A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation [1970 ed.] 0387903429, 9780387903422

Citation preview

H. B. Griffiths

P.J. Hilton

A Comprehensive Textbook of

Classical Mathematics A Contemporary Interpretation

Springer Science+Business Media, LLC

H. B. Griffiths University of Southampton Southampton, S09 5 NH England

P. J. Hilton Case Western Reserve University Cleveland, Ohio 44106 USA

AMS Subject Classifications: 00.01, OOA05

Library of Congress Cataloging in Publication Data

Griffiths, Hubert Brian. A comprehensive textbook of classical mathematics. Bibliography: p. Includes index. I. Hilton, Peter l. Mathematics-1961· J ohn, joint author. II. Title. 78-15692 510 QA37.2.G75 1978

Ali rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1970 by H. B. Griffiths and P. J. Hilton

Originally pub1ished by Springer· Ver1ag New York in 1970.

987654321 ISBN 978-0-387-90342-2

ISBN 978-1-4612-6321-0 (eBook)

DOI 10.1007/978-1-4612-6321-0

To Catherine and Margaret

Introduction to the Springer-Verlag Edition

This edition does not differ in content from its predecessor, but whereas the first edition appeared in 1970, the book is now addressed to an audience at a period when mathematics itself, the study of mathematics, and attitudes to both, have all undergone very profound and rapid change-especially compared with the years during which the book was conceived and written. An important (but by no means the sole) influence on these changes has been the increasing penetration into Mathematics programs, of information theory, numerical analysis, and computer science. We excluded such topics from the book, because we did not have any special contribution to make to their discussion-although we explicitly recognised their interest and importance. Today further areas of mathematics have come into prominence largely as a result of the key role played by the computer in society and in education. But, although these subjects have become of very great importance, they have not displaced the more classical subjects which form the content of this book. Moreover there are still, of course, vast areas of application of mathematicsfor example, differential equations-which centre more on the classical subjects than on the more modern ones. Thus we would claim that the mathematics treated in this book has not lost its relevance, either within mathematics or with a view to applications to the real world; it still has to be learned, even though mathematics has grown apace and economic realities are forcing greater stress on applications than a decade ago. Indeed, we ourselves were pointing out applications throughout the book as we wrote it. Our next remarks are directed at American readers. When this book was written, it was not our thought that the distribution of the book would be largely confined to Europe and Canada (but this seems to have happened as a result of certain unpredictable changes of policy in the publishing house which originally handled our book)_ Now a significant difference between practice in America and the European system, is that at American universities, undergraduate courses are always accompanied by assigned textbooks, whereas in the European system, texts may be recommended, but very rarely form an integral part of the course. Thus we included for American readers, on page xviii of the original Introduction (pagexviiiin this edition), a list of five courses which can be constructed by suitable selection of material from the text. Each of these courses could find a home in an American university or college program. Thus for example, Course 2 (Algebra) incorporates the v

VI

INTRODUCTION

arithmetic of the integers, linear algebra, an introduction to group theory, the theory of polynomial functions and polynomial equations, and some Boolean algebra. It could be supplemented, of course, by material from other chapters. Again, Course 5 (Calculus) aiscusses the differential and integral calculus more or less from the beginnings of these theories, and proceeds through functions of several real variables, functions of a complex variable, and topics of real analysis such as the implicit function theorem. We would, however, like to make a further point with regard to the appropriateness of our text in course work. We emphasized in the Introduction to the original edition that, in the main, we had in mind the reader who had already met the topics once and wished to review them in the light of his (or her) increased knowledge and mathematical maturity. We therefore believe that our book could form a suitable basis for American graduate courses in the mathematical sciences, especially those prerequisites for a Master's degree. It is the view of many who are currently seriously considering the question of the appropriate education for students of the mathematical sciences entering the American job market now or in the foreseeable future/ that the mathematics which they will use in industry is not likely to be in itself of a very sophisticated technical nature, but that such students would need to have a very mature attitude towards that mathematics in order to use it and adapt it in an effective and flexible manner. Thus we contend that a second, perhaps more systematic, look at undergraduate mathematics might be the most appropriate type of course for students entering the job market with a Master's degree. Similar considerations, with appropriate naming of University degrees, are wholly relevant to the European system also, where too many University courses force advanced, abstract Mathematics, on to students who possess a totally inadequate (because neglected) foundation of basic, Classical Mathematics. We might also argue that a course from this book will be valuable for the student going on to become a professional mathematician, but for such a student there would naturally be a need of very substantial supplementation. The present edition, then, is identical with the original except for the elimination of errors in the original text which have been drawn to our attention by readers. To those readers we are, of course, extremely grateful. A few additional calculus exercises have also been added.

lThis has been argued, for example, by members of the United States National Research Council Committee on Applied Mathematics Training, of which one of us (P.J.H.) is chairman.

