⊆(F, such that (G, A)⊆(F A). Theorem 4.4.3 A FSMS (F, A) in F S M S A ( F, A) is an open FSMS if and only if (F, A) is a neighbourhood of each FSMS (G, A) contained in (F, A).

4.4 Neighbourhoods and Neighbourhood Systems

63

Proof Let (F, A) be an open FSMS and (G, A) be any FSMS contained in (F, A). Since we have (G, A) ⊆ (F, A) ⊆ (F, A), it follows that (F, A) is a neighbourhood of (G, A). Conversely, let (F, A) be a neighbourhood for every FSMSs contained in it. Since ˜ A), there exist an open FSMS (H, A) such that (F, A)⊆(F, ˜ ˜ (F, A)⊆(H, A)⊆(F, A) Hence (F, A) = (H, A) and (F, A) is open. Definition 4.4.4 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (F, A) be a FSMS in FSAM S ( F, A). The family of all neighbourhoods of (F, A) is called the neighbourhood system of (F, A) and is denoted by N(F,A) . Theorem 4.4.5 Let ((F, A), τ ) be a fuzzy soft multi topological space. If N(F,A) is the neighbourhood system of a FSMS (F, A). Then (i) N(F, A) is nonempty and (F, A) belong to each member of N(F,A) (ii) The intersection of any two members of N(F,A) belong to N(F, A) . (iii) Each FSMS contains a member of N(F, A) belong to N(F, A) . Proof (i) If (H, A) ∈ N(F,A) , then there exists a fuzzy soft multi-open set (G, A) ∈ τ such ˜ ˜ ˜ that (F, A)⊆(G, A)⊆(H, A); hence (F, A)⊆(H, A). Note (F, A) ∈ N(F, A) and since (F, A) is an open set containing (F, A); so N(F,A) is nonempty. (ii) Let (G, A) are (H, (F, ( and ) ( A) ) two neighbourhoods ) A), so there exist two ( of ' ' ' G , A , H , A such that , A ⊆ (G, A) and (F, A) ⊆ A) ⊆ G open sets (F, ( ' ) H , A ⊆ (H, A). ( ) ( ' ) ) ( ' ) ( ˜ H , A ⊆ (G, A)∩(H, ˜ ˜ H , A is Hence, (F, A) ⊆ G ' , A ∩ A) and G ' , A ∩ ˜ open. Thus, (G, A)∩(H, A) is a neighbourhoods of (F, A). (iii) Let (G, A) is(a neighbourhood of (F, A) and A) ) ) ⊆ (H, A), so there exists ( (G, ' , A ⊆ (G, A). By hypothesis an open set G ' , A , such that (F, A) ⊆ G ( ' ) so , A ⊆ A) ⊆ (H, A), which implies A) ⊆ G (G, (G, A) ⊆ (H, A), (F, ) ( that (F, A) ⊆ G ' , A ⊆ (H, A) and hence (H, A) is a neighbourhood of (F, A). Definition 4.4.6 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (F, A) be a FSMS in S A ( F, A). A collection B(F,A) ⊆ F S M S A ( F, A) of subsets all containing the FSMS (F, A) is called a neighbourhood basis of (F, A) if (1) Every element of B(F, A) is a neighbourhood of (F, A). (2) Every neighbourhood of (F, A) contains an element of B(F, A) as a subset. Definition 4.4.7 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (F, A) be a FSMS in F S M S A (F, A). Then the union of all fuzzy soft multi open sets contained in (F, A) is called the interior of (F, A) and is denoted by int(F, A) ˜ and defined by int(F, A) = ∪{(G, A)|(G, A) is an open set contained in (F, A)}.

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Example 4.4.8 Let us consider the fuzzy soft multi topology τ1 as in the Example 4.2.2 and let . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , (F, A) = e1 , 0.2 0.4 0.9 0.5 0.8 0.6 0.6 0.9 0.7 . ) .. ( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , e2 , , , , , , , 0.8 0.9 1 0.8 0.8 0.8 0.5 0.7 0.6 be a FSMS, then int (F, A) = U˜ {(G, A) : (G, A) is an open set contained in (F, A)} ˜ 4 , A) = (F1 , A)∪(F .( (. )) . h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } = e1 , , , , , , , , , , 0.2 0.4 0.8 0.5 0.8 0.5 0.4 0.8 0.7 . { )). ( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , . e2 , 0.7 0.7 1 0.8 0.8 0.6 0.3 0.5 0.4 Since (F1 , A) and (F4 , A) are two fuzzy soft multi-open sets contained in (F, A). Theorem 4.4.9 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (F, A) be a FSMS in FF M S ( F, A). Then (i) int(F, A) is an open and int(F, A) is the largest open FSMS contained in (F, A). (ii) The FSMS(F, A) is open if and only if (F, A) = int(F, A). Proof[i]. Since ˜ int(F, A) = ∪{(G, A)|(G, A) is an open set contained in (F, A)}, we have that int(F, A) is itself an interior FSMS of int(F, A). Then there exists an open FSMS ˜ ˜ A)⊆int(F, A). Hence, (H, A) = int(F, A). Thus, (H, A) such that int(F, A)⊆(H, int(F, A) is open and int(F, A) is the largest open FSMS contained in (F, A). [ii]. Let (F, A) be an open FSMS. Since int(F, A) is an interior FSMS of (F, A), we have (F, A) = int(F, A). Conversely, if (F, A) = int(F, A) then (F, A) is open. Proposition 4.4.10 For any two FSMSs (F, A) and (G, A) in a fuzzy soft multi topological space ((F, A), τ ) on (F, A), ˜ ˜ (i) (G, A)⊆(F, A) ⇒ int(G, A)⊆int(F, A) (ii) int(. A ) = . A and int(F, A) = (F, A) (iii) int(int(F, A)) = int(F, A) ∼ ( ) ˜ (iv) int (F, A)∩(G, A) = int(F, A) int(G, A) ˜ ˜ int (F, A)∪int(G, ˜ (v) int((F, A)∪(G, A))⊇ A) ˜ Proof (i) Since (G, A)⊆F, A), implies all the open FSMS contained in (G, A) also contained in (F, A). Therefore.

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65

{( ' )|( ' ) G , A | G , A is an open set contained in (G, A)}. )|( ) {( ˜ F ' , A | F ' , A is an open set contained in (F, A)}. ⊆ This {( implies )|( ) ∼ ˜ G ' , A | G ' , A is an open set contained in (G, A)}. )|( ) {( ˜∪ ˜ F ' , A | F ' , A is an open set contained in (F, A)}. ⊆ So int(G, A) . int(F, A). (ii) Straightforward. (iii) int(int(F, A)) ˜ = ∪{(G, A)|(G, A) is an open set contained in int(F, A)} and since int(F, A) is the largest open FSMS contained in int(F, A), therefore int(int(F, A)) = int(F, A). (iv) Since int((G, A)) ⊆ (G, A) and int((F, A)) ⊆ (F, A), we have, ˜ ˜ ˜ int(G, A)∩int(F, A)⊆(G, A)⊆(F, A). ( ) ˜ ˜ Hence, int(G, A)∼int(F, ˜ A)⊆int A) . (G, A)∩(F, Again since ˜ ˜ ˜ ˜ (G, A)∩(F, A)⊆(G, A)and(G, A)∩(F, A)⊆(F, A), ( ) ˜ we have int (G, A)∩(F, A) ⊆ int(G, A) ( ) ˜ ˜ A) ⊆int(F, A). and int (G, A)∩(F, ( ) ˜ ˜ ˜ A) ⊆int(G, A)∩int(F, A). So int (G, A)∩(F, ˜ Using (1) and (2), we get int((G, A)∼(F, ˜ A)) = int(G, A)∩int(F, A). ˜ ˜ ˜ A) and (F, A)⊆(G, (v) Since(G, A) ⊆ (G, A)∪(F, A)∪(F, A), ∼

˜ int((G, A) (F, A)). SO int(G, A) ⊆ int((G, A) (F, A)) and int(F, A)⊆ ∼( ) ˜ ˜ Hence, int(G, A)∪int(F, A) int (G, A)∪(F, A) . Definition 4.4.11 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (F, A) be a FSMS in F S M S A (F, A). Then the intersection of all closed FSMS containing (F, A) is called the closure of (F, A) and is denoted by cl(F, A) and defined by ˜ cl(F, A) = ∩{(G, A)|(G, A) is a closed set containing (F, A)}. Observe first that Cl(F, A) is a fuzzy soft multi-closed set since it is the intersection of fuzzy soft multi-closed sets. Furthermore, cl(F, A) is the smallest fuzzy soft multi-closed set containing (F, A). Example 4.4.12 Let us consider the fuzzy soft multi topology τ1 as in the Example 4.2.2 and let.

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. )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , e1 , , , , , , , (F, A) = 0.8 0.6 0.1 0.5 0.2 0.4 0.4 0.1 0.3 . )). ( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , e2 , 0.2 0.1 0 0.2 0.2 0.2 0.5 0.3 0.4 be a FSMS, then cl(F, A) = n{(G, ˜ A) : (G, A) is a closed set containing (F, A)} c˜ = (F1 , A) ∩(F4 , A)c . ) .( . h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , ) , = e1 , { 0.8 0.6 0.2 0.5 0.2 0.5 0.6 0.2 0.3 ( (. . . . { ) . h1 h2 h3 h4 c1 c2 C3 v1 v2 } e2 , , , , , , , , , . 0.3 0.3 0 0.2 0.2 0.4 0.7 0.5 0.6 Since (F1 , A)c and (F4 , A)c are only two fuzzy soft multi closed sets in τ1 containing (F, A). Proposition 4.4.13 For any two FSMSs (F, A) and (G, A) in a fuzzy soft multi topological space ((F, A), τ ) on (F, A), (i) (ii) (iii) (iv) (v) (vi) (vii)

Cl. A = . A and cl(F, A) = (F, A). (F, A) . Cl(F, A). (F, A) is a fuzzy soft multi closed set if and only if (F, A) = Cl(F, A). Cl(Cl(F, A)) = Cl(F, A). (G,( A) ⊆ (F, A) ⇒ ) Cl(G, A) . Cl(F, A). ˜ ˜ Cl (F, A)∩(G, A) ⊆ Cl(F, A)∩Cl(G, A). ( ) ˜ ˜ Cl (F, A)∪(G, A) = Cl(F, A)∪Cl(G, A).

˜ A)|(G, A) Proof Proofs of (i)–(iii) are straightforward. (iv) cl(cl(F, A)) = ∪{(G, is a closed set containing cl(F, A)} and since cl(F, A) is the smallest closed FSMS containing cl(F, A), therefore cl(cl(F, A)) = cl(F, A). ˜ (v) Since (G, A)⊆(F, A), implies all the closed sets containing (F, A) also Therefore, contained A). (G, )|( ) {( ∩˜ G ' , A | G ' , A is a closed set containing (G, A)}. {( ' )|( ' ) ˜∩ ˜ F , A | F , A is a closed set containing (F, A)}. ⊆ So cl(G, A) ⊆ cl(F, A). ∼ ˜ ˜ (vi) Since (G, A)(∼ = (G, A)∩(F, )A) and (F, A) = (G, ( A)∩(F, A). ) ∼ ˜ ˜ So cl(G, A) ∼ cl A) ∩(F, A) and cl(F, A) cl A)∩(F, A) . (G, (G, = = ( ) ∼ ˜ ˜ Hence, cl(G, A)∩cl(F, A) = cl (G, A)∩(F, A) . (vii) Since cl(G, A) ∼ = (G, A) and cl(F, A) ∼ = (F, A), we have ∼ ˜ ˜ cl(G, A)∪cl(F, A) = (G, A)∪(F, A).

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67

This implies ( ) ˜ ˜ cl(G, A)∪cl(F, A) ∼ A) . = cl (G, A)∪(F, Again, since ˜ A) ∼ (G, A)∪(F, = (G, A) ˜ and (G,(A)∪(F, A).(F, ˜ )A), ( ) ˜ ˜ so cl (G, A)∪(F, A) ∼ A) ρcl(F, ˜ A). = (G, A) and cl (G, A)∪(F, Therefore, ( ) ˜ ˜ cl (G, A)∪(F, A) ∼ A). = c(G, A)∪cl(F, ( ) ˜ ˜ Using (1) and (2), we get cl (G, A)∪(F, A) = cl(G, A)∪cl(F, A). Theorem 4.4.14 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and let (F, A) be a fuzzy soft multiset in F S M S A (F, A). Then ) ( (i) (cl(F, A))c = int (F, A)c ) ( (ii) (int(F, A))c = cl (F, A)c Proof (i) (cl(F, A))c ˜ = (∩{(G, A)|(G, A) is a closed set containing (F, A)})c { } = U˜ (G, A)c |(G, A) is a closed set containing (F, A) ) ( = int (F, A)c . The other can be proved similarly. Definition 4.4.15 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (F, A) be a FSMS in FMS A ( F, A). Then we defined a FSMS associated with (F, A) over (F, A) is denoted by (cl(F), A) and defined by cl(F)(e) = cl(F(e)), where cl(F(e)) is the closer of F(e) in τe for each e ∈ A. Proposition 4.4.16 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (F, A) be a FSMS in F S M S A (F, A). Then (cl(F), A) ⊆ cl(F, A). Proof For any e ∈ A, Cl(F(e)) is the smallest closed set in (U, τe ), which contains F(e). Moreover, if cl(F, A) = (G, A), then G(e) is also closed set in (U, τe ) containing F(e). This implies that cl(F)(e) = cl(F(e)) ⊆ G(e). Thus, ˜ A). (cl(F), A)⊆cl(F, | } { = ∪˜ (G, A)c |(G, A)c is an open set contained in (F, A)c Corollary 4.4.17 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (F, A) be a FSMS in FSMS A ( F, A). Then (cl(F), A) = cl(F, A) if and only if (cl(F), A)c ∈ τ .

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Proof If (cl(F), A) = cl(F, A), then (cl(F), A) = cl(F, A) is a fuzzy soft multi closed set and so (cl(F), A)c ∈ τ . Conversely, if (cl(F), A)c ∈ τ then (cl(F), A) is a fuzzy soft multi closed set ˜ containing (F, A). By proposition 4.4.16 (cl(F), A)⊆cl(F, A) and by the definition of fuzzy soft multi closure of (F, A), any fuzzy soft multi closed set over (F, A) which contains (F, A) will contain cl(F, A). This implies that cl(F, A) ⊆ (cl(F), A). Thus, (cl(F), A) = cl(F, A).

4.5 Fuzzy Soft Multi Subspace Topology Here we introduce fuzzy soft multi subspace topology and study the basic properties. Theorem 4.5.1 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, { A) and (F, A) be a FSMS } in F S M S A (F, A). Then the collection τ(F, A) = ˜ A)|(G, A) ∈ τ is a fuzzy soft multi topology on the FSMS(F, A). (F, A)∩(G, ˜ Proof [O1 ] Since . A , (F, A) ∈ τ , therefore, (F, A) = (F, A)∩(F, A) and φ˜ (F, A) = ˜ ˜ φ , therefore, , A) ∈ τ A) ∩. (F, A (F, A) (F, (F,A) [O1 ]. {( k ) } [O2 ] Let F , A |k ∈ K be an arbitrary (family )of fuzzy soft multi( open )sets in τ(F,A) , then for each k ∈ K , there exist G k , A ∈ τ such that F k , A = ) ( (F, A)∩˜ G k , A . Now ( ( ( ( ( ) )) )) ˜ G k , A = (F, A)∩˜ U˜ k∈K G k , A U˜ k∈K F k , A = U˜ k∈K (F, A)∩ ( k ) ( ) and since U˜ k∈K U˜)k∈K F k , A ∈ τ(F, A) ( 1G ,) A ∈ (τ ⇒ [O3 ] Let F , A and F 2 , A ( are the ( multi ) two fuzzy soft ) open sets in ( τ(F, A)), then for each i = 1, 2, there exist G i , A ∈ τ such that F i , A = (F, A)∼ ˜ Gi , A . ( ) ( 2 ) ( ( )) ( ( )) ˜ F , A = (F, A)∩˜ G 1 , A ∩˜ (F, A)∼ Now F 1 , A ∩ ˜ G2, A (( ) ( 2 )) ˜ G ,A = (F, A)∼ ˜ G1, A . ( ( ) ( 2 ) ) ( 2 ) ˜ G , A ∈ τ , thus F 1 , A ∼ and since G 1 , A ∩ ˜ F , A ∈ τ(F,A) . Thus, τ(F, A) is a fuzzy soft multi topology on the fuzzy multi set (F, A). Definition 4.5.2 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (F, A) be a FSMS in F S M S A (F, A). Then the fuzzy soft multi topology. { } ˜ τ(F, A) = ((F, A)∩(G, A)|(G, A) ∈ τ is called fuzzy soft multi subspace ) topology and (F, A), τ(F, A) is called fuzzy soft multi topological subspace of ((F, A), τ ).

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69

Example 4.5.3 Let us consider the fuzzy soft multi topological space ((F, A), τ1 ) given in example 4.2 and let a FSMS be . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , e1 , (F, A) = 0.3 0.5 0.8 0.5 0.7 0.5 0.6 0.5 0.7 . { ) (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , }. (e2 , 0.5 0.7 1 0.6 0.7 0.5 0.3 0.5 0.7 Then ˜ A .(F, A) = (F, A)∩. . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , = e1 , 0 0 0 0 0 0 0 0 0 . ). (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , (e2 , , , , , , , 0 0 0 0 0 0 0 0 0 ˜ 1 , A) (G 1 , A) = (F, A)∩(F .( (. . )) h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } = e1 , , , , , , , , , , 0.2 0.4 0.8 0.5 0.7 0.5 0.4 0.5 0.7 . ). (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , (e2 , 0.5 0.7 1 0.6 0.7 0.5 0.3 0.5 0.4 ˜ 2 , A) (G 2 , A) = (F, A)∩(F .( (. . )) h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } = e1 , , , , , , , , , , 0.3 0.3 0.7 0.5 0.7 0.5 0.6 0.5 0.7 ( (. . )). h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } e2 , , , , , , , , , , 0.5 0.7 1 0.6 0.7 0.5 0.3 0.5 0.6 ˜ 3 , A) (G 3 , A) = (F, A)∩(F .( (. . )) h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } = e1 , , , , , , , , , 0.3 0.4 0.8 0.5 0.7 0.5 0.6 0.5 0.7 . { ). (. } { c1 c2 c3 v1 v2 } h1 h2 h3 h4 , , , , , , , , , (e2 , 0.5 0.7 1 0.6 0.7 0.5 0.3 0.5 0.6 ˜ 4 , A) (G 4 , A) = (F, A)∩(F .( (. . )) h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } = e1 , , , , , , , , , , 0.2 0.3 0.7 0.5 0.7 0.5 0.4 0.5 0.7 . ). (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , (e2 , 0.5 0.7 1 0.6 0.7 0.5 0.3 0.5 0.4 { } Then τ1(F, A) = .(F, A) , (F, A), (G 1 , (A), (G 2 , A), (G ) 3 , A), (G 4 , A) is a fuzzy soft multi subspace topology for τ1 and (F, A), τ1(F,A) is called a fuzzy soft multi

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subspace of ((F, A), τ1 ). Theorem 4.5.4. Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and let B be a fuzzy soft multi basis for τ and (F, A) be a FSMS in F S M S A (F, A). Then the family QF,A) = {(F, A)Q ∩(G, A)|(G, A) ∈ D.} is a fuzzy soft multi basis for fuzzy soft multi subspace topology τ(F, A) . Proof Let (H, A) ∈ τ(F, A) , then there exists a FSMS (G, A) ∈ τ , such that ˜ A). Since B is a base for τ , so there exists sub collection (H, A) = (F, A)∩(G, {(G i , A)|i ∈ I } of D such that (G, A) = ∪˜ i∈I (G i , A). Therefore ˜ A) (H, A) = (F, A)∩(G, ) ( ˜ ∪˜ i∈I (G i , A) = (F, A)∩ ( ) ˜ i∈I (F, A)∩(G ˜ i , A) . =∪ ˜ i , A) ∈ QF,A) , which implies that QF,A) is a fuzzy soft multi Since (F, A)∩(G basis for the fuzzy soft multi subspace topology τ(F, A) . Example 4.5.5 If we consider the fuzzy { soft multi subspace topology τ1(F,A) as in} example 4.5.3, then the sub family B = .(F,A) ,{(F, A), (G1 , A), (G2 , A), (G4 , A)} of τ1(F,A) is a basis for the topology τ1(F,A) and S = .(F, A) , (F, A), (G 1 , A), (G 2 , A) is a sub basis for the topology τ1(F,A) . Let ((F, Theorem (( ' ) 4.5.6 ) (( ' A),)τ ) ' )be a fuzzy soft multi topological subspace of ' F F , A , τ be a fuzzy soft multi topological subspace of , A , τ and let (( '' ) '' ) F , A , τ . Then A), τ ) is also ((F, ) (soft'' multi ) topological subspace of (( '' ) '' ) ( 'a fuzzy , A ⊆ F , A , ((F, A), τ ) is a fuzzy soft F , A , τ . Proof Since A) ⊆ F (F, (( ) ) '' = τ . Let (F, A) ∈ τ , now topological subspace of F '' , A , τ '' if and only if τ(F, A)(( ) ) ' since ((F, A), τ ) be a fuzzy soft topological subspace of F ' , A , τ ' , i.e. τ(F ' , A) = ( ' ) (( ) ) ) ( ' ' ' ˜ τ , so there exist F , A ∈ τ such that (F, A) = (F, A)∩ F , A . But F , A , τ ' (( '' ) '' ) ( '' ) be a fuzzy soft topological subspace of F , A , τ , so there exist F , A ∈ τ '' ( ) ( ) ( '' ) ) ( ˜ F , A . Thus, (F, A) = (F, A)∩˜ F ' , A . such that F ' , A = F ' , A ∩ ( ) ( '' ) = (F, A)∼ ˜ F ', A ∼ ˜ F ,A ( ' ) ˜ F' , A . = (F, A)∩ ( ) ) ( '' ˜ F ' , A , so F ' , A ∈ τ '' . Accordingly, τ ⊆ τ(F, Since (F, A)⊆ A) . ' '' Now assume (G, A) ∈ τ(F, , i.e. there exists A) ∈ τ ' such (H, A) ( ' ) ' ' ˜ ˜ A) ∈ τ(F;A) = τ ' , so A). But F , A ∩(H, that (G, A) = (F, A)∩(H, ) (( ' ) ' ˜ A) ∈ τ(F, A) = τ . (F, A)∩˜ F , A ∩(H, ) (( ' ) ˜ ˜ ˜ Since (F, A)∩ F , A ∩(H, A) = (F, A)∩(H, A) = (G, A), we have (G, A) ∈ ' ' τ . Accordingly, τ(F,A) ⊆ τ and thus the theorem is proved.

Chapter 5

Fuzzy Soft Multi Points and Their Sequences

The concepts of fuzzy soft multi points will be taught in this chapter, and their basic features will be examined. We also develop a new sequence of fuzzy soft multi points, and their basic features in fuzzy soft multi topological spaces are examined. Also introduced are the notions of subsequence, convergence sequence and cluster fuzzy soft multi points of fuzzy soft multi points. Cluster analysis, or clustering, is the problem of arranging a set of objects so that objects in the same group (called a cluster) are more similar to each other than objects in other groups (clusters). It is the basic goal of exploratory data mining, and it is a common statistical data analysis technique utilised in a variety of domains like machine learning, pattern recognition, image analysis, information retrieval and bioinformatics. Some key characteristics of the notions discussed above are investigated.

5.1 Fuzzy Soft Multi Points and Their Properties ∩ Let i∈I Ui = φ and let { {Ui : i ∈} I } be a collection of universes, such that . EUi : i ∈ I be a collection of sets of parameters. Let U = i∈I F S(Ui ) where . F S(Ui ) denotes the set of all fuzzy subsets of Ui , E = i∈I EUi and A ⊆ E. We consider an absolute FSMS (F, A) over U and F S M S A (F, A) denote the family of all fuzzy soft multi subsets of (F, A) in which all the parameter set A are the same. Throughout this chapter, (F, A) refers to an initial universal FSMS with fixed parameter set A. Definition 5.1.1 A fuzzy soft multiset (FSMS)(F, A) ∈ FSMSF A ( F, A) is called a fuzzy soft multi point in ) A), denoted by e(F,A) , if for the element e ∈ A, F(e) /= ϕ ( (F, and ∀e' ∈ A − {e}, F e' = ϕ. Example 5.1.2 Let us consider there are three universes U1 ,{U2 and U3 . Let}U1 = {h 1 , h 2 , h 3 , h 4 }, U2 = {c1 , c2 , c3 } and U3 = {v1 , v2 }. Let EU1 , EU2 , EU3 be a collection of sets of decision parameters related to the above universes, where © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_5

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{ } EU1 = eU1 ,1 = expensive, eU1 ,2 = cheap, eU1 ,3 = wooden , { } EU2 = eU2 ,1 = expensive, eU2 ,2 = cheap, eU2 ,3 = sporty , { } EU3 = eU3 ,1 = expensive, eU3 ,2 = cheap, eU3 ,3 = in Kuala Lumpur . Let U =

.3

i=1 F S(Ui ),

E=

.3

i=1 E Ui

and

) )} { ( ( A = e1 = eU1 ,1 , eU2 ,1 , eU3 ,1 , e2 = eU1 ,1 , eU2 ,2 , eU3 ,1 . Then the FSMS . )). .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , e1(F,A) = e1 , 0.5 0.3 0 0 0.7 0.4 0 0 0 is a fuzzy soft multi point. Definition 5.1.3 A fuzzy soft multi point e(F,A) is said to be in the FSMS (G, A) , ˜ ˜ denoted by e(F,A) E(G, A), if (F, A)⊆(G, A). Example 5.1.4 The fuzzy soft multi point e1(F,A) as in the Example 5.1.2, in the FSMS . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 } { v1 v2 } , , , , , , , , e1 , (G, A) = 0.5 0.7 0 0 0.9 0.6 0.6 0 ( (. )). . { h1 h2 h3 h4 c1 c2 c3 } { v1 v2 } , , , , , , , e2 , , , 0 0 0.3 0.4 0.9 0 1 0 0.2 ˜ A), i.e. e(F, A) ∈(G, Proposition 5.1.5 Let e(F,A) be a fuzzy soft multi point and (G, A) be the FSMS in ˜ /˜ A)c . A), then e(F,A) ∈(G, FSMS A ( F, A). If e(F,A) ∈(G, Proof The proof is straightforward. Remark 5.1.6 The converse of the above proposition is not true in general. Example 5.1.7 Let us consider there are three universes U1 ,{U2 and U3 . Let}U1 = {h 1 , h 2 , h 3 , h 4 }, U2 = {c1 , c2 , c3 } and U3 = {v1 , v2 }. Let EU1 , EU2 , EU3 be a collection of sets of decision parameters related to the above universes, where { } EU1 = eU1, 1 = expensive, eU1, 2 = cheap, eU1, 3 = wooden , { } EU2 = eU2, 1 = expensive, eU2, 2 = cheap, eU2, 3 = sporty , { } EU3 = eU3, 1 = expensive, eU3, 2 = cheap, eU3, 3 = in Kuala Lumpur . .3 .3 Let U{ = i=1 ( F S(Ui ), E = ) i=1 EU( i and )} A = e1 = eU1 ,1 , eU2 ,1 , eU3 ,1 , e2 = eU1 ,1 , eU2 ,2 , eU3 ,1 .

