Set Theory for Physicists [Concise ed.] 1643276476, 9781643276472

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Set Theory for Physicists

Set Theory for Physicists Nicolas A Pereyra University of Texas, Rio Grande Valley, Texas, USA

Morgan & Claypool Publishers

Copyright ª 2019 Morgan & Claypool Publishers All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, or as expressly permitted by law or under terms agreed with the appropriate rights organization. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency, the Copyright Clearance Centre and other reproduction rights organizations. Rights & Permissions To obtain permission to re-use copyrighted material from Morgan & Claypool Publishers, please contact [email protected]. ISBN ISBN ISBN

978-1-64327-650-2 (ebook) 978-1-64327-647-2 (print) 978-1-64327-648-9 (mobi)

DOI 10.1088/2053-2571/ab126a Version: 20190501 IOP Concise Physics ISSN 2053-2571 (online) ISSN 2054-7307 (print) A Morgan & Claypool publication as part of IOP Concise Physics Published by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA, 94901, USA IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

I wish to dedicate this work to my son Gabriel Arturo Pereyra, who is its main motivation, and to my wife Sijham Ghaleb Bahri, for her unconditional love and support.

Contents Preface

ix

Acknowledgments

x

Author biography

xi

Equality ‘=’

1-1

Reference

1-2

2

Fundamental properties of sets

2-1

2.1 2.2 2.3 2.4 2.5 2.6 2.7

What is a set? Defining a set Equality of sets A set can be an element of another set A set cannot contain itself A set can be empty (null) Subsets 2.7.1 Definition of subsets 2.7.2 A ⊂ A 2.7.3 ∅ ⊂ A

2-1 2-2 2-2 2-4 2-4 2-5 2-5 2-5 2-5 2-6

3

Set operators

3-1

3.1 3.2

What is a set operator? The ∪ operator (UNION) 3.2.1 The commutative property of ∪ 3.2.2 The associative property of ∪ 3.2.3 The ∪ operator and the empty set ∅ The ∩ operator (INTERSECTION) 3.3.1 The commutative property of ∩ 3.3.2 The associative property of ∩ 3.3.3 The ∩ operator and the empty set ∅ Mixed properties of ∪ and ∩ The \ operator (SET SUBTRACTION) 3.5.1 The \ operator and the empty set ∅ Mixed properties of \, ∪ and ∩ The × operator (CARTESIAN PRODUCT) 3.7.1 The × operator and the empty set ∅

3-1 3-1 3-2 3-2 3-2 3-2 3-3 3-3 3-4 3-4 3-5 3-5 3-6 3-8 3-8

1

3.3

3.4 3.5 3.6 3.7

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Set Theory for Physicists

4

Universal set systems

4-1

4.1 4.2 4.3

What is a universal set? Complement Properties of the complement

4-1 4-2 4-2

5

Relations and functions

5-1

5.1 5.2 5.3

What is a relation? One-to-one relations What is a function?

5-1 5-1 5-2

6

Equivalence relations and classes

6-1

6.1 6.2

What is an equivalence relation? What is an equivalence class?

6-1 6-1

7

Mathematical theory

7-1

7.1 7.2

Axiomatic definitions and definitions Axioms and theorems

7-1 7-2

Appendix A

A-1

viii

Preface As we discussed in the book Logic for Physicists, modern natural sciences (physics, chemistry, biology, …) are based upon the understanding of nature through models. The validity of a model is, in turn, measured by its capacity to accurately describe and predict natural phenomena in the simplest manner possible. The experiments designed to test the validity of a given model must, therefore, be readily reproducible in different laboratories. To precisely define the results that will be observed in different laboratories, modern science is led to express the results in numbers that represent the measured quantities of the experiments. Thus, two separate laboratories with identical experimental setups, can compare their results by simply comparing the numbers obtained by measuring the preestablished quantities associated with the experiments. The natural sciences are thus unavoidably led to incorporate numbers and, therefore, mathematics. Mathematics becomes more than just a useful tool, it becomes an essential intrinsic part of the models with which we describe, understand, and attempt to predict nature. Mathematical theories, in turn, consist of a series of initial propositions, and the deduction of additional propositions, that depend on elements that belong to a given set. The process of derivation/deduction of properties/propositions is called logic. The general properties of elements and sets is called set theory. For example, geometry consists of the study of propositions about points that belongs to the set space. Straight lines, segments, planes, etc are series of points that in turn form subsets of space. This book builds upon the previous book Logic for Physicists, where we studied the basic elements of logic. In this book we will study the fundamentals of set theory.

ix

Acknowledgments I wish to thank my family for their support. I am grateful to the many students and colleagues that I have interacted with over the years. Also I wish to thank the staff at IOP Publishing and at Morgan & Claypool Publishers for their assistance and support.

x

Author biography Nicolas A Pereyra Dr Pereyra pursued his undergraduate studies in physics in Caracas at the Universidad Central de Venezuela, where he graduated in 1991. His graduate studies in physics took place at the University of Maryland at College Park, where he obtained his MS in 1995 and his PhD in 1997. Currently Dr Pereyra is an Associate Professor in Astrophysics at the Physics and Astronomy Department of the University of Texas Rio Grande Valley. Dr Pereyraʼs research work has been largely in the development of computational models of physical systems. His research has included the computer simulation of transport of oil through pipelines, the computer simulation of accretion disk winds in cataclysmic variables, earth atmospheric wind simulations, solar loop simulations, simulations of cool fronts in the interstellar medium, and radiative transfer simulations. Currently Dr Pereyra is working on the development of computational models of accretion disk winds in QSOs.

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Set Theory for Physicists Nicolas A Pereyra

Chapter 1 Equality ‘=’

Before discussing the concept of sets, we will first discuss the concept of equality. Given a group of elements one can define an ‘equality’ among the elements by establishing a condition for equality. That is, to define an equality, one must be, in principle, given two elements x and y, be able to establish whether or not they satisfy the condition of equality. If x and y satisfy the condition of equality, then one states that:

x=y

(1.1)

that is, x and y are equal. If x and y do not satisfy the condition of equality, then one states that:

¬(x = y )

(1.2)

x≠y

(1.3)

or equivalently

that is, x and y are not equal. In addition to being able to establish whether or not two elements x and y of a given group are equal, an equality must also satisfy the following four properties: • Reflexivity:

x=x

(1.4)

that is, every element must be equal to itself. • Symmetry:

x=y ⟺ y=x

(1.5)

that is, a given element x is equal to a given element y if and only if the element y is equal to the element x.

