Introduction to Real Analysis
INTRODUCTION TO REAL ANALYSIS
Fourth Edition
Robert G. Bartle Donald R. Sherbert
University of Illinois, Urbana-Champaign
John Wiley & Sons, Inc.
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Library of Congress Cataloging-in-Publication Data Bartle, Robert Gardner, 1927-
Introduction to real analysis / Robert G. Bartle, Donald R. Sherbert. – 4th ed.
p. cm.
Includes index.
ISBN 978-0-471-43331-6 (hardback)
1. Mathematical analysis. 2. Functions of real variables. I. Sherbert, Donald R., 1935- II. Title.
QA300.B294 2011
515–dc22 2010045251
Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
A TRIBUTE
This edition is dedicated to the memory of Robert G. Bartle, a wonderful friend and colleague of forty years. It has been an immense honor and pleasure to be Bob’s coauthor on the previous editions of this book. I greatly miss his knowledge, his insights, and especially his humor.
November 20, 2010
Urbana, Illinois Donald R. Sherbert
To Jan, with thanks and love.
PREFACE
The study of real analysis is indispensable for a prospective graduate student of pure or applied mathematics. It also has great value for any student who wishes to go beyond the routine manipulations of formulas because it develops the ability to think deductively, analyze mathematical situations and extend ideas to new contexts. Mathematics has become valuable in many areas, including economics and management science as well as the physical sciences, engineering, and computer science. This book was written to provide an accessible, reasonably paced treatment of the basic concepts and techniques of real analysis for students in these areas. While students will find this book challenging, experience has demonstrated that serious students are fully capable of mastering the material.
The first three editions were very well received and this edition maintains the same spirit and user-friendly approach as earlier editions. Every section has been examined.
Some sections have been revised, new examples and exercises have been added, and a new section on the Darboux approach to the integral has been added to Chapter 7. There is more material than can be covered in a semester and instructors will need to make selections and perhaps use certain topics as honors or extra credit projects.
To provide some help for students in analyzing proofs of theorems, there is an appendix on ‘‘Logic and Proofs’’ that discusses topics such as implications, negations, contrapositives, and different types of proofs. However, it is a more useful experience to learn how to construct proofs by first watching and then doing than by reading about techniques of proof.
Results and proofs are given at a medium level of generality. For instance, continuous functions on closed, bounded intervals are studied in detail, but the proofs can be readily adapted to a more general situation. This approach is used to advantage in Chapter 11 where topological concepts are discussed. There are a large number of examples to illustrate the concepts, and extensive lists of exercises to challenge students and to aid them in understanding the significance of the theorems.
Chapter 1 has a brief summary of the notions and notations for sets and functions that will be used. A discussion of Mathematical Induction is given, since inductive proofs arise frequently. There is also a section on finite, countable and infinite sets. This chapter can used to provide some practice in proofs, or covered quickly, or used as background material and returning later as necessary.
Chapter 2 presents the properties of the real number system. The first two sections deal with Algebraic and Order properties, and the crucial Completeness Property is given in Section 2.3 as the Supremum Property. Its ramifications are discussed throughout the remainder of the chapter.
In Chapter 3, a thorough treatment of sequences is given, along with the associated limit concepts. The material is of the greatest importance. Students find it rather natural though it takes time for them to become accustomed to the use of epsilon. A brief introduction to Infinite Series is given in Section 3.7, with more advanced material presented in Chapter 9.
vii
Chapter 4 on limits of functions and Chapter 5 on continuous functions constitute the heart of the book. The discussion of limits and continuity relies heavily on the use of sequences, and the closely parallel approach of these chapters reinforces the understanding of these essential topics. The fundamental properties of continuous functions on intervals are discussed in Sections 5.3 and 5.4. The notion of a gauge is introduced in Section 5.5 and used to give alternate proofs of these theorems. Monotone functions are discussed in Section 5.6.
The basic theory of the derivative is given in the first part of Chapter 6. This material is standard, except a result of Caratheodory is used to give simpler proofs of the Chain Rule and the Inversion Theorem. The remainder of the chapter consists of applications of the Mean Value Theorem and may be explored as time permits.