5.1 Fuzzy Soft Multi Points and Their Properties

73

We consider a fuzzy soft multi point e1(F,A)

. )). .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , = e1 , 0.5 0.3 0 0 0.7 0.4 0 0 0

and a FSMS . )) .( . . h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , e1 , (G, A) = 0.3 0.9 0.5 0.5 0.9 0.6 0.5 0.6 0.5 ( (. )). . { c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 e2 , , , , , , , , , . 0.5 0.5 0.3 0.4 0.9 0.5 1 0.5 0.2 Then e1(F,A) ∈ / (G, A) and also e1(F,A) ∈ / (G, A)c e1(F, A) ∈ / (G, A)c . )) .( . . h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , (G, A) = e1 , 0.7 0.1 0.5 0.5 0.1 0.4 0.5 0.4 0.5 ( (. . { )). h1 h2 h3 h4 c1 c2 c3 } { v1 v2 } , e2 , , , , , , , , . 0.5 0.5 0.7 0.6 0.1 0.5 0 0.5 0.8 c

Definition 5.1.8 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (G, A) be a FSMS in F S M S A (F, A). The fuzzy soft multi point e(F,A) in FSMS A ( F, A) is called a fuzzy soft multi interior point of a FSMS (G, A) if there ˜ A) ⊆ (G, A). exists a fuzzy soft multi open set (H, A), such that e(F, A) ∈(H, Example 5.1.9 Let us consider there are three universes U1 ,{U2 and U3 . Let}U1 = {h 1 , h 2 , h 3 , h 4 }, U2 = {c1 , c2 , c3 } and U3 = {v1 , v2 }. Let EU1 , EU2 , EU3 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

E U1

E U2

eU1 ,1 = modern eU1 ,2 = quooden = eU1 ,3 = in green surroundings, ⎪ ⎪ ⎪ eU1 ,4 = ocheap ⎪ ⎪ ⎩ eU1 ,5 = in good repair, ⎧ ⎫ ⎪ ⎪ eU2 ,1 = qeautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,2 = new model, ⎨ ⎬ , = eU2 ,3 = osporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = oblack, e U2 ,4 ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

74

5 Fuzzy Soft Multi Points and Their Sequences

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

E U3

Let U =

.3 i=1

eU3 ,1 = expensive, eU3 ,2 = ocheap, = eU3 ,3 = in Kuala Lumpur, ⎪ ⎪ ⎪ eU3 ,4 = majestic ⎪ ⎪ ⎩ eU3 ,5 = qeautiful,

F S(Ui ), E =

.3 i=1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.

EUi and

) )} { ( ( A = e1 = eU1 ,1 , eU2 ,1 , eU3 ,1 , e2 = eU1 ,1 , eU2 ,2 , eU3 ,1 , Suppose that . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , .A = e1 , 0 0 0 0 0 0 0 0 0 . )). ( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , e2 , 0 0 0 0 0 0 0 0 0 . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , , (F, A) = e1 , 1 1 1 1 1 1 1 1 1 ( (. . ). h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } e2 , , , , , , , , , , 1 1 1 1 1 1 1 1 1 . { )) .( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , e1 , (F1 , A) = 0.2 0.4 0.8 0.5 0.8 0.5 0.4 0.8 0.7 . { ). ( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , e2 , 0.7 0.7 1 0.8 0.8 0.6 0.3 0.5 0.4 . { )) .( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , e1 , (F2 , A) = 0.3 0.3 0.7 0.7 0.8 0.6 0.6 0.9 0.7 . { )). ( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , e2 , 0.8 0.9 1 0.8 0.8 0.8 0.5 0.7 0.6 (F3 , A) = (F1 , A) ∼ (F2 , A) = .( (. . )) h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } e1 , , , , , , , , , 0.3 0.4 0.8 0.7 0.8 0.6 0.6 0.9 0.7 . { )). ( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , e2 , 0.8 0.9 1 0.8 0.8 0.8 0.5 0.7 0.6 ˜ 2 , A) (F4 , A) = (F1 , A)∩(F .( (. . )) h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } = e1 , , , , , , , , , 0.2 0.3 0.7 0.5 0.8 0.5 0.4 0.8 0.7 . { )). ( (. c1 c2 c3 } { v1 v2 } h1 h2 h3 h4 , , , , , , , , e2 , 0.7 0.7 1 0.8 0.8 0.6 0.3 0.5 0.4

5.2 Sequence of Fuzzy Soft Multi Points

75

Now we consider the fuzzy soft multi topology τ1 = {. A , (F, A), (F1 , A), (F2 , A), (F3 , A), (F4 , A)} of FSMS A (F, A) and let . )) .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , e1 , , , , , , , (G, A) = 0.3 0.6 0.8 0.5 0.8 0.5 0.7 0.8 0.7 ( (. . ). h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } e2 , , , , , , , , , , 0.7 0.7 1 0.8 0.9 0.6 0.7 0.7 0.5 be a FSMS, then . )). .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , e1(F,A) = e2 , 0.1 0.3 0.8 0.5 0.6 0.5 0.3 0.5 0.4 is a fuzzy soft multi interior point of the FSMS (G, A), since there exist fuzzy soft ˜ 1 , A) ⊆ (G, A), but multi open set (F1 , A), such that e2(F, A) ∈(F e2(F, A)

. )). .( (. h 1 h 2 h 3 h 4 { c1 c2 c3 } { v1 v2 } , , , , , , , , is not a fuzzy = e2 , 0 0 0 0 0.9 0.3 0.5 0.3 0.2

soft multi interior point of the FSMS (G, A), since there does not exist a fuzzy soft ˜ ˜ A)⊆(G, A). multi open set (H, C), such that e2(F,A) ∈(H, Proposition 5.1.10 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and (G, A) be an open FSMS in FSMS A ( F, A). Then every fuzzy soft multi point ˜ A) is a fuzzy soft multi interior point. e(F,A) ∈(G, Proof The proof is straightforward. '

Definition 5.1.11 Two fuzzy soft multi points e(F,A) and e(G,A) are said to be distinct if and only if e and e' are distinct, i.e. e /= e' .

5.2 Sequence of Fuzzy Soft Multi Points Here we introduce a new sequence of fuzzy soft multi points in fuzzy soft multi topological spaces and study their basic properties. Definition 5.2.1 Let ((F, A), τ ) be the fuzzy soft multi topological space and N be the set of all natural numbers. A sequence of fuzzy soft multi { points} in e(Fn , A) or is a mapping from N to FSMS and is denoted by F, A) A), τ ( ((F, ) A } { e(Fn , A) : n = 1, 2, . . . .

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5 Fuzzy Soft Multi Points and Their Sequences

Example 5.2.2 Let us consider there { are three } universes U1 and U2 . Let U1 = {h 1 , h 2 , h 3 } and U2 = {c1 , c2 }. Let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where { } EU1 = eU 1,1 = expensive, eU1 ,2 = cheap, eU1 ,3 = wooden , { } EU2 = eU2 ,1 = expensive, eU2 ,2 = cheap, eU2 ,3 = sporty . Let U =

.2 i=1

F S(Ui ), E =

.2 i=1

EUi and

) )} { ( ( A = e1 = eU1 ,1 , eU2 ,1 , e2 = eU1 ,1 , eU2 ,2 , If we chose for n = 1, 2, 3, . . . . . .). . .( (. h2 h3 c1 c2 h1 , , , , e(Fn , A) = e1 , 2/5n 3/4n 0 0 5/7n { } then e(Fn , A) : n = 1, 2, . . . forms a sequence of fuzzy soft multi points. Definition{ 5.2.3 Let f be mapping over the set of{ positive } } integers. Then the sequence e(G n , A) is a subsequence of a sequence e(Fn , A) if and only if there is a map f such that e(G i , A) = e( F f (i ) , A) and for each integer m, there is an integer n o such that f (i ) ≥ m whenever i ≥ n o . Example 5.2.4 Let us consider } two universes U1 and U2 . Let U1 = {h 1 , h 2 , { there are h 3 } and U2 = {c1 , c2 }. Let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where { } EU1 = eU 1,1 = expensive, eU1 ,2 = cheap, eU1 ,3 = wooden , { } EU2 = eU2 ,1 = expensive, eU2 ,2 = cheap, eU2 ,3 = sporty . Let U =

.2 i=1

F S(Ui ), E =

.2 i=1

EUi and

) ) )} { ( ( ( A = e1 = eU1 ,1 , eU2 ,1 , e2 = eU1 ,1 , eU2 ,2 , e3 = eU1 ,2 , eU2 ,1 . If we chose n = 1, 2, 3, . . . . . .). .( (. c1 c2 h1 h2 h3 , , , , , e(Fn , A) = e1 , 2/5n 0.5 0.7 1/2n 3/7n . . .)). .( (. c1 c2 h1 h2 h3 , , , , , e(G n ,A) = e1 , 1/5n 0.3 0.5 1/3n 2/7n { } { } then e(G n ,A) is a subsequence of the sequence e(Fn , A) .

5.2 Sequence of Fuzzy Soft Multi Points

77

{ } Definition 5.2.5 A sequence e(Fn , A) of fuzzy soft multi points is said to be increasing sequence if and only if for each positive integer n, (Fn , A) . (Fn+1 , A), i.e. if (F1 , A) ⊆ (F2 , A) ⊆ (F3 , A) ⊆ (F4 , A) ⊆ { } Definition 5.2.6 A sequence e(Fn , A) of fuzzy soft multi points is said to be .

decreasing sequence if and only if for each positive integer n, (Fn , A) ⊇(Fn+1 , A) if (F1 , A) . (F2 , A) . (F3 , A) . (F4 , A) . · · · { } Definition 5.2.7 A sequence e(Fn , A) of fuzzy soft multi points is said to be monotonic if and only if the sequence is either an increasing or decreasing sequence. Example 5.2.8 Let us consider { there are } two universes U1 and U2 . Let U1 = {h 1 , h 2 , } }. {c E h 3 and U2 = 1 , c2 Let U1 , EU2 be a collection of sets of decision parameters related to the above universes, where { } EU1 = eU 1,1 = expensive, eU1 ,2 = cheap, eU1 ,3 = wooden , { } EU2 = eU2 ,1 = expensive, eU2 ,2 = cheap, eU2 ,3 = sporty . .2 .2 and A Ui i=1 F S(U ) = ( ) i ), E( = i=1 E)} { Let ( U e1 = eU1 ,1 , eU2 ,1 , e2 = eU1 ,1 , eU2 ,2 , e3 = eU1 ,2 , eU2 ,1 . If we chose n = 1, 2, 3 . . . . . .)). .( (. c1 h1 h2 h3 c2 , , e(Fn , A) = e1 , , , , 1 − 2/5n 0.5 0.7 1 1 − 3/7n . . .)). .( (. c1 c2 h1 h2 h3 , , , , . e(G n , A) = e1 , 1/5n 0.3 0.5 1/3n 2/7n

=

{ } { } Then the sequence e(Fn , A) is increasing sequence and the sequence e(G n , A) is decreasing sequence of fuzzy soft multi points. { } Definition 5.2.9 The complement of a fuzzy soft multi sequence e(Fn ,A) in a fuzzy { }C soft multi topological space ((F, A), τ ) is denoted by e(Fn , A) and is defined by } { { }C e(Fn , A) = e( FnC , A) , where c denotes the fuzzy complement. Example 5.2.10 Let us consider there { are two } universes U1 and U2 . Let Ul = {h 1 , h 2 , h 3 } and U2 = {c1 , c2 }. Let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where { } EU1 = eU 1,1 = expensive, eU1 ,2 = cheap, eU1 ,3 = wooden , { } EU2 = eU2 ,1 = expensive, eU2 ,2 = cheap, eU2 ,3 = sporty . .2 .2 Let U{ = i=1 )} ( F S(Ui ),)E = (i=1 EUi and) ( A = e1 = eU1 ,1 , eU2 ,1 , e2 = eU1 ,1 , eU2 ,2 , e3 = eU1 ,2 , eU2 ,1 .

78

5 Fuzzy Soft Multi Points and Their Sequences

If we chose n = 1, 2, 3 . . . . . .)). .( (. h2 h3 c1 c2 h1 , , , , . e(Fn , A) = e1 , 2/5n 3/4n 0 0 5/7n } { } { }C { Then the complement of e(Fn , A) is e(Fn , A) = e( FnC ,A) , where for n = 1, 2, 3, . . . . . . .( (. . . .). . h1 c1 h2 h3 c2 e1 , e( Fnc , A) = , , , , . 1 − 2/5n 1 − 3/4n 1 1 1 − 5/7n

5.3 Convergence Sequence of Fuzzy Soft Multi Points { } Definition 5.3.1 A sequence e(Fn , A) of fuzzy soft multi points is said to be eventually contained in a FSMS(F, A) if and only if there is a positive integer m such ˜ A). that, ∀n ≥ m implies e(Fn , A) ∈(F, { } Definition 5.3.2 A sequence e(Fn ,A) of fuzzy soft multi points in a fuzzy soft multi topological space ((F, A), τ ) is said to be convergent and converge to a fuzzy soft multi point e(F, A) if it is eventually contained { }in each neighbourhood of the FSMS (F, A) and we say that the sequence e(Fn ,A) has the limit (F, A) and we write limn→∞ (Fn , A) = (limn→∞ Fn , A) = (F, A) or (Fn , A) → (F, A) as n → ∞ or simply Fn → F as n → ∞. Example 5.3.3 Let us consider { there are } two universes U1 and U2 . Let U1 = {h 1 , h 2 , h 3 } and U2 = {c1 , c2 }. Let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where { } EU1 = eU 1,1 = expensive, eU1 ,2 = cheap, eU1 ,3 = wooden , { } EU2 = eU2 ,1 = expensive, eU2 ,2 = cheap, eU2 ,3 = sporty . Let U =

.2 i=1

F S(Ui ), E =

.2 i=1

EUi and

) ) )} { ( ( ( A = e1 = eU1 ,1 , eU2 ,1 , e2 = eU1 ,1 , eU2 ,2 , e3 = eU1 ,2 , eU2 ,1 , If we chose n = 1, 2, 3, . . . . . .)). .( (. h2 h3 c1 c2 h1 , , , , . e(Fn , A) = e1 , 2/5n 3/4n 0 0 5/7n Then

5.4 Cluster Fuzzy Soft Multi Point and Its Properties

79

limn→∞ e(Fn ,A) .( (. . . .)). h2 h3 c1 c2 h1 e1 , , , , , = lim n→∞ 2/5n 3/4n 0 0 5/7n . { )). .( (. } c1 c2 h1 h2 h3 , , , , . = e1 , 0 0 0 0 0 Theorem 5.3.4 If the neighbourhood system of each FSMS in a soft topological space ((F,{ A), τ )}is countable, then a FSMS (F, A) is open if and only if each sequence e(Fn ,A) of fuzzy soft multi points converges to a fuzzy soft multi point e(G,A) contained in (F, A) is eventually contained in (F, A). { } Proof Since (F, A) is open, (F, A) is a neighbourhood of (G, A). Hence, e(Fn ,A) is eventually contained in (F, A). ˜ A) let Conversely, for each e(G, A) ∈(F, e(G 1 , A) , e(G 2 ,A) , . . . , e(G n ,A) , . . . be the fuzzy soft multi points and system of (G, A). Let (G 1 , A), (G 2 , A), . . . , (G n , A), { . . . be the neighbourhood } n ˜ i=1 {(G i , A)}. Then e(Hn , A) : n = 1, 2, . . . is a sequence of fuzzy soft (Hn , A) = ∩ {multi points which is }eventually contained in each neighbourhood of (G, A), i.e. e(Hn , A) : n = 1, 2, . . . converges to e(G, A) . Hence, there is an m such that for n ≥ m, (Hn , A) ⊆ (F, A). Thus (Hn , A) are neighbourhoods of (G, A). This implies (F, A) is a neighbourhood of (G, A) and by Theorem 2.11, (F, A) is a fuzzy soft multi open set. { } Theorem 5.3.5 If a FSMS (F, A) is open FSMS, then each sequence e(Fn , A) . of fuzzy soft multi points which converges to a fuzzy soft multi point e(G, A) contained in the FSMS (F, A) is eventually contained in (F, A). ˜ Proof Since the FSMS (F, A) is open and e(G, A) ⊆(F, A) i.e. ˜ ˜ the FSMS is a neighbourhood of ⊆(F, A), A) A) and (G, A){⊆(F, A) (F, (G, } it is eventually contained A), since e(Fn ,A) converges to a fuzzy soft multi net (G, { } in each neighbourhood of the FSMS (G, A). Hence, e(Fn , A) is eventually contained in the FSMS (F, A).

5.4 Cluster Fuzzy Soft Multi Point and Its Properties Here, we define cluster fuzzy soft multi point of a sequence of fuzzy soft multi points and study its basic properties. { } Definition 5.4.1 A sequence e(Fn , A) of fuzzy soft multi points is said to be frequently contained in a FSMS (F, A) if and only if for each positive integer n, ˜ A). there is a positive integer m, such that ∀n ≥ m implies e(Fn , A) ∈(F,

80

5 Fuzzy Soft Multi Points and Their Sequences

Definition 5.4.2 A fuzzy soft multi point e(F, A) in a fuzzy{ soft topological space } e is a cluster fuzzy soft multi point of a sequence f the sequence A), τ ((F, ) (Fn , A) { } e(Fn , A) is frequently contained in every neighbourhood of (F, A). Theorem 5.4.3 If the neighbourhood system of each FSMS in a fuzzy soft multi topological space ((F, { A), τ }) is countable, then for e(F, A) is a cluster fuzzy soft multi point of a sequence e(Fn ,A) there is a subsequence converging to e(F, A) . Proof Let e(K 1 ,A) , e(K 2 , A) , . . . , e(K n , A) , . . . be fuzzy soft multi points and of (F, A) and let (K 1 , A), (K 2 , A), . . . , (K n , A), . . .{ be neighbourhood system } n ˜ i=1 {(K i , A)}. Then e(Fn , A) : n = 1, 2, . . . is a sequence such that (L n , A) = . contained in each neighbour(Fn+1 , A) ⊆ (Fn , A){ for each n and is eventually } hood of (F, A), i.e. e(Fn , A) : n = 1, 2, . . . converges ) . For every posi( to e(F,A) i, choose f such that f ≥ i and , A ⊆ (L i , A). Then F tive integer (i ) (i ) f (i) { } } { e( F f (i ) , A) : i = 1, 2, . . . is a subsequence of the sequence e(Fn , A) : n = 1, 2, . . . and which converges to e(F, A) . { } Theorem 5.4.4 Let (F, A) be a cluster FSMS of a sequence e(Fn , A) of fuzzy soft multi points and { (F, }A) contained in a FSMS(G, A). If (G, A) is open FSMS, then the sequence e(Fn , A) of fuzzy soft multi points is frequently contained in (G, A). Proof Since the FSMS (G, A) is open and (F, A) is contained in a FSMS(G, A). Hence the FSMS(G, A) is a neighbourhood { } of (F, A). Also, since the FSMS(F, A) is a cluster FSMS of a sequence e(Fn , A){ of fuzzy soft multi points, so by the } contained in every definition of cluster FSMS the sequence e(Fn , A){ is frequently } neighbourhood of the FSMS(F, A) and hence, e(Fn ,A) is frequently contained in the FSMS(G, A).

Chapter 6

Fuzzy Soft Multi Compactness and Separation Axioms

In this chapter, we introduce the concepts of fuzzy soft multi compactness and some basic properties of these concepts are explored. It is seen that if ((F, A), τ ) be a fuzzy soft multi compact space and if (F,( A) be a fuzzy)soft multi closed sets in τ , then the closed fuzzy soft multi subspace (F, A), τ(F,A) of ((F, A), τ ) is fuzzy soft multi compact space. The concepts of fuzzy soft multi separation axioms are introduced. The notion of fuzzy soft multi T0 —space, fuzzy soft multi T1 —space, fuzzy soft multi T2 —space, fuzzy soft multi regular space and fuzzy soft multi normal space are to be introduced and their basic properties are investigated. Also, it is shown that every fuzzy soft multi subspace of a fuzzy soft multi Ti —space, i = 0, 1, 2 is fuzzy soft multi Ti —space, i = 0, 1, 2 and every fuzzy soft multi closed subspace of a fuzzy soft multi normal space is a fuzzy soft multi normal.

6.1 Fuzzy Soft Multi Compact Spaces Definition 6.1.1 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A) and let (F, A) be any fuzzy soft multiset (FSMS) inFSMS A ( F, A). Then a subfamily . of F S M S A (F, A) is called a fuzzy soft multi cover for (F, A) if and only if (F, A) ⊆ ˜ ∪{(G, A) : (G, A) ∈ .} and we say that . covers (F, A). If a subcollection of fuzzy soft multi-cover . also covers (F, A), then it is called a fuzzy soft multi subcover of . for (F, A). If the members of fuzzy soft multi cover . are open, then . is called fuzzy soft multi open cover. If the members of fuzzy soft multi cover . are finite in number, then it is called the finite fuzzy soft multi cover. Definition 6.1.2 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A). A FSMS (F, A) in F S M S A ( F, A) is called fuzzy soft multi compact set if and only if every fuzzy soft multi open cover of (F, A) has a finite fuzzy soft multi subcover. Definition 6.1.3 A fuzzy soft multi topological space ((F, A), τ ) is called fuzzy soft multi compact space if and only if (F, A) is fuzzy soft multi compact. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_6

81

82

6 Fuzzy Soft Multi Compactness and Separation Axioms

Definition 6.1.3 Let ((F, A), τ ) be a fuzzy soft multi topological space on (F, A). A subfamily . of F S M S A (F, A) has the finite intersection property if and only if the interaction of any finite subcollection of . is not null fuzzy soft multi set. Theorem 6.1.4 A fuzzy soft multi topological space ((F, A), τ ) is fuzzy soft multi compact space if and only if every family of closed fuzzy soft multi subsets with finite intersection property has a non-null intersection. Proof Let ((F, A), τ ) be a fuzzy soft multi compact space and let {(Fk , A) : k ∈ K } be an arbitrary family of fuzzy soft multi closed sets in τ with finite intersection property. ˜ complements, ( If possible,)c let ∩k∈K (Fk , A) = . A , then by taking { } c c ˜ ˜ .k∈K (Fk , A) = (. A ) , i.e. Uk∈K (Fk , A) = (F, A). So that (Fk , A)c : k ∈ K forms a fuzzy soft multi open cover for{ (F, A). Since (F, A) is } compact, there is a finite fuzzy soft multi sub-cover (Fi , A)c : i = 1, 2, . . . , n , such that ( n )c n (F, A) =∼i=1 (Fi , A)c . Then by taking complements, (F, A)c = ∼i=1 (Fi , A)c , n ˜ i=1 i.e. . A = ∩ (Fi , A). Thus {(Fk , A) : k ∈ K } does not have the finite intersection ˜ k∈K (Fk , A) /= . A . property, which contrary to our assumption. Hence . Conversely, let every family of closed fuzzy soft multi subsets in ((F, A), τ ) with finite intersection property has a non-null intersection in ((F, A), τ ). Now suppose that ((F, A), τ ) is not fuzzy soft multi compact space. Then there is a fuzzy soft multi open cover {(G k , A) : k ∈ K } of (F, A) that has no finite fuzzy soft(multi subcover, )c n n ˜ i=1 i.e. (F, A) /= U˜ i=1 (G i , A), then by taking complements, (F, A)c /= ∪ (G i , A) , n ˜ i=1 i.e. . A /= . (G i , A)c , which implies {(G k , A) : k ∈ K } has the finite intersection property. But by fuzzy soft multi cover property (F, A) = ∪˜ k∈K (G k , A), then by ˜ k∈K (G k , A)c = . A , i.e. the intersection of all members of taking complements, . the family of fuzzy soft multi closed sets is a null FSMS, which contradicts the given condition. Hence ((F, A), τ ) is fuzzy soft multi compact space.

Theorem 6.1.5 Let ((F, A), τ ) be a fuzzy soft multi compact space and let (F, (A) be a fuzzy )soft multi closed set inτ . Then the closed fuzzy soft multi subspace (F, A), τ(F, A) of ((F, A), τ ) is fuzzy soft multi compact space. Proof Let ((F, A), τ ) be a fuzzy soft {( multi )compact }space and let (F, A) be a fuzzy soft multi closed set in τ . (Let F k , Ak ): k ∈ K be an arbitrary family of fuzzy soft multi closed( sets in) (F, A), τ(F, A) with the finite intersection propin) ((F, A),}τ ); erty. Then the FSMSs F k , Ak for each k ∈ K are closed FSMSs {( since (F, A) is a fuzzy soft multi closed set in τ . Thus F k , Ak : k ∈ K is a family of fuzzy soft multi closed sets in ((F, A), τ ), possessing finite intersection ( property ) and as ((F, A), τ ) is fuzzy soft multi compact, it follows that ˜ k∈K F k , Ak /= . A (by Theorem 6.1.5). This implies that the closed fuzzy soft ∩ ( ) multi subspace (F, A), τ(F, A) of ((F, A), τ ) is fuzzy soft multi compact space.

6.2 Fuzzy Soft Multi Separation Axioms

83

6.2 Fuzzy Soft Multi Separation Axioms Here, we introduce and study various separation axioms for fuzzy soft multi topological space. Definition 6.2.1 A fuzzy soft multi topological space ((F, A), τ ) is said to be a fuzzy ' soft multi T0 —space if for every pair of distinct fuzzy soft multi points e(F,A) , e(G,A) , ∃ a fuzzy soft multi open set containing one but not the other. Example 6.2.2 A discrete fuzzy soft multi topological space is a fuzzy soft multi T0 —space since every e(F,A) is a fuzzy soft multi open set in the discrete space. Theorem 6.2.3 A fuzzy soft multi subspace of a fuzzy soft multi T0 —space is fuzzy soft multi T0 ( ) Proof Let (F, A), τ(F, A) be a fuzzy soft multi subspace of a fuzzy soft multi T0 ' space ((F, A), τ ) and let e(G,A) , e(H,A) be two distinct fuzzy soft multi points of (F, A). Then these fuzzy soft multi points are also in (F, A) ⇒ ∃ a fuzzy soft multi open set (S, A) containing one fuzzy soft multi point, but not the other ⇒ ˜ A), where (S, A) ∈ τ is a fuzzy soft multi open set in τ(F, A) containing (F, A)∩(S, one fuzzy soft multi point but not the other. Definition 6.2.4 A fuzzy soft multi topological space ((F, A), τ ) is said to be a fuzzy ' soft multi T 1 —space if for every distinct pair of fuzzy soft multi points e(F,A) , e(G,A) ˜ A) and of (F, A), ∃ fuzzy soft multi open sets (S, A) and (H, A) such that e(F,A) ∈(S, ' ' ˜ ˜ ˜ / A); e(G,A) ∈(H, / A) and e(G,A) ∈(S, / A). e(F,A) ∈(H, Theorem 6.2.5 A fuzzy soft multi subspace of a fuzzy soft multi T 1 —space is fuzzy soft multi T1 —space. Proof The proof is straightforward. Definition 6.2.6 A fuzzy soft multi topological space ((F, A), τ ) is said to be a fuzzy soft multi T2 —space if and only if for distinct fuzzy soft multi points e(F,A) , ' e(G,A) of (F, A), ∃ disjoint fuzzy soft multi open sets (S, A) and (H, A) such that ' ˜ ˜ A) and e(G,A) ∈(H, A). e(F,A) ∈(S, Theorem 6.2.7 A fuzzy soft multi subspace of a fuzzy soft multi T2 —space is fuzzy soft multi T2 —space. Proof The proof is straightforward. Theorem 6.2.8 A fuzzy soft multi topological space ((F, A), τ ) is fuzzy soft multi ' T2 —space if and only if for distinct fuzzy soft multi points e(F,A) , e(G,A) of (F, A), ∃ ' ' a fuzzy soft multi open set (S, A)containinge(F,A) but not e(G,A) such that e(G,A) ∈ / Cl(S, A).