INTRODUCTION

1.

Origins and Purpose of the Book

This book owes its origin to a course of lectures given by the authors, with the aid of some of their colleagues, while members of the mathematics department of the University of Birmingham, England. The course was designed for grammar school teachers of mathematics working in the Birmingham area. Although it included individual lectures on the uses of 'modern mathematics' in engineering, the theory of intelligence, statistics, and computer science, our primary purpose was to convey something of the climate of thought in a contemporary mathematics department of a university, and to discuss its relation with school mathematics. We hoped that our mutual discussions would help both schools and universities to prepare their pupils better for careers in mathematics, with an eye on the growing shortage of mathematicians. The lectures took place on Wednesday afternoons during term time and the sessions lasted between two and three hours each. Such co-operation (and in-service training) was highly unusual at the time (1961-2) but fortunately is now less rare. We would like to say here how much we gained from the give and take of our arguments with the teachers in our audience; such contacts are likely to be of immense value to the universities, not only with regard to mathematics. It was certainly not possible to provide in the course a comprehensive coverage of the material relevant to sixth-form and first-year undergraduate courses in 'pure' mathematics. We therefore decided to take certain topics selected from classical mathematics and, within these topics, to treat the underlying concepts and theories from a modern standpoint. In referring to a 'modern' standpoint we do not wish to imply that we subscribe to any particular view of how to teach mathematics today. Rather, we intend to convey the impression that we tried to treat these topics from the point of view of working mathematicians, adopting the language and standards of presentday mathematics. The same intention informs this book. However, the book is, naturally, much more than a record of the course. Although we have not included any topics which did not find mention in the course, nevertheless we have given much more attention to each topic than was possible-or would have been desirable-in the course itself. Our hope all along in the writing of the book has been that students (of any age!) who have reached roughly the level of completing one year of specialist mathematical vii

viii

INTRODUCTION

study at a university may be able to take up this book and read it without requiring further instruction through lectures or course work (see p. xviii). The topics themselves are more or less represented by the titles of the various parts of the book, with the exception that Parts I, II, and VIII may all be subsumed as treating the topic Set Theory and Foundations. Thus, apart from this topic, we discuss Arithmetic; Geometry; Algebra; Number Systems; Calculus. We do not imply by this choice that no other parts of mathematics are appropriate to a reader at the level at which he may wish to review the topics which we treat here, but we chose them as the central part of our course because they formed a background from classical (pure) mathematics which teachers in schools have in common with teachers.in universities. It has not been our primary intention to offer readers their first contact with the concepts and ideas we treat. In the spirit of the course of lectures itself, we wish to encourage the reader to look at rather familiar ideas a second time, with a view to fitting them into the framework of present-day, contemporary mathematical thought; and we hope thus to enable the reader to see how certain key ideas recur again and again and give a real unity to apparently separate parts of his early mathematical experience. Perhaps we might believe, in certain cases, that what we have written is the best way to introduce and develop an idea, but this forms no part of our claim nor of our motive in writing the book. Now, in selecting our topics, we have been very much aware of the importance of presenting probability and statistics, and information theory, at this stage; and we would certainly hope that a student would by this time have some acquaintance with numerical analysis and computer science. Also, we fully recognise the vital role which computers are playing in our society, and we appreciate the effect that role must surely have on the whole of education; nevertheless, we have not taken account of it in this book. To have done so would not only have further lengthened an already outsize text; it would also have led to a very different organization of the material. Further, these excluded topics are not so likely to be already familiar to the reader as those included; moreover if they are familiar, then it is reasonably likely that the treatment accorded them on the reader's first introduction is not so very different from that which we ourselves would give. (It seems a good rule for authors to remain silent if they have nothing new to say.) However, we have often emphasized the algorithmic nature of certain techniques, because it is usually helpful, when presenting a topic, to get a student to pretend that he is clarifying the ideas ready to explain them to a computing machine. (This is only an extension of the notion that the best way of understanding Pythagoras's Theorem is to pretend that one is Pythagoras himself, searching-and struggling-for a proof.) It is our hope that the informed reader will recognize where the availability of computers can lead to an insightful approach to a piece of mathematics and, if he is a teacher, will communicate this awareness to his students.