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Set Theory for Physicists

• Transitivity:

(x = y ) ∧ (y = z ) ⟹ (x = z )

(1.6)

that is, if a given element x is equal to a given element y and the element y is equal to a given element z, then the element x must also be equal to the element z. • Substitutivity:

x = y ⟹ P (x ) = P (y )

(1.7)

that is, if two given elements x and y are equal, then when substituted into a given element-dependent expression the corresponding resulting values must also be equal. An example of equality, previously studied in the book Logic for Physicists [1], is the equality between two propositions. Two propositions are equal if and only if they have the same logical value. Since, as we studied previously, a proposition must have one and only one logical value (either true or false), it follows that one can, in principle, always establish whether or not two propositions have the same logical value, that is whether or not the two propositions are equal. In addition, the equality between two propositions also satisfy the above four properties: • The equality between propositions satisfies reflexivity, that is a given proposition P has the same logical value than itself, thus a proposition P is always equal to itself. • The equality between propositions satisfies symmetry, that is the logical value of a given proposition P is equal to the logical value of a given proposition Q, if and only if, the logical value of Q is equal to the logical value of P. • The equality between propositions satisfies transitivity, that is if the logical value of a given proposition P is equal to the logical value of a given proposition Q and the logical value of Q is equal to the logical value of a given proposition R, then the logical value of P must also be equal to the logical value of R. • Finally, the equality between propositions satisfies substitutivity, that is if the logical value of a given proposition P is equal to the logical value of a given proposition Q, then when each is separately substituted into a given expression in propositional algebra, the corresponding resulting logical values will also be equal.

Reference [1] Pereyra N A 2018 Logic for Physicists (Bristol, UK: IOP Publishing)

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Set Theory for Physicists Nicolas A Pereyra

Chapter 2 Fundamental properties of sets

2.1 What is a set? A set is a well-defined group of elements. For example, consider the set S:1

A ≡ {bicycle , tricycle , car}

(2.1)

thus the set A consists of (or contains) three elements: bicycle, tricycle, and car. To state that a given element x belongs to a given set A, one may use the symbol ‘∈’:

x ∈ A ≡ ‘x is an element of set A’

(2.2)

that is, ‘x belongs to A’ or ‘x is in A’. For example, one can state that

bicycle ∈ A

(2.3)

tricycle ∈ A

(2.4)

that is, bicycle belongs to A. One can also state that

that is, tricycle belongs to A. Additionally one can state that

(2.5)

car ∈ A

that is, car belongs to A. One may also consider elements that do not belong to a given set, for example one can state

1

The mathematical symbol ‘≡’ denotes: is equal, by definition, to.

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Set Theory for Physicists

¬(train ∈ A)

(2.6)

train ∉ A

(2.7)

or equivalently

that is, train does not belong to A.

2.2 Defining a set In order for a set A to be well-defined, one must be able, given an arbitrary element x, to state whether or not the element x belongs to set A. A given set can thus be defined in two ways: first, by explicitly stating which elements belong to the given set. For example, consider the set B defined as:

B ≡ {0, 2, 4, 6, 8, 10}

(2.8)

From the above definition one can state that, for instance:

7∉B

(2.9)

that is the number 7 does not belong to set B. In turn, one could also state that

4∈B

(2.10)

that is the number 4 does belong to set B. Second, a set can be defined by clearly stating a condition that must be fulfilled by an element in order to belong to the given set. For example, consider the set C defined as the set of all the even natural numbers less than eleven, that is

x ∈ C ⟺ (x is even) ∧ (x < 11)

(2.11)

For any given natural number x, one can determine whether the condition ‘(x is even) ∧ (x < 11)’ is true or not. Therefore, for any given natural number x, one can determine whether or not x belongs to set C. Thus C is a well-defined set. Also note that in this case set B and set C are actually the same set defined in two different ways.

2.3 Equality of sets Given two sets A and B, they are stated to be equal if and only if it holds that any element x that belongs to A also belongs to B and any element x that belongs to B also belongs to A, that is

A = B ⟺ (x ∈ A ⟺ x ∈ B )

(2.12)

We will now prove that the above definition of equality among sets is valid by showing that the following three properties hold: • Reflexivity:

A=A

2-2

(2.13)

Set Theory for Physicists

• Symmetry:

A=B⟺B=A

(2.14)

(A = B ) ∧ (B = C ) ⟹ (A = C )

(2.15)

• Transitivity:

Also, note that further below as we define different operations on sets, the property of substitutivity will have to hold for each new set operator. Proof of reflexivity: equation (2.12);

A = A ⟺ (x ∈ A ⟺ x ∈ A ) A = A ⟺ T; A = A.

that is, every set is equal to itself. Proof of symmetry: equation (2.12);

A = B ⟺ (x ∈ A ⟺ x ∈ B ) A = B ⟺ (x ∈ B ⟺ x ∈ A); A=B⟺B=A

equation (2.12).

that is, a given set A is equal to a given set B if and only if the set B is equal to the set A. Proof of transitivity: • first case: (A = B ) ∧ (B = C ) ⟹ (x ∈ A ⟹ x ∈ C ) (A = B ) ∧ ( B = C ) ⟺ ( x ∈ A ⟺ x ∈ B ) ∧ (x ∈ B ⟺ x ∈ C ) (A = B ) ∧ ( B = C ) ⟺ ( x ∈ A ⟹ x ∈ B ) ∧ (x ∈ B ⟹ x ∈ A ) ∧ (x ∈ B ⟹ x ∈ C ) ∧ (x ∈ C ⟹ x ∈ B ); (A = B ) ∧ (B = C ) ⟹ (x ∈ A ⟹ x ∈ B ) ∧ (x ∈ B ⟹ x ∈ C ); (A = B ) ∧ (B = C ) ⟹ (x ∈ A ⟹ x ∈ C ).

equation (2.12);

• second case: (A = B ) ∧ (B = C ) ⟹ (x ∈ C ⟹ x ∈ A) A = B ∧ B = C ⟺ ( x ∈ A ⟺ x ∈ B ) ∧ (x ∈ B ⟺ x ∈ C )

2-3

equation (2.12);

Set Theory for Physicists

(A = B ) ∧ ( B = C ) ⟺ ( x ∈ A ⟹ x ∈ B ) ∧ (x ∈ B ⟹ x ∈ A ) ∧ (x ∈ B ⟹ x ∈ C ) ∧ (x ∈ C ⟹ x ∈ B ); (A = B ) ∧ (B = C ) ⟹ (x ∈ B ⟹ x ∈ A) ∧ (x ∈ C ⟹ x ∈ B ); (A = B ) ∧ (B = C ) ⟹ (x ∈ C ⟹ x ∈ A).

Considering the first and second cases: (A = B ) ∧ (B = C ) ⟹ (x ∈ A ⟹ x ∈ C ) ∧ (x ∈ C ⟹ x ∈ A); (A = B ) ∧ (B = C ) ⟹ (x ∈ A ⟺ x ∈ C ); ( A = B ) ∧ (B = C ) ⟹ ( A = C )

equation (2.12).

that is, if a given set A is equal to a given set B and the set B is equal to a given set C, then set A must also be equal to set C.