In Chapter 7, the Riemann integral is defined in Section 7.1 as a limit of Riemann sums. This has the advantage that it is consistent with the students’ first exposure to the integral in calculus, and since it is not dependent on order properties, it permits immediate generalization to complex- and vector-values functions that students may encounter in later courses. It is also consistent with the generalized Riemann integral that is discussed in Chapter 10. Sections 7.2 and 7.3 develop properties of the integral and establish the Fundamental Theorem of Calculus. The new Section 7.4, added in response to requests from a number of instructors, develops the Darboux approach to the integral in terms of upper and lower integrals, and the connection between the two definitions of the integral is established. Section 7.5 gives a brief discussion of numerical methods of calculating the integral of continuous functions.
Sequences of functions and uniform convergence are discussed in the first two sections of Chapter 8, and the basic transcendental functions are put on a firm foundation in Sections 8.3 and 8.4. Chapter 9 completes the discussion of infinite series that was begun in Section 3.7. Chapters 8 and 9 are intrinsically important, and they also show how the material in the earlier chapters can be applied.
Chapter 10 is a presentation of the generalized Riemann integral (sometimes called the
‘‘Henstock-Kurzweil’’ or the ‘‘gauge’’ integral). It will be new to many readers and they will be amazed that such an apparently minor modification of the definition of the Riemann integral can lead to an integral that is more general than the Lebesgue integral. This relatively new approach to integration theory is both accessible and exciting to anyone who has studied the basic Riemann integral.
Chapter 11 deals with topological concepts. Earlier theorems and proofs are extended to a more abstract setting. For example, the concept of compactness is given proper emphasis and metric spaces are introduced. This chapter will be useful to students continuing on to graduate courses in mathematics.
There are lengthy lists of exercises, some easy and some challenging, and ‘‘hints’’ to many of them are provided to help students get started or to check their answers. More complete solutions of almost every exercise are given in a separate Instructor’s Manual, which is available to teachers upon request to the publisher.
It is a satisfying experience to see how the mathematical maturity of the students increases as they gradually learn to work comfortably with concepts that initially seemed so mysterious. But there is no doubt that a lot of hard work is required on the part of both the students and the teachers.
Brief biographical sketches of some famous mathematicians are included to enrich the historical perspective of the book. Thanks go to Dr. Patrick Muldowney for his photograph of Professors Henstock and Kurzweil, and to John Wiley & Sons for obtaining portraits of the other mathematicians.
Many helpful comments have been received from colleagues who have taught from earlier editions of this book and their remarks and suggestions have been appreciated. I wish to thank them and express the hope that they find this new edition even more helpful than the earlier ones.
November 20, 2010
Urbana, Illinois Donald R. Sherbert
THE GREEK ALPHABET
A a Alpha N n Nu
B b Beta X j Xi
G g Gamma O o Omicron
D d Delta P p Pi
E e Epsilon P r Rho
Z z Zeta S s Sigma
H h Eta T t Tau
Q u Theta Y y Upsilon
I i Iota F w Phi
K k Kappa X x Chi
L l Lambda C c Psi
M m Mu V v Omega
PREFACE ix
CONTENTS
CHAPTER 1 PRELIMINARIES 1
1.1 Sets and Functions 1 1.2 Mathematical Induction 12 1.3 Finite and Infinite Sets 16
CHAPTER 2 THE REAL NUMBERS 23
2.1 The Algebraic and Order Properties ofR 23 2.2 Absolute Value and the Real Line 32 2.3 The Completeness Property ofR 36 2.4 Applications of the Supremum Property 40 2.5 Intervals 46
CHAPTER 3 SEQUENCES AND SERIES 54
3.1 Sequences and Their Limits 55 3.2 Limit Theorems 63
3.3 Monotone Sequences 70
3.4 Subsequences and the Bolzano-Weierstrass Theorem 78 3.5 The Cauchy Criterion 85
3.6 Properly Divergent Sequences 91 3.7 Introduction to Infinite Series 94
CHAPTER