INTRODUCTION

2.

ix

Mode of Presentation

Our style of presentation reflects our intention to provide a review (in the sense of a survey starting from a familiar background) of the material treated. We present a great deal of theory and we present it as a definitive body of knowledge. Naturally we are concerned, often very explicitly, with the question of motivation for the introduction of a new concept, for a generalization, or for attributing significance to a theorem; but we adopt a didactic style in the actual enunciations of definitions and results, and we apparently leave the student little area of choice for the development of his own approach to the material. We do not forget, however, that 'the most important existence theorem in mathematics is the existence of people'. And our didacticism is not, we repeat, due to any prejudice against 'discovery methods' in mathematics; nor does it imply that we fail to recognize that the creative aspect of mathematics is at least as important as the use of mathematics for systematizing our knowledge. But, where a topic has already become familiar through use and applications at a less sophisticated level (or, in the language of Goals for School Mathematicst, at the premathematicallevel), it remains to elucidate the nature of the mathematical ideas involved and to organize the knowledge gained. This requires, of course, a systematic style of presentation and a careful organization of the material to bring out its interrelations with other parts of mathematics, so that the pattern of thought may the more clearly emerge. This was, essentially, the rationale underlying the course as originally given, and the transfer from the lecture room to the printed page has tended to emphasize the style in which the topics have been treated. We very much hope that teachers will use our material as a quarry, so to speak, of material from which they can select topics for treatment by bold classroom techniques, along the lines of the remarkable ATM book Some Lessons in Mathematics (listed as [36a] in the Bibliography at the end of this book). Before they can effect this transformation they need to have the mathematical material in 'professional' form, and that form is what we intend the book to have. On the other hand, mathematical ideas are not communicated from mathematician to mathematician in extremely precise form; indeed, it is often true that a published article, while perfectly precise and formally correct, may fail to convey the essential nature of the result in question just because the author has neglected to include in his article an informal description of the motivation underlying the problem or his chosen line of solution. The implication for mathematical exposition would seem to be to use the 'spiral approach' recommended in the Goals pamphlet. Thus ideas should first be introduced informally, and precise definitions and proofs should later be

t The report of the Cambridge Conference on School Mathematics, published by Houghton Miffiin (1964). This report should be read by all interested in mathematics education.

X

INTRODUCTION

provided for the key concepts, after which informality becomes once more appropriate. We have attempted to implement this principle. Moreover, in conformity with the spiral approach, we believe that topics should reappear in any well-planned syllabus (if we may, for brevity, use such an authoritarian phrase); typically a topic, treated in the first place for its own sake, will be reconsidered later as part of a broader generalization or as exemplifying a feature somewhat too subtle to be understood and appreciated at the initial exposure. Where a new concept first makes its appearance we discuss it informally and intuitively; we then introduce a degree of precision unusual in a text at this level in order to isolate and identify the concept in question; and, once we have been precise, we take the view that we and the reader have both earned the right to be informal in the interest of the free flow of ideas and of thought. Since we regard this book principally as a review, we proceed somewhat rapidly through the first stage to the point where we feel it appropriate to adopt a precision of language and notation; nevertheless we hope that we pass sufficiently rapidly to the less formal style to avoid the charge of pedantry.

3.