2.4 A set can be an element of another set One can define a set that contains elements that are sets themselves. For example, consider the following sets whose elements are transport vehicles:

A ≡ {bicycle , tricycle , car}

(2.16)

B ≡ {canoe , raft , boat}

(2.17)

C ≡ {helicopter , airplane}

(2.18)

S ≡ {A , B , C }

(2.19)

One can define a set S

or equivalently

S ≡ {{bicycle , tricycle , car}, {canoe , raft , boat}, {helicopter , airplane}} (2.20) That is, S is the set whose elements are A, B, and C. Thus S is a set that contains three elements, and each of these elements are in turn sets of transport vehicles. Therefore, one can state, for example

A∈S

(2.21)

2.5 A set cannot contain itself For any set A, it always holds that

A∉A

2-4

(2.22)

Set Theory for Physicists

2.6 A set can be empty (null) A set can be such that it contains no elements. Such a set is said to be empty or to be a null set, and is denoted by the symbol ∅. That is, for any element x, it always holds that

x∉∅

(2.23)

2.7 Subsets 2.7.1 Definition of subsets Given two sets A and B, it will be stated that set A is a subset of set B if and only if any element in A is also in B, that is

A ⊂ B ⟺ (x ∈ A ⟹ x ∈ B )

(2.24)

For example, given the three sets A, B, C

A ≡ {bicycle , tricycle , car}

(2.25)

B ≡ {bicycle , car}

(2.26)

C ≡ {car , truck}

(2.27)

B⊂A

(2.28)

one can state that

since all the elements that are in B are also in A. One can also state that

C⊄A

(2.29)

since not all elements in C are also in A. In particular, the element truck is in set C but not in set A. 2.7.2 A ⊂ A Any given set A is a subset of itself, that is it always holds that

A⊂A

(2.30)

Proof: equation (2.24);

A ⊂ A ⟺ (x ∈ A ⟹ x ∈ A ) A ⊂ A ⟺ T; A ⊂ A.

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Set Theory for Physicists

2.7.3 ∅ ⊂ A The empty set ∅ is a subset of any set, that is it always holds that

∅⊂A

(2.31)

Proof: equation (2.24);

∅ ⊂ A ⟺ (x ∈ ∅ ⟹ x ∈ A )

since for the empty set ∅ it is always false (F) than any given element x belongs to it equation (2.23);

∅ ⊂ A ⟺ (F ⟹ x ∈ A ) ∅ ⊂ A ⟺ T; ∅ ⊂ A.

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IOP Concise Physics

Set Theory for Physicists Nicolas A Pereyra

Chapter 3 Set operators

3.1 What is a set operator? A set operator defines a new set S, from one or more initial sets A, B, ….

3.2 The ∪ operator (UNION) Given two sets A and B, the union of A and B is denoted by

A∪B and defined by the condition

x ∈ A ∪ B ⟺ (x ∈ A) ∨ (x ∈ B )

(3.1)

that is, an element x belongs to A ∪ B if and only if it belongs to A, or it belongs to B, or it belongs to both A and B. For the set operator ∪, it holds that

A ∪A = A

(3.2)

Proof: equation (3.1);

x ∈ A ∪ A ⟺ (x ∈ A ) ∨ (x ∈ A ) x ∈ A ∪ A ⟺ x ∈ A; A∪A=A

equation (2.12).

Also, note that in particular

∅∪∅=∅

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(3.3)

ª Morgan & Claypool Publishers 2019

Set Theory for Physicists

3.2.1 The commutative property of ∪ The set operator ∪ is commutative, that is

A∪B=B∪A

(3.4)

Proof: equation (3.1);

x ∈ A ∪ B ⟺ (x ∈ A ) ∨ (x ∈ B ) x ∈ A ∪ B ⟺ (x ∈ B ) ∨ (x ∈ A); x∈A∪B⟺x∈B∪A A∪B=B∪A

equation (3.1); equation (2.12).

3.2.2 The associative property of ∪ The set operator ∪ is associative, that is

A ∪ (B ∪ C ) = (A ∪ B ) ∪ C

(3.5)

Proof: x ∈ A ∪ (B ∪ C ) ⟺ (x ∈ A ) ∨ (x ∈ B ∪ C ) x ∈ A ∪ (B ∪ C ) ⟺ (x ∈ A) ∨ ((x ∈ B ) ∨ (x ∈ C )) x ∈ A ∪ (B ∪ C ) ⟺ ((x ∈ A) ∨ (x ∈ B )) ∨ (x ∈ C ); x ∈ A ∪ (B ∪ C ) ⟺ (x ∈ A ∪ B ) ∨ (x ∈ C ) x ∈ A ∪ (B ∪ C ) ⟺ x ∈ (A ∪ B ) ∪ C A ∪ (B ∪ C ) = ( A ∪ B ) ∪ C

equation (3.1); equation (3.1); equation (3.1); equation (3.1); equation (2.12).

3.2.3 The ∪ operator and the empty set ∅ The union of a set A with the empty set ∅, is the set A itself, that is

A∪∅=A

(3.6)

Proof: equation (3.1); equation (2.23);

x ∈ A ∪ ∅ ⟺ (x ∈ A) ∨ (x ∈ ∅) x ∈ A ∪ ∅ ⟺ (x ∈ A ) ∨ F x ∈ A ∪ ∅ ⟺ x ∈ A; A∪∅=A

equation (2.12).

3.3 The ∩ operator (INTERSECTION) Given two sets A and B, the intersection of A and B is denoted by

A∩B and defined by the condition 3-2

Set Theory for Physicists

x ∈ A ∩ B ⟺ (x ∈ A) ∧ (x ∈ B )

(3.7)

that is, an element x belongs to A ∩ B if and only if it belongs both to A and B. For the set operator ∩, it holds that

A∩A=A

(3.8)

Proof: equation (3.7);

x ∈ A ∩ A ⟺ (x ∈ A ) ∧ (x ∈ A ) x ∈ A ∩ A ⟺ x ∈ A; A∩A=A

equation (2.12).

Also, note that in particular

∅∩∅=∅

(3.9)

3.3.1 The commutative property of ∩ The set operator ∩ is commutative, that is

A∩B=B∩A

(3.10)

Proof: equation (3.7);

x ∈ A ∩ B ⟺ (x ∈ A ) ∧ (x ∈ B ) x ∈ A ∩ B ⟺ (x ∈ B ) ∧ (x ∈ A); x∈A∩B⟺x∈B∩A A∩B=B∩A

equation (3.7); equation (2.12).