84

6 Fuzzy Soft Multi Compactness and Separation Axioms '

Proof Let ((F, A), τ ) be fuzzy soft multi T2 —space and e(F,A) , e(G,A) be distinct fuzzy soft points in (F, A). So ∃ distinct fuzzy soft open sets (S, A) and (H, A) such ' ˜ ˜ ˜ A), e(F,A) ∈(H, A) ⇒ (H, A)⊂(S, A)c and (S, A)c is fuzzy soft multi that e(G,A) ∈(S, ' ' / (S, A)c ⇒ e(G,A) closed ⇒ cl(H, A) . Cl((S, A)c ) = (S, A)c and since e(G,A) ∈ ' ∈Cl(H, /˜ A). So (H, A) is a fuzzy soft open set containing e(F,A) but not e(G,A) , such ' /˜ A). that e(G,A) ∈Cl(H, '

Conversely, take a pair of distinct fuzzy soft points e(F,A) and e(G,A) of (F, A), ∃ ' ' a fuzzy soft open set (S, A) containing e(F,A) but not e(G,A) such that e(G,A) ∈ / ' c C ˜ A)) ⇒ (S, A) and (Cl(S, A)) are disjoint fuzzy soft cl(S, A) ⇒ e(G,A) ∈(cl(S, ' open set containing e(F,A) and e(G,A) respectively. Definition 6.2.9 A fuzzy soft multi topological space ((F, A), τ ) is said to be a fuzzy soft multi regular space if for every fuzzy soft multi point e(F, A) and fuzzy soft multi closed set (K , A) not containing e(F,A) , ∃ disjoint fuzzy soft multi open ˜ ˜ A) and (K , A)⊂(H, sets (S, A), (H, A) such that e(F,A) ∈(S, A). A fuzzy soft multi regular T1 —space is called a fuzzy soft multi T3 —space. Theorem 6.2.10 A fuzzy soft multi topological space ((F, A), τ ) in which every fuzzy soft multi point is fuzzy soft multi closed, is fuzzy soft multi regular if and only if for a fuzzy soft multi open set (S, A) containing a fuzzy soft multi pointe(F,A) , there exists a fuzzy soft multi open set (H, A) containing e(F,A) such that cl(H, A) . (S, A). Proof Take a fuzzy soft multi open set (S, A) containing e(F,A) in a regular fuzzy soft multi topological space ((F, A), τ ). Then (S, A)c is fuzzy soft multi closed. By hypothesis, ∃ disjoint fuzzy soft )multi open sets (H, A) and (K , A) such ˜ ˜ , A . Now, (H, A) and (K , A). are disjoint, so that e(F,A) ⊆(H, A) and (S, A)c ⊆K c (H, A) ⊆ (K , A) ⇒ Cl(H, A) . (K , A)c ⇒ Cl(H, A) . (S, A). Conversely, assume the hypothesis. Take a fuzzy soft multi closed set (K , A) not containing a fuzzy soft multi point e(F,A) ∈(K /˜ , A). Then (K , A)c is a fuzzy soft multi open set containing the e(F,A) ⇒ ∃ a fuzzy soft multi open set (S, A) containing ˜ (Cl(S, A))c ⇒ (Cl(S, A))c is a e(F,A) such that cl(S, A) ⊆ (K , A)c ⇒ (K , A)⊂ fuzzy soft multi open set containing (K , A) and (S, A)∼(cl(S, ˜ A))c = . A . Definition 6.2.11 A fuzzy soft multi topological space ((F, A), τ ) is said to be a fuzzy soft multi normal space if for every pair of disjoint fuzzy soft multi closed sets (K , A) and (L , A), ∃ disjoint fuzzy soft multi open sets(S, A), (H, A) such that (K , A) . (S, A) and (L , A) ⊆ (H, A). Theorem 6.2.12 A fuzzy soft multi topological space ((F, A), τ ) is fuzzy soft multi normal if and only if for any fuzzy soft multi closed set (K , A) and fuzzy soft multi open set (H, A) containing(K , A), there exists a fuzzy soft multi open set (S, A) ˜ such that (K , A)⊆(S, A) and cl(S, A) ⊆ (H, A).

6.2 Fuzzy Soft Multi Separation Axioms

85

Proof Let ((F, A), τ ) be fuzzy soft multi normal space and (K , A) be a fuzzy soft multi closed set and (H, A) be a fuzzy soft multi open set containing (K , A) ⇒ (K , A) and (H, A)c are disjoint fuzzy soft multi closed sets ⇒ ∃ disjoint fuzzy soft ˜ (S, A) and (H, A)c ⊆(G, ˜ multi open sets (S, A), (G, A) such that (K , A). A). Now c c ˜ cl((G, A) ) = (G, A)c Also, (H, A)c ⊆(G, ˜ A) ⇒ (S, A) ⊆ (G, A) ⇒ cl(S, A)⊆ ˜ A) ⇒ cl(S, A) ⊆ (H, A). (G, A)c ⊆(H, Conversely, let (L , A) and (K , A) be any disjoint pair fuzzy soft multi closed sets ⇒ (L , A) ⊆ (K , A)c , then by hypothesis there exists a fuzzy soft multi open set ˜ , A)c ⇒ (K , A)⊆(cl(S, ˜ ˜ A))c ⇒ A) and cl(S, A)⊆(K (S, A) such that (L , A)⊆(S, c (S, A) and (cl(S, A)) are disjoint fuzzy soft multi open sets such that (L , A) ⊆ ˜ A))c . (S, A) and (K , A)⊆(cl(S, Theorem 6.2.13 A fuzzy soft multi closed subspace of a fuzzy soft multi normal space is a fuzzy soft multi normal. Proof The proof is straightforward.

Chapter 7

Fuzzy Soft Multiset Based Applications

In this chapter, we have proposed an algorithm (Algorithm 5) to solve fuzzy soft multi set (FSMS) based decision-making problems using Feng’s algorithm (Algorithm 2), which is more stable and more feasible than the Alkhazaleh-Salleh Algorithm. The feasibility of our proposed algorithm in practical applications is illustrated by a numerical example. Also, we modify Feng’s Algorithm and obtain a new algorithm (Algorithm 6) by introducing decision values for solving fuzzy soft set based decision-making problems and using this modified Feng’s Algorithm (Algorithm 6) we solve the FSMS based decision-making. Finally, we have introduced the notion of fuzzy multi-valued information system in FSMS theory. Here we shall present the application of FSMS in information system and show that every FSMS is a fuzzy multi valued information system.

7.1 An Adjustable Approach Based on Feng’s Algorithm Feng’s Algorithm Using Choice Values The details of Feng’s Algorithm [1] for solving a decision-making problem based on a fuzzy soft set are as follows:

7.2 Algorithm 2 (Feng’s Algorithm) Step 1 Input the fuzzy soft set  = (F, A). Step 2 Input a threshold fuzzy set λ : A → [0, 1] (or select a threshold value t ∈ [0, 1] or select mid-level decision criterion or select top-level decision criterion) for solving decision-making problems.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_7

87

88

7 Fuzzy Soft Multiset Based Applications

Step 3 Obtain the level soft set L( ; λ) of  with respect to the threshold fuzzy set λ (or L( ; t) or L( ; mid ) or ( ; max)). Step 4 Present the level soft set L( ; λ) (or L( ; t); or L( ; mid); or L( ; max)) as in tabular form and also, obtain the choice value Si of u i ∈ U, ∀i. Step 5 The final optimal decision to be select u k if Sk = maxi Si . Step 6 If k has more than one value, then any one of u k may be chosen.

Application of FSMSs in Decision-Making Here, we propose an algorithm (Algorithm 5) for FSMSs based decision-making problems, using Feng’s Algorithm [1], as described above. In the following, we have to show our algorithm (Algorithm 5).

7.3 Algorithm 5 Step 1 Input the (resultant) FSMS (F, A). Step 2 Apply Algorithm 2 (Feng’s Algorithm) to the first FSMS part in (F, A), to obtain the decision Sk1 . Step 3 Modify the FSMS (F, A), by taking all values in each row, where the choice value of Sk1 is maximum and changing the values in the other rows by 0 (zero), to get (F, A)1 . Step 4 Apply Algorithm 2 (Feng’s Algorithm) to the second FSMS part in (F, A)1 , to obtain the decision Sk2 . Step 5 Modify the FSMS (F, A)1 , by taking the first two parts fixed and applying the method as in Step 3 to the next part, to get (F, A)2. Step 6 Apply Algorithm 2 (Feng’s Algorithm) to the third FSMS part in (F, A)2 , to obtain the decision Sk3 .   Step 7 Finally, we have the optimal decision for decision-maker is Sk1 , Sk2 , Sk3 .

Application in Decision-Making Problems Let us consider three universes U1 = {h 1 , h 2 , h 3 , h 4 }, U2 = {c1 , C2 , c3 } and U3 = {v1 , v2 , v3 } are sets of houses, cars and hotels respectively and let   EU1 = eU1 ,1 , eU1 ,2 , eU1 ,3 , eU1 ,4 , eU1 ,5 , eU1 ,6 , eU1 ,7 , eU1 ,8 , eU1 ,9 , eU1 ,10  EU2 = eU2 ,1 , eU2 ,2 , eU2 ,3 , eU2 ,4 , eU2 ,5 , eU2 ,6 , eU2 ,7 , eU2 ,8 , eU2 ,9 , eU2 ,10 EU3 = eU3 ,1 , eU3 ,2 , eU3 ,3 , eU3 ,4 , eU3 ,5 , eU3 ,6 , eU3 ,7 , eU3 ,8 , eU3 ,9 , eU3 ,10 be the sets of respective decision parameters related to the above three universes.

7.3 Algorithm 5

Let U =

89

3

i=1 F S(Ui ),

E=

3

i=1 E Ui

and A ⊆ E, such that

     A = a1 = eU1 ,8 , eU2 ,1, eU3 ,5 ,a2 = eU1 ,5 , eU2 ,2 ,eU3 ,3 ,   , a3 = eU1 ,4 , eU2 ,6 , eU3 ,1 ,a4 = eU1 ,1 , eU2 ,8 , eU3 ,10 , a5 = eU1 ,7 , eU2 ,4 , eU3 ,2  a6 = eU1 ,1 , eU2 ,8 , eU3 ,10 , a7 = eU1 ,2 , eU2 ,3 , eU3 ,6 , a8 = eU1 ,9 , eU2 ,7 , eU3 ,9 . Assume that, Mr. X wants to buy a house, a car and rent a hotel with respect to the three sets of decision parameters as in above. Suppose the resultant FSMS (F, A) given in Table 7.1. First, we apply Algorithm 2 (Feng’s Algorithm) to the U1 —FSMS part in (F, A) to obtain the decision from the first FSMS part U1 . Now we represent the U1 —FSMS part in (F, A) as in Table 7.2. In Table 7.3, we see that the maximum choice value (sk ) is s3 = 6 and scored by h 3 . So we modify the FSMS (F, A), by taking all values in each row are fixed, where the choice value of h 3 is maximized and changing the values in other rows by 0 (zero), to get (F, A)1 as in Table 7.4. We apply Algorithm 2 (Feng’s Algorithm) to the U2 —FSMS part in (F, A)1 , to obtain the decision from U2 —FSMS part in (F, A). Now we represent the U2 —FSMS part in (F, A)1 as in Table 7.5. In Table 7.6, we see that the maximum choice value (sk ) is s1 = 3 and scored by C1 . Therefore, we modify the FSMS (F, A)1 by taking all values in each row are Table 7.1 FSMS (F, A) Ui U1

U3

a1

a2

a3

a4

a5

a6

a7

a8

h1

0.3

0.8

1

0.8

0.4

0.9

1

0.8

h2

0.4

0.9

0.8

0.6

0.6

0.6

0.9

0.7

h3

0.9

0.3

0.7

0.1

0.8

0.7

0.8

1

h4

0.7

0.8

0

0.5

0.7

0.5

0.4

0.9

U2

0.8

0.8

0.8

0.5

1

0.8

0.8

0.8

c1

0.6

0.8

0.6

0.3

0.9

0.8

0.8

0.8

c2

0.6

0.5

0.3

0.1

0.9

0.5

0.5

0.5

v1

0.9

0.7

0.5

0.5

0.8

0.8

0.5

0.8

v2

0.7

0.6

0.5

0.3

0.5

0.8

0.6

0.9

v3

0.9

0.5

0.7

0.4

0.4

1

0.8

0.9

Table 7.2 U1 —FSMS part of (F, A)

U1

a1

a2

a3

a4

a5

a6

a7

a8

h1

0.3

0.8

1

0.8

0.4

0.9

1

0.8

h2

0.4

0.9

0.8

0.6

0.6

0.6

0.9

0.7

h3

0.9

0.3

0.7

0.1

0.8

0.7

0.8

1

h4

0.7

0.8

0

0.5

0.7

0.5

0.4

0.9

90

7 Fuzzy Soft Multiset Based Applications

Table 7.3 Mid-level soft set of the U1 —FSMS part in (F, A), with choice values U1

a1

a2

a3

a4

a5

a6

a7

a8

Choice value (sk )

h1

0

1

1

1

0

1

1

0

s1 = 5

h2

0

1

1

1

0

0

1

0

s2 = 4

h3

1

0

1

0

1

1

1

1

s3 = 6

h4

1

1

0

1

1

0

0

1

s4 = 5

Table 7.4 FSMS (F, A)1 Ui U1

U2

U3

a1

a2

a3

a4

a5

a6

a7

a8

h1

0.3

0.8

1

0.8

0.4

0.9

1

0.8

h2

0.4

0.9

0.8

0.6

0.6

0.6

0.9

0.7

h3

0.9

0.3

0.7

0.1

0.8

0.7

0.8

1

h4

0.7

0.8

0

0.5

0.7

0.5

0.4

0.9

c1

0.8

0

0

0

1

0

0

0.8

c2

0.6

0

0

0

0.9

0

0

0.8

c3

0.6

0

0

0

0.9

0

0

0.5

v1

0.9

0

0

0

0.8

0

0

0.8

v2

0.7

0

0

0

0.5

0

0

0.9

v3

0.9

0

0

0

0.8

0

0

0.9

Table 7.5 U2 —FSMS part in (F, A)1 U2

a1

a2

a3

a4

a5

a6

a7

a8

c1

0.8

0

0

0

1

0

0

0.8

c2

0.6

0

0

0

0.9

0

0

0.8

c3

0.6

0

0

0

0.9

0

0

0.5

Table 7.6 Mid-level soft set of the U2 —FSMS part in (F, A)1 , with choice values U2

a1

a2

a3

a4

a5

a6

a7

a8

Choice value (sk )

c1

1

0

0

0

1

0

0

1

s1 = 3

c2

0

0

0

0

0

0

0

1

s2 = 1

c3

0

0

0

0

0

0

0

0

s3 = 0

fixed, where the choice value of C1 is maximized and changing the values in other rows by 0 (zero), to get (F, A)2 as in Table 7.7. Similarly, we apply Algorithm 2 (Feng’s Algorithm) to the U3 —FSMS part in (F, A)2 , to obtain the decision from U3 —FSMS part in (F, A). Now, we represent the U3 —FSMS part in (F, A)2 as in Table 7.8.

7.3 Algorithm 5

91

Table 7.7 FSMS (F, A)2 Ui U1

U2 U3

a1

a2

a3

a4

a5

a6

a7

a8

h1

0.3

0.8

1

0.8

0.4

0.9

1

0.8

h2

0.4

0.9

0.8

0.6

0.6

0.6

0.9

0.7

h3

0.9

0.3

0.7

0.1

0.8

0.7

0.8

1

h4

0.7

0.8

0

0.5

0.7

0.5

0.4

0.9

c1

0.8

0

0

0

1

0

0

0.8

c2

0.6

0

0

0

0.9

0

0

0.8

c3

0.6

0

0

0

0.9

0

0

0.5

v1

0.9

0

0

0

0.8

0

0

0.8

v2

0.7

0

0

0

0.5

0

0

0.9

v3

0.9

0

0

0

0.4

0

0

0.9

Table 7.8 U3 —FSMS part in (F, A)2 v1

0.9

0

0

0

0.8

0

0

0.8

v2

0.7

0

0

0

0.5

0

0

0.9

v3

0.9

0

0

0

0.4

0

0

0.9

Table 7.9 Mid-level soft set of the U3 —FSMS part in (F, A)2 , with choice values U3

a1

a2

a3

a4

a5

a6

a7

a8

Choice value (sk )

v1

1

0

0

0

1

0

0

0

s1 = 2

v2

0

0

0

0

0

0

0

1

s2 = 1

v3

1

0

0

0

0

0

0

1

s3 = 2

In Table 7.9, we see that the maximum choice value (sk ) is 2, scored by v1 and v2 . Thus, the final optimal decision for decision-maker Mr. X is (h 3 , C1 , v1 ) or (h 3 , C1 , v3 ), i.e. Mr. X may chose (h 3 , C1 , v1 ) or (h 3 , C1 , v3 ). Remark 7.1.4 In step (7) of our algorithm (Algorithm 5), if there are too many optimal choices obtained, then decision-maker may go back to step (2) as in our algorithm (Algorithm 5) and replace the level soft set (decision criterion) that he/she once used to adjust the final optimal decision in the FSMS based decision-making problems.

92

7 Fuzzy Soft Multiset Based Applications

7.4 Advantages of Our Algorithm (Algorithm 5) are as Follows (1) From the above illustration, we have seen that our algorithm (Algorithm 5) is too simple and less computation than Alkhazaleh-Salleh Algorithm [2]. Because instead of computing comparison tables and calculating scores as in AlkhazalehSalleh Algorithm [2], we have to consider only choice values of objects from the level soft sets of FSMS parts in the FSMS. (2) Also, our algorithm (Algorithm 5) is an adjustable algorithm, because the level soft set (decision rule) used by decision-makers, which are changeable. For example, if we take top-level decision criterion in step (2) of our algorithm (Algorithm 5), then we have the choice value of each object in the top-level soft set of FSMS parts in the FSMS, if we take another decision rule such as the mid-level decision criterion, then we have choice values from the mid-level soft set of FSMS parts in the FSMS.

7.5 Modified Feng’s Algorithm Here, we modify Feng’s Algorithm and obtain a new algorithm (Algorithm 6) by introducing decision values for solving fuzzy soft set based decision-making problems and using this modified Feng’s Algorithm (Algorithm 6) we solve the FSMS based decision-making. Definition 7.2.1 Suppose  = (F, A) be a fuzzy soft set over U , where A ⊆ E and E is the parameter set. Let L( ; λ) = (Fλ , A) be the level soft set of the fuzzy soft set  with respect to the fuzzy set λ then the decision value can be obtained as di =



μ F(e) (u i ) · μ Fλ (e) (u i ), u i ∈ U.

e∈A

Example 7.2.2 Let us consider three universes U1 = {h 1 , h 2 , h 3 , h 4 }, U2 = {C1 , C2 , C3 } and U3 = {v1 , v2 , v3 } are sets of houses, cars and hotels respectively and let   EU1 = eU1 ,1 , eU1 ,2 , eU1 ,3 , eU1 ,4 , eU1 ,5 , eU1 ,6 , eU1 ,7 , eU1 ,8 , eU1 ,9 , eU1 ,10 , EU2 = eU2 ,1 , eU2 ,2 , eU2 ,3 , eU2 ,4 , eU2 ,5 , eU2 ,6 , eU2 ,7 , eU2 ,8 , eU2 ,9 , eU2 ,10, EU3 = eU3 ,1 , eU3 ,2 , eU3 ,3 , eU3 ,4 , eU3 ,5 , eU3 ,6 , eU3 ,7 , eU3 ,8 , eU3 ,9 , eU3 ,10 be the sets of respective decision parameters related to the above three universes. 3 3 Let U = i=1 F S(Ui ), E = i=1 EUi and A ⊆ E, such that

7.6 Algorithm 6 (Modified Feng’s Algorithm)

93

Table 7.10 FSMS (F, A) Ui U1

U2

U3

a1

a2

a3

a4

a5

a6

a7

a8

h1

0.3

0.8

1

0.8

0.4

0.9

1

0.8

h2

0.4

0.9

0.8

0.6

0.6

0.6

0.9

0.7

h3

0.9

0.3

0.7

0.1

0.8

0.7

0.8

1

h4

0.7

0.8

0

0.5

0.7

0.5

0.4

0.9

c1

0.8

0.8

0.8

0.5

1

0.8

0.8

0.8

c2

0.6

0.8

0.6

0.3

0.9

0.8

0.8

0.8

c3

0.6

0.5

0.3

0.1

0.9

0.5

0.5

0.5

v1

0.9

0.7

0.5

0.5

0.8

0.8

0.5

0.8

v2

0.7

0.6

0.5

0.3

0.5

0.8

0.6

0.9

v3

0.9

0.5

0.7

0.4

0.4

1

0.8

0.9

Table 7.11 Mid-level soft set of the U1 —FSMS part in (F, A), with decision values U1

a1

a2

a3

a4

a5

a6

a7

a8

Choice value (sk )

Decision value (d k )

h1

0

1

1

1

0

1

1

0

s1 = 5

d1 = 4.5

h2

0

1

1

1

0

0

1

0

s2 = 4

d2 = 3.2

h3

1

0

1

0

1

1

1

1

s3 = 6

d3 = 4.9

h4

1

1

0

1

1

0

0

1

s4 = 5

d4 = 3.6

     A = a1 = eU1 ,8 , eU2 ,1, eU3 ,5 ,a2 = eU1 ,5 , eU2 ,2 ,eU3 ,3   , a3 = eU1 ,4 , eU2 ,6 , eU3 ,1 ,a4 = eU1 ,1 , eU2 ,8 , eU3 ,10 , a5 = eU1 ,7 , eU2 ,4 , eU3 ,2  a6 = eU1 ,1 , eU2 ,8 , eU3 ,10 , a7 = eU1 ,2 , eU2 ,3 , eU3 ,6 , a8 = eU1 ,9 , eU2 ,7 , eU3 ,9 . Assume that, Mr. X wants to buy a house, a car and rent a hotel with respect to the three sets of decision parameters as in above. Suppose the resultant FSMS (F, A) given in Table 7.10. If we consider the FSMS as in Table 7.10, the mid-level soft set of the U1 —FSMS part in (F, A), with decision values can be obtained as in Table 7.11.

Modified Feng’s Algorithm Using Decision Values The details of Modified Feng’s Algorithm for solving decision-making problems based on fuzzy soft set are as follows:

7.6 Algorithm 6 (Modified Feng’s Algorithm) Step 1 Input the fuzzy soft set  = (F, A). Step 2 Input a threshold fuzzy set λ : A → [0, 1].

94

7 Fuzzy Soft Multiset Based Applications

Step 3 Obtain the level soft set L( ; λ) of  with respect to the threshold fuzzy set λ. Step 4 Present the level soft set L( ; λ) as in tabular form and also, obtain the decision value di of u i ∈ U, ∀i. Step 5 The final optimal decision to be select u k if dk = maxi di . Step 6 If k has more than one value, then any one of u k may be chosen.

Application of FSMSs Here, we propose another algorithm (Algorithm 7) for FSMSs based decisionmaking, using Modified Feng’s Algorithm, as described above. In the following, we have to show our algorithm (Algorithm 7).

7.7 Algorithm 7 Step 1 Input the (resultant) FSMS (F, A). Step 2 Apply Algorithm 3 (Modified Feng’s Algorithm) to the first FSMS part in (F, A), to obtain the decision Sd1 . Step 3 Modify the FSMS (F, A), by taking all values in each row, where the choice value of Sk1 is maximum and changing the values in the other rows by 0 (zero), to get (F, A)1 . Step 4 Apply Algorithm 1 (Modified Feng’s Algorithm) to the second FSMS part in (F, A)1 , to obtain the decision Sd2 . Step 5 Modify the FSMS (F, A)1 , by taking the first two parts fixed and applying the method as in Step 3 to the next part, to get (F, A)2 . Step 6 Apply Algorithm 1 (Modified Feng’s Algorithm) to the third FSMS part in (F, A)2 , to obtain the decision Sd3 .   Step 7 Finally, we have the optimal decision for decision-maker is Sk1 , Sk2 , Sk3 .

Application in Decision-Making Problems Let us consider three universes U1 = {h 1 , h 2 , h 3 , h 4 }, U2 = {c1 , c2 , c3 } and U3 = {v1 , v2 , v3 } are sets of houses, cars and hotels respectively and let  EU1 = eU1 ,1 , eU1 ,2 , eU1 ,3 , eU1 ,4 , eU1 ,5 , eU1 ,6 , eU1 ,7 , eU1 ,8 , eU1 ,9 , eU1 ,10 , EU2 = eU2 ,1 , eU2 ,2 , eU2 ,3 , eU2 ,4 , eU2 ,5 , eU2 ,6 , eU2 ,7 , eU2 ,8 , eU2 ,9 , eU2 ,10, EU3 = eU3 ,1 , eU3 ,2 , eU3 ,3 , eU3 ,4 , eU3 ,5 , eU3 ,6 , eU3 ,7 , eU3 ,8 , eU3 ,9 , eU3 ,10 be the sets of respective 3 decision parameters 3 related to the above three universes. Let U = i=1 F S(Ui ), E = i=1 EUi and A ⊆ E, such that      A = a1 = eU1 ,8 , eU2 ,1 , eU3 ,5 , a2 = eU1 ,5 , eU2 ,2 , eU3 ,3 ,       a3 = eU1 ,4 , eU2 ,6 , , a4 = eU1 ,1 , eU2 ,8 , eU3 ,10 , a5 = eU1 ,7 , eU2 ,4 , eU3 ,2 ,

7.7 Algorithm 7

95

      a6 = eU1 ,1 , eU2 ,8 , eU3 ,10 , a7 = eU1 ,2 , eU2 ,3 , eU3 ,6 , a8 = eU1 ,9 , eU2 ,7 , eU3 ,9 . Assume that, Mr. X wants to buy a house, a car and rent a hotel with respect to the three sets of decision parameters as in above. Suppose the resultant FSMS (F, A) given in Table 7.12. If we consider the FSMS as in Table 7.12, the mid-level soft set of the U1 - FSMS part in (F, A), with decision values can be obtained as in Table 7.13. In Table 7.13, we see that the maximum decision value (dk ) is s3 = 4.9 and scored by h 3 . So we modify the FSMS (F, A), by taking all values in each row are fixed, where the choice value of h 3 is maximized and changing the values in other rows by 0 (zero), to get (F, A)1 as in Table 7.14. In Table 7.15, we see that the maximum decision value (dk ) is d1 = 2.6 and scored by C1 . Therefore, we modify the FSMS (F, A)1 by taking all values in each row are fixed, where the choice value of C1 is maximized and changing the values in other rows by 0 (zero), to get (F, A)2 as in Table 7.16. Similarly, we apply Algorithm 6 (Modified Feng’s Algorithm) to the U3 —FSMS part in (F, A)2 , to obtain the decision from the U3 —FSMS part in (F, A). Now, we represent the U3 —FSMS part in (F, A)2 as in Table 7.17. In Table 7.17, we see that the maximum decision value (dk ) is 1.8, scored by v3 . Table 7.12 FSMS (F, A) Ui U1

U2

U3

a1

a2

a3

a4

a5

a6

a7

a8

h1

0.3

0.8

1

0.8

0.4

0.9

1

0.8

h2

0.4

0.9

0.8

0.6

0.6

0.6

0.9

0.7

h3

0.9

0.3

0.7

0.1

0.8

0.7

0.8

1

h4

0.7

0.8

0

0.5

0.7

0.5

0.4

0.9

c1

0.8

0.8

0.8

0.5

1

0.8

0.8

0.8

c2

0.6

0.8

0.6

0.3

0.9

0.8

0.8

0.8

c3

0.6

0.5

0.3

0.1

0.9

0.5

0.5

0.5

v1

0.9

0.7

0.5

0.5

0.8

0.8

0.5

0.8

v2

0.7

0.6

0.5

0.3

0.5

0.8

0.6

0.9

v3

0.9

0.5

0.7

0.4

0.4

1

0.8

0.9

Table 7.13 Mid-level soft set of the U1 —FSMS part in (F, A), with decision values U1

a1

a2

a3

a4

a5

a6

a7

a8

Choice value (sk )

Decision value (dk )

h1

0

1

1

1

0

1

1

0

s1 = 5

d1 = 4.5

h2

0

1

1

1

0

0

1

0

s2 = 4

d2 = 3.2

h3

1

0

1

0

1

1

1

1

s3 = 6

d3 = 4.9

h4

1

1

0

1

1

0

0

1

s4 = 5

d4 = 3.6

96

7 Fuzzy Soft Multiset Based Applications

Table 7.14 FSMS (F, A)1 Ui U1

U1 U3

a1

a2

a3

a4

a5

a6

a7

a8

h1

0.3

0.8

1

0.8

0.4

0.9

1

0.8

h2

0.4

0.9

0.8

0.6

0.6

0.6

0.9

0.7

h3

0.9

0.3

0.7

0.1

0.8

0.7

0.8

1

h4

0.7

0.8

0

0.5

0.7

0.5

0.4

0.9

c1

0.8

0

0

0

1

0

0

0.8

c2

0.6

0

0

0

0.9

0

0

0.8

c3

0.6

0

0

0

0.9

0

0

0.5

v1

0.9

0

0

0

0.8

0

0

0.8

v2

0.7

0

0

0

0.5

0

0

0.9

v3

0.9

0

0

0

0.4

0

0

0.9

Table 7.15 Mid-level soft set of the U2 —FSMS part in (F, A)1 , with choice values U2

a1

a2

a3

a4

a5

a6

a7

a8

Choice value (sk )

Decision value (dk )

c1

1

0

0

0

1

0

0

1

s1 = 3

d1 = 2.6

c2

0

0

0

0

0

0

0

1

s2 = 1

d2 = 0.8

c3

0

0

0

0

0

0

0

0

s3 = 0

d3 = 0

Table 7.16 FSMS (F, A)2 Ui U1

U1 U3

a1

a2

a3

a4

a5

a6

a7

a8

h1

0.3

0.8

1

0.8

0.4

0.9

1

0.8

h2

0.4

0.9

0.8

0.6

0.6

0.6

0.9

0.7

h3

0.9

0.3

0.7

0.1

0.8

0.7

0.8

1

h4

0.7

0.8

0

0.5

0.7

0.5

0.4

0.9

c1

0.8

0

0

0

1

0

0

0.8

c2

0.6

0

0

0

0.9

0

0

0.8

c3

0.6

0

0

0

0.9

0

0

0.5

v1

0.9

0

0

0

0.8

0

0

0.8

v2

0.7

0

0

0

0.5

0

0

0.9

v3

0.9

0

0

0

0.4

0

0

0.9

Table 7.17 Mid-level soft set of the U3 —FSMS part in (F, A)2 , with decision values U3

a1

a2

a3

a4

a5

a6

a7

a8

Choice value (sk )

Decision value (dk )

v1

1

0

0

0

1

0

0

0

s1 = 2

d1 = 1.7

v2

0

0

0

0

0

0

0

1

s2 = 1

d2 = 0.9

v3

1

0

0

0

0

0

0

1

s3 = 2

d3 = 1.8

7.8 Application of FSMS Theory in Information Systems

97

Thus, the final optimal decision for decision-maker Mr.X is (h 3 , C1 , v3 ), i.e. Mr. X may chose (h 3 , C1 , v3 ). Remark 7.2.6 In the step 7 of our algorithm (Algorithm 7), if there are too many optimal choices obtained, then decision-maker may go back to the step 2 as in our algorithm (Algorithm 7) and replace the level soft set (decision criterion) that he/she once use to adjust the final optimal decision in the FSMS based decision-making problems.