The Spiral Approach versus Pedantry: Notational Difficulties

While we certainly do not wish to be rigid we also believe that, in departing from precise terminology, the student should know just what it is from which he is departing! The appearance of pedantry will perhaps be particularly conspicuous in matters of notation. At first we insist, for example, that a function is simply denoted by a symbol 'f' and that we must (for practical computational reasons) specify the domain and range off; moreover, after the introductory 'chat', a function first appears in its precise form as a set, suitably restricted, of ordered pairs taken from an appropriate Cartesian product. We completely repudiate the attitude that so formalistic a view should be maintained when one comes to work with functions, as in the differential calculus (say); and we adopt and encourage more conventional and traditional notation and terminology such as 'f(x)', 'dyfdx', 'y is a function of x '. Similarly we emphasize the role of the identity function, calling it ld or even 1, irrespective of its domain, so that the reader may get used to thinking of it as a very important and ubiquitous mathematical concept; but it may, and usually should, become a very inconspicuous element in most calculations. Similarly we lay great stress initially on the composition of functions, but here an extra purpose is served. For we believe that a real confusion is created unless the student understands at the outset the nature of function-composition and the distinction between such a composition and the important product operation in the ring of real-valued functions. The notation f- 1 is really dangerous for the function inverse to f (with respect to function-composition), when the student knows that x- 1 = lfx. It is not possible to work with a notation that is entirely free of ambiguity-such a

INTRODUCTION

xi

notation would be intolerably cumbersome; but at least a teacher must do his duty to his student by explaining precisely-at the appropriate turn on the spiral-where the danger of misunderstanding lies, and why a potentially misleading notation is adopted. In this case, to avoid early misunderstanding, ·we use the symboljl> for f- 1 until we feel the time is ripe to revert to f- 1 • We have, rather reluctantly, continued the traditional practice of writing the function symbol to the left of the variable (thus, we write f(x)). We are well aware of the advantages to be gained by writing the function symbol on the right, but we have thought it prudent not to introduce notational innovations beyond those which may be regarded as standard, if not universal, practice. Undoubtedly it would be most unusual to find the right-hand convention adopted in a text at this level and we do not wish to present the reader with a further, and somewhat gratuitous, difficulty in bridging the gap between this book and others he may encounter before, during, or immediately after contact with our terminology and notation. Nevertheless, since we have placed considerable emphasis on flexibility, it would be retrograde not to recommend to the reader that he prepare himself for the use of either con-, vention. At the risk of underlining the obvious, let us point out that the notational problem comes most clearly into focus when discussing composition of functions (in Chapter 2). For, given

S~T~U

'

the composite function from S to U seems to demand the symbol fog or, simply, fg. However, with the left-hand convention, we are forced to write go f, or gf; for, if xis in S, then the image of x under the composite function is g(f(x)). Certainly most mathematicians use the left-hand convention (this, of course, accounts for our own choice in this book), but it should be remembered that the convention arose before so much emphasis was placed, as it is today, on composition of functions. It is often an amusing experience to see how different mathematicians, giving a lecture, cope with the anomaly referred to above. Many, in writing 'fg', actually write the 'g' first (that is, first in time!) and the 'f' afterwards! Some even go as far as to read the symbol 'fg' as 'gee eff'! In the spirit of the spiral approach we do not always recommend a single point of view towards a given topic, nor even a single designation for a particular concept. Thus, for example, we describe both the Cauchy-sequence procedure and the Dedekind-section procedure for completing the rationals to the reals. One of us believes that the former has greater algebraic appeal and the latter greater geometric appeal, while the other believes the opposite: but both agree that it would be folly to insist here on a single viewpoint. Similarly we describe a function f: S -7 T variously as invertible or as an equivalence if it admits a function g: T -7 S, such that gf is the identity on S and fg the identity on T. Again, the term 'invertible' brings out the algebraic flavor

xii

INTRODUCTION

while the term 'equivalence' brings out the set-theoretic flavor. In other cases, too, we try to cater for the various uses of a concept and the varying tastes of students of mathematics; and to prepare them for reading texts which use differing terminology, notations and founts of type. Particular notational features are, of course, always explained in the text when they are introduced. However a few should be mentioned here in the introduction. We use the symbol • to indicate that a proof has been completed; whereas we use the symbol D to indicate that no proof is being given of the statement thus adorned (although a proof exists!). The neologism 'iff' is used to mean 'if and only if', since this phrase is of such frequent occurrence; but we often use the symbols '~ ', '., p. E IR; where >. -f + ,_,.. g denotes that function whose value at xis >.-j(x) + wg(x). A second operator on C is the function Q: C -7 C, given by Q(f) = g, where g is that member of C whose value at x e 1R is the definite integral 2.5.9

g(x) =

fox f(t) dt.