3.3.2 The associative property of ∩ The set operator ∩ is associative, that is

A ∩ (B ∩ C ) = (A ∩ B ) ∩ C

(3.11)

Proof: x ∈ A ∩ (B ∩ C ) ⟺ (x ∈ A ) ∧ (x ∈ B ∩ C ) x ∈ A ∩ (B ∩ C ) ⟺ (x ∈ A) ∧ ((x ∈ B ) ∧ (x ∈ C )) x ∈ A ∩ (B ∩ C ) ⟺ ((x ∈ A) ∧ (x ∈ B )) ∧ (x ∈ C ); x ∈ A ∩ (B ∩ C ) ⟺ (x ∈ A ∩ B ) ∧ (x ∈ C ) x ∈ A ∩ (B ∩ C ) ⟺ x ∈ (A ∩ B ) ∩ C A ∩ (B ∩ C ) = ( A ∩ B ) ∩ C

3-3

equation (3.7); equation (3.7); equation (3.7); equation (3.7); equation (2.12).

Set Theory for Physicists

3.3.3 The ∩ operator and the empty set ∅ The intersection of a set A with the empty set ∅, is the empty set ∅, that is

A∩∅=∅

(3.12)

Proof: x∈A∩∅⟺ x∈A∩∅⟺ x∈A∩∅⟺ x∈A∩∅⟺ A∩∅=∅

equation (3.7); equation (2.23);

(x ∈ A) ∧ (x ∈ ∅) (x ∈ A ) ∧ F F; x∈∅

equation (2.23); equation (2.12).

3.4 Mixed properties of ∪ and ∩ We will now prove the following two properties:

A ∩ (B ∪ C ) = (A ∩ B ) ∪ (A ∩ C )

(3.13)

A ∪ (B ∩ C ) = (A ∪ B ) ∩ (A ∪ C )

(3.14)

Proof of the first property: x ∈ A ∩ (B ∪ C ) ⟺ (x ∈ A) ∧ (x ∈ (B ∪ C )) x ∈ A ∩ (B ∪ C ) ⟺ (x ∈ A) ∧ ((x ∈ B ) ∨ (x ∈ C )) x ∈ A ∩ (B ∪ C ) ⟺ ((x ∈ A) ∧ (x ∈ B )) ∨ ((x ∈ A) ∧ (x ∈ C )); x ∈ A ∩ (B ∪ C ) ⟺ (x ∈ A ∩ B ) ∨ (x ∈ A ∩ C ) x ∈ A ∩ (B ∪ C ) ⟺ x ∈ (A ∩ B ) ∪ (A ∩ C ) A ∩ (B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )

equation (3.7); equation (3.1); equation (3.7); equation (3.1); equation (2.12).

Proof of the second property: x ∈ A ∪ (B ∩ C ) ⟺ (x ∈ A) ∨ (x ∈ (B ∩ C )) x ∈ A ∪ (B ∩ C ) ⟺ (x ∈ A) ∨ ((x ∈ B ) ∧ (x ∈ C )) x ∈ A ∪ (B ∩ C ) ⟺ ((x ∈ A) ∨ (x ∈ B )) ∧ ((x ∈ A) ∨ (x ∈ C )); x ∈ A ∪ (B ∩ C ) ⟺ (x ∈ A ∪ B ) ∧ (x ∈ A ∪ C ) x ∈ A ∪ (B ∩ C ) ⟺ x ∈ (A ∪ B ) ∩ (A ∪ C ) A ∪ (B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )

3-4

equation (3.1); equation (3.7); equation (3.1); equation (3.7); equation (2.12).

Set Theory for Physicists

3.5 The \ operator (SET SUBTRACTION) Given two sets A and B, the set subtraction of A and B is denoted by

A\B and defined by the condition

x ∈ A \ B ⟺ (x ∈ A) ∧ (x ∉ B )

(3.15)

For the set operator \ , it holds that

(3.16)

A\A=∅ Proof:

equation (3.15);

x ∈ A \ A ⟺ (x ∈ A ) ∧ (x ∉ A ) x ∈ A \ A ⟺ (x ∈ A) ∧ ¬ (x ∈ A); x ∈ A \ A ⟺ F; ¬(x ∈ A \ A); x ∉ A \ A; A\A=∅

equation (2.23).

3.5.1 The \ operator and the empty set ∅ The set subtraction of a set A minus the empty set ∅, is the set A, that is

A\∅=A

(3.17)

Proof: equation (3.15); equation (2.23);

x ∈ A \ ∅ ⟺ (x ∈ A) ∧ (x ∉ ∅) x ∈ A \ ∅ ⟺ (x ∈ A ) ∧ T x ∈ A \ ∅ ⟺ x ∈ A; A\∅=A

equation (2.12).

Also, the set subtraction of the empty set ∅ minus a set A, is the empty set ∅, that is

∅\A = ∅

(3.18)

Proof: equation (3.15); equation (2.23);

x ∈ ∅ \ A ⟺ (x ∈ ∅) ∧ (x ∉ A) x ∈ ∅ \ A ⟺ F ∧ (x ∉ A ) x ∈ ∅ \ A ⟺ F; x ∉ ∅ \ A; ∅\A=∅

equation (2.23).

3-5

Set Theory for Physicists

3.6 Mixed properties of \, ∪ and ∩ Given three sets A, B, and C, the following property holds

C \ (A ∪ B ) = (C \ A) ∩ (C \ B )

(3.19)

Proof: x∈C x∈C x∈C x∈C x∈C x∈C x∈C x∈C x∈C x∈C x∈C C \ (A

\ (A ∪ B ) ⟺ (x ∈ C ) ∧ (x ∉ A ∪ B ) \ (A ∪ B ) ⟺ (x ∈ C ) ∧ ¬ (x ∈ A ∪ B ); \ (A ∪ B ) ⟺ (x ∈ C ) ∧ ¬ ((x ∈ A) ∨ (x ∈ B )) \ (A ∪ B ) ⟺ (x ∈ C ) ∧ ( ¬ (x ∈ A) ∧ ¬ (x ∈ B )); \ (A ∪ B ) ⟺ ((x ∈ C ) ∧ (x ∈ C )) ∧ ( ¬ (x ∈ A) ∧ ¬ (x ∈ B )); \ (A ∪ B ) ⟺ (x ∈ C ) ∧ (x ∈ C ) ∧ ¬ (x ∈ A) ∧ ¬ (x ∈ B ); \ (A ∪ B ) ⟺ (x ∈ C ) ∧ ¬ (x ∈ A) ∧ (x ∈ C ) ∧ ¬ (x ∈ B ); \ (A ∪ B ) ⟺ ((x ∈ C ) ∧ ¬ (x ∈ A)) ∧ ((x ∈ C ) ∧ ¬ (x ∈ B )); \ (A ∪ B ) ⟺ ((x ∈ C ) ∧ (x ∉ A)) ∧ ((x ∈ C ) ∧ (x ∉ B )); \ (A ∪ B ) ⟺ (x ∈ C \ A ) ∧ (x ∈ C \ B ) \ (A ∪ B ) ⟺ x ∈ (C \ A ) ∩ (C \ B ) ∪ B ) = (C \ A ) ∩ (C \ B )

equation (3.15); equation (3.1);

equation (3.15); equation (3.7); equation (2.12).