7.8 Application of FSMS Theory in Information Systems Here we define some basic supporting tools in information systems and also application of FSMSs in the information systems are presented and discussed. Definition 7.3.1 A fuzzy multi-valued information system is a quadruple Inf system = (X, A, f, V ) where  X is a nonempty finite set of objects, A is a nonempty finite set of attribute, V = a∈A Va , where V is the domain (a fuzzy set,) set of attribute, which has multi value and f : X × A → V is a total function such that f (x, a) ∈ Va for every (x, a) ∈ X × A. Proposition 7.3.2 If (F, A) is a FSMS over universe U , then (F, A) is a fuzzy multi-valued information system. Proof Let {Ui : i ∈ I } be a collection of universes such that i∈I Ui = φ and let {E i : i ∈ I } be a collection of sets of parameters. Let U  = i∈I F S(Ui ) where F S(Ui ) denotes the set of all fuzzy subsets of U , E = i i∈I E Ui and A ⊆ E.  Let (F, A) be a FSMS over U and X = i∈I Ui . We define a mapping f where f : X × A → V , defined as f (x, a) = x/μ F(a) (x).  Hence V = a∈A Va where Va is the set of all counts of in F(a) and ∪ represent the classical set union. Then the fuzzy multi-valued information system (X, A, f, V ) represents the FSMS(F, A).

Application in Information System Let us consider three universes U1 = {h 1 , h 2 , h 3 , h 4 , h 5 }, U2 = {c1 , c2 , c3 , c4 } and U3 = {v1 , v2 , v3 } be the sets of “houses,” “cars,” and “hotels”, respectively. Suppose Mr. X has a budget to buy a house, a car and rent a venue to hold a wedding celebration. Let us consider an intuitionistic FSMS (F, A) which describes “houses,” “cars,” and “hotels” that Mr. X is considering for accommodation purchase, transand a venue to hold a wedding celebration, respectively. Let portation purchase,  EU1 , EU2 , EU3 be a collection of sets of decision parameters related to the above universes, where

98

7 Fuzzy Soft Multiset Based Applications

  EU1 = eU1 ,1 , eU1 ,2 , eU1 ,3 , eU1 ,4 , eU1 ,5 , eU1 ,6 , eU1 ,7 , eU1 ,8 , eU1 ,9 , eU1 ,10 , EU2 = eU2 ,1 , eU2 ,2 , eU2 ,3 , eU2 ,4 , eU2 ,5 , eU2 ,6 , eU2 ,7 , eU2 ,8 , eU2 ,9 , eU2 ,10, EU3 = eU3 ,1 , eU3 ,2 , eU3 ,3 , eU3 ,4 , eU3 ,5 , eU3 ,6 , eU3 ,7 , eU3 ,8 , eU3 ,9 , eU3 ,10 be the sets of respective decision parameters to the above three universes. related 3 3 Let U = i=1 F S(Ui ), E = i=1 EUi and A ⊆ E, such that        A = a1 = eU1 ,8 , eU2 ,1 , eU3 ,5 , a2 = eU1 ,5 , eU2 ,2 , eU3 ,3 , a3 = eU1 ,4 , eU2 ,6 , eU3 ,1 ,       a4 = eU1 ,1 , eU2 ,8 , eU3 ,10 , a5 = eU1 ,7 , eU2 ,4 , eU3 ,2 , a6 = eU1 ,1 , eU2 ,8 , eU3 ,10 . Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of choice parameters. Let F(a1 ) = ({h 1 /0.2, h 2 /0.4, h 3 /0, h 4 /0, h 5 /0}, {c1 /0.8, c2 /0.1, c3 /0, c4 /1}, {v1 /0.8, v2 /0.7, v3 /0}), F(a2 ) = ({h 1 /0.3, h 2 /0.5, h 3 /0, h 4 /0.4, h 5 /1}, {c1 /0.9, c2 /0.6, c3 /1, c4 /0}, {v1 /0.6, v2 /0.6, v3 /0}), F(a3 ) = ({h 1 /0.5, h 2 /0.7, h 3 /1, h 4 /0.4, h 5 /0}, {c1 /0.6, c2 /0.1, c3 /0, c4 /1}, {v1 /0, v2 /0.7, v3 /0.7}), F(a4 ) = ({h 1 /0.4, h 2 /0.5, h 3 /0.1, h 4 /0, h 5 /1}, {c1 /0.8, c2 /0, c3 /0.8, c4 /1}, {v1 /1, v2 /0.7, v3 /0}), F(a5 ) = ({h 1 /0.7, h 2 /0.5, h 3 /0.1, h 4 /0.4, h 5 /1}, {c1 /0, c2 /0.1, c3 /0.7, c4 /1}, {v1 /1, v2 /0.8, v3 /0}), F(a6 ) = ({h 1 /0.6, h 2 /0.4, h 3 /0.8, h 4 /1, h 5 /0.1}, {c1 /0, c2 /0.1, c3 /1, c4 /1}, {v1 /1, v2 /0, v3 /1}). Then the FSMS (F, A) defined above describes the conditions of some “house”, “car” and “hotel” in a state. Then the quadruple (X, A, f, V ) corresponding to the FSMS given above 3 is a fuzzy multi-valued information system. Where X = i=1 Ui and A is the set of parameters in the FSMS and Va1 = {h 1 /0.2, h 2 /0.4, h 3 /0, h 4 /0, h 5 /0 , c1 /0.8, c2 /0.1, c3 /0, c4 /1, v1 /0.8, v2 /0.7, v3 /0}, Va2 = {h 1 /0.3, h 2 /0.5, h 3 /0, h 4 /0.4, h 5 /1 , c1 /0.9, c2 /0.6, c3 /1, c4 /0, v1 /0.6, v2 /0.6, v3 /0}, Va3 = {h 1 /0.5, h 2 /0.7, h 3 /1, h 4 /0.4, h 5 /0, c1 /0.6, c2 /0.1, c3 /0, c4 /1, v1 /0, v2 /0.7, v3 /0.7}, Va4 = {h 1 /0.4, h 2 /0.5, h 3 /0.1, h 4 /0, h 5 /1, c1 /0.8, c2 /0, c3 /0.8, c4 /1, v1 /1, v2 /0.7, v3 /0}, Va5 = {h 1 /0.7, h 2 /0.5, h 3 /0.1, h 4 /0.4, h 5 /1, c1 /0, c2 /0.1, c3 /0.7, c4 /1, v1 /1, v2 /0.8, v3 /0}, Va6 = {h 1 /0.6, h 2 /0.4, h 3 /0.8, h 4 /1, h 5 /0.1 , c1 /0, c2 /0.1, c3 /1, c4 /1, v1 /1, v2 /0, v3 /1}.For the pair (h 1 , a1 ) we have f (h 1 , a1 ) = 0.2, for (h 2 , a1 ), we have f (h 2 , a1 ) = 0.4. Continuing in this way we obtain the values of other pairs.

References

99

Table 7.18 An information table X

a1

a2

a3

a4

a5

a6

h1

0.2

0.3

0.5

0.4

0.7

0.6

h2

0.4

0.5

0.7

0.5

0.5

0.4

h3

0

0

1

0.1

1

0.8

h4

0.4

0.4

0.4

0

0.4

1

h5

0

1

0

1

1

0.1

c1

0.8

0.9

0.6

0.8

0

0

c2

0.1

0.6

0.1

0

0.1

0.1

c3

0

1

0

0.8

0.7

1

c4

1

0

1

1

1

1

v1

0.8

0.6

0

1

1

1

v2

0.7

0.6

0.7

0.7

0.8

0

v3

0

0

0.7

0

0

1

Therefore, according to the result above, it is seen that FSMSs are fuzzy soft multivalued information systems. Nevertheless, it is obvious that fuzzy soft multivalued information systems are not necessarily FSMSs. We can construct an information table representing FSMS (F, A) defined above as follows. Table 7.18.

References 1. Atmaca, S., Zorlutuna, I.: On topological structures of fuzzy parametrized soft sets. Sci. World J. Article ID 164176, 8 2. Alkhazaleh, S., Salleh, A.R.: Fuzzy soft multi sets theory. In: Abstract and Applied Analysis, vol. 2012, 20 pages, Hindawi Publishing Corporation (2012)

Chapter 8

Generalization of Fuzzy Soft Multisets

In this chapter, we introduce the concept of intuitionistic fuzzy soft multi set (IFSMS) as a generalization of fuzzy soft multiset (FSMS) and we review some operations in IFSMS theory in a different approach and show that De Morgan’s types of results hold in IFSMS theory for these operations in our way. Basic supporting tools in the information systems are also defined. Applications of IFSMSs in the information systems are presented and discussed. Also, we show that every IFSMS is an intuitionistic fuzzy multi valued information system.

8.1 Intuitionistic Fuzzy Soft Multi Set ∩ Definition { 8.1.1 Let} {Ui : i ∈ I } be a collection of universes, such that. i∈I Ui = φ and let EUi : i ∈ I be a collection of sets of parameters. Let U = i∈I. IFS(Ui ) where IFS(Ui ) denotes the set of all intuitionistic fuzzy subsets of Ui , E = i∈I EUi and A ⊆ E. A pair (F, A) is called an IFSMS over U, where F is a mapping given by F : A → U . ( ) Definition 8.1.2 For any IFSMS (F, A), a pair eUi , j , FeUi , j is called a Ui —IFSMS part ∀eUi , j ∈ ak and FeUi , j ⊆ F(A) is an intuitionistic fuzzy approximate value set, where ak ∈ A, k ∈ {1, 2, 3, . . . , m}, i ∈ {1, 2, 3, . . . , n} and j ∈ {1, 2, 3, . . . , r }. Definition 8.1.3 An IFSMS (F, A) over U is called an absolute IFSMS, denoted by ( ) (F, A)U , if eUi , j , FeUi , j = Ui , ∀i. Definition 8.1.4 A null IFSMS . A over U, is an IFSMS in which all the IFSMS parts equals φ. Definition 8.1.5 Union of two IFSMSs (F, A) and (G, B) over U is an IFSMS (H, D) where D = A ∪ B and ∀e ∈ D, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_8

101

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8 Generalization of Fuzzy Soft Multisets

⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ ∪(F(e), G(e)), if e ∈ A ∩ B ) ( ˜ eU , ∀i ∈ {1, 2, 3, .., m}, j ∈ {1, 2, 3, .., n} where U (F(e), G(e)) = FeUi , j ∪G i, j with U˜ as an intuitionistic fuzzy union and is written as (F, A)∼(G, ˜ B) = (H, D). Definition 8.1.6 Intersection of two IFSMSs (F, A) and (G, B) over U is an IFSMS (H, D) where D = A ∪ B and ∀e ∈ D, ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ ∩(F(e), G(e)), if e ∈ A ∩ B ) ˜ eU , ∀i ∈ {1, 2, 3, . . . , m}, j ∈ {1, 2, 3, .., n} Feei, j ∩G i, j ˜ with ∼ ˜ as an intuitionistic fuzzy intersection and is written as (F, A).(G, B) = (H, D). where

∩

(F(e), G(e)) =

(

Definition 8.1.7 The complement of an IFSMS (F, A) over U is denoted by (F, A)c and is defined by (F, A)c = (Fc , A), where Fc : A → U is a mapping given by Fc (α) = c( F(α)), ∀α ∈ A, where c is an intuitionistic fuzzy complement.

8.2 Results on IFSMSs Here, we review some operations on IFSMSs in a different approach and show that De Morgan’s types of results hold in IFSMS theory for these ∩ operations in our way. {U } Let : i ∈ I be a collection of universes, such that i i∈I Ui = φ and let { } . EUi : i ∈ I be a collection of sets of parameters. Let U = i∈I IFS(U . i ) where IFS(Ui ) denotes the set of all intuitionistic fuzzy subsets of Ui , E = i∈I EUi and A ⊆ E. Definition 8.2.1 A pair (F, A) is called an IFSMS over U, where F is a mapping given by F : A → U , such that ∀e ∈ A, (. F(e) =

u : u ∈ Ui μ F(e) (u), v F(e) (u)

.

) :i ∈I .

Definition 8.2.2 For any IFSMS (F, A), a Ui —IFSMS part of (F, A) over U, is of the form. . . u : u ∈ Ui , e ∈ A . μ F(e) (u), v F(e) (u)

8.2 Results on IFSMSs

103

Definition 8.2.3 The union of two IFSMSs (F, A) and (G, B) over a common universe U is an IFSMS (H, C), where C = A ∪ B and ∀e ∈ C, u ∈ U ⎧ if e ∈ A − B ⎨ μ F(e) (u), μ H (e) (u) = μG(e) (u), if ) e∈B−A ⎩ ( m μ F(e) (u), μG(e) (u) , if e ∈ A ∩ B and ⎧ if e ∈ A − B ⎨ v F(e) (u), v H (e) (u) = vG(e) (u), if ) e∈B−A ⎩ ( m v F(e) (u), vG(e) (u) , if e ∈ A ∩ B. ˜ We write (F, A)∩(G, B) = (H, C). Definition 8.2.4 The intersection of two IFSMSs(F, A) and (G, B) over a common universe U is an IFSMS (H, C), where C = A ∪ B and ∀e ∈ C, u ∈ U ⎧ if e ∈ A − B ⎨ μ F(e) (u), μ H (e) (u) = μG(e) (u), if ) e∈B−A ⎩ ( m μ F(e) (u), μG(e) (u) , if e ∈ A ∩ B and ⎧ if e ∈ A − B ⎨ v F(e) (u), v H (e) (u) = vG(e) (u), if ) e∈B−A ⎩ ( m v F(e) (u), vG(e) (u) , if e ∈ A ∩ B ˜ We write (F, A)∩(G, B) = (H, C). Definition 8.2.5 The complement of an IFSMS (F, A) over U is denoted by (F, A)c c c and is defined ({by (F, A) = (F , A),}where ) u : u ∈ Ui : i ∈ I , ∀e ∈ A. F c (e) = v F(e) (u),μ F(e) (u) Proposition 8.2.6 For two IFSMSs (F, A) and (G, B) over U, then we have [i] ((F, A) ∼ (G, B)),

∼

∼

∼

[i] ((F, A) ∪(G, B))c ⊆(F, A)c ∪(G, B)c , ∼ ∼ [ii] (F, A)c ∩(G, B)c ∼ = ((F, A) ∩(G, B))c . ∼

Proof [i] Let (F, A) ∪ (G, B) = (H, C), where C = A ∪ B and ∀e ∈ C, ⎧ if e ∈ A − B ⎨ μ F(e) (u), μ H (e) (u) = μG(e) (u), if ) e∈B−A ⎩ ( m μ F(e) (u), μG(e) (u) , if e ∈ A ∩ B

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8 Generalization of Fuzzy Soft Multisets

and ⎧ if e ∈ A − B ⎨ v F(e) (u), v H (e) (u) = vG(e) (u), if ) e∈B−A ⎩ ( m v F(e) (u), vG(e) (u) , if e ∈ A ∩ B ˜ Thus ((F, A)∪(G, B))c = (H, C)c = (Hc , C), where C = A ∪ B and ∀e ∈ C, ⎧ if e ∈ A − B ⎨ v F(e) (u), μ H c (e) (u) = v H (e) (u) = vG(e) (u), if ) e∈B−A ⎩ ( m v F(e) (u), vG(e) (u) , if e ∈ A ∩ B and ⎧ if e ∈ A − B ⎨ μ F(e) (u), c v H (e) (u) = μ H (e) (u) = μG(e) (u), ) if e ∈ B − A ⎩ ( m μ F(e) (u), μG(e) (u) , if e ∈ A ∩ B ˜ c , B) = (K , D). Where D = A ∪ B and ˜ Again, (F, A)c ∪(G, B)c = (F c , A)∪(G ∀e ∈ D, ⎧ if e ∈ A − B ⎨ v F(e) (u), μ K (e) (u) = vG(e) (u), if ) e∈B−A ⎩ ( m v F(e) (u), vG(e) (u) , if e ∈ A ∩ B i.e. μ H c (e) (u) ≤ μ K (e) (u) and ⎧ if e ∈ A − B ⎨ μ F(e) (u), v K (e) (u) = μG(e) (u), if ) e∈B−A ⎩ ( m μ F(e) (u), μG(e) (u) , if e ∈ A ∩ B i.e. v H c (e) (u) ≥ v K (e) (u). We see that C = D and ∀e ∈ C, H c (e) ⊆ K (e). ∼

˜ Thus ((F, A) ∪ (G, B))c ⊆ (F, A)c ∪(G, B)c . The other can be proved similarly. Proposition 8.2.7 If (F, A) and (G, A) are two IFSMSs in I F S M S A ( F, A), then we have the following ˜ ˜ (i) ((F, A)∪(G, A))c = (F, A)c ∩(G, A)c , c c ˜ ˜ (ii) ((F, A) ∩(G, A)) = (F, A) ∪(G, A)c . Proof (i) Let (F, A) ∼ (G, B) = (H, C), where C = A ∪ B and ∀e ∈ C,

8.2 Results on IFSMSs

105

⎧ if e ∈ A − B ⎨ μ F(e) (u), μ H (e) (u) = μG(e) (u), if ) e∈B−A ⎩ ( m μ F(e) (u), μG(e) (u) , if e ∈ A ∩ B and ⎧ if e ∈ A − B ⎨ v F(e) (u), v H (e) (u) = vG(e) (u), if ) e∈B−A ⎩ ( m v F(e) (u), vG(e) (u) , if e ∈ A ∩ B Thus ((F, A)U˜ (G, B))c = (H, C)c = (H c , C), where C = A ∪ B and ∀e ∈ C ⎧ if e ∈ A − B ⎨ v F(e) (u), μ H c (e) (u) = v H (e) (u) = vG(e) (u), if ) e∈B−A ⎩ ( m v F(e) (u), vG(e) (u) , if e ∈ A ∩ B and ⎧ if e ∈ A − B ⎨ μ F(e) (u), v H c (e) (u) = μ H (e) (u) = μG(e) (u), if ) e∈B−A ⎩ ( m μ F(e) (u), μG(e) (u) , if e ∈ A ∩ B ˜ c , B) = (K , D). Where D = A ∪ B ˜ Also, let (F, A)c ∩(G, B)c = (F c , A)∩(G and ∀e ∈ D ⎧ if e ∈ A − B ⎨ v F(e) (u), μ K (e) (u) = vG(e) (u), if ) e∈B−A ⎩ ( m v F(e) (u), vG(e) (u) , if e ∈ A ∩ B = μ H c (e) (u), and ⎧ if e ∈ A − B ⎨ μ F(e) (u), v K (e) (u) = μG(e) (u), if ) e∈B−A ⎩ ( m μ F(e) (u), μG(e) (u) , if e ∈ A ∩ B = v H c (e) (u) We see that C = D and ∀e ∈ C, H c (e) = K (e). Hence the result. The other can be proved similarly. Definition 8.2.8 An IFSMS (F, A) over U is called an absolute IFSMS, denoted by (F, A)U , if ∀e ∈ A, μ F(e) (u) = 1 and v F(e) (u) = 0, ∀u ∈ Ui , i ∈ I .

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8 Generalization of Fuzzy Soft Multisets

Definition 8.2.9 A null IFSMS (F, A)U over U is an IFSMS in which ∀e ∈ A, μ F(e) (u) = 0 and v F(e) (u) = 1, ∀u ∈ Ui , i ∈ I . Proposition 8.2.10 If (F, A) be any IFSMS in over U , then (i) (ii) (iii) (iv)

˜ (F, A)∩(G, B)φ = (G, B)φ , ˜ (F, A)∩(H, C)U = (F, A), (F, A) ∼ (G, B)φ = (F, A), (F, A) ∼ (H, C)U = (H, C)U.

8.3 A Study on IFSMSs Definition 8.3.1 An IFSMS (F, A) over U is called an IFSM-subset of an IFSMS (G, B) if (a) A ⊆ B and (b) ∀e ∈ A, F(e) ⊆ G(e) ⇐⇒ μ F(e) (u) ≤ μG(e) (u) and v F(e) (u) ≥ vG(e) (u), ∀u ∈ Ui , i ∈ I and this relationship is denoted by ˜ (F, A)⊆(G, B). Definition 8.3.2 The restricted union of two IFSMSs(F, A) and (G, B) over U is an IFSMS(H, C) where C = A ∩ B and ∀e ∈ C, H (e) = =

. (.

(F(e), G(e)) u ( { } { }) : u ∈ Ui m μ F(e) (u), μG(e) (u) , m v F(e) (u), vG(e) (u)

.

) :i ∈I .

and is written as (F, A)U˜ R (G, B) = (H, C). Definition 8.3.3 The extended union of two IFSMSs (F, A) and (G, B) over U is an IFSMS (H, D), where D = A ∪ B and ∀e ∈ D, ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩U (F(e), G(e)), if e ∈ A ∩ B where U (F(e), G(e)) = (.

u ( { } { }) : u ∈ Ui m μ F(e) (u), μG(e) (u) , m v F(e) (u), vG(e) (u)

.

) :i ∈I

˜ E (G, B) = (H, D). and is written as (F, A)∪ Definition 8.3.4 The restricted intersection of two IFSMSs (F, A) and (G, B) over U is an IFSMS (H, D) where D = A ∩ B and ∀e ∈ D,

8.3 A Study on IFSMSs

H (e) =

107

(.

∩(F(e), G(e)) =

u ( { } { }) : u ∈ Ui m μ F(e) (u), μG(e) (u) , m v F(e) (u), vG(e) (u)

.

) :i ∈I

˜ R (G, B) = (H, D). and is written as (F, A)∩ Definition 8.3.5 The extended intersection of two IFSMSs (F, A) and (G, B) over U is an IFSMS (H, D), where D = A ∪ B and ∀e ∈ D, ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ ∩(F(e), G(e)), if e ∈ A ∩ B where ∩(F(e), G(e)) = (.

u ( { } { }) : u ∈ Ui m μ F(e) (u), μG(e) (u) , m v F(e) (u), vG(e) (u)

.

) :i ∈I

˜ E (G, B) = (H, D). and is written as (F, A)∩ Proposition 8.3.6 For two IFSMSs (F, A) and (G, B) over U, then we have ( )c ∼ ∼ ∼ [i] (F, A) ∪ E (G, B) ⊆(F, A)c ∪ E (G, B)c , ( )c ∼ ∼ [ii] (F, A)c ∩ E (G, B)c ⊆ (F, A) ∩ E (G, B) . Proof [i] Let (F, A)∪˜ E (G, B) = (H, C), where C = A ∪ B and ∀e ∈ C, ⎧ if e ∈ A − B ⎨ F(e), H (e) = G(e), if e ∈ B − A ⎩ U(F(e), G(e)), if e ∈ A ∩ B where U (F(e), G(e)) = (.

u ( { } { }) : u ∈ Ui max μ F(e) (u), μG(e) (u) , min v F(e) (u), vG(e) (u)

.

) :i ∈I .

( )c Thus (F, A)U˜ E (G, B) = (H, C)c = (H c , C), where C = A ∪ B and ∀e ∈ C, ⎧ c if e ∈ A − B ⎨ F (e), H c (e) = Gc (e), if e ∈ B − A ⎩ U ( (F(e), G(e)))c , if e ∈ A ∩ B

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8 Generalization of Fuzzy Soft Multisets

where (U (F(e), G(e)))c = (.

u ( { } { }) : u ∈ Ui min v F(e) (u), vG(e) (u) , max μ F(e) (u), μG(e) (u)

.

) :i ∈I .

Again, (F, A)c U˜ E (G, B)c = (F c , A)U˜ E (G c , B) = (K , D). Where D = A ∪ B and ∀e ∈ D, ⎧ c if e ∈ A − B ⎨ F (e), c K (e) = G (e), if e ∈ B − A ⎩U c (F (e), G c (e)), if e ∈ A ∩ B where ( ) U F c (e), G c (e) (. =

u ( { } { }) : u ∈ Ui m v F(e) (u), vG(e) (u) , m μ F(e) (u), μG(e) (u)

.

) :i ∈I .

We see ∈ C, H c (e) ⊆ K (e). ( that C = D and )∀e c ˜ Thus (F, A)∪˜ E (G, B) ⊆(F, A)c ∪˜ E (G, B)c . The other can be proved similarly. Proposition 8.3.7 For two I F S M Ss(F, A) and (G, B) over U , then we have ( )c ∼ ∼ ∼ [i] (F, A) ∪ R (G, B) ⊆(F, A)c U R (G, B)c , )c ∼( ∼ ∼ [ii] (F, A)c ∩ R (G, B)c ⊆ (F, A) ∩ R (G, B) . Proof Straightforward Proposition 8.3.8 For two I F S M Ss(F, A) and (G, B) over U, then we have ( )c ∼ ∼ [i] (F, A) ∩ R (G, B) ⊆ (F, A)c ∪ E (G, B)c , ∼

[ii] (F, A)c . R (G, B)c ⊆ ((F, A) ∼ E (G, B))c . Proof Straightforward Proposition 8.3.9 (De Morgan Laws) For two IFSMSs (F, A) and (G, B) over U, then we have ( )c ∼ ∼ [i] (F, A) ∪ R (G, B) = (F, A)c . R (G, B)c , ( )c ∼ ∼ [ii] (F, A) . R (G, B) = (F, A)c ∪ R (G, B)c .