Notice that C can be defined analogously for any interval

(a, b) = {x e !Ria < x < b}, and it is not 'just' a set but has other features which make it what is technically known as a 'real algebra'. If ef denotes the set of all intervals of the form (a, b), then the assignment of CU) to each j e ef, is itself afunction C with domain ef and range the family of all real algebras. The detailed discussion of the image of C is part of the subject called 'functional analysis'. (See also Chapter 26.)

2.6 Notation and Abuse of Language Our next remarks are concerned with notation and some points arising therefrom, which may strike the reader initially as pedantic. First, if, in the equation 2.5.9 for g(x), we had written Q(f) for g (which is whatg stands for), then we would obtain the cumbersome formula Q(f)(x). It is usual, therefore, often to omit certain parentheses, and to write Qfinstead of Q(f). However, if/

§ 2.6

NOTATION AND ABUSE OF LANGUAGE

25

itself had to be replaced by a complicated formula, we might then write Q(f)ix = g(x). In certain branches, the notation xf is also used for f(x); e.g., when n EN, factorial n is written n! ( = 1·2· · · · · n). After this practical matter, consider the following more subtle one, which takes up the remark following 2.2.1. 2.6.1 Suppose thatf: A~ B is a given function, and that Cis a set such that f(A) £ C £ B. Then we have a new function g: A~ C defined by setting g(a) = f(a) for each a EA. For brevity, we often refer to the functions as 'j' and 'g', respectively, and even regard them as equal. But an ambiguity can occur: for suppose C = f(A) =I= B. Then using the brief forms we would say 'f is not onto', whereas 'g is onto'. Thus two 'equal' things have different properties! In some contexts, however, such ambiguities do not lead to confusion, and the brevity is worthwhile, so there we allow the abuse of language 'f = g '. When the distinction has to be made, we may denote g by f: A~ C, which is still a recognizably different formula fromf: A~ B, and so avoids the memory-strain that' regarding' things can impose. 2.6.2 The same conventions apply to the use of the identity function (2.5.5). Thus, if C £ B,. there are three functions, ide: C ~ C, id 8 : B ~ B, and the inclusion map j: C ~ B given by j(c) = c. We shall often refer to all three as 'the identity function id on C '. The upshot of this discussion is that we allow convenient abbreviations in contexts where ambiguities are not serious, now that we have the language to resolve these ambiguities when required. These logical distinctions must be grasped, since they are essential in many, even quite elementary, branches of mathematics. 2.6.3 EXAMPLE. Let ~R+ = {x E !Rjx ~ 0}, and let us temporarily denote by Sq: IR ~ IR the function such that Sq (x) = x 2 for each x E R Then Sq is not one-one, since Sq (1) = Sq ( -1), yet 1 =I= -1 (for example). Also, the image, Sq (IR), is IR +, so taking account of the above remarks, we see that the function Sq: IR ~ lR + is onto. If we restrict Sq to IR + (that is, we consider the function f: IR + ~ IR + given by f(x) = x 2 ), then it is known that f is an equivalence (see 28.6). We calljthe restriction ofjto IR+, and write (i)f= Sqj!R+. 2.6.4 More generally, for any function g: A~ B, if C £ A, we define the restriction of g to C to be the function h: C ~ B, given by

h(c)

=

g(c)

for each c E C;

and we indicate the dependence of h on C and g, by using the notation

(i) h = giC.

26

FUNCTIONS: DESCRIPTIVE THEORY

CH. 2

2.6.5 EXAMPLE. The sine function, sin: ~-+ ~ (explained, e.g., in 29.8) is not onto, because in particular 2 =I= sin x for any x E ~. Its image is known to be the interval

A= {2} = 0. Do the analogous problems when f is replaced respectively by sin l\0, 7T/6) and sin l\0, 'TT/7). [The relationship between sinP {x} and sin - l x, suspected by the reader, is discussed below, in 2.9,6.] (iv) With a general/: X->- Y, and D s; Y prove that, if also E s; Y, then P(D u E) = fP(D) u JP(E); JP(D (\ E) = fP(D) (\ JP(E). (v) Show by examples that, in (iv), (1) JP(D) can be empty, although D ::j:. 0; (2) s; D} . h . equa1"1ty. (3) f(JP(D)) JP(j(A) 2 A w1t out necessan"1y h avmg (4) If fP(f(A)) = A for all A s; X, prove that f is one-one, and prove the converse of this result. (5) Ifj(JP(D)) = D for all D s; Y, prove that/ is onto, and prove the converse of this result. (6) Prove that JP( Y) = X, fP( Y - D) = X - fi'(D).