Given three sets A, B, and C, the following property holds

C \ (A ∩ B ) = (C \ A) ∪ (C \ B )

(3.20)

Proof: x∈C x∈C x∈C x∈C x∈C x∈C x∈C x∈C C \ (A

\ (A ∩ B ) ⟺ (x ∈ C ) ∧ (x ∉ A ∩ B ) \ (A ∩ B ) ⟺ (x ∈ C ) ∧ ¬ (x ∈ A ∩ B ); \ (A ∩ B ) ⟺ (x ∈ C ) ∧ ¬ ((x ∈ A) ∧ (x ∈ B )) \ (A ∩ B ) ⟺ (x ∈ C ) ∧ ( ¬ (x ∈ A) ∨ ¬ (x ∈ B )); \ (A ∩ B ) ⟺ ((x ∈ C ) ∧ ¬ (x ∈ A)) ∨ ((x ∈ C ) ∧ ¬ (x ∈ B )); \ (A ∩ B ) ⟺ ((x ∈ C ) ∧ (x ∉ A)) ∨ ((x ∈ C ) ∧ (x ∉ B )); \ (A ∩ B ) ⟺ (x ∈ C \ A ) ∨ (x ∈ C \ B ) \ (A ∩ B ) ⟺ x ∈ (C \ A ) ∪ (C \ B ) ∩ B ) = (C \ A ) ∪ (C \ B )

equation (3.15); equation (3.7);

equation (3.15); equation (3.1); equation (2.12).

Given three sets A, B, and C, the following property holds

C \ (B \ A) = (C ∩ A) ∪ (C \ B )

(3.21)

Proof: x ∈ C \ (B \ A ) ⟺ (x ∈ C ) ∧ (x ∉ B \ A ) x ∈ C \ (B \ A) ⟺ (x ∈ C ) ∧ ¬ (x ∈ B \ A); x ∈ C \ (B \ A) ⟺ (x ∈ C ) ∧ ¬ ((x ∈ B ) ∧ (x ∉ A))

3-6

equation (3.15); equation (3.15);

Set Theory for Physicists

x∈C x∈C x∈C x∈C x∈C x∈C x∈C x∈C x∈C C \ (B

\ (B \ (B \ (B \ (B \ (B \ (B \ (B \ (B \ (B \ A)

\ A) ⟺ (x ∈ C ) ∧ ¬ ((x ∈ B ) ∧ ¬ (x ∈ A)); \ A) ⟺ (x ∈ C ) ∧ ( ¬ (x ∈ B ) ∨ ¬ ( ¬ (x ∈ A))); \ A) ⟺ (x ∈ C ) ∧ ( ¬ (x ∈ B ) ∨ (x ∈ A)); \ A) ⟺ (x ∈ C ) ∧ ((x ∈ A) ∨ ¬ (x ∈ B )); \ A) ⟺ ((x ∈ C ) ∧ (x ∈ A)) ∨ ((x ∈ C ) ∧ ¬ (x ∈ B )); \ A) ⟺ (x ∈ C ∩ A) ∨ ((x ∈ C ) ∧ ¬ (x ∈ B )) \ A) ⟺ (x ∈ C ∩ A) ∨ ((x ∈ C ) ∧ (x ∉ B )); \ A ) ⟺ (x ∈ C ∩ A ) ∨ (x ∈ C \ B ) \ A ) ⟺ x ∈ (C ∩ A ) ∪ (C \ B ) = (C ∩ A ) ∪ (C \ B )

equation (3.7); equation (3.15); equation (3.1); equation (2.12).

Given two sets A and C, the following property holds

C \ (C \ A) = C ∩ A

(3.22)

Proof: equation (3.21); equation (3.16); equation (3.6).

C \ (C \ A ) = (C ∩ A ) ∪ (C \ C ) C \ (C \ A ) = (C ∩ A ) ∪ ∅ C \ (C \ A ) = C ∩ A

Given three sets A, B, and C, the following property holds

(B \ A) ∩ C = (B ∩ C ) \ A

(3.23)

Proof: x ∈ (B x ∈ (B x ∈ (B x ∈ (B x ∈ (B (B \ A )

\ A ) ∩ C ⟺ (x ∈ B \ A ) ∧ (x ∈ C ) \ A) ∩ C ⟺ ((x ∈ B ) ∧ (x ∉ A)) ∧ (x ∈ C ) \ A) ∩ C ⟺ ((x ∈ B ) ∧ (x ∈ C )) ∧ (x ∉ A); \ A ) ∩ C ⟺ (x ∈ B ∩ C ) ∧ (x ∉ A ) \ A ) ∩ C ⟺ x ∈ (B ∩ C ) \ A ∩ C = (B ∩ C ) \ A

equation (3.7); equation (3.15); equation (3.7); equation (3.15); equation (2.12).

Given three sets A, B, and C, the following property holds

(B ∩ C ) \ A = B ∩ (C \ A)

(3.24)

Proof: x x x x

∈ ∈ ∈ ∈

(B (B (B (B

∩ C ) \ A ⟺ (x ∈ (B ∩ C )) ∧ (x ∉ A) ∩ C ) \ A ⟺ ((x ∈ B ) ∧ (x ∈ C )) ∧ (x ∉ A) ∩ C ) \ A ⟺ (x ∈ B ) ∧ ((x ∈ C ) ∧ (x ∉ A)); ∩ C ) \ A ⟺ (x ∈ B ) ∧ (x ∈ C \ A )

3-7

equation (3.15); equation (3.7); equation (3.15);

Set Theory for Physicists

equation (3.7); equation (2.12).

x ∈ (B ∩ C ) \ A ⟺ x ∈ B ∩ (C \ A ) (B ∩ C ) \ A = B ∩ (C \ A )

Given three sets A, B, and C, the following property holds

(B \ A) ∪ C = (B ∪ C ) \ (A \ C )

(3.25)

Proof: x ∈ (B x ∈ (B x ∈ (B x ∈ (B x ∈ (B x ∈ (B x ∈ (B x ∈ (B x ∈ (B x ∈ (B x ∈ (B (B \ A )

\ A ) ∪ C ⟺ (x ∈ B \ A ) ∨ (x ∈ C ) \ A) ∪ C ⟺ ((x ∈ B ) ∧ (x ∉ A)) ∨ (x ∈ C ) \ A) ∪ C ⟺ ((x ∈ B ) ∨ (x ∈ C )) ∧ ((x ∉ A) ∨ (x ∈ C )); \ A) ∪ C ⟺ (x ∈ B ∪ C ) ∧ ((x ∉ A) ∨ (x ∈ C )) \ A) ∪ C ⟺ (x ∈ B ∪ C ) ∧ ( ¬ (x ∈ A) ∨ (x ∈ C )); \ A) ∪ C ⟺ (x ∈ B ∪ C ) ∧ ( ¬ (x ∈ A) ∨ ¬ ( ¬ (x ∈ C ))); \ A) ∪ C ⟺ (x ∈ B ∪ C ) ∧ ¬ ((x ∈ A) ∧ ¬ (x ∈ C )); \ A) ∪ C ⟺ (x ∈ B ∪ C ) ∧ ¬ ((x ∈ A) ∧ (x ∉ C )); \ A ) ∪ C ⟺ (x ∈ B ∪ C ) ∧ ¬ (x ∈ A \ C ) \ A) ∪ C ⟺ (x ∈ B ∪ C ) ∧ (x ∉ A \ C ); \ A ) ∪ C ⟺ x ∈ (B ∪ C ) \ (A \ C ) ∪ C = (B ∪ C ) \ (A \ C )

equation (3.1); equation (3.15); equation (3.1);

equation (3.15); equation (3.15); equation (2.12).