8.3 A Study on IFSMSs

109

∼

Proof [i] Let (F, A)∪ R (G, B) = (H, C), where C = A ∩ B and ∀e ∈ C, H (e) = (. =

.

(F(e), G(e))

u ( { } { }) : u ∈ Ui m μ F(e) (u), μG(e) (u) , m v F(e) (u), vG(e) (u)

.

) :i ∈I .

( )c Thus (F, A)U˜ R (G, B) = (H, C)c = (H c , C), where C = A ∩ B and ∀e ∈ C, (. H (e) = c

u ( { } { }) : u ∈ Ui m v F(e) (u), vG(e) (u) , m μ F(e) (u), μG(e) (u)

.

) :i ∈I .

˜ R (G c , B) = (K , D). Where D = A ∩ B ˜ R (G, B)c = (F c , A)∩ Again, (F, A)c ∩ and ∀e ∈ D, .( ) F c (e), G c (e) K (e) = . ) (. u ( { } { }) : u ∈ Ui : i ∈ I . = m v F(e) (u), vG(e) (u) , m μ F(e) (u), μG(e) (u) We see that C = D and ∀e ∈ C, H c (e) = K (e). Hence the result. The other can be proved similarly. Proposition 8.3.10 (De Morgan Laws) For two IFSMSs (F, A) and (G, B) over U , then we have ( )c ∼ ∼ [i] (F, A) ∪ E (G, B) = (F, A)c ∩ E (G, B)c , )c ( ∼ ∼ [ii] (F, A) ∩ E (G, B) = (F, A)c ∪ E (G, B)c . Proof Straightforward Definition 8.3.11 If (F, A) and (G, B) be two IFSMSs over U , then “(F, A) AND (G, B)” is an IFSMS denoted by (F, A)∧(G, B) and is defined by (F, A)∧(G, B) = (H, A × B), where H is mapping given by H : A × B → U and ∀(a, b) ∈ A × B, H (a, b) = (. =

.

(F(a), G(b))

u ( { } { }) : u ∈ Ui m μ F(a) (u), μG(b) (u) , m v F(a) (u), vG(b) (u)

.

) :i ∈I .

Definition 8.3.12 If (F, A) and (G, B) be two IFSMSs over U , then “F, A) OR (G, B)” is an IFSMS denoted by (F, A)∨(G, B) and is defined by (F, A)∨(G, B) = (K , A × B), where K is mapping given by K : A × B → U and ∀(a, b) ∈ A × B

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8 Generalization of Fuzzy Soft Multisets

K (a, b) = (. =

.

(F(a), G(b))

u ( { } { }) : u ∈ Ui m μ F(a) (u), μG(b) (u) , m v F(a) (u), vG(b) (u)

.

) :i ∈I .

Proposition 8.3.13 For two IFSMSs (F, A) and (G, B) over U, we have the following [i] ((F, A) ∧ (G, B))c = (F, A)c ∨ (G, B)c , [ii] ((F, A) ∨ (G, B))c = (F, A)c ∧ (G, B)c . Proof [i] Let (F, A) ∧ (G, B) = (H, A × B), where ∀a ∈ A and ∀b ∈ B, H (a, b) = (. =

.

(F(a), G(b))

u ( { } { }) : u ∈ Ui m μ F(a) (u), μG(b) (u) , m v F(a) (u), vG(b) (u)

.

) :i ∈I .

Thus ((F, A)∧(G, B))c = (H, A× B)c = (H c , A × B), where ∀(a, b) ∈ A× B, H c (a, b) = (H (a, b))c (. =

u ( { } { }) : u ∈ Ui m v F(a) (u), vG(b) (u) , m μ F(a) (u), μG(b) (u)

.

) :i ∈I .

Again, let (F, A)c ∨(G, B)c = (F c , A)∨(G c , B) = (K , A × B), where ∀(a, b) ∈ A × B, ( ) K (a, b) = U F c (a), G c (b) (. =

u ( { } { }) : u ∈ Ui m v F(a) (u), vG(b) (u) , m μ F(a) (u), μG(b) (u)

Thus it follows that ((F, A) ∧ (G, B))c = (F, A)c ∨ (G, B)c . The other can be proved similarly.

.

) :i ∈I

Chapter 9

Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

Soft set theory and multi set theory are important mathematical techniques for dealing with ambiguous concepts. We introduce the concept of relations in IFSMS and create a new operation product of two intuitionistic fuzzy soft multi sets (IFSMSs). It is necessary to define the terms null relation and absolute relation. We also examine reflexive, symmetric, and transitive intuitionistic fuzzy soft multi relations, as well as several important and desirable characteristics. The relationship between an intuitionistic fuzzy soft multi relation (IFSMR) and its inverse is investigated. We also look at the qualities of IFSMRs’ weakly-reflexivity and wreflexivity. In our work, we have presented several novel findings as well as illustrative instances. The chapter also tries to build an intuitionistic fuzzy soft multi topological space (IFSM-topological space), and it is demonstrated that an IFSM-topological space yields a parameterized topological space. Some fundamental features of IFSM-topological spaces are examined, as well as the parameterized family of topological spaces created by an IFSM-topological space. Finally, we discussed fuzzy soft multi compactness and looked at some of its features. These concepts are discussed with some examples and theorems.

9.1 Intuitionistic Fuzzy Soft Multi Relations Here, we define a new operation product of two IFSMSs and present the concept of relations in IFSMSs. The notions of null relation and absolute relation are to be defined. Definition 9.1.1 The product (F, A) × (G, B) of two IFSMSs (F, A) and (G, B) over U is an IFSMS (H, A × B) where H is mapping given by H : A × B → U and ∀(a, b) ∈ A × B,

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_9

111

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9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

H (a, b) = ∩ (F(a), G(b)) () =

u ( { } { }) : u ∈ Ui m μF(a) (u), μG(b) (u) , m vF(a) (u), vG(b) (u)

.

) :i∈I .

Example 9.1.2 { } Let us consider two universes U1 = {h1 , h2 , h3 }, U2 = {c1 , c2 }. Let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎪ eU1 ,1 = modern, ⎪ ⎪ ⎪ ⎪ eU1 ,2 = wooden, ⎨ EU1 = eU1 ,3 = in green surroundings, ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎩ eU1 ,5 = in good repair, ⎧ ⎫ ⎪ ⎪ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = new model, e ⎨ U2 ,2 ⎬ EU2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition. Let U = and 9.3)

.2

i=1 FS(Ui ), E

=

.2

i=1 EUi

and A, B ⊆ E, such that (Tables 9.1, 9.2

) )} { ( ( A = a1 = eU1 ,1 , eU2 ,1 , a2 = eU1 ,2 , eU2 ,2 and { ) )} ( ( B = b1 = eU1 ,2 , eU2 ,1 , a2 = eU1 ,1 , eU2 ,2 . Definition 9.1.3 Let (F, A) be an IFSMS over U. Then an intuitionistic fuzzy soft multi relation (IFSMR) R on (F, A) is an IFSM-subset of the product set (F, A) × (F, A) and is defined as a pair (R, A × A), where R is mapping given by R : A × A → U. The collection of all IFSMRs R on (F, A) over U is denoted by IFSMRU (F, A). We denote an IFSMR(R, A × A) ∈ IFSMR(F, A) as simply R.

Table 9.1 The IFSMS (F, A)

a1

a2

h1

(0.7, 0.2)

(0.5, 0.4)

h2

(0.3, 0.6)

(0.1, 0.8)

h3

(0.7, 0.2)

(0.3, 0.6)

c1

(0.7, 0.2)

(0.3, 0.6)

c2

(0.4, 0.5)

(0.4, 0.5)

9.1 Intuitionistic Fuzzy Soft Multi Relations

113

Table 9.2 The IFSMS (G, B) b1

b2

h1

(0.5, 0.4)

(0.3, 0.6)

h2

(0.7, 0.2)

(0.5, 0.4)

h3

(0.3, 0.6)

(0.5, 0.4)

c1

(0.4, 0.5)

(0.7, 0.2)

c2

(0.7, 0.2)

(0.4, 0.5)

Table 9.3 The product set (H, A × B) (a1 , b1 )

(a1 , b2 )

(a2 , b1 )

(a2 , b2 )

h1

(0.5, 0.4)

(0.3, 0.6)

(0.5, 0.4)

(0.3, 0.6)

h2

(0.3, 0.6)

(0.3, 0.6)

(0.1, 0.8)

(0.1, 0.8)

h3

(0.3, 0.6)

(0.5, 0.4)

(0.3, 0.6)

(0.3, 0.6)

c1

(0.4, 0.5)

(0.7, 0.2)

(0.3, 0.6)

(0.3, 0.6)

c2

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

Table 9.4 An IFSMR R h1

(a1 , b1 )

(a1 , b2 )

(a2 , b1 )

(a2 , b2 )

(0.5, 0.4)

(0.4, 0.5)

(0.3, 0.6)

(0.2, 0.7)

h2

(0.3, 0.6)

(0.1, 0.8)

(0.1, 0.8)

(0.1, 0.8)

h3

(0.3, 0.6)

(0.3, 0.6)

(0.3, 0.6)

(0.3, 0.6)

c1

(0.4, 0.5)

(0.1, 0.8)

(0.1, 0.8)

(0.3, 0.6)

c2

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

Example 9.1.4 Consider the IFSMS(F, A) given in Example 3.2, then a relation (R, A × A) on (F, A) is given in Table 9.4 is an IFSMR. Definition 9.1.5 Let R1 , R2 ∈ IFSMU (F, A), then we define for (a, b) ∈ A × A, (i) R1 ≤ R2 iff R1 (a, b) ⊆ R2 (a, b), for (a, b) ∈ A × A. (ii) R1 ∨ R2 as (R1 ∨ R2 )(a, b) = R1 (a, b) ∪ R2 (a, b), where ∪ denotes the intuitionistic fuzzy union. (iii) R1 ∧ R2 as (R1 ∧ R2 )(a, b) = R1 (a, b) ∩ R2 (a, b), where ∩ denotes the intuitionistic fuzzy intersection. (iv) RC1 as RC1 (a, b) = C[R1 (a, b)], where C denotes the intuitionistic fuzzy complement. Result 9.1.6 let R1 , R2 , R3 ∈ IFSMR(F, A). Then the following properties hold: (a) (R1 ∨ R2 )C = R1 C ∧ R2 C . (b) (R1 ∧ R2 )C = R1 C ∨ R2C . (c) R1 ∨ (R2 ∨ R3 ) = (R1 ∨ R2 ) ∨ R3 .

114

9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

(d) R1 ∧ (R2 ∧ R31 ) = (R1 ∧ R2 ) ∧ R3 . (e) R1 ∧ (R2 ∨ R3 ) = (R1 ∧ R2 ) ∨ (R1 ∧ R3 ). (f) R1 ∨ (R2 ∧ R3 ) = (R1 ∨ R2 ) ∧ (R1 ∨ R3 ). Definition 9.1.7 A null relation R. ∈ IFSMRU (F, A) is defined as R. = (R. , A × A). and an absolute relation RU ∈ IFSMRU (F, A) is defined as RU = (RU , A × A)U . Remark 9.1.8 For any R ∈ IFSMRU (F, A), we have. [i] [ii] [iii] [iv]

R ∨ R. = R. R ∧ R. = R. . R ∨ RU = Ru . R ∧ Ru = R.

9.2 Various Types of IFSMRs Definition 9.2.1 An IFSMR R ∈ IFSMRU (F, A) is said to be reflexive if μR(a,a) (u) = 1 and vR(a,a) (u) = 0, ∀u ∈ Ui , ∀i ∈ I and ∀a ∈ A. Example 9.2.2 { Let us} consider two universes U1 = {h1 , h2 , h3 }, U2 = {c1 , c2 }. Let EU1 , EU2 be a collection{ of sets of decision parameters related } } { to the above universes, where EU1 = eU1 ,1 , eU1 ,2 , eU1 ,3 , EU1 = eU2 ,1 , eU2 ,2 . .2 .2 = )} i=1 EUi and A ⊆ E, such that A = i=1)FS(Ui(), E {Let U( = a1 = eU1 ,1 , eU2 ,1 , b = eU1 ,2 , eU2 ,2 (Table 9.5). Then the relation R given in Table 9.6 is a reflexive IFSMR on (F, A). Definition 9.2.3 An IFSMRR ∈ IFSMRU (F, A) is said to be symmetric if μR(a,b) (u) = μR(b,a) (u) and vR(a,b) (u) = vR(b,a) (u), ∀u ∈ Ui , ∀i ∈ I and ∀a, b ∈ A Example 9.2.4 Consider the IFSMS (F, A) given in Example 9.2.2. Then the relation R as in Table 9.7 is a symmetric IFSMR on (F, A). Table 9.5 The IFSMS (F, A) h1

a

b

(1, 0)

(1, 0)

h2

(1, 0)

(1, 0)

h3

(1, 0)

(1, 0)

c1

(1, 0)

(1, 0)

c2

(1, 0)

(1, 0)

9.2 Various Types of IFSMRs

115

Table 9.6 A reflexive IFSMR R h1

(a, a)

(a, b)

(b, a)

(b, b)

(1, 0)

(0.4, 0.2)

(0.3, 0.6)

(1, 0)

h2

(1, 0)

(0.1, 0.7)

(0.1, 0.5)

(1, 0)

h3

(1, 0)

(0.7, 0.2)

(0.4, 0.6)

(1, 0)

c1

(1, 0)

(0.1, 0.5)

(0.5, 0.1)

(1, 0)

c2

(1, 0)

(0.4, 0.5)

(0.5, 0.4)

(1, 0)

Table 9.7 A symmetric IFSMR R R:

(a, a)

(a, b)

(b, , a)

(b, b)

h1

(0.5.0.3)

(0.4, 0.3)

(0.4, 0.3)

(0.2, 0.7)

h2

(0.3, 0.6)

(0.1, 0.7)

(0.1, 0.7)

(0.1, 0.5)

h3

(0.3, 0.4)

(0.3, 0.5)

(0.3, 0.5)

(0.3, 0.7)

c1

(0.4, 0.5)

(0.1, 0.7)

(0.1, 0.7)

(0.3, 0.1)

c2

(0.4, 0.2)

(0.4, 0.1)

(0.4, 0.1)

(0.4, 0.4)

Definition 9.2.5 Let R1 , R2 ∈ IFSMRU (F, A) be two IFSMRs on (F, A). Then the composition of R1 and R2 , denoted by R1 oR2 , is defined by R1 oR2 = (R1 oR2 , A × A) where R1 oR2 : A × A → U is defined as ∀u ∈ Ui , ∀i ∈ I and ∀a, b, c ∈ A. { ( )} μR1 OR2 (a,b) (u) = max m μR1 (a,c) (u), μR2 (c,b) (u) , c

and

{ ( )} vR1 OR2 (a,b) (u) = min m vR1 (a,c) (u), vR2 (c,b) (u) . c

Definition 9.2.6 An IFSMR R ∈ IFSMRU (F, A) is said to be transitive if oR ⊆ R. Example 9.2.7 Consider the IFSMS (F, A) given in Example 9.2.2. Then the relation as in Table 9.8 is a transitive IFSMR on (F, A).

Table 9.8 A transitive IFSMR R (a, a)

(a, b)

(b, a)

(b, b)

h1

(0.5, 0.4)

(0.4, 0.5)

(0.3, 0.6)

(0.4, 0.5)

h2

(0.3, 0.6)

(0.1, 0.8)

(0.1, 0.8)

(0.1, 0.8)

h3

(0.3, 0.6)

(0.3, 0.6)

(0.3, 0.6)

(0.3, 0.6)

c1

(0.4, 0.5)

(0.1, 0.8)

(0.1, 0.8)

(0.3, 0.6)

c2

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

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9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

9.3 Inverse of IFSMRs Here, we study some essential and valuable properties of IFSMRs and examine the existing relations between an IFSMR and its inverse. Definition 9.3.1 Let R ∈ IFSMRU (F, A) be an IFSMR on (F, A). Then R−1 is defined as ∀a, b ∈ A, R−1 (a, b) = R(b, a) if and only if μR−1 (a,b) (u) = μR(b,a) (u) and vR−1 (a,b) (u) = vR(b,a) (u), ∀u ∈ Ui , ∀i ∈ I . Example 9.3.2 If we consider an IFSMR R as in Table 9.9, then the inverse of R can be represented as in Table 9.10. Proposition 9.3.3 Inverse of an IFSMR is also an IFSMR. Proof Let R ∈ IFSMRU (F, A). Then ∀a, b ∈ A, R−1 (a, b) = R(b, a) ⊆ F(b) ∩ F(a) = F(a) ∩ F(b) ⇒ R−1 (a, b) ⊆ F(a) ∩ F(b), ∀a, b ∈ A. Thus R−1 is an IFSM-subset of the product set (F, A) × (F, A). Hence R−1 is an IFSMR on (F, A). Proposition 9.3.4 Let R1 , R2 ∈ IFSMRU (F, A). Then Table 9.9 An IFSMR R (a, a)

(a, b)

(b, , a)

(b, b)

h1

(0.6, 0.4)

(0.3, 0.6)

(0.4, 0.5)

(0.5, 0.4)

h2

(0.3, 0.6)

(0.1, 0.8)

(0.1, 0.8)

(0.4, 0.5)

h3

(0.5, 0.3)

(0.3, 0.6)

(0.3, 0.6)

(0.3, 0.6)

c1

(0.4, 0.5)

(0.1, 0.8)

(0.1, 0.8)

(0.3, 0.6)

c2

(0.5, 0.3)

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

Table 9.10 R−1 (a, a)

(a, b)

(b, a)

(b, b)

h1

(0.6, 0.4)

(0.4, 0.5)

(0.3, 0.6)

(0.5, 0.4)

h2

(0.3, 0.6)

(0.1, 0.8)

(0.1, 0.8)

(0.4, 0.5)

h3

(0.5, 0.3)

(0.3, 0.6)

(0.3, 0.6)

(0.3, 0.6)

c1

(0.4, 0.5)

(0.1, 0.8)

(0.1, 0.8)

(0.3, 0.6)

c2

(0.5, 0.3)

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

9.3 Inverse of IFSMRs

117

)−1 ( [1]. R−1 = RI . l −1 [2]. RI ≤ R2 ⇒ R−1 I ≤ R2 .

Proof Let R1 , R2 ∈ IFSMRU (F, A). Then ∀a, b ∈ A, ( )−1 ( −1 )−1 [1] R−1 (a, b) = R−1 = RI . 1 I (b, a) = RI (a, b). Hence RI −1 −1 [2] Rl (a, b) ⊆ R2 (a, b) ⇒ R−1 ≤ R−1 a) ⊆ R a) ⇒ R (b, (b, 2 2 . l l Proposition 9.3.5 Let R1 , R2 ∈ IFSMRU (F, A). Then −1 −1 −1 (a) (R1 ∨ R2 )−1 = R−1 = R−1 1 ∨ R2 (b) (R1 ∧ R2 ) 1 ∧ R2 . Proof Let R1 , R2 ∈ IFSMRU (F, A). Then ∀a, b ∈ A, (a) (R1 ∨ R2 )−1 (a, b) = (R1 ∨ R2 )(b, a) = R1 (b, a) ∨ R2 (b, a) −1 = R−1 l (a, b) ∨ R2 (a, b) ( −1 ) = R1 ∨ R−1 2 (a, b). −1 Hence (R1 ∨ R2 )−1 = R−1 1 ∨ R2 . (b)

(R1 ∧ R2 )−1 (a, b) = (R1 ∧ R2 )(b, a) = R1 (b, a) ∧ R2 (b, a) −1 = R−1 1 (a, b) ∧ R2 (a, b) ( −1 ) = R1 ∧ R−1 2 (a, b). −1 Hence (R1 ∧ R2 )−1 = R−1 1 ∧ R2 . −1 Proposition 9.3.6 If R1 , R2 ∈ IFSMRU (F, A), then (RI oR2 )−1 = R−1 2 oRI

Proof Let R1 , R2 ∈ IFSMRU (F, A). Then ∀a, b ∈ A, −1 μ(R−1 (u) 2 R1 )(a,b) { ( )} −1 = maxc min μR−1 μ (u), (u) R (a,c) (c,b) 2 1 { ( )} = maxc min μR2 (c,a) (u), μR1 (b,c) (u) { ( )} = maxc min μR1 (b,c) (u), μR2 (c,a) (u) = μR1 OR2 (b,a) (u) = μ(R1 OR2 )−1 (b,a) (u), ∀u ∈ Ui , ∀i ∈ I .

118

9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

Similarly, −1 v(R−1 (u) 2 OR1 )(a,b) { ( )} = min m vR−1 (u), vR−1 (u) (a,c) (c,b) 2 1 c { ( )} = min m vR2 (c,a) (u), vR1 (b,c) (u) c { ( )} = min m vR1 (b,c) (u), vR2 (c,a) (u)

c

= vR1 OR (b, a) = v(R1 OR2 )−1 (b,a) (u), ∀u ∈ Ui , ∀i ∈ I . −1 Hence (R1 oR2 )−1 = R−1 2 OR1

Proposition 9.3.7 R ∈ IFSMRU (F, A) is reflexive if and only if R−1 is reflexive. Proof Let R ∈ IFSMU (F, A) is reflexive. Then ∀(a, b) ∈ A × A μR−1 (a,a) (u) = μR(a,a) (u) = 1 and νR−1 (a,a) (u) = vR(a,a) (u) = 0, ∀u ∈ Ui , ∀i ∈ I . Implies that R−1 is reflexive IFSMR on (F, A). Conversely, let R−1 be reflexive IFSMR on (F, A). Then ∀(a, b) ∈ A × A, μR(a,a) (u) = μR−1 (a,a) (u) = 1 and vR(a,a) (u) = vR−1 (a,a) (u) = 0, ∀u ∈ Ui , ∀i ∈ I . Implies that R is reflexive IFSMR on (F, A). Proposition 9.3.8 R ∈ IFSMU (F, A) is symmetric if and only if R−1 is symmetric. Proof Let R ∈ IFSMRU (F, A). Then ∀(a, b) ∈ A × A μR−1 (a,b) (u) = μR(b,a) (u) = μR(a,b) (u) = μR−1 (b,a) (u) and vR−1 (a,b) (u) = vR(b,a) (u) = vR(a,b) (u) = vR−1 (b,a) (u), ∀u ∈ Ui , ∀i ∈ I . Thus R−1 (a, b) = R(b, a) = R(a, b) = R−1 (b, a). Hence R−1 is symmetric IFSMR on (F, A). Conversely, let R−1 be symmetric IFSMR on (F, A). Then ∀(a, b) ∈ A×, A μR(a,b) (u) = μR−1 (b,a) (u) = μR−1 (a,b) (u) = μR(b,a) (u) and vR(a,b) (u) = vR−1 (b,a) (u) = vR−1 (a,b) (u) = vR(b,a) (u), ∀u ∈ Ui , ∀i ∈ I .

9.3 Inverse of IFSMRs

119

Thus R(a, b) = R−1 (b, a) = R−1 (a, b) = R(b, a). Hence R is symmetric IFSMR on (F, A). Proposition 9.3.9 R ∈ IFSMU (F, A) is symmetric if and only if R−1 = R. Proof Let R ∈ IFSMRU (F, A) is symmetric. Then ∀(a, b) ∈ A × A. μR−1 (a,b) (u) = μR(b,a) (u) = μR(a,b) (u) and vR−1 (a,b) (u) = vR(b,a) (u) = vR(a,b) (u), ∀u ∈ Ui , ∀i ∈ I . Thus R−1 (a, b) = R(b, a) = R(a, b) and hence R−1 = R. Conversely, let R−1 = R. Then ∀(a, b) ∈ A × A, μR(a,b) (u) = μR−1 (a,b) (u) = μR(b,a) (u) and. vR(a,b) (u) = vR−1 (a,b) (u) = vR(b,a) (u), ∀u ∈ Ui , ∀i ∈ I . Thus R(a, b) = R−1 (a, b) = R(b, a). Hence R is symmetric IFSMR on (F, A). Proposition 9.3.10 If R ∈ IFSMRU (F, A) is symmetric, then RoR is also symmetric. Proof Let R ∈ IFSMRU (F, A) is symmetric. Then by the above Proposition 9.3.8, we have R−1 = R. Now (RoR)−1 = R−1 oR−1 = RoR and hence RoR is symmetric. Proposition 9.3.11 If R1 , R2 ∈ IFSMRU (F, A) is symmetric, then R1 oR2 is symmetric if and only if R1 OR2 = R2 OR1 . −1 Proof R1 , R2 ∈ IFSMRU (F, A) is symmetric implies R−1 1 = R1 and R2 = R2 −1 −1 −1 Now (R1 oR2 ) = R2 oR1 . −1 So R1 oR2 is symmetric implies R1 oR2 = (R1 oR2 )−1 = R−1 2 oR1 = R2 oR1 . −1 −1 −1 Conversely, (R1 OR2 ) = R2 OR1 = R2 OR1 = R1 OR2 . So R1 OR2 is symmetric.

Proposition 9.3.12 If R ∈ IFSMRU (F, A) is transitive if and only if R−1 is transitive. Proof Let R ∈ IFSMRU (F, A) is transitive. Then ∀(a, b) ∈ A × A μR−1 (a,b) (u) = μR(b,a) (u) ≥ μROR(b,a) (u) { ( )} = maxc min μR(b,c) (u), μR(c,a) (u) { ( )} = maxc min μR(c,a) (u), μR(b,c) (u) { ( )} = maxc min μR−1 (a,c) (u), μR−1 (c,b) (u) = μR−1 OR−1 (b,a) (u), ∀u ∈ Ui , ∀i ∈ I ⇒ μR−1 (a,b) (u) ≥ μR−1 R−1 (b,a) (u), ∀u ∈ Ui , ∀i ∈ I . Similarly,

120

9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

vR−1 (a,b) (u) = vR(b,a) (u) ≤ vROR(b,a) (u) { ( )} = minc max vR(b,c) (u), vR(c,a) (u) { ( )} = minc max vR(c,a) (u), vR(b,c) (u) { ( )} = minc max vR−1 (a,c) (u), vR−1 (c,b) (u) = vR−1 R−1 (b,a) (u), ∀u ∈ Ui , ∀i ∈ I ⇒ vR−1 (a,b) (u) ≤ vR−1 OR−1 (b,a) (u), ∀u ∈ Ui , ∀i ∈ I . Hence R−1 is transitive. Conversely, let R−1 is transitive. Then ∀(a, b) ∈ A × A, μR(a,b) (u) = μR−1 (b,a) (u) ≥ μ(R−1 OR−1 )(b,a) (u) { ( )} = max m μR−1 (b,c) (u), μR−1 (c,a) (u) c { ( )} = max m μR−1 (c,a) (u), μR−1 (b,c) (u) c { ( )} = max m μR(a,c) (u), μR(c,b) (u) c

and vR(a,b) (u) = vR−1 (b,a) (u) ≤ v(R−1 R−1 )(b,a) (u) { ( )} = min m vR−1 (b,c) (u), vR−1 (c,a) (u) c { ( )} = min m vR−1 (c,a) (u), vR−1 (b,c) (u) c { ( )} = min m vR(a,c) (u), vR(c,b) (u) c

=

vROR(b,a) (u), ∀u ∈ U

⇒

vR(a,b) (u) ≤ vROR(b,a) (u), ∀u ∈ Ui , ∀i ∈ I .