2. 7 Composition of Functions The reasons mentioned in 1.5 for assembling new sets from old apply also to functions. From two given functions

f: A-+B,

g: B-+ C

§ 2.7

27

COMPOSITION OF FUNCTIONS

we form a new function denoted by gof:A~c

called the composite of g and f (read: 'g circle f '), and defined by the rule that 2.7.1 (go f)( a) = g(f(a)), for all a EA. In calculus, go f is often called a 'function of a function', but the reader should by now be able to see why this is a misleading description. Notice also that fog will not even be defined, if A i= C. 2.7.2 EXAMPLES. On using the id function of Example 2.5.5, the equation (i) of Example 2.5.4 shows that h o h = id1• If f: A~ B is the 'father' function of Example 2.1.1, then (f of)(a) denotes the paternal grandfather of a. The function f: A~ B in 2.6.1 is jog. In 2.6.4, the restriction g[ C of g: A~ B, isg o k, where k: c~ A is the inclusion map. Given any f: A~ B, we have 2.7.3

thus id functions behave like a unity element in algebra. For this reason we shall often use the symbol 1 to denote an identity function (with or without a subscript, as explained in 2.6.2). 2.7.4 EXAMPLE. Consider the functions log: IR+ ~ 1R and sin: IR ~ -1, 1). Then sino log: IR+ ~ \-1, 1) has at each x E IR+ the value sin (log x), whereas logo sin (x) is not even defined for all x [e.g., when x = 37Tj2]; and when it is defined, sin (log x) i= log (sin x), in general.


itself is commonly denoted by f- 1 even when f is not invertible. We advise the inexperienced reader to use the notation jl> until he can confidently avoid the pitfalls. 2.9.7 CAUTION. When A is a set of numbers, do not confuse f- 1{x) with {f(x))- 1 (the reciprocal of f(x)). With the first we have taken j, then its inverse, and evaluated at x; with the second we have taken f, evaluated at x, and then taken the reciprocal. Inverting and evaluating f are two operations which do not commute. As an example, sin -I (1) = TT/2 = 1.57 · · ·, while (sin t)- 1 = 1.18 .... We may conveniently summarize the results 2.9.1-2.9.4 in the following theorem, which we call the Algebraic Inversion Theorem. 2. 9.8 THEOREM. A function f: A --o. B is an equivalence iff 3g: B --o. A satisfying the equations gf = lA, fg = 18 • Such a g is then unique; it is the inverse of f. As an application, suppose that f: A --o. B, g: B --o. C are invertible; then go f: A --o. Cis an equivalence by 2.8.2; so it has an inverse. Let us prove that, in this case 2.9.9 Before beginning the proof we remark that we are really establishing that the composition of two invertible functions is again invertible. Thus the proof is formal, or algebraic, and does not rely on the nature of functions but only on their laws of composition 2.7.3, 2.7.6. Proof.

We leave the reader to supply reasons for the following equalities:

u-1g-l)(gf) = 1 -1cg-1g)f

= ~-1v = J-1J = t.

Similarly, (gf)(j-lg- = 1. Hence gf has J-lg- 1 as an inverse; and by 2.9.4, an inverse is unique if it exists, so we are forced to write 1)

Exercise 7

•

(i) Given functions A --4. B ~ C (not necessarily equivalences), prove that the composite of gl>: pC-+ pB and jl>: pB -+ pA is (g of) I>: pC-+ pA, and (g oj)l> = jl> o gl>. Prove also that (lA)I> is the identity on pA. Hence prove that iff is invertible, so isfl>. {Hint: consider (ff- 1 )1> and (f- 1j)l>. See also 6.2.9.}

32

FUNCTIO"'S: DESCRIPTIVE THEORY

CH.