3.7 The × operator (CARTESIAN PRODUCT) Given two sets A and B, the Cartesian product of A and B is denoted by

A×B and it is a set of ordered pairs defined by the condition

(x , y ) ∈ A × B ⟺ (x ∈ A) ∧ (y ∈ B )

(3.26)

that is, an ordered pair (x , y ) belongs to A × B if and only if the first element x belongs to A and the second element y belongs to B. 3.7.1 The × operator and the empty set ∅ The Cartesian product of a set A times the empty set ∅, is the empty set ∅, that is

A×∅=∅

(3.27)

Proof: (x , y ) ∈ A × (x , y ) ∈ A × (x , y ) ∈ A × (x , y ) ∉ A × A×∅=∅

equation (3.26); equation (2.23);

∅ ⟺ (x ∈ A) ∧ (y ∈ ∅) ∅ ⟺ (x ∈ A ) ∧ F ∅ ⟺ F; ∅;

equation (2.23).

3-8

Set Theory for Physicists

Also, the Cartesian product of the empty set ∅ times a set A, is the empty set ∅, that is

∅×A=∅

(3.28)

Proof: (x , y ) ∈ ∅ × (x , y ) ∈ ∅ × (x , y ) ∈ ∅ × (x , y ) ∉ ∅ × ∅×A=∅

equation (3.26);

A ⟺ (x ∈ ∅) ∧ (y ∈ A) A ⟺ F ∧ (y ∈ A); A ⟺ F; A;

equation (2.23).

3-9

IOP Concise Physics

Set Theory for Physicists Nicolas A Pereyra

Chapter 4 Universal set systems

4.1 What is a universal set? A set U is referred to as a universal set when we are considering only elements that belong to set U and only sets that are subsets of U. In this section we will assume that we are working within a universal set system, with the universal set denoted as U. Let A be a set of the universe U, it holds that

A∪U=U

(4.1)

Proof: x ∈ (A ∪ U ) x ∈ (A ∪ U ) x ∈ (A ∪ U ) x ∈ (A ∪ U ) A∪U=U

equation (3.1);

⟺ (x ∈ A ) ∨ (x ∈ U ) ⟺ (x ∈ A ) ∨ T ; ⟺ T; ⟺ x ∈ U;

equation (2.12).

Also, let A be a set of the universe U, it holds that

A∩U=A

(4.2)

Proof: equation (3.7);

x ∈ (A ∩ U ) ⟺ (x ∈ A ) ∧ (x ∈ U ) x ∈ (A ∩ U ) ⟺ (x ∈ A ) ∧ T ; x ∈ (A ∩ U ) ⟺ (x ∈ A); A∩U=A

doi:10.1088/2053-2571/ab126ach4

equation (2.12).

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4.2 Complement Within a universal set system, the complement of a set A is defined as

A∁ ≡ U \ A

(4.3)

x ∈ A∁ ⟺ (x ∈ U ) ∧ (x ∈ A)

(4.4)

That is

4.3 Properties of the complement Let A be a set of the universe U, the following property holds

(A∁ )∁ = A

(4.5)

Proof: (A∁)∁ (A∁)∁ (A∁)∁ (A∁)∁ (A∁)∁

= = = = =

equation equation equation equation equation

U \(A∁) U \(U \A) U∩A A∩U A

(4.3); (4.3); (3.22); (3.10); (4.2).

Given the universe U, the following property holds

(4.6)

∅∁ = U Proof: ∅∁ = U \∅ ∅∁ = U

equation (4.3); equation (3.17).

Given the universe U, the following property holds

(4.7)

U∁ = ∅ Proof: U ∁ = U \U U∁ = ∅

equation (4.3); equation (3.16).

Let A and B be two sets of the universe U, the following property holds

(A ∪ B )∁ = A∁ ∩ B ∁

4-2

(4.8)

Set Theory for Physicists

Proof: (A ∪ B )∁ = U \(A ∪ B ) (A ∪ B )∁ = (U \A) ∩ (U \B ) (A ∪ B )∁ = A∁ ∩ B ∁

equation (4.3); equation (3.19); equation (4.3).

Let A and B be two sets of the universe U, the following property holds

(A ∩ B )∁ = A∁ ∪ B ∁

(4.9)

Proof: equation (4.3); equation (3.20); equation (4.3).

(A ∩ B )∁ = U \(A ∩ B ) (A ∩ B )∁ = (U \A) ∪ (U \B ) (A ∩ B )∁ = A∁ ∪ B ∁

Let A be a set of the universe U, the following property holds

(4.10)

A ∪ A∁ = U Proof: A A A A A A A

∪ A∁ = A∁ ∪ A ∪ A∁ = (U \A) ∪ A ∪ A∁ = (U ∪ A)\(A\A) ∪ A∁ = (U ∪ A)\∅ ∪ A∁ = U ∪ A ∪ A∁ = A ∪ U ∪ A∁ = U

equation equation equation equation equation equation equation

(3.4); (4.3); (3.25); (3.16); (3.17); (3.4); (4.1).

Let A be a set of the universe U, the following property holds

(4.11)

A ∩ A∁ = ∅ Proof: A A A A

equation equation equation equation

∩ A∁ = A ∩ (U \A) ∩ A∁ = (A ∩ U )\A ∩ A∁ = A\A ∩ A∁ = ∅

(4.3); (3.24); (4.2); (3.16).

Let A and B be two sets of the universe U, the following property holds

A ⊂ B ⟺ B ∁ ⊂ A∁

4-3

(4.12)

Set Theory for Physicists

Proof: A A A A A A A A A A A A

⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂

B B B B B B B B B B B B

⟺ ⟺ ⟺ ⟺ ⟺ ⟺ ⟺ ⟺ ⟺ ⟺ ⟺ ⟺

(x ∈ A ⟹ x ∈ B ) ¬ (x ∈ A ∧ ¬ (x ∈ B )); ¬ (x ∈ A ∧ x ∈ B ); ¬ (x ∈ B ∧ x ∈ A); ¬ (x ∈ B ∧ ¬ ( ¬ (x ∈ A))); ¬ (x ∈ B ∧ ¬ (x ∈ A)); (x ∈ B ⟹ x ∈ A); (T ∧ x ∈ B ⟹ T ∧ x ∈ A ) ((x ∈ U ) ∧ (x ∈ B ) ⟹ (x ∈ U ) ∧ (x ∈ A)); ((x ∈ U \B ) ⟹ (x ∈ U \A)) ((x ∈ B ∁) ⟹ (x ∈ A∁)) B ∁ ⊂ A∁

equation (2.24);

equation (3.15); equation (4.3); equation (2.24).