Hence R is transitive. Proposition 9.3.13 If R ∈ IFSMU (F, A) is transitive, then ROR is transitive. Proof Let R ∈ IFSMU (F, A) is transitive. Then ∀(a, b) ∈ A × A { ( )} μ(ROR)(b,a) (u) = max m μR(a,c) (u), μR(c,b) (u) c { ( )} ≥ max m μ(ROR)(a,c) (u), μ(ROR)(c,b) (u) c

= μ((ROR)O(ROR))(a,b) (u) ⇒ μ(ROR)(a,b) (u) ≥ μ((ROR)O(ROR))(a,b) (u), ∀u ∈ Ui , ∀i ∈ I , and

9.4 Weakly-Reflexive and w-reflexive IFSMRs

121

{ ( )} v(ROR)(b,a) (u) = min m vR(a,c) (u), vR(c,b) (u) c { ( )} ≤ min m v(ROR)(a,c) (u), v(ROR)(c,b) (u) c

= v((ROR)O(ROR))(a,b) (u) ⇒ v(ROR)(a,b) (u) ≤ v((ROR)O(ROR))(a,b) (u), ∀u ∈ Ui , ∀i ∈ I . Hence RoR is transitive.

9.4 Weakly-Reflexive and w-reflexive IFSMRs Here, we investigate the weakly-reflexivity and w-reflexivity properties of IFSMRs. Definition 9.4.1 An IFSMR R ∈ IFSMRU (F, A) is said to be weakly-reflexive if μR(a,a) (u) ≥ μR(a,b) (u) and vR(a,a) (u) ≤ vR(a,b) (u), ∀u ∈ Ui , ∀i ∈ I and ∀a, b ∈ A. Example 9.4.2 The IFSMR R as in Table 9.11 is weakly-reflexive. Proposition 9.4.3 If an IFSMR R ∈ IFSMRU (F, A) is symmetric and transitive, then R is weakly-reflexive. Proof Let R ∈ IFSMRU (F, A) is symmetric and transitive. Then ∀a, b ∈ A μR(a,a) (u) ≥ μ(ROR)(a,a) (u), [since R is transitive] { ( )} = max m μR(a,b) (u), μR(b,a) (u) b { ( )} [ ] ≥ max m μR(a,b) (u), μR(a,b) (u) , since R is symmetric b { } = max μR(a,b) (u) b

≥ μR(a,b) (u) ⇒ μR(a,a) (u) ≥ μR(a,b) (u), ∀u ∈ Ui , ∀i ∈ I , Table 9.11 Weakly-reflexive IFSMR R (a, a)

(a, b)

(b, a)

(b, b)

h1

(0.6, 0.4)

(0.4, 0.5)

(0.3, 0.6)

(0.5, 0.4)

h2

(0.3, 0.6)

(0.1, 0.8)

(0.1, 0.8)

(0.4, 0.5)

h3

(0.5, 0.3)

(0.3, 0.6)

(0.3, 0.6)

(0.3, 0.6)

c1

(0.4, 0.5)

(0.1, 0.8)

(0.1, 0.8)

(0.3, 0.6)

c2

(0.5, 0.3)

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

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9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

and vR(a,a) (u) ≤ v(ROR)(a,a) (u), [since R is transitive] { ( )} = min m vR(a,b) (u), vR(b,a) (u) b { ( )} [ ] ≤ min m vR(a,b) (u), vR(a,b) (u) , since R is symmetric b { } = min vR(a,b) (u) b

≤ vR(a,b) (u) ⇒ vR(a,a) (u) ≤ vR(a,b) (u), ∀u ∈ Ui , ∀i ∈ I . Hence R is weakly-reflexive. Proposition 9.4.4 If an IFSMR R ∈ IFSMRU (F, A) is weakly-reflexive and symmetric, then R−1 is weakly-reflexive. Proof Let R ∈ IFSMRU (F, A) is weakly-reflexive and symmetric. Then ∀a, b ∈ A μR−1 (a,a) (u) = μR(a,a) (u) ≥ μR(a,b) (u), [since R is weakly − reflexive] [ ] = μR(b,a) (u), since R is symmetric = μR−1 (a,b) (u) ⇒ μR−1 (a,a) (u) ≥ μR−1 (a,b) (u), ∀u ∈ Ui , ∀i ∈ I , and vR−1 (a,a) (u) = vR(a,a) (u) [ ] ≤ vR(a,b) (u), since R is weakly − reflexive [ ] = vR(b,a) (u), since R is symmetric = vR−1 (a,b) (u) ⇒ vR−1 (a,a) (u) ≥ vR−1 (a,b) (u), ∀u ∈ Ui , ∀i ∈ I . Hence R−1 is weakly-reflexive. Definition 9.4.5 An IFSMR R ∈ IFSMRU (F, A) is said to be w-reflexive if μR(a,a) (u) ≥ μF(a) (u) and vR(a,a) (u) ≤ vF(a) (u), ∀u ∈ Ui , ∀i ∈ I and ∀a, b ∈ A. Example 9.4.6 We select an IFSMS (F, A) as in Table 9.1. Then the IFSMR R on (F, A) as in Table 9.12, is w-reflexive. Proposition 9.4.7 If R1 , R2 ∈ IFSMRU (F, A) be two w-reflexive IFSMRs. Then R1 ∨ R2 ≤ R1 oR2 .

9.4 Weakly-Reflexive and w-reflexive IFSMRs

123

Table 9.12 w-reflexive IFSMR R (a, a)

(a, b)

(b, a)

(b, b)

h1

(0.6, 0.3)

(0.4, 0.5)

(0.3, 0.6)

(0.5, 0.4)

h2

(0.5, 0.4)

(0.1, 0.8)

(0.1, 0.8)

(0.7, 0.3)

h3

(0.7, 0.2)

(0.3, 0.6)

(0.3, 0.6)

(0.3, 0.6)

c1

(0.4, 0.4)

(0.1, 0.8)

(0.1, 0.8)

(0.3, 0.6)

c2

(0.6, 0.2)

(0.4, 0.5)

(0.4, 0.5)

(0.4, 0.5)

Proof ∀a, b ∈ A, we have { ( )} μ(R1 oR2 )(a,b) (u) = max m μR1 (a,c) (u), μR2 (c,b) (u) c ( ) ≥ m μR1 (a,a) (u), μR2 (a,b) (u) ( ) ≥ m μF(a) (u), μR2 (a,b) (u) , [as R1 is W − reflexive] { } Again, μR2 (a,b) (u) ≤ m μF(a) (u), μF(b) (u) ≤ μF(a) (u) SO μ(R1 oR2 )(a,b) (u) ≥ μR2 (a,b) (u). Similarly, it can be proved that μ(R1 oR2 )(a,b) (u) ≥ μR1 (a,b) (u), ∀u ∈ Ui , ∀i ∈ I . Hence, μ(R1 oR2 )(a,b) (u) ≥ μR1 (a,b) (u) ∨ μR2 (a,b) (u), ∀u ∈ Ui , ∀i ∈ I . Similarly, we can prove that v(R1 OR2 )(a,b) (u) ≤ VR1 (a,b) (u) ∨ VR2 (a,b) (u), ∀u ∈ Ui , ∀i ∈ I . Thus, R1 ∨ R2 ≤ R1 OR2 . Proposition 9.4.8 If an IFSMR R ∈ IFSMRU (F, A) is a w-reflexive, then R ≤ RoR. Proof ∀a, b ∈ A, we have { ( )} μ(RoR)(a,b) (u) = max m μR(a,c) (u), μR(c,b) (u) c ( ) ≥ m μR(a,a) (u), μR(a,b) (u) ( ) ≥ m μF(a) (u), μR(a,b) (u) , [as R is W − reflexive] { } Again, μR(a,b) (u) ≤ m μF(a) (u), μF(b) (u) ≤ μF(a) (u) SO μ(RoR)(a,b) (u) ≥ μR(a,b) (u), ∀u ∈ Ui , ∀i ∈ I . Similarly, we can show that { ( )} v(RoR)(a,b) (u) = min m μR(a,c) (u), μR(c,b) (u) c

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9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

( ) m vR(a,a) (u), vR(a,b) (u) ( ) ≤ m vF(a) (u), vR(a,b) (u) , [as R is w − reflexive] { } Again, vR(a,b) (u) ≥ m vF(a) (u), vF(b) (u) ≥ vF(a) (u) so v(RoR)(a,b) (u) ≤ vR(a,b) (u), ∀u ∈ Ui , ∀i ∈ I .

≤

Thus R ≤ RoR. Proposition 9.4.9 If an IFSMR R ∈ IFSMRU (F, A) is w-reflexive and symmetric, then R−1 is weakly-reflexive. Proof Let R ∈ IFSMRU (F, A) is w-reflexive. Then ∀a, b ∈ A μR−1 (a,a) (u) = μR(a,a) (u) ≥ μF(a) (u) and vR−1 (a,a) (u) = vR(a,a) (u) ≤ vF(a) (u), ∀u ∈ Ui , ∀i ∈ I . Hence R−1 is w-reflexive.

9.5 IFSM-Topological Spaces We consider an absolute IFSMS (F, A) over U and IFSMsA (F, A) denote the family of all IFSM-subsets of (F, A) in which all the parameter set A are the same. Throughout this chapter, (F, A) refers to an initial universal IFSMS with fixed parameter set A. Definition 9.5.1 A subfamily τ of IFSMsA (F, A) is called intuitionistic fuzzy soft multi topology on (F, A), if the following axioms are satisfied: [O1 ]..A , (F, A) ∈ τ, ~i∈I (Fi , A) ∈ τ, [O2 ].{(Fi , A) : i ∈ I } ⊆ τ ⇒ U ∩(G, A) ∈ τ. [O3 ]. If (F, A), (G, A) ∈ τ , then (F, A)~ Then the pair ((F, A), τ ) is called IFSM-topological space. The members of τ are called IFSM-open sets (simply open sets), and the conditions [O1 ], [O2 ] and [O3 ] are called the axioms for IFSM-open sets. are three universes U1 and U2 . Let U1 = Example 9.5.2 Let us consider there { } {h1 , h2 , h3 }, U2 = {c1 , c2 } and EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where

9.5 IFSM-Topological Spaces

125

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

eU1 ,1 = modern, eU1 ,2 = wooden, EU1 = eU1 ,3 = in green surroundings, ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎩ eU1 ,5 = in good repair, ⎧ ⎫ ⎪ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ ⎪e ⎪ ⎪ ⎪ ⎨ U2 ,2 = new model, ⎬ EU2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition. .2 = E ) ( i=1 P(Ui ),)} { Let ( U a1 = eU1 ,1 , eU2 ,1 , a2 = eU1 ,2 , eU2 ,2 ,

=

.2 i=1

EUi

and

A

=

. . .)) .( (. h2 h3 c1 c2 h1 , , , , , .A = e1 , (0, 1) (0, 1) (0, 1) (0, 1) (0, 1) ( (. . . .) . h2 h3 c1 c2 h1 e2 , , , , , , (0, 1) (0, 1) (0, 1) (0, 1) (0, 1) . . .)) .( (. h2 h3 c1 c2 h1 , , , , , e1 , (F, A) = (1, 0) (1, 0) (1, 0) (1, 0) (1, 0) ( (. . . .)). h2 h3 c1 c2 h1 e2 , , , , , , (1, 0) (1, 0) (1, 0) (1, 0) (1, 0) . . .) .( (. h2 h3 c1 c2 h1 , , , , , e1 , (F1 , A) = (0.2, 0.7) (0.4, 0.5) (0.8, 0.1) (0.8, 0.1) (0.5, 0.5) ( (. . . .)). h2 h3 c1 c2 h1 e2 , , , , , , (0.7, 0.2) (0.7, 0.2) (1, 0) (0.8, 0.1) (0.6, 0.3) . . .) .( (. h2 h3 c1 c2 h1 , , , , , e1 , (F2 , A) = (0.3, 0.6) (0.3, 0.6) (0.7, 0.2) (0.8, 0.1) (0.6, 0.3) ( (. . . .)). h2 h3 c1 c2 h1 e2 , , , , , , (0.8, 0.1) (0.9, 0.1) (1, 0) (0.8, 0.1) (0.8, 0.1) ∪(F2 , A) (F3 , A) = (F1 , A)~ . . .)) .( (. c1 h1 h2 h3 c2 , = e1 , , , , (0.3, 0.6) (0.4, 0.5) (0.8, 0.1) (0.8, 0.1) (0.6, 0.3) . . .)). ( (. h1 c1 h2 h3 c2 , , , , , e2 , (0.8, 0.1) (0.9, 0.1) (1, 0) (0.8, 0.1) (0.8, 0.1) (F4 , A) = =

(F1 , A) ∩ (F2 , A) .( (. . . .) h2 h3 c1 c2 h1 e1 , , , , , , (0.2, 0.7) (0.3, 0.6) (0.7, 0.2) (0.8, 0.1) (0.5, 0.5)

126

9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

( (. e2 ,

. . .). h2 h3 c1 c2 h1 , , , , . (0.7, 0.2) (0.7, 0.2) (1, 0) (0.8, 0.1) (0.6, 0.3)

Then we observe that the subfamily τ1 ={.A , (F, A), (F1 , A), (F2 , A), (F3 , A), (F4 , A)} of IFSMsA (F, A) is an IFSM-topology on (F, A), since it satisfies the necessary three axioms [O1 ], [O2 ] and [O3 ] and ((F, A), τ1 ) is an IFSM-topological space. Definition 9.5.3 Let D denote the family of all IFSM-subsets of (F, A). Then we observe that D satisfies all the axioms for topology on (F, A). This topology is called discrete IFSM-topology and the pair ((F, A), Q) is called a discrete IFSMtopological space. Definition 9.5.4 As every IFSM-topology on (F, A) must contain the sets .A and (F, A), so the family I = {.A , (F, A)}, forms an IFSM-topology on (F, A). This topology is called indiscrete IFSM-topology and the pair ((F, A), I ) is called an indiscrete IFSM-topological space. Definition 9.5.5 Let ((F, A), τ ) be an IFSM-topological space over (F, A). An IFSMsubset (F, A) of (F, A) is called IFSM-closed if its complement (F, A)c is a member of τ . Example 9.5.6 Let us consider Example 9.5.2, then the IFSM-closed sets in ((F, A), τ1 ) are .( (. . . .)) h1 c1 h2 h3 c2 .C = e , , , , , , 1 A (1, 0) (1, 0) (1, 0) (1, 0) (1, 0) . . .)). ( (. h2 h3 c1 c2 h1 , , , , = (F, A), e2 , (1, 0) (1, 0) (1, 0) (1, 0) (1, 0) . . .)) .( (. c1 h1 h2 h3 c2 C , , e1 , (F, A) = , , , (0, 1) (0, 1) (0, 1) (0, 1) (0, 1) . . .)). ( (. h1 c1 h2 h3 c2 , , , , = .A , e2 , (0, 1) (0, 1) (0, 1) (0, 1) (0, 1) . . .) .( (. c1 h1 h2 h3 c2 , , e1 , , , , (F1 , A)C = (0.7, 0.2) (0.5, 0.4) (0.1, 0.8) (0.1, 0.8) (0.5, 0.5) . . .)). ( (. c1 h1 h2 h3 c2 , , , , , e2 , (0.2, 0.7) (0.2, 0.7) (0, 1) (0.1, 0.8) (0.3, 0.6) .( (. . . .)) h1 c1 h2 h3 c2 e1 , , , , , , (F2 , A)C = (0.6, 0.3) (0.6, 0.3) (0.2, 0.7) (0.1, 0.8) (0.3, 0.6) . . .). (. ( c1 h1 h2 h3 c2 , ,, e2 , , , , (0.1, 0.8) (0.1, 0.9) (0, 1) (0.1, 0.8) (0.1, 0.8)

. . .) .( (. h2 h3 c1 c2 h1 , , , , , e1 , (F3 , A)C = (0.6, 0.3) (0.5, 0.4) (0.1, 0.8) (0.1, 0.8) (0.3, 0.6) ( (. . . .)). h2 h3 c1 c2 h1 e2 , , , , , , (0.1, 0.8) (0.1, 0.9) (0, 1) (0.1, 0.8) (0.1, 0.8)

9.5 IFSM-Topological Spaces

127

. . .) .( (. h2 h3 c1 c2 h1 , , , , , e1 , (F4 , A)C = (0.7, 0.2) (0.6, 0.3) (0.2, 0.7) (0.1, 0.8) (0.5, 0.5) ( (. . . .)). h2 h3 c1 c2 h1 e2 , , , , , . (0.2, 0.7) (0.2, 0.7) (0, 1) (0.1, 0.8) (0.3, 0.6) Proposition 9.5.7 Let ((F, A), τ ) be an IFSM-topological space over (F, A). Then. [i]. The intersection of an arbitrary collection of IFSM-closed sets is an IFSMclosed set over (F, A). [ii]. The union of any two IFSM-closed sets is an IFSM-closed set over (F, A). Proof Straightforward. Definition 9.5.8 Let τ be the IFSM-topology on (F, A). An IFSMS (F, A) in IFSMsA ( F, A) is a neighbourhood of an IFSMS (G, A) if and only if there exists ˜ , A)⊆(F, ˜ A). an IFSM-open set (H , A) ∈ τ such that (G, A)⊆(H Definition 9.5.9 Let ((F, A), τ ) be an IFSM-topological space on (F, A) and (F, A) be an IFSMS in IFSMsA (F, A). Then the union of all IFSM-open sets contained in (F, A) is called the interior of (F, A) and is denoted by int(F, A) and defined by ˜ int(F, A) = ∪{(G, A) | (G, A) is an open set contained in (F, A)}. Definition 9.5.10 Let ((F, A), τ ) be an IFSM-topological space on (F, A) and (F, A) be an IFSMS in IFSMsA (F, A). Then the intersection of all closed IFSMS containing (F, A) is called the closure of (F, A) and is denoted by cl(F, A) and defined by ˜ cl(F, A) = ∩{(G, A) | (G, A) is a closed set containing (F, A)}. Theorem 9.5.11 Let ((F, A), τ ) be an IFSM-topological space on (F, A) and let (F, A) be an intuitionistic fuzzy soft multiset in IFSMsA ( F, A). Then [i]. (cl(F, A))c = int((F, A)c ), [ii].(int(F, A))c = cl((F, A)c ). Proof Straightforward. Theorem 9.5.12 Let ((F, A), τ ) be an IFSM-topological space on (F, A) and (F, A) be an IFSMS { } in IFSMsA (F, A). Then the collection τ(F,A) = ∩(G, A)|(G, A) ∈ τ is an IFSM-topology on the IFSMS(F, A). (F, A)~ Proof Straightforward. Definition 9.5.13 Let ((F, A), τ ) be an IFSM-topological space on (F, A) and fuzzy soft (F, A) be an intuitionistic | multiset }in IFSMsA (F, A). Then the IFSM{ |(G, A) ∈ τ is called IFSMsubspace topology ˜ τ = A) ∩(G, A) topology (F, (F,A) ( ) and (F, A), τ(F,A) is called IFSM-topological subspace of ((F, A), τ ).

128

9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

9.6 Parameterized Topological Space Induced by an IFSM-Topological Space Proposition 9.6.1 Let ((F, A), τ ) be an IFSM-topological space over (F, A). Then the collection τe = {F(e) : (F, A) ∈ τ } for each e ∈ A, defines a topology on F(e). Proof [O1 ] Since .A , (F, A) ∈ τ implies that ϕ, F(e) ∈ τe , for each e ∈ E. ~i∈I (Fi , A) ∈ [O2 ] Let {Fi (e) : i ∈ I } ⊆ τe , for some {(Fi , A) : i ∈ I } ⊆ τ . Since U ~i∈I Fi (e) ∈ τe , for each e ∈ E. τ , so U ∩(G, A) ∈ [O3 ] Let F(e), G(e) ∈ τe , fore some (F, A), (G, A) ∈ τ . Since (F, A)~ ∩G(e) ∈ τe , for each e ∈ E. τ , so F(e)~ Thus τe defines a topology on F(e) for each e ∈ E. Definition 9.6.2 Let ((F, A), τ ) be an IFSM-topological space over (F, A). Then the topology τe = {F(e) : (F, A) ∈ τ } for each e ∈ A, is called parameterized topology and (F(e), τe ) is called parameterized topological space. Example 9.6.3 Let us consider the IFSM-topology τ1 = {.A , (F, A), (F1 , A), (F2 , A), (F3 , A), (F4 , A)} as in the example: 3.3., It can be easily seen that τe1 = {ϕ, F(e1 ), F1 (e1 ), F2 (e1 ), F3 (e1 ), F4 (e1 )}, and τe2 = {ϕ, F(e2 ), F1 (e2 ), F2 (e2 ), F3 (e2 ), F4 (e2 )} are parameterized topologies on F(e1 ) and F(e2 ) respectively, where . . .) h2 h3 c2 c1 h1 , , , , , F(e1 ) = (1, 0) (1, 0) (1, 0) (1, 0) (1, 0) . . .) (. h2 h3 c1 c2 h1 , , , , , F(e2 ) = (1, 0) (1, 0) (1, 0) (1, 0) (1, 0) . . .) (. h2 h3 c1 c2 h1 , , , , , F1 (e1 ) = (0.2, 0.7) (0.4, 0.5) (0.8, 0.1) (0.8, 0.1) (0.5, 0.5) . . .) (. h2 h3 c2 c1 h1 , , , , , F1 (e2 ) = (0.7, 0.2) (0.7, 0.2) (1, 0) (0.8, 0.1) (0.6, 0.3) . . .) (. h2 h3 c1 c2 h1 , , , , , F2 (e1 ) = (0.3, 0.6) (0.3, 0.6) (0.7, 0.2) (0.8, 0.1) (0.6, 0.3) . . .) (. h2 h3 c1 c2 h1 , , , , , F2 (e2 ) = (0.8, 0.1) (0.9, 0.1) (1, 0) (0.8, 0.1) (0.8, 0.1) . . .) (. h2 h3 c1 c2 h1 , , , , , F3 (e1 ) = (0.3, 0.6) (0.4, 0.5) (0.8, 0.1) (0.8, 0.1) (0.6, 0.3) . . .) (. h2 h3 c1 c2 h1 , , , , , F3 (e2 ) = (0.8, 0.1) (0.9, 0.1) (1, 0) (0.8, 0.1) (0.8, 0.1) . . .) (. h2 h3 c1 c2 h1 , , , , , F4 (e1 ) = (0.2, 0.7) (0.3, 0.6) (0.7, 0.2) (0.8, 0.1) (0.5, 0.5) . . .) (. h2 h3 c1 c2 h1 , , , , . F4 (e2 ) = (0.7, 0.2) (0.7, 0.2) (1, 0) (0.8, 0.1) (0.6, 0.3) (.

9.7 IFSM-Compact Spaces

129

Definition 9.6.4 Let ((F, A), τ ) be an IFSM-topological space on (F, A) and (F, A) be an IFSMS in IFSMsA (F, A). Then we defined an IFSMS associated with (F, A) over (F, A) is denoted by (cl(F), A) and defined by cl(F)(e) = cl(F(e)), where cl(F(e)) is the closer of F(e) in τe for each e ∈ A. Proposition 9.6.5 Let ((F, A), τ ) be an IFSM-topological space on (F, A) and ~cl(F, A). (F, A) be an IFSMS in IFSMsA (F, A). Then (cl(F), A)⊆ Proof For any e ∈ A, cl(F(e)) is the smallest closed set in (U , τe ), which contains F(e). Moreover if cl(F, A) = (G, A), then G(e) is also closed set in (U , τe ) containing F(e). This implies that cl(F)(e) = cl(F(e)) ⊆ G(e). Thus (cl(F), A) ⊆ cl(F, A). Corollary 9.6.6 Let ((F, A), τ ) be an IFSM-topological space on (F, A) and (F, A) be an IFSMS in IFSMsA (F, A). Then (cl(F), A) = cl(F, A) if and only if (cl(F), A)c ∈ τ . Proof If (cl(F), A) = cl(F, A), then (cl(F), A) = cl(F, A) is an IFSM-closed set and so (cl(F), A)c ∈ τ . Conversely if (cl(F), A)c ∈ τ then (cl(F), A) is an IFSM-closed set containing (F, A). By proposition 4.6., (cl(F), A) ⊆ cl(F, A) and by the definition of IFSMclosure of (F, A), any IFSM-closed set over (F, A), which contains (F, A) will ~(cl(F), A). Thus (cl(F), A) = cl(F, A). contain cl(F, A). This implies that cl(F, A)⊆

9.7 IFSM-Compact Spaces Definition 9.7.1 Let ((F, A), τ ) be an IFSM-topological space on (F, A) and let (F, A) be any intuitionistic fuzzy soft multiset in IFSMsA (F, A). Then a subfamily . of IFSMsA (F, A) is called an IFSM-cover for (F, A) if and only if (F, A) ⊆ ~ {(G, A) : (G, A) ∈ .} and we say that . covers (F, A). U If a sub-collection of IFSM-cover . also covers (F, A), then it is called an IFSMsubcover of . for (F, A).If the members of IFSM-cover . are open, then . is called IFSM-open cover. If the members of IFSM-cover . are finite in number, then it is called the finite IFSM-cover. Definition 9.7.2 Let ((F, A), τ ) be an IFSM-topological space on (F, A). An intuitionistic fuzzy soft multiset (F, A) in IFSMsA (F, A) is called IFSMcompact set if and only if every IFSM-open cover of (F, A) has a finite IFSMsubcover. Example 9.7.3 If we consider an IFSM-topological space ((F, A), τ1 ) as in Example 9.5.2, then every intuitionistic fuzzy soft multiset (F, A) in IFSMsA (F, A) is IFSMcompact since every IFSM-open cover of (F, A) is finite.

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Definition 9.7.4 An IFSM-topological space ((F, A), τ ) is called IFSMcompact space if and only if (F, A) is IFSM-compact. Example 9.7.5 The IFSM-topological space ((F, A), τ1 ) as in Example 4.2, is IFSMcompact space since every IFSM-open cover of (F, A) is finite. Example 9.7.6 Every indiscreet IFSM-topological space on (F, A) are IFSMcompact spaces, since it has only one IFSM-open cover . = {.A , (F, A)}, which is finite. Definition 9.7.7 Let ((F, A), τ ) be an IFSM-topological space on (F, A). A subfamily . of IFSMsA (F, A) has the finite intersection property if and only if the interaction of any finite subcollection of . is not null IFSMS. Theorem 9.7.8 An IFSM-topological space ((F, A), τ ) is IFSM-compact space if and only if every family of closed IFSM-subsets with finite intersection property has a non-null intersection. Proof Let ((F, A), τ ) be an IFSM-compact space and let {(Fk , A) : k ∈ K} be an arbitrary family of IFSM-closed sets in τ with finite intersection property. ( )c ∩k∈K (Fk , A) = If possible, let ~ ∩k∈K (Fk , A) = .A , then by taking complements, ~ { } ~k∈K (Fk , A)c = (F, A). So that (Fk , A)c : k ∈ K forms an IFSM(.A )c , i.e. U is a finite IFSM-subcover for (F, A). Since {open cover } (F, A) is compact, there n ~i=1 (Fi , A)c : i = 1, 2, . . . , n , such that (F, A) = U (Fi , A)c . Then by taking complements, ) ( n ~i=1 (Fi , A)c c , i.e. .A = . ~ni=1 (Fi , A). Thus (F, A)c = U {(Fk , A) : k ∈ K} does not have the finite intersection property, which is ~k∈K (Fk , A) /= .A . contrary to our assumption. Hence . Conversely, let every family of closed IFSM-subsets in ((F, A), τ ) with finite intersection property has a non-null intersection. Now suppose that ((F, A), τ ) is not IFSM-compact space. Then there is an IFSM-open cover {(Gk , A) : k ∈ K} of ~n then by taking (F, A) that has no finite IFSM-subcover, ( n )c i.e. (F, A) /= Un i=1 (Gi , A), c ~ ~ complements, (F, A) /= ∪i=1 (Gi , A) , i.e. .A /= ∩i=1 (Gi , A)c , which implies {(Gk , A) : k ∈ K} has the finite intersection property. But by the IFSM-cover property ∪k∈K (Gk , A), then by taking complements, ~ ∩k∈K (Gk , A)c = .A , i.e. the (F, A) = ~ intersection of all members of the family of IFSM-closed sets is a null intuitionistic fuzzy soft multiset, which contradicting the given condition. Hence ((F, A), τ ) is IFSM-compact space. Theorem 9.7.9 Let ((F, A), τ ) be an IFSM-compact(space and let) (F, A) be an IFSM-closed sets in τ . Then the closed IFSM-subspace (F, A), τ(F,A) of ((F, A), τ ) is IFSM-compact space.