2

(ii) Following Example 2.9.6, suppose we regard A, B as subsets of pA, pB, as suggested in Exercise S(ii). Show that if j: A~ B is an equivalence, then its inverse is the restriction to B of f~: pB ~ pA. More precisely, to avoid the 'regarding' process let sA: A~ pA denote the embedding of A in pA, and similarly for s8 . Show that the diagram 'commutes', i.e., that f~ o s8 = sAo j- 1 • This equation is the precise form of the equation PIE= j- 1 •

B~A)

SB

1 1 SA

pB~pA

2.9.10 exp: IR

In Examples 2.5.6, 2.6.7 the functions log: IR+ IR+ are inverses of each other, because of the equations

EXAMPLE.

~

~

IR,

log (exp x) = x exp (log x) = x so log o exp = id, exp o log = id, where, in the spirit of 2.6.2, we omit the subscripts IR and IR+ from the first and second occurrences of id, respectivelyt. In 2.6.3, we remarked that the function Sq[IR + of 2.6.3(i) is an equivalence: IR + ~ IR +. Hence, by the Inversion Theorem, it has an inverse, usually called 'the positive square root'; and of course Sq( y;x) = x = y;(Sq (x)). In Example 2.5.8, the 'fundamental theorem of calculus' (30.1.2) gives D o Q = ide; but Q o D =f'. ide, since (Q o D)(f) is that function whose value at xis

Lx f'(t) dt

which is not f(x) unless f(O) = 0. for any jE C, so Q o D =/= ide.

2.10

=

f(x) - f(O),

Thus, with this exception, (Q

D)(f) =f f

Equivalent Sets

It is convenient to abbreviate the statement

2.10.1

o

f:

A~

'f: A ~ B is an equivalence' to!

B.

To express the fact that there exists some equivalence between A and B we write 2.10.2 saying that A is equivalent to B.

t Another excuse is: 'for typographical reasons', (because of the expense of setting up mathematical type). Notice that we write id or 1 for the identity function. t When writing symbols like ~ or ::::::, lecturers often emit the sound 'twiddles'.

§ 2.11

33

COUNTING

The statement '/is an equivalence' is often, and naturally, confused with 'A is equivalent (to B)'. To avoid this confusion we will often say '/ is invertible', or 'f is a bijection', instead of 'f is an equivalence'; we recall Theorem 2. 9.8.

Exercise 8 Prove the following: if A ::::: B if A ::=:: B

2.11

A::::: A; then B ::::: A; and B ::=:: C, then

A

::=::

C.

Counting

Let N,. denote the subset {1, 2, ... , n} of N, consisting of the first n natural numbers; N 0 is therefore empty. Then to say of a set A that we have counted it and found it to have n members, is to say that we have constructed an equivalence for the act of counting consists in pairing off each element in A with one in N,., until A is exhausted. The last integer to be paired is n. Thus we make the definition:

#A=

n (read: the number of elements in N,.. Thus A = 0 iff# A = 0. If there does not exist such a bijection, we write

2.11.1

DEFINITION.

We write

A is n) iff there exists a bijection/:

A~

#A=

oo, andt call A infinite; otherwise A is finite. Caution. The fact that we speak of 'the' number of elements in A, implies that A cannot have both n elements and m elements, with n #- m. Commonsense, and experience with counting, both combine to assure us that this view is reasonable. However, experience also reminds us of occasions when we have inaccurately counted a set twice and obtained different answers; and we have no experience at all of counting really large sets, like the total number of possible games of chess having fewer than 50 moves. To set these doubts at rest we shall, in Chapter 7, prove: 2.11.2

THEOREM.

It is impossible for b1jections h: A;::::; N 11 ,

k: A ;::::; N 111 ,

to exist simultaneously.

t

Note that we do not define the symbol oo on its own.

11

#-

Ill)

34

FUNCTIONS: DESCRIPTIVE THEORY

Hence the integer #A is uniquely defined. then, we have: 2.11.3 PROPOSITION. of elements.

CH.

2

Assuming this result temporarily,

Two finite sets are equivalent, iff they have the same number

Proof. Suppose f: A~ Nn and g: A~ B. Then g- 1 : B ~A (by 2.9.1); so fg- 1 : B ~ Nn by 2.8.2. Hence #A = n = #B. Conversely, if #A = #B = n, then there exist bijections p: A ~ Nn, q: B ~ Nn; so q- 1p: A ~ B .