Let A and B be two sets of the universe U, the following property holds

(4.13)

A\ B = A ∩ B ∁ Proof: A ∩ B ∁ = A ∩ (U \ B ) A ∩ B ∁ = (A ∩ U )\B A ∩ B ∁ = A\B A\B = A ∩ B ∁

equation (4.3); equation (3.24); equation (4.2);

Let A and B be two sets of the universe U, the following property holds

(A \ B )∁ = A∁ ∪ B

(4.14)

Proof: (A\B )∁ (A\B )∁ (A\B )∁ (A\B )∁ (A\B )∁ (A\B )∁

= = = = = =

equation equation equation equation equation equation

U \(A\B ) (U ∩ B ) ∪ (U \ A ) (B ∩ U ) ∪ (U \ A ) B ∪ (U \ A ) B ∪ A∁ A∁ ∪ B

(4.3); (3.21); (3.10); (4.2); (4.3); (3.4).

Let A and B be two sets of the universe U, the following property holds

A∁ \ B ∁ = B \ A

4-4

(4.15)

Set Theory for Physicists

Proof: A∁ \B ∁ = A∁ \B ∁ = A∁ \B ∁ = A∁ \B ∁ =

equation equation equation equation

A∁ ∩ (B ∁)∁ A∁ ∩ B B ∩ A∁ B \A

4-5

(4.13); (4.5); (3.10); (4.13).

IOP Concise Physics

Set Theory for Physicists Nicolas A Pereyra

Chapter 5 Relations and functions

5.1 What is a relation? Given two sets A and B, a relation R between A and B is a subset of the Cartesian product A × B, that is

(5.1)

R⊂A×B

Let a be an element of set A and let b be an element of set B, one states that a is related through R to b (aRb) if and only if the pair (a,b) belongs to relation R, that is

aRb ⟺ (a , b) ∈ R

(5.2)

The domain of R is the subset of A whose elements are related to at least one element of set B, that is

a ∈ domain of R ⟺ ∃b aRb

(5.3)

The range of R is the subset of B whose elements are related to at least one element of set A, that is

b ∈ range of R ⟺ ∃a aRb

(5.4)

5.2 One-to-one relations A relation R is a one-to-one relation if and only if every element of set A is related to one and only one element of set B and in turn every element of set B is related to one and only one element of set A. That is, R is a one-to-one relation if and only if the following four conditions hold:

∀a ∃b aRb

doi:10.1088/2053-2571/ab126ach5

5-1

(5.5)

ª Morgan & Claypool Publishers 2019

Set Theory for Physicists

aRb1 ∧ aRb2 ⟹ b1 = b2

(5.6)

∀b ∃a aRb

(5.7)

a1Rb ∧ a2Rb ⟹ a1 = a2

(5.8)

5.3 What is a function? Given sets A and B, a function f from A to B is a relation Rf between A and B such that every element of A is related to one and only one element of B. That is, f is a function from A to B, if and only if the following two conditions hold:

∀a ∃b aRf b

(5.9)

aRf b1 ∧ aRf b2 ⟹ b1 = b2

(5.10)

Standard notations for denoting a function f from A to B are

f :A→B

(5.11)

f A → B

(5.12)

b = f (a )

(5.13)

If a function f from A to B is also a one-to-one relation between A and B, then we can define the inverse function f −1 from B to A such that

bR f −1a ⟺ aRf b

(5.14)

Given a function f from A to B, and given its inverse function f −1 from B to A, it follows that

f −1 (f (a )) = a

(5.15)

f (f −1 (b)) = b

(5.16)

and that

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IOP Concise Physics

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Chapter 6 Equivalence relations and classes

6.1 What is an equivalence relation? Given a set A, an equivalence relation ∼ in A is a relation between A and A such that the following three conditions hold: • Reflexivity:

a∼a

(6.1)

that is, every element must be equivalent to itself. • Symmetry:

(6.2)

a∼b ⟺ b∼a

that is, a given element a is equivalent to a given element b if and only if the element b is equivalent to the element a. • Transitivity:

(a ∼ b ) ∧ (b ∼ c ) ⟹ (a ∼ c )

(6.3)

that is, if a given element a is equivalent to a given element b and the element b is equivalent to a given element c, then the element a must also be equivalent to the element c. Note that unlike equality ‘=’ (see chapter 1), equivalence ‘∼’ does not have to satisfy the condition of substitutivity.

6.2 What is an equivalence class? Given a set A and an equivalence relation ∼ in A, the equivalence relation will ‘partition’ the set A into a series of subsets of A such that each subset contains elements of A that are equivalent to each other. These resulting subsets are referred to as equivalence classes.

doi:10.1088/2053-2571/ab126ach6

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Set Theory for Physicists

An equivalence class can be denoted with an element contained in that class in between square brackets

[a ]

(6.4)

Given elements x and y of a set A, and an equivalence relation ∼ in A, it follows that

[x ] ⊂ A ∧ [ y ] ⊂ A

(6.5)

x ∼ y ⟺ [x ] = [ y ]

(6.6)

[x ] ∩ [ y ] = ∅ ⟺ ¬ (x ∼ y )

(6.7)

Equivalence classes are not just well-defined mathematical constructs, they are also very important and useful. For example, as we will discuss in future books, using equivalence classes we can construct and prove the general properties of integer numbers from the natural numbers and the properties of natural numbers. Using equivalence classes we can also construct and prove the general properties of rational numbers from the integer numbers and the properties of integer numbers. In turn, using equivalence classes again, we can construct and prove the general properties of real numbers from the rational numbers and the properties of rational numbers. Since real numbers and their properties are a fundamental part modern natural sciences (physics, chemistry, biology, …) we are led here and in future books to discuss equivalence classes.

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Chapter 7 Mathematical theory

7.1 Axiomatic definitions and definitions An axiomatic definition is the establishment of the existence of a series of elements without proof. It is considered self-evident that the given elements exist. For example, in the theory of the set  of natural numbers, the existence of natural numbers may be taken to be self-evident and can described as:

 = {0, 1, 2, 3, 4, …}.

(7.1)

Note that we are not defining natural numbers from previous concepts, but rather we have taken their existence to be obvious or self-evident. That is, natural numbers, as described above, are an axiomatic definition. Intuitively, a natural number represents a ‘whole quantity’, thus we can speak of having ‘two pencils’ or of there being ‘eight chairs’ in a given room. Similarly, we can take as self-evident that an addition operation ‘+’ exists, that is that given two arbitrary natural numbers a and b one can always apply the addition operator and obtain a third natural number c:

a + b = c.