9.8 IFSM-Points and Separation Axioms

131

Proof Let ((F,{( space and let (F, A) be an IFSMclosed A), τ ) be) an IFSM-compact } k k F : k ∈ K τ . Let , A be an arbitrary family of IFSM-closed sets in sets in ( ) with the finite intersection property. Then intuitionistic fuzzy soft (F, A), τ(F,A) ( ) multisets F k , Ak for each k ∈ K are closed intuitionistic fuzzy } in { ( soft)|multisets ((F, A), τ ); since (F, A) is an IFSM-closed sets in τ . Thus F k , Ak |k ∈ K is a family of soft multi closed sets in ((F, A), τ ), possessing finite intersection ( ) prop˜ k∈K F k , Ak /= φ ~ erty and as ((F, A), τ ) is IFSM-compact space, it follows that . ( ) (by theorem 5.8). This implies that the closed soft multi subspace (F, A), τ(F,A) of ((F, A), τ ) is an IFSMcompact space.

9.8 IFSM-Points and Separation Axioms Here, we introduce IFSM-points and study various separation axioms for IFSMtopological space. Definition 9.8.1 An intuitionistic fuzzy soft multiset (F, A) ∈ IFSMsA (F, A) is called an IFSM-point(in )(F, A), denoted by e(F,A) , if for the element e ∈ A, F(e) /= ϕ and ∀e' ∈ A − {e}, F e' = ϕ. Example 9.8.2 Let us consider there are three universes U1 and U2 . Let U1 = { } {h1 , h2 , h3 }, U2 = {c1 , c2 } and EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎫ ⎪ ⎪ eU1 ,1 = modern, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU1 ,2 = wooden, ⎬ EU1 = eU1 ,3 = in green surroundings, , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ = in good repair, U ,5 ⎧1 ⎫ ⎪ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU2 ,2 = new model, ⎬ . EU2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = black, e U ⎪ U2 ,4 ⎪ ⎪ ⎪ ⎩ ⎭ eU2 ,5 = in good condition. ) )} { ( ( . . Let U = 2i=1 P(Ui ), E= 2i=1 EUi and A = a1 = eU1 ,1 , eU2 ,1 , a2 = eU1 ,2 , eU2 ,2 Then the intuitionistic fuzzy soft multiset e(F,A) =

.( (. e1 ,

is an IFSM-point.

. . .). h2 h3 c1 c2 h1 , , , , (0.2, 0.7) (0.4, 0.5) (0.8, 0.1) (0.8, 0.1) (0.5, 0.5)

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Definition 9.8.3 Two IFSM-points e(F,A) and e(G,A) are said to be distinct if and only ' ' if e and e are distinct, i.e. e /= e . Definition 9.8.4 An IFSM-topological space ((F, A), τ ) is said to be an IFSM' T0 −space if for every pair of distinct fuzzy soft multi points e(F,A) , e(G,A) , ∃ an IFSM-open set containing one but not the other. Example 9.8.5 A discrete IFSM-topological space is an IFSM- T0 − space since every e(F,A) is an IFSM-open set in the discrete space. Theorem 9.8.6 An IFSM-subspace of an IFSM- T0 −space is IFSM-T0 . ( ) Proof Let (F, A), τ(F,A) be an IFSM-subspace of an IFSM- T0 − space ((F, A), τ ) ' and let e(G,A) , e(H,A) be two distinct IFSM- points of (F, A). Then these IFSM- points are also in (F, A) ⇒ ∃ an IFSM- open set (S, A). ˜ Containing one IFSM-point but not the other ⇒ (F, A)∩(S, A) where (S, A) ∈ τ is an IFSM- open set in τ(F,A) containing one IFSM- point but not the other. Definition 9.8.7 An IFSM-topological space ((F, A), τ ) is said to be an IFSM T1 ' space if for every distinct pair of IFSM-points e(F,A) , e(G,A) of (F, A), ∃ IFSM-open ' ˜ ˜ , A) and A) and e(F,A) ∈(H sets (S, A) and (H , A) such that e(F,A) ∈(S, /˜ , A); e(G,A) ∈(H ' /˜ A). Theorem 9.8.8 An IFSM-subspace of an IFSM- T1 − space is IFSMe(G,A) ∈(S, T1 − space. Proof The proof is straightforward. Definition 9.8.9 An IFSM-topological space ((F, A), τ ) is said to be an IFSM T2 − ' space if and only if for distinct IFSM-points e(F,A) , e(G,A) of (F, A), ∃ disjoint IFSM' ˜ ˜ , A). A) and e(G,A) ∈(H open sets (S, A) and (H , A) such that e(F,A) ∈(S, Theorem 9.8.10 An IFSM-subspace of an IFSM-T2 −space is IFSM-T2 −space. Proof The proof is straightforward. Theorem 9.8.11 An IFSM-topological space ((F, A), τ ) is IFSM-T2 -space if and ' only if for distinct IFSM-points e(F,A) , e(G,A) of (F, A), ∃ an IFSM-open set(S, A) ' ' containing e(F,A) but not e(G,A) such that e(G,A) ∈ / Cl(S, A). ' Proof Let ((F, A), τ ) be IFSM- T2 − space and e(F,A) , e(G,A) be distinct intuitionistic fuzzy soft points in (F, A). So ∃ distinct intuitionistic fuzzy soft open sets (S, A) ' ~ ∈(S, A), e(F,A)~ ∈(H , A) ⇒ (H , A) . (S, A)c and (S, A)c and (H , A) such that e(G,A) ∼

' ⊂(S, A)c ⇒ is IFSM-closed ⇒ cl(H , A) ⊆ ll((S, A)c ) = (S, A)c and since e(G,A) /=

' ∈Cl(H /˜ , A). So (H , A) is an intuitionistic fuzzy soft open set containing e(F,A) e(G,A) ' ' ~ ∈Cl(H / , A). but not e(G,A) , such that e(G,A)

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133 '

Conversely, take a pair of distinct intuitionistic fuzzy soft points e(F,A) and e(G,A) of (F, A), ∃ an intuitionistic fuzzy soft open set (S, A) containing e(F,A) but not ' ' ' ˜ A))c ⇒ (S, A) and (cl(S, A))c are e(G,A) such that e(G,A) ∈ / cl(S, A) ⇒ e(G,A) ∈(cl(S, ' disjoint intuitionistic fuzzy soft open set containing e(F,A) and e(G,A) respectively. Definition 9.8.12 An IFSM-topological space ((F, A), τ ) is said to be an IFSMregular space if for every fuzzy soft multi point e(F,A) and IFSM-closed set (K, A) not ˜ A) containing e(F,A) , ∃ disjoint IFSM-open sets (S, A), (H , A) such that e(F,A) ∈(S, ˜ and (K, A)⊆(H , A). An IFSM-regular T1 -space is called an IFSM-T3 -space, Theorem 9.8.13 An IFSM-topological space ((F, A), τ ) in which every IFSM-point is IFSM-closed, is IFSM-regular if and only if for an IFSM-open set (S, A) containing an IFSM-point e(F,A) , there exists an IFSM-open set (H , A) containing e(F,A) such ˜ A). that cl(H , A)⊆(S, Proof Take an IFSM-open set (S, A) containing e(F,A) in a regular IFSMtopological space ((F, A), τ ). Then (S, A)c is IFSM-closed. By hypothesis, ∃ disjoint IFSM-open ˜ ˜ , A) and (S, A)c ⊆(K, sets (H , A) and (K, A) such that e(F,A) ∈(H A). Now, (H , A) and ˜ (K, A) are disjoint, so (H , A)C˜ (K, A)c ⇒ cl(H , A)⊆(K, A)c ⇒ cl(H , A)˜(S, A). Conversely, assume the hypothesis. Take an IFSM-closed set (K, A) not containing /˜ A). Then (K, A)c is an IFSM-open set containing the an IFSM-point e(F,A) ∈(K, ˜ e(F,A) ⇒ ∃ an IFSM-open set (S, A) containing e(F,A) such that cl(S, A)⊆(K, A)c ⇒ c c ˜ (K, A)⊆(cl(S, A)) ⇒ (cl(S, A)) is an IFSM-open set containing (K, A) and ˜ (S, A)∩(cl(S, A))c = .A . Definition 9.8.14 An IFSM-topological space ((F, A), τ ) is said to be an IFSMnormal space if for every pair of disjoint IFSM-closed sets (K, A) and (L, A), ∃ disjoint IFSM-open sets (S, A), (H , A) such that (K, A) . (S, A) and (L, A) . (H , A). Theorem 9.8.15 An IFSM-topological space ((F, A), τ ) is IFSM-normal if and only if for any IFSM-closed set (K, A) and IFSM-open set (H , A) containing (K, A), there ˜ , A). exists an IFSM-open set (S, A) such that (K, A) ⊆ (S, A) and cl(S, A)⊆H Proof Let ((F, A), τ ) be IFSM-normal space and (K, A) be an IFSM-closed set and (H , A) be an IFSM-open set containing (K, A) ⇒ (K, A) and (H , A)c are disjoint IFSM-closed sets ⇒ ∃ disjoint IFSM-open sets (S, A), (G, A) such that ~cl((G, A)c ) ~(S, A) and (H , A)c ⊆ ~(G, A)c ⇒ cl(S, A)⊆ ~(G, A). Now (S, A)⊆ (K, A)⊆ c c~ c ~ ~ = (G, A) Also, (H , A) ⊆(G, A) ⇒ (G, A) ⊆(H, A) ⇒ cl(S, A)⊆(H , A). Conversely, let (L, A) and (K, A) be any disjoint pair of IFSM-closed sets ⇒ (L, A) ⊆ (K, A)c , then by hypothesis there exists an IFSM-open set (S, A) such that (L, A) ⊆ (S, A) and Cl(S, A) ⊆ (K, A)c ⇒ (K, A) ⊆ (Cl(S, A))c ⇒ (S, A) and (Cl(S, A))c are disjoint IFSM-open sets such that (L, A) ⊆ (S, A) and (K, A) ⊆ (Cl(S, A))c .

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Theorem 9.8.16 An IFSM-closed subspace of an IFSM-normal space is an IFSMnormal. Proof The proof is straightforward.

9.9 Sequences of IFSMSs Here we introduce a new sequence of IFSMSs in IFSM-topological spaces and study their basic properties. Definition 9.9.1 Let ((F, A), τ ) be the IFSM-topological space on (F, A) and N be the set of all natural numbers. A sequence of IFSMSs in ((F, A), τ ) is a mapping from N to IFSMSA (F, A) and is denoted by {(Fn , A)} or {(Fn , A) : n = 1, 2, 3, . . .}. Example { 9.9.2 Let } us consider two universes U1 = {h1 , h2 } and U2 = {c1 , c2 }. Let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎫ ⎪ ⎪ eU1 ,1 = modern, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = wooden, e ⎨ ⎬ U1 ,2 EU1 = eU1 ,3 = in green surroundings, , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ eU1 ,5 = in good repair, ⎧ ⎫ ⎪ ⎪ e = beautiful, ⎪ ⎪ ⎪ U2 ,1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU2 ,2 = new model, ⎬ EU2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎪ ⎩e ⎭ = in good condition. U2 ,5 Let U =

.2 i=1

FS(Ui ), E =

.2 i=1

EUi and

) )} { ( ( A = a1 = eU1 ,1 , eU2 ,1 , a2 = eU1 ,2 , eU2 ,2 . If we chose n = 1, 2, 3, . . . . . .)) .( (. c1 h1 h2 c2 , , e1 , , , (Fn , A) = (2/5n, 3/5n) (1/n, 2/4n) (1/2n, 1/2n) (5/7n, 1/7n) . . .). (. ( c1 h1 h2 c2 , , , , e2 , (1/5n, 3/5n) (2/3n, 1/3n) (1/2n, 1/2n) (1/7n, 1/7n)

then {(Fn , A) : n = 1, 2, . . .} forms a sequence of IFSMSs.

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135

Definition 9.9.3 A sequence {(Fn , A)} of IFSMSs is said to be eventually contained in an IFSMS (F, A) if and only if there is a positive integer m such that, n ≥ m ~(F, A). implies (Fn , A)⊆ Definition 9.9.4 A sequence {(Fn , A)} of IFSMSs in an IFSM-topological space ((F, A), τ ) is said to be convergence and converge to an IFSMS (F, A) if it is eventually contained in each neighbourhood of the IFSMS (F, A) and we say that the sequence {(Fn , A)} has the limit (F, A) and we write limn→∞ (Fn , A) = (limn→∞ Fn , A) = (F, A) or (Fn , A) → (F, A) as n → ∞ or simply Fn → F as n → ∞. Example 9.9.5 If we consider an IFSM-sequence {(Fn , A)} as in Example 9.9.2, then lim (Fn , A) . . .) .( (. c1 h1 h2 c2 , , , , e1 , = lim n→∞ (2/5n, 3/5n) (1/n, 2/4n) (1/2n, 1/2n) (5/7n, 1/7n) . . .)). .( (. c1 h1 h2 c2 , , , , = e2 , (1/5n, 3/5n) (2/3n, 1/3n) (1/2n, 1/2n) (1/7n, 1/7n) . . .) ( . . .). (. . c1 c1 h1 h1 h2 c2 h2 c2 , , e2 , , . = , , , , (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) n→∞

Theorem 9.9.6 If the neighbourhood system of each IFSMS in an IFSM-topological space ((F, A), τ ) is countable, then an IFSMS (F, A) is open if and only if each sequence {(Fn , A)} of IFSMSs which converges to an IFSMS (G, A) contained in (F, A) is eventually contained in (F, A). Proof Since (F, A) is an IFSM-open set in ((F, A), τ ), (F, A) is a neighbourhood of (G, A). Hence, {(Fn , A)} is eventually contained in (F, A). Conversely, for each ~(F, A), let (G1 , A), (G2 , A), . . . , (Gn , A), . . . be the neighbourhood system (G, A)⊆ n ∩i=1 (Gi , A), then {(Hn , A) : n = 1, 2, . . .} is a sequence of (G, A) and let (Hn , A) = ~ of IFSMSs which is eventually contained in each neighbourhood of (G, A). Hence, there is an m such that for n ≥ m, (Hn , A) ⊆ (F, A). Thus (Hn , A) are neighborhoods of (G, A). This implies (F, A) is a neighbourhood of (G, A) and hence (F, A) is an IFSM-open set. Theorem 9.9.7 If an IFSMS (F, A) is open, then each IFSM-sequence {(Fn , A)} that converges to an IFSMS (G, A) contained in (F, A) is eventually contained in (F, A). Proof Since (F, A) is open and (G, A) ⊆ (F, A) ⊆ (F, A), (F, A) is a neighbourhood of (G, A) and since {(Fn , A)} converges to a fuzzy soft multi set (G, A), it is eventually contained in each neighbourhood of (G, A). Hence, {(Fn , A)} is eventually contained in (F, A). Definition 9.9.8 Let f be mapping over the set of positive integers. Then the sequence {(Gn , A)} is a subsequence of a sequence {(Fn , A)} if and only if there is a map f

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9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

( ) such that (Gi , A) = Ff (i) , A and for each integer m, there is an integer no such that f (i) ≥ m whenever i ≥ no . Example 9.9.9 { } Let us consider two universes U1 = {h1 , h2 } and U2 = {c1 , c2 }. Let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎫ ⎪ ⎪ eU1 ,1 = modern, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU1 ,2 = wooden, ⎬ EU1 = eU1 ,3 = in green surroundings, , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎪ ⎩e ⎭ = in good repair, U1 ,5 ⎧ ⎫ ⎪ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU2 ,2 = new model, ⎬ . EU2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition. Let U =

.2 i=1

FS(Ui ), E =

.2 i=1

EUi and

) )} { ( ( A = a1 = eU1 ,1 , eU2 ,1 , a2 = eU1 ,2 , eU2 ,2 . We chose for n = 1, 2, 3 . . ., . . .)) .( (. c1 h1 h2 c2 , , e1 , , , (Fn , A) = (2/5n, 1/2n) (3/4n, 1/5n) (1/n, 3/7n) (5/7n, 1/7n) . . .). ( (. c1 h1 h2 c2 , , , , e2 , (1/3n, 2/5n) (2/3n, 2/3n) (1/2n, 1/4n) (1/3n, 2/3n)

and .( (. . . .) h1 h2 c1 c2 e1 , , , , , (Gn , A) = (1/5n, 1/2n) (1/2n, 1/2n) (1/5n, 1/2n) (1/2n, 1/5n) . . .). (. ( c1 h1 h2 c2 , , e2 , , , (1/4n, 3/5n) (1/2n, 1/2n) (1/3n, 1/3n) (1/4n, 2/3n)

then {(Gn , A)} is a subsequence of the sequence {(Fn , A)}. Definition 9.9.10 The complement of an IFSM-sequence {(Fn , A)} in an IFSMtopoτ ) is denoted by {(Fn , A)}C and is defined by {(Fn , A)}C = } ((F, {( CA), )} {logical space C (Fn , A) = Fn , A .

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137

{(Fn}, A)} {( Example 9.9.11 If we consider an IFSM-sequence as in Example 9.9.2, { )} then complement of {(Fn , A)} is {(Fn , A)}C = (Fn , A)C = FnC , A , where for n = 1, 2, 3, . . . . . . . . .)) .( (. ( c ) c1 h1 h2 c2 , , e1 , Fn , A = , , (3/5n, 2/5n) (2/4n, 1/n) (1/2n, 1/2n) (1/7n, 5/7n) . . .)). ( (. h1 c1 h2 c2 , , , . e2 , (3/5n, 1/5n) (1/3n, 2/3n) (1/2n, 1/2n) (1/7n, 1/7n)

Definition 9.9.12 A sequence {(Fn , A)} of IFSMSs is said to be increasing sequence if and only if for each positive integer n, (Fn , A) ⊆ (Fn+1 , A), i.e. ~(F2 , A)⊆ ~(F3 , A)⊆ ~. (F1 , A)⊆ Definition 9.9.13 A sequence {(Fn , A)} of IFSMSs is said to be decreasing sequence if and only if for each positive integer n, (Fn , A) . (Fn+1 , A), i.e. ∨

.(F2 , A)~ .(F3 , A) ⊇ . . . . . .. (F1 , A)~ Definition 9.9.14 A sequence {(Fn , A)} of IFSMSs is said to be monotonic if and only if the sequence is either increasing or decreasing sequence. Example 9.9.15 { } Let us consider two universes U1 = {h1 , h2 } and U2 = {c1 , c2 }. Let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where ⎧ ⎫ ⎪ ⎪ eU1 ,1 = modern, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU1 , = wooden, ⎬ EU1 = eU1,3 = in green surroundings, , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U1 ,5 = in good repair, ⎧ ⎫ ⎪ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = new model, e ⎨ U2 ,2 ⎬ . EU2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition. . . Let U{ = (2i=1 FS(Ui ), )E = 2i=1 ( EUi and. )} A = a1 = eU1 ,1 , eU2 ,1 , a2 = eU1 ,2 , eU2 ,2 . We chose for n = 1, 2, 3 . . . .( (. . . .)) h1 h2 c1 c2 e1 , , , , , (Fn , A) = (1 − 1/5n, 1/2n) (1 − 3/4n, 1/5n) (1/2, 3/7n) (5/7, 1/7n) . . .)). ( (. h2 c1 c2 h1 , , e2 , , , (1/3, 2/5n) (1 − 2/3n, 2/3n) (1 − 1/2n, 1/4) (1 − 1/3n, 2/3n)

138

9 Algebraic and Topological Structures on Intuitionistic Fuzzy Soft Multisets

and . . .) .( (. h2 c1 c2 h1 , , , , e1 , (Gn , A) = (1/5n, 1 − 1/2n) (1/2n, 1/2) (1/5n, 1 − 1/2n) (1/2n, 1/5) ( . . .). (. h2 c1 c2 h1 e2 , , , , , (1/4n, 3/5) (1/2n, 1 − 1/2n) (1/3n, 1 − 1/3n) (1/4n, 2/3)

then the sequence {(Fn , A)} is increasing sequence and the sequence {(Gn , A)} is decreasing sequence. Definition 9.9.16 A sequence {(Fn , A)} of IFSMSs is said to be frequently contained in an IFSMS (F, A) if and only if, for each positive integer n, there is a positive integer m such that, n ≥ m implies (Fn , A) ⊆ (F, A). Definition 9.9.17 An IFSMS (F, A) in an IFSM-topological space ((F, A), τ ) is a cluster IFSMS of a sequence {(Fn , A)} if the sequence {(Fn , A)} is frequently contained in every neighbourhood of (F, A). Theorem 9.9.18 If the neighbourhood system of each IFSMS in an IFSMtopological space ((F, A), τ ) is countable, then for (F, A) is a cluster IFSMS of a sequence {(Fn , A)} there is a subsequence converging to (F, A). Proof Let (K1 , A), (K2 , A), . . . , (Kn , A), . . . be neighbourhood system of (F, A) and n ∩i=1 {(Ki , A)}. Then {(Ln , A) : n = 1, 2, . . .} is a sequence of IFSMSs let (Ln , A) = ~ such that (Ln+1 , A) ⊆ (Ln , A) for each n and is eventually contained in each neighinteger i, choose f}(i) such that f (i) ≥ i and bourhood ) of (F, A). For every positive ) ( {( Ff (i) , A ⊆ (Li , A) and hence Ff (i) , A : i = 1, 2, . . . is a subsequence of the sequence {(Fn , A) : n = 1, 2, . . .}, which converges to (F, A). Theorem 9.9.19 Let (F, A) be a cluster IFSMS of an IFSM-sequence {(Fn , A)} and (F, A) contained in an IFSMS (G, A). If (G, A) is open, then the IFSM sequence is frequently contained in (G, A). Proof Since (G, A) is open and (F, A) is contained in an IFSMS(G, A). Hence (G, A) is a neighbourhood of (F, A). Also, since (F, A) be a cluster IFSMS of an IFSMsequence {(Fn , A)} so by the definition of cluster IFSMS the sequence {(Fn , A)} is frequently contained in every neighbourhood of (F, A) and hence, {(Fn , A)} is frequently contained in (G, A).

Chapter 10

Application of Intuitionistic Fuzzy Soft Multisets

Distributed computing and internet technology research has progressed swiftly since it may be applied to a variety of fields like as software engineering, therapeutic science, financial matters, circumstances, and construction, among others. Soft set theory’s applications, particularly in information systems, have been discovered to be of fundamental importance. The applicability of IFSMSs (IFSMSs) in real-life decision-making situations is discussed here. We describe another innovative method that employs decision rules (thresholds) to tackle real-world decision-making challenges. To demonstrate the applicability of our methodology in practical applications, an example case is used. We will also discuss the use of IFSMS in information systems and demonstrate that every IFSMS is an intuitionistic fuzzy multi-valued information system.

10.1 An Adjustable Approach Based on IFSMSs We begin a novel algorithm designed for solving intuitionistic fuzzy soft set (IFSset) based decision-making problems, which was presented in [1]. Jiang et al.’s Algorithm Jiang et al. [1] used the following adjustable approach to IFS-set based decisionmaking by using level soft sets.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Mukherjee and A. K. Das, Essentials of Fuzzy Soft Multisets, https://doi.org/10.1007/978-981-19-2760-7_10

139

140

10 Application of Intuitionistic Fuzzy Soft Multisets

10.2 Algorithm 3 (Jiang’s Algorithm) Step1. Input the (resultant) IFS-set . = (F, A). Step2. Input a threshold IF-set λ : A → [0, 1]×[0, 1] (or give a threshold value (s, t) ∈ [0, 1]×[0, 1]; ; or choose the mid-level decision rule; or choose the topbottom-level decision rule, or choose the top-top-level decision rule or choose the bottom-bottom-level decision rule) for decision-making. Step3. Compute the LS-set L(. ; λ) of . with respect to the threshold IF-set λ (or the (s, t)-level soft set L(. ; s, t); or the mid level soft set L(. ; mid); or the top– bottom level soft set L(. ; ; topbottom); or the top-top level soft set L(. ; toptop ), or the bottom-bottom level soft set L(. ; bottombottom )). Step4. Present the LS-set L(. ; λ) (or L(. ; s, t) or L(. ; mid); or L(. ; topbottom ); or L(. ; toptop ); or L(. ; bottombottom )) in tabular form and compute the choice value Si of u i ∈ U, ∀i. Step5. The optimal decision is to select u k if Sk = maxi Si . Step6. If k has more than one value then any one of u k may be chosen. Application of IFSMSs in Decision-Making In this segment, we show our calculation for decision-making in view of IFSMSs. By considering proper reduct IFS-sets and level soft sets of IFS-sets, IFSMSs based decision-making can be changed over into just crisp soft sets based decision-making. Firstly, by figuring the reduct IFS-set, an IFSMS is changed over into an IFSMS part and then the IFSMS part is further changed over into a crisp soft set by utilizing level soft sets of IFS-sets. At last, decision-making is performed on the crisp soft set. The points of interest of our calculation are recorded underneath.

10.3 Algorithm 8 Step1. Input the (resultant) IFSMS(F, A). Step2. Apply Jiang’s algorithm to the first IFSMS part in (F, A) to get the decision Sk1 . Step3. Redefine the IFSMS (F, A) by keeping all values in each row where Sk1 is maximized and replacing the values in the other rows by zero, to get (F, A)1 . Step4. Apply Jiang’s algorithm to the second IFSMS part in (F, A)1 to get the decision Sk2 . .

10.3 Algorithm 8

141

Step5. Redefine the IFSMS (F, A)1 by keeping the first and second parts and applying the method in step (3) to the third part. Step6. Apply Jiang’s algorithm to the third IFSMS part in (F, A)2 to get the decision Sk3 . ( ) Step7. Continuing in this way we get the decision Sk1 , Sk2 , Sk3 , . . . .. . Remark 10.1.3 If there are too many optimal choices in Step 7, we may go back to the second step and change the decision rule (threshold) such that few optimal choices remain in the end. Application of IFSMSs in Decision-Making Let us consider three universes U1 = {h 1 , h 2 , h 3 , h 4 }, U2 = {c1 , c2 , c3 } and U3 = {v1 , v2 , v3 } be the sets of houses, cars and hotels, respectively. Suppose Mr. X has a budget to buy a house, a car and rent a venue to hold a wedding celebration. Let us consider an IFSMS (F, A) which describes houses, cars and hotels that Mr. X is considering for accommodation purchase, transportation purchase, and a venue to { } hold a wedding celebration, respectively. Let EU1 , EU2 , EU3 be a collection of sets of decision parameters related to the above universes, where

E U1

E U2

E U3

⎧ ⎪ ⎪ eU1 ,1 = ⎪ ⎪ ⎪ ⎨ eU1 ,2 = = eU1 ,3 = ⎪ ⎪ ⎪ eU1,4 = ⎪ ⎪ ⎩e U1 ,5 = ⎧ ⎪ eU2,1 = ⎪ ⎪ ⎪ ⎪ ⎨ eU2 ,2 = = eU2 ,3 = ⎪ ⎪ ⎪ eU2,4 = ⎪ ⎪ ⎩e U2,5 = ⎧ ⎪ eU3 ,1 = ⎪ ⎪ ⎪ ⎪ e ⎨ U3 ,2 = = eU3 ,3 = ⎪ ⎪ ⎪ eU3 ,4 = ⎪ ⎪ ⎩e U3 ,5 =

modern, wooden, in green surroundings, cheap, in good repair, ⎫ ⎪ beautiful, ⎪ ⎪ ⎪ ⎪ new model, ⎬ , sporty, ⎪ ⎪ ⎪ black ⎪ ⎪ in good condition, ⎭ ⎫ ⎪ expensive , ⎪ ⎪ ⎪ ⎪ cheap, ⎬ in Kuala Lumpur, . ⎪ ⎪ ⎪ majestic, ⎪ ⎪ ⎭ beautiful.