•

Remarks. The proof uses the method, common in real life, of counting the set B by comparing it with a known set A. Also, we have shown that it is impossible for B to be simultaneously infinite and equivalent to the set A. The method of proof shows too that if h, k existed in 2.11.2, then hk -l: Nm ~ Nn, n i= m, and it is basically this statement that we prove to be impossible in 7.1.3. We shall also prove there that every subset of a finite set is finite; this statement is not trivial, since it must be shown to hold by using Definition 2.11.1, and not other usages of the word 'finite'. A consequence is that, for each finite universe '¥/, #is a function: p'¥/ ---7 l +, where l + is the set N U {0} of non-negative integers. Exercise 9 (i) Prove that N contains proper subsets B (i.e., B # N) such that B ~ N. (ii) If X is a set such that X ~ N, then X is said to be denumerably (or countably) infinite. Prove that the following sets are denumerably infinite: the even integers, any infinite subset of N, the rational numbers, the set of all points (p, q) in the plane such that p, q E N. {In the last, count off the points in a diagram like Fig. 5.3 by going along successive diagonals like the one shown there.} 00

§(iii) Use the same diagram to show that if Xn ~ N, n

= 1, 2, ... , then U Xn n=l

~ N.

Hence prove that the set of all polynomials with integer coefficients is equivalent to N. Then show that the set of all algebraic numbers (i.e., real roots of such polynomials) is denumerably infinite.

Let us now obtain some formulae for counting finite subsets of a finite universe 0//. First, since f\J 0 ~ N 0 = 0, then, as remarked in 2.11.1,

#0 =

2.11.4

0.

The following result is used constantly in ordinary life, as a Venn diagram shows, but we must prove it from the definition of#· 2.11.5

THEOREM.

If A, B s;:; '¥/, and A n B

=

0, then

#(A v B)= #A+ #B.

§ 2.11

35

COUNTING

Proof. Let #A = n, #B = m. The formula 2.11.5 obviously holds when n or m is zero, by 2.11.4; hence suppose 0 < n and 0 < m. By Definition 2.11.1, there exist bijections and we construct a bijection c: AU B

c(x)

~

Nn+m by the rules:

if X E-A,

a(x) = { b(x) + n

if

X

E

B.

We leave the reader to verify that the function c is in fact one-one and onto; the fact that A n B = 0 ensures no ambiguity in the definition of c. But then, by Definition 2.11.1, the existence of c gives #(AU B)= #Nn+m

and 2.11.5 follo,vs.

•

2.11.6 COROLLARY. explained in 1. 7):

If

=n+m =#A+ #B.

X~

A

~ ~.

then (by using the notation A -X

(i) #A = #X + #(A - X) (ii) #X::::; #A.

{A is the union of the two disjoint sets X, A - X, so (i) is an application of 2.11.6; (ii) follows from (i), since #(A - X) ;::, 0.} II We can now prove a generalization of 2.11.6 which is again fairly obvious from a Venn diagram; it reduces to 2.11.5 when An B = 0, by 2.11.4. 2.11.7

THEOREM.

If A, B

~

ql, then

#(A u B) = #A

+ #B

- #(A n B).

Proof. A u B = [(A - (A n B)) u (A n B)] u [(A n B) u (B - (A n B))], = (A - (A n B)) u (A n B) u (B - (A n B)), applying the rule 1.7.7 with X= An B. Now Au B has been expressed as the union of three subsets, any pair of which are disjoint. Thus, we can apply 2.11.5 twice to get

+ #(A n B) + #(B - (A n B)) B)] + #(A n B) + [#B - #(A n B)],

#(A u B) = #(A - (A n B)) =

[#A - #(A n

using 2.11.6(i). Rearrangement gives 2.11. 7. • Notice the purely algebraic nature of this proof, which is different from that of the proof of 2.11.5. In a purely algebraic manner, also, we can extend 2.11. 7 to a union of more sets, as follows. 3

36

FUNCTIONS: DESCRIPTIVE THEORY

2.11.8

CH.2

THEOREM.

#(A v B v C) = #A

+ #B + #C

- #(A n B) - #(B n C) - #(C n A)+ #(An B n C),

and more generally #(A 1 v · · · v An)

=

n

L #At - L #(At n A i