(7.2)

We refer to a as the first addend, to b as the second addend, and c is referred to as the sum. That is, the first and second addends are the two arguments of the addition operation and the sum is the result obtained once the addition operation has been evaluated. Intuitively, c represents the total ‘whole quantity’, when the ‘whole quantities’ of a and b are put together. A definition (rather than an axiomatic definition) is the establishment of the existence of a series of elements based on previous established elements and properties (rather than to accept the existence of the elements without proof as in axiomatic definitions). For example, in the theory of the set  of natural numbers, one may define the multiplication operation based on the addition operation:

doi:10.1088/2053-2571/ab126ach7

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Set Theory for Physicists

0 · a≡0 1·a≡a ⌢ +a n · a ≡ a + a + n times

(for n > 1)

Thus, the multiplication operation in natural numbers, as described above, is a definition.

7.2 Axioms and theorems An axiom is a property of the elements of a given set that is taken as self-evident, that is, it is accepted as true without proof (similar to axiomatic definitions). For example, in the theory of the set  of natural numbers, the commutative property of the addition operator may be taken as an axiom, that it is assumed as self-evident that

a+b=b+a

(7.3)

for any two natural numbers a and b. That is, one accepts without proof that the sum of two natural numbers is independent of the order of the addends. Intuitively, the commutative property of addition in  states that when one puts together two whole quantities a and b, the resulting total whole quantity c is the same regardless if one puts a together with b or b together with a. A theorem, on the other hand, is the establishment of a property of a series of elements through proof (rather than to accept the property without proof as in axioms). For example, in the theory of the set  of natural numbers, based on the properties of addition and the definition of multiplication, one can prove the distributive property of multiplication with respect to addition, that is one can prove that

a · (b + c ) = (a · b) + (a · c ). To illustrate this we present a proof below: • first case (a = 0):

a a a a a

· · · · ·

(b (b (b (b (b

+ + + + +

c ) = 0 · (b + c ) c) = 0 c) = 0 + 0 c ) = (0 · b) + (0 · c ) c ) = (a · b ) + (a · c )

a a a a

· · · ·

(b (b (b (b

+ + + +

c ) = 1 · (b + c ) c) = b + c c ) = (1 · b) + (1 · c ) c ) = (a · b ) + (a · c )

• second case (a = 1):

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(7.4)

Set Theory for Physicists

• third case (a > 1):

a · (b + c ) = (b + c ) + (b + c ) + a times ⌢ + (b + c ) a · (b + c ) = (b + b + a times ⌢ + b) + (c + c + a times ⌢ + c) a · (b + c ) = (a · b ) + (a · c )

Thus, we have proven that equation (7.4) is true. Therefore, equation (7.4) is a theorem (rather than an axiom) since it was proven from previously established elements and properties (rather taken to be self-evident).

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IOP Concise Physics

Set Theory for Physicists Nicolas A Pereyra

Appendix A A.1 General equations of set theory A = B ⟺ (x ∈ A ⟺ x ∈ B ) x ∈∅ A ⊂ B ⟺ (x ∈ A ⟹ x ∈ B ) A⊂A ∅⊂A x ∈ A ∪ B ⟺ (x ∈ A ) ∨ (x ∈ B ) A∪A=A A∪B=B∪A A ∪ (B ∪ C ) = ( A ∪ B ) ∪ C A∪∅=A x ∈ A ∩ B ⟺ (x ∈ A ) ∧ (x ∈ B ) A∩A=A A∩B=B∩A A ∩ (B ∩ C ) = ( A ∩ B ) ∩ C A∩∅=∅ A ∩ (B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A ∪ (B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) x ∈ A \ B ⟺ (x ∈ A ) ∧ (x ∈ B ) A\A = ∅ A\∅ = A ∅\A = ∅ C \(A ∪ B ) = (C \A) ∩ (C \B ) C \(A ∩ B ) = (C \A) ∪ (C \B ) C \(B \A) = (C ∩ A) ∪ (C \B ) C \(C \A) = C ∩ A (B \A) ∩ C = (B ∩ C )\A (B ∩ C )\A = B ∩ (C \A) (B \A) ∪ C = (B ∪ C )\(A\C ) (x , y ) ∈ A × B ⟺ (x ∈ A ) ∧ (y ∈ B )

doi:10.1088/2053-2571/ab126ach8

A-1

equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation equation

(2.12) (2.23) (2.24) (2.30) (2.31) (3.1) (3.2) (3.4) (3.5) (3.6) (3.7) (3.8) (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) (3.26)

ª Morgan & Claypool Publishers 2019

Set Theory for Physicists

A.2 Universal set systems

A∪U=U A∩U=A A∁ ≡ U \A (A∁)∁ = A ∅∁ = U U∁ = ∅ (A ∪ B )∁ = A∁ ∩ B ∁ (A ∩ B )∁ = A∁ ∪ B ∁ A ∪ A∁ = U A ∩ A∁ = ∅ A ⊂ B ⟺ B ∁ ⊂ A∁ A\B = A ∩ B ∁ (A\B )∁ = A∁ ∪ B A∁ \B ∁ = B \A

equation equation equation equation equation equation equation equation equation equation equation equation equation equation

(4.1) (4.2) (4.3) (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) (4.15)

A.3 Relations and functions A.3.1 Relations R⊂A×B aRb ⟺ (a , b ) ∈ R a ∈ domain of R ⟺ ∃b aRb b ∈ range of R ⟺ ∃a aRb

equation equation equation equation

(5.1) (5.2) (5.3) (5.4)

equation equation equation equation

(5.5) (5.6) (5.7) (5.8)

A.3.2 One-to-one relations ∀a ∃b aRb aRb1 ∧ aRb2 ⟹ b1 = b2 ∀b ∃a aRb a1Rb ∧ a2Rb ⟹ a1 = a2

A-2

Set Theory for Physicists

A.3.3 Functions ∀a ∃b aRf b

equation (5.9)

aRf b1 ∧ aRf b2 ⟹ b1 = b2

equation (5.10)

Standard notations: f :A→B f A→B b = f (a )

equation (5.11) equation (5.12)

bR f −1a ⟺ aRf b

equation (5.14)

f −1 (f (a ))

equation (5.15) equation (5.16)

equation (5.13)

Inverse functions:

=a f (f −1 (b )) = b

A.4 Equivalence relations and classes A.4.1 Equivalence relations a∼a a∼b ⟺ b∼a a∼b∧b∼c⟹a∼c

equation (6.1) equation (6.2) equation (6.3)

A.4.2 Equivalence classes [x ] ⊂ A ∧ [ y ] ⊂ A x ∼ y ⟺ [x ] = [y ] [x ] ∩ [y ] = ∅ ⟺ ¬( x ∼ y )

A-3

equation (6.5) equation (6.6) equation (6.7)