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10 Application of Intuitionistic Fuzzy Soft Multisets

Let U =

.3 i=1

F S(Ui ), E =

.3

⎧ ⎪ a1 ⎪ ⎪ ⎪ ⎪ ⎨ a2 A = a3 ⎪ ⎪ ⎪ a4 ⎪ ⎪ ⎩a 5

i=1

EUi and A ⊆ E, such that

) ⎫ ( = (eU1 ,1 , eU2 ,5 , eU3 ,1 ), ⎪ ⎪ ⎪ ⎪ = (eU1 ,4 , eU2 ,2 , eU3 ,3 ), ⎪ ⎬ = (eU1 ,2 , eU2 ,3 , eU3 ,2 ), . ⎪ ⎪ = (eU1 ,3 , eU2 ,4 , eU3 ,5 ), ⎪ ⎪ ⎪ = eU1 ,5 , eU2 ,1 , eU3 ,4 , ⎭

Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of choice parameters. Let the resultant IFSMS be (F, A) as in Table 10.1. Now we apply Jiang’s algorithm to the first IFSMS part in (F, A) to take the decision from the availability set U1 . The tabular representation of the first resultant IFSMS part will be as in Table 10.2. Now we calculate the mid level soft set of U1− IFSMS part in (F, A), with choice values (sk ) as in Table 10.3. From Table 10.3, it is clear that the maximum choice value is 4, scored by h 5 . Now we redefine the IFSMS(F, A) by keeping all values in Table 10.1 IFSMS(F, A)IFSMS(F, A) Ui U1

U2

U3

a1

a2

a3

a4

a5

h1

(0.2, 0.5)

(0.8, 0.2)

(0.9, 0.1)

(0.8, 0.2)

(0.5, 0.4)

h2

(0.5, 0.4)

(0.7, 0.3)

(0.6, 0.3)

(0.4, 0.6)

(0.7, 0.3)

h3

(0.6, 0.4)

(0.5, 0.3)

(0.7, 0.2)

(0.6, 0.1)

(0.6, 0.1)

h4

(0.7, 0.1)

(0.7, 0.1)

(0, 1)

(0.5, 0.5)

(0.7, 0.2)

h5

(0.8, 0.1)

(0.8, 0.3)

(1, 0)

(0.7, 0.2)

(0.9, 0.1)

c1

(0, 7, 0.2)

(0.8, 0.2)

(0.8, 0.2)

(0.5, 0.5)

(1, 0)

c2

(0.8, 0.2)

(0.8, 0.1)

(0.6, 0.3)

(0.6, 0.3)

(0.9, 0.1)

c3

(0.6, 0.3)

(0.5, 0.5)

(0.7, 0.3)

(0.7, 0.3)

(0.9, 0.1)

c4

(0.8, 0.1)

(0.7, 0.3)

(0.8, 0.2)

(0.6, 0.2)

(0.8, 0.2)

v1

(0.9, 0.1)

(0.7, 0.3)

(0.6, 0.3)

(0.5, 0.5)

(0.8, 0.2)

v2

(0.7, 0.3)

(0.6, 0.3)

(0.6, 0.2)

(0.4, 0.3)

(0.5, 0.5)

v3

(0.9, 0)

(0.8, 0.2)

(0.5, 0.5)

(0.5, 0.4)

(0.6, 0.4)

Table 10.2 U U1 —IFSMS part of (F, A) U1

a1

a2

a3

a4

a5

h1

(0.2, 0.5)

(0.8, 0.2)

(0.9, 0.1)

(0.8, 0.2)

(0.5, 0.4)

h2

(0.5, 0.4)

(0.7, 0.3)

(0.6, 0.3)

(0.4, 0.6)

(0.7, 0.3)

h3

(0.6, 0.4)

(0.5, 0.3)

(0.7, 0.2)

(0.6, 0.1)

(0.6, 0.1)

h4

(0.7, 0.1)

(0.7, 0.1)

(0, 1)

(0.5, 0.5)

(0.7, 0.2)

h5

(0.8, 0.1)

(0.8, 0.3)

(1, 0)

(0.7, 0.2)

(0.9, 0.1)

10.3 Algorithm 8

143

each row where h1 is maximized and replacing the values in the other rows by zero, to get (F, A)1 (Table 10.4). Now we apply Jiang’s algorithm to the second IFSMS part in (F, A)1 to take the decision from the availability set U2 . From Table 10.5, it is clear that the maximum choice value is 3, scored by c1 . Now we redefine the IFSMS (F, A)1 by keeping all values in each row where c1 is maximized and replacing the values in the other rows by zero, to get (F, A)2 (Table 10.6). Table 10.3 The mid level soft set of U1 -IFSMS part in (F, A), with choice values (Sk ) U1

a1

a2

a3

a4

a5

sk

h1

0

1

1

1

0

s1 = 3

h2

0

0

0

0

0

s2 = 0

h3

0

0

1

1

0

s3 = 2

h4

1

1

0

0

1

s4 = 3

h5

1

0

1

1

1

s5 = 4

Table 10.4 The IFSMS (F, A)1 Ui U1

U2

U3

h1

a1

a2

a3

a4

a5

(0.2, 0.5)

(0.8, 0.2)

(0.9, 0.1)

(0.8, 0.2)

(0.5, 0.4)

h2

(0.5, 0.4)

(0.7, 0.3)

(0.6, 0.3)

(0.4, 0.6)

(0.7, 0.3)

h3

(0.6, 0.4)

(0.5, 0.3)

(0.7, 0.2)

(0.6, 0.1)

(0.6, 0.1)

h4

(0.7, 0.1)

(0.7, 0.1)

(0, 1)

(0.5, 0.5)

(0.7, 0.2)

h5

(0.8, 0.1)

(0.8, 0.3)

(1, 0)

(0.7, 0.2)

(0.9, 0.1)

c1

(0, 7, 0.2)

0

(0.8, 0.2)

0

(1, 0)

c2

(0.8, 0.2)

0

(0.6, 0.3)

0

(0.9, 0.1)

c3

(0.6, 0.3)

0

(0.7, 0.3)

0

(0.9, 0.1)

c4

(0.8, 0.1)

0

(0.8, 0.2)

0

(0.8, 0.2)

v1

(0.9, 0.1)

0

(0.6, 0.3)

0

(0.8, 0.2)

v2

(0.7, 0.3)

0

(0.6, 0.2)

0

(0.5, 0.5)

v3

(0.9, 0)

0

(0.5, 0.5)

0

(0.6, 0.4)

Table 10.5 Mid level soft set of U2 —IFSMS part in (F, A)1 , with choice values (Sk ) U2

a1

a2

a3

a4

a5

sk

c1

1

0

1

0

1

s1 = 3

c2

1

0

0

0

1

s2 = 2

c3

0

0

0

0

1

S3 = 1

c4

1

0

1

0

0

S4 = 2

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10 Application of Intuitionistic Fuzzy Soft Multisets

Table 10.6 The IFSMS (F, A)2 Ui U1

U2

U3

h1

a1

a2

a3

a4

a5

(0.2, 0.5)

(0.8, 0.2)

(0.9, 0.1)

(0.8, 0.2)

(0.5, 0.4)

h2

(0.5, 0.4)

(0.7, 0.3)

(0.6, 0.3)

(0.4, 0.6)

(0.7, 0.3)

h3

(0.6, 0.4)

(0.5, 0.3)

(0.7, 0.2)

(0.6, 0.1)

(0.6, 0.1)

h4

(0.7, 0.1)

(0.7, 0.1)

(0, 1)

(0.5, 0.5)

(0.7, 0.2)

h5

(0.8, 0.1)

(0.8, 0.3)

(1, 0)

(0.7, 0.2)

(0.9, 0.1)

c1

(0, 7, 0.2)

0

(0.8, 0.2)

0

(1, 0)

c2

(0.8, 0.2)

0

(0.6, 0.3)

0

(0.9, 0.1)

c3

(0.6, 0.3)

0

(0.7, 0.3)

0

(0.9, 0.1)

c4

(0.8, 0.1)

0

(0.8, 0.2)

0

(0.8, 0.2)

v1

0

0

(0.6, 0.3)

0

(0.8, 0.2)

v2

0

0

(0.6, 0.2)

0

(0.5, 0.5)

v3

0

0

(0.5, 0.5)

0

(0.6, 0.4)

Table 10.7 Mid-level soft set of U3 —IFSMS part in (F, A)2 , with choice values (Sk ) U3

a1

a2

a3

a4

a5

sk

v1

0

0

1

0

1

s1 = 2

v2

0

0

1

0

0

s2 = 1

v3

0

0

0

0

0

s3 = 0

Now we apply Jiang’s algorithm to the third IFSMS part in (F, A)2 to take the decision from the availability set U3 . From Table 10.7, it is clear that the maximum choice value (sk ) is 2, by v1 . Then from the above results, the decision for Mr. X is (h5 , c1 , v1 ). Remark 10.2.2 In Algorithm 8, one may go back to the second step and change the threshold (or decision rule) that he once use so as to adjust the final optimal decision, especially when there are too many “optimal choices” to be chosen. To illustrate the basic idea of Algorithm 8, let us consider the following example. Example 10.2.3 Let us consider three universes U1 = {h 1 , h 2 , h 3 , h 4 , h 5 }, U2 = {c1 , C2 , C3 , C4 } and U3 = {v1 , v2 , v3 } be the sets of houses, cars and hotels, respectively. Suppose Mr. X has a budget to buy a house, a car and rent a venue to hold a wedding celebration. Let us consider an IFSMS(F, A) which describes houses, cars and hotels that Mr. X is considering for accommodation purchase, transportation purchase, and a venue to hold a wedding celebration, respectively. { } Let EU1 , EU2 , EU3 be a collection of sets of decision parameters related to the above universes, where

10.3 Algorithm 8

145

E U1

E U2

E U3

Let U =

.3 i=1

⎧ ⎪ ⎪ eU1 ,1 = ⎪ ⎪ ⎪ ⎨ eU1 ,2 = = eU1,3 = ⎪ ⎪ ⎪ eU1 ,4 = ⎪ ⎪ ⎩e U1 ,5 = ⎧ ⎪ eU2 ,1 = ⎪ ⎪ ⎪ ⎪ ⎨ eU2 ,2 = = eU2 ,3 = ⎪ ⎪ ⎪ eU2 ,4 = ⎪ ⎪ ⎩e U2 ,5 = ⎧ eU3 ,3 = ⎪ ⎪ ⎨ eU3 ,4 = = ⎪ = e ⎪ ⎩ U3 ,5 eU3 ,1 =

F S(Ui ), E =

.3

⎧ ⎪ a1 ⎪ ⎪ ⎪ ⎪ ⎨ a2 A = a3 ⎪ ⎪ ⎪ a4 ⎪ ⎪ ⎩a 5

i=1

modern, wooden, in green surroundings, cheap, in good repair, ⎫ ⎪ beautiful ⎪ ⎪ ⎪ ⎪ new model ⎬ sporty ⎪ ⎪ ⎪ black ⎪ ⎪ in good condition , ⎭ ⎫ in Kuala Lumpur, ⎪ ⎪ ⎬ majestic ⎪ beautiful ⎪ ⎭ expensive EUi and A ⊆ E, such that

)⎫ ( = (eU1 ,1 , eU2 ,5 , eU3 ,1 ) ⎪ ⎪ ⎪ ⎪ = (eU1 ,4 , eU2 ,2 , eU3 ,3 ) ⎪ ⎬ = (eU1 ,2 , eU2 ,3 , eU3 ,2 ) . ⎪ ⎪ = (eU1 ,3 , eU2 ,4 , eU3 ,5 ) ⎪ ⎪ ⎪ = eU1 ,5 , eU2 ,1 , eU3 ,4 ⎭

Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of choice parameters. Let the resultant IFSMS be (F, A) as in Table 10.8. If we deal with this problem by mid-level decision rule, then the optimal decision is to (h3 , c1 , v1 ). At the same time, if we deal with this problem by top–bottom-level decision rule, we shall use the top–bottom-threshold of U1 —IFSMS part in (F, A) and thus we the top-level soft set of U1 —IFSMS part in (F, A) with choice values with tabular representation is in Table 10.9. From Table 10.9, it is clear that the maximum choice value is 3, scored by h 5 . Now we redefine the fuzzy soft multi set (F, A) by keeping all values in each row where h 3 is maximized and replacing the values in the other rows by zero, to get (F, A)1 (Table 10.10). Now we apply Jiang’s algorithm to the second IFSMS part in (F, A)1 to take the decision from the availability set U2 . From Table 10.11, it is clear that the maximum choice value is 2, scored by c1 and c4 , so we can choose any one of c1 and c4 .

146

10 Application of Intuitionistic Fuzzy Soft Multisets

Table 10.8 IFSMS(F, A) Ui U1

U2

U3

h1

a1

a2

a3

a4

a5

(0.2, 0.5)

(0.8, 0.2)

(0.9, 0.1)

(0.8, 0.2)

(0.5, 0.4)

h2

(0.5, 0.4)

(0.7, 0.3)

(0.6, 0.3)

(0.4, 0.6)

(0.7, 0.3)

h3

(0.6, 0.4)

(0.5, 0.3)

(0.7, 0.2)

(0.6, 0.1)

(0.6, 0.1)

h4

(0.7, 0.1)

(0.7, 0.1)

(0, 1)

(0.5, 0.5)

(0.7, 0.2)

h5

(0.8, 0.1)

(0.8, 0.3)

(1, 0)

(0.7, 0.2)

(0.9, 0.1)

c1

(0, 7, 0.2)

(0.8, 0.2)

(0.8, 0.2)

(0.5, 0.5)

(1, 0)

c2

(0.8, 0.2)

(0.8, 0.1)

(0.6, 0.3)

(0.6, 0.3)

(0.9, 0.1)

c3

(0.6, 0.3)

(0.5, 0.5)

(0.7, 0.3)

(0.7, 0.3)

(0.9, 0.1)

c4

(0.8, 0.1)

(0.7, 0.3)

(0.8, 0.2)

(0.6, 0.2)

(0.8, 0.2)

v1

(0.9, 0.1)

(0.7, 0.3)

(0.6, 0.3)

(0.5, 0.5)

(0.8, 0.2)

v2

(0.7, 0.3)

(0.6, 0.3)

(0.6, 0.2)

(0.4, 0.3)

(0.5, 0.5)

v3

(0.9, 0)

(0.8, 0.2)

(0.5, 0.5)

(0.5, 0.4)

(0.6, 0.4)

Table 10.9 Top–bottom level soft set of U1 —IFSMS part in (F, A), with choice values U1

a1

a2

a3

a4

a5

Choice value (s k )

h1

0

0

0

0

0

s1 = 0

h2

0

0

0

0

0

S2 = 0

h3

0

0

0

0

0

S3 = 0

h4

0

0

0

0

0

S4 = 0

h5

1

0

1

0

1

S5 = 3

Table 10.10 The IFSMS (F, A)1 a1

Ui U1

U2

U3

a2

a3

a4

a5

h1

(0.2, 0.5)

(0.8, 0.2)

(0.9, 0.1)

(0.8, 0.2)

(0.5, 0.4)

h2

(0.5, 0.4)

(0.7, 0.3)

(0.6, 0.3)

(0.4, 0.6)

(0.7, 0.3)

h3

(0.6, 0.4)

(0.5, 0.3)

(0.7, 0.2)

(0.6, 0.1)

(0.6, 0.1)

h4

(0.7, 0.1)

(0.7, 0.1)

(0, 1)

(0.5, 0.5)

(0.7, 0.2)

h5

(0.8, 0.1)

(0.8, 0.3)

(1, 0)

(0.7, 0.2)

(0.9, 0.1)

c1

(0, 7, 0.2)

0

(0.8, 0.2)

0

(1, 0)

c2

(0.8, 0.2)

0

(0.6, 0.3)

0

(0.9, 0.1)

c3

(0.6, 0.3)

0

(0.7, 0.3)

0

(0.9, 0.1)

c4

(0.8, 0.1)

0

(0.8, 0.2)

0

(0.8, 0.2)

v1

(0.9, 0.1)

0

(0.6, 0.3)

0

(0.8, 0.2)

v2

(0.7, 0.3)

0

(0.6, 0.2)

0

(0.5, 0.5)

v3

(0.9, 0)

0

(0.5, 0.5)

0

(0.6, 0.4)

10.4 Case I

147

Table 10.11 The top-level soft set of U2 -IFSMS part in (F, A)1 , with choice values U2

a1

a2

a3

a4

a5

Choice value (s k )

c1

0

0

1

0

1

s1 = 2

c2

0

0

0

0

0

s1 = 0

c3

0

0

0

0

0

s1 = 0

c4

1

0

1

0

0

s1 = 2

10.4 Case I If we chose c1 , then we redefine the IFSMS(F, A)1 by keeping all values in each row where c1 is maximized and replacing the values in the other rows by zero, to get(F, A)2 as in Table 10.12. Now we apply Jiang’s algorithm to the third IFSMS part in (F, A)2 to take the decision from the availability set U3 . From Table 10.13, it is clear that the maximum choice value (sk ) is 1, by v1 and v2 . Then from the above results, the decision for Mr. X is (h3 , c1 , v1 ) or (h5 , c1 , v2 ). Table 10.12 The IFSMS (F, A)2 (F, A)2 a1

a2

a3

a4

a5

h1

(0.2, 0.5)

(0.8, 0.2)

(0.9, 0.1)

(0.8, 0.2)

(0.5, 0.4)

h2

(0.5, 0.4)

(0.7, 0.3)

(0.6, 0.3)

(0.4, 0.6)

(0.7, 0.3)

h3

(0.6, 0.4)

(0.5, 0.3)

(0.7, 0.2)

(0.6, 0.1)

(0.6, 0.1)

h4

(0.7, 0.1)

(0.7, 0.1)

(0, 1)

(0.5, 0.5)

(0.7, 0.2)

h5

(0.8, 0.1)

(0.8, 0.3)

(1, 0)

(0.7, 0.2)

(0.9, 0.1)

c1

(0, 7, 0.2)

0

(0.8, 0.2)

0

(1, 0)

c2

(0.8, 0.2)

0

(0.6, 0.3)

0

(0.9, 0.1)

c3

(0.6, 0.3)

0

(0.7, 0.3)

0

(0.9, 0.1)

Ui U1

U3

c4

(0.8, 0.1)

0

(0.8, 0.2)

0

(0.8, 0.2)

v1

0

0

(0.6, 0.3)

0

(0.8, 0.2)

v2

0

0

(0.6, 0.2)

0

(0.5, 0.5)

v3

0

0

(0.5, 0.5)

0

(0.6, 0.4)

Table 10.13 The top-level soft set of U3 —IFSMS part in (F, A)2 , with choice values U3

a1

a2

a3

a4

a5

Choice value (sk )

v1

0

0

0

0

1

s1 = 1

v2

0

0

1

0

0

s1 = 1

v3

0

0

0

0

0

s1 = 0

148

10 Application of Intuitionistic Fuzzy Soft Multisets

10.5 Case II If we chose c4 , then we redefine the IFSMS(F, A)1 by keeping all values in each row where c4 is maximized and replacing the values in the other rows by zero, to get (F, A)2 as in Table 10.14. Now we apply Jiang’s algorithm to the third IFSMS part in (F, A)2 to take the decision from the availability set U3 . From Table 10.15, it is clear that the maximum choice value (sk ) is 1, by v2 and v3 . Then from the above results, the decision for Mr.X is (h5 , c4 , v2 ) or (h5 , c4 , v3 ). Remark 10.2.4 From these above two cases, we see that the optimal decision for two different decision rules, the optimal decisions are different for the decision-maker.

Table 10.14 The IFSMS (F, A)2 U3 U1

U2

v1

a1

a2

a3

a4

a5

h1

(0.2, 0.5)

(0.8, 0.2)

(0.9, 0.1)

(0.8, 0.2)

(0.5, 0.4)

h2

(0.5, 0.4)

(0.7, 0.3)

(0.6, 0.3)

(0.4, 0.6)

(0.7, 0.3)

h3

(0.6, 0.4)

(0.5, 0.3)

(0.7, 0.2)

(0.6, 0.1)

(0.6, 0.1)

h4

(0.7, 0.1)

(0.7, 0.1)

(0, 1)

(0.5, 0.5)

(0.7, 0.2)

h5

(0.8, 0.1)

(0.8, 0.3)

(1, 0)

(0.7, 0.2)

(0.9, 0.1)

c1

(0, 7, 0.2)

0

(0.8, 0.2)

0

(1, 0)

c2

(0.8, 0.2)

0

(0.6, 0.3)

0

(0.9, 0.1)

c3

(0.6, 0.3)

0

(0.7, 0.3)

0

(0.9, 0.1)

c4

(0.8, 0.1)

0

(0.8, 0.2)

0

(0.8, 0.2)

(0.9, 0.1)

0

(0.6, 0.3)

0

0

v2

(0.7, 0.3)

0

(0.6, 0.2)

0

0

v3

(0.9, 0)

0

(0.5, 0.5)

0

0

Table 10.15 The top-level soft set of U3 —IFSMS part in (F, A)2 , with choice values U3

a1

a2

a3

a4

a5

Choice value (sk )

v1

0

0

0

0

0

s1 = 0

v2

0

0

1

0

0

s1 = 1

v3

1

0

0

0

0

s1 = 1

10.5 Case II

149

10.5.1 Advantages The benefits of Algorithm 8 are for the most part twofold. In the first place, it is anything but difficult to see that this calculation includes generally less reckonings. We just need to consider (established) decision estimations of articles in level soft sets part. The purpose behind this effortlessness is that in Algorithm 8 we manage crisp soft sets as opposed to the starting IFSMS, by considering certain membership levels. This makes our calculation less complex and simpler for application in reasonable issues. Second, Algorithm 8 can be seen as a movable way to deal with IFSMS based decision-making on the grounds that the last ideal choice is in connection to the thresholds on participation values and non-enrollment values or at the end of the day, the choice criteria utilized by leaders. For example, on the off chance that we pick the top–bottom-level decision rule in the second stride of Algorithm 8, we should consider the decision estimation of every article in the topbottom-level soft set, if another choice basis, for example, the mid-level choice guideline is utilized; we might consider decision values in the mid-level soft set. When all is said in done, the decision estimation of an article in mid-level choice standard need not correspond with the worth in top–bottom-level decision rule. Therefore, the ideal items dictated by the mid-level choice decision rule maybe not quite the same as those chosen by top–bottom-level guidelines. As was said above, numerous choice-making issues are basically humanistic and subjective in nature; henceforth for choice making in an uncertain domain, there really does not exist an exceptional or uniform foundation. This flexible element makes Algorithm 8 productive as well as more suitable for some certifiable applications.

10.5.2 Application of IFSMS Theory in Information Systems Definition 10.3.1 An intuitionistic fuzzy multi-valued information system is a quadruple I n f system = (X, A, f, V ), where X is a nonempty finite set of objects, A is a nonempty finite set of attributes, V = ∪a∈A Va , where V is the domain (an intuitionistic fuzzy set,) set of attributes, which has multi value and f : X × A → V is a total function such that f (x, a) ∈ Va for every (x, a) ∈ X × A. Proposition 10.3.2 If (F, A) is an IFSMS over universe U , then (F, A) is an intuitionistic fuzzy multi-valued information system. . Proof Let {Ui : i ∈ I } be a collection of universes such that Ui = φ and let .i∈I {E i : i ∈ I } be a collection of sets of parameters. Let U = i∈I I F. S(Ui ) where IFS(Ui ) denotes the set of all intuitionistic fuzzy subsets of U , E = i i∈I E Ui and . A ⊆ E. Let (F, A) be an IFSMS over U and X = Ui We define a mapping f where f : X × A → V , defined as.

i∈I

150

10 Application of Intuitionistic Fuzzy Soft Multisets

x ). f (x, a) = ( μ F(a) (x), v F(a) (x) Hence V = ∪a∈A Va , where Va is the set of all counts of in F(a) and ∪ represent the classical set union. Then the intuitionistic fuzzy multi-valued information system (X, A, f, V ) represents the IFSMS(F, A).IFSMS(F, A). Example 10.3.3 Let there are two universes U1 = {h 1 , h 2 , h 3 } and { us consider } U2 = {c1 , C2 }. Let EU1 , EU2 be a collection of sets of decision parameters related to the above universes, where. ⎧ ⎪ e = modern, ⎪ ⎪ U1 ,1 ⎪ ⎪ ⎨ eU1 ,2 = wooden, EU1 = eU1 ,3 = in green surroundings, ⎪ ⎪ ⎪ ⎪ eU1 ,4 = cheap, ⎪ ⎩e ⎧ U1 ,5 = in good repair, ⎫ ⎪ eU2 ,1 = beautiful, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ eU2 ,2 = new model, ⎬ , EU2 = eU2 ,3 = sporty, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ eU2 ,4 = black, ⎪ ⎪ ⎪ ⎪ ⎩e ⎭ U2 ,5 = in good condition. Let U =

.2 i=1

F S(Ui ), E =

.2 i=1

EUi and A ⊆ E, such that

) ) )} { ( ( ( A = a1 = eU1 ,1 , eU2 ,5 , a2 = eU1 ,4 , eU2 ,2 , a3 = eU1 ,2 , eU2 ,3 . Let F(e1 ) = F(e2 ) F(e3 ) =

({

} {

})

h1 c1 , , h2 , h3 , , c2 (0.2,0.7) (0.4,0.5) (0,1) } { (0.8,0.1) (0,1) }) ({ h2 h3 h1 , c1 , c2 , , (0.7,0.2) , (1,0) = (0,1) } { (0,1) (0.6,0.3) }) ({ h2 h3 c1 c2 h1 , (0.5,0.3) , (0.6,0.3) . , , (0,1) (0.8,0.1) (0,1)

Then the IFSMS (F, A) defined above describes the conditions of some “house” and “car” in a state. Then the quadruple (X, A, f, V ) corresponding to the IFSMS given above is an intuitionistic fuzzy multi-valued information system. 2 Where X = ∪i=1 Ui and A is the set of parameters in the IFSMS and {

}

h2 h3 c1 c2 h1 , , (0.4,0.5) , (0,1) , (0.8,0.1) , (0,1) (0.2,0.7) } { h1 h2 h3 C1 C2 Ve2 = (0,1) , (0.7,0.2) , (1,0) , (0,1) , (0.6,0.3) , } { h2 h3 C1 C2 h1 Ve3 = (0,1) . , (0.8,0.1) , (0,1) , (0.5,0.3) , (0.6,0.3)

Ve1 =

Reference Table 10.16 The information table representing IFSMS (F, A)

151 e1

e2

e3

h1

(0.2, 0.7)

(0, 1)

(0, 1)

h2

(0.4, 0.5)

(0.7, 0.2)

(0.8, 0.1)

h3

(0, 1)

(1, 0)

(0.8, 0.1)

c1

(0, 1)

(0, 1)

(0.5, 0.3)

c2

(0, 1)

(0.6, 0.3)

(0.6, 0.3)

For the pair (h 1 , e1 ) we have f (h 1 , e1 ) = (0.2, 0.7), for (h 2 , e1 ), we have f (h 2 , e1 ) = (0.4, 0.5). Continuing in this way we obtain the values of other pairs. Therefore, according to the result above, it is seen that IFSMSs are intuitionistic fuzzy soft multi-valued information systems. Nevertheless, it is obvious that intuitionistic fuzzy soft multi-valued information systems are not necessarily IFSMSs. We can construct an information table representing IFSMS (F, A) defined above as in Table 10.16.

Reference 1. Babitha, K.V., John, S.J.: On soft multi sets. Ann. Fuzzy Math. Inform. 5, 35–44 (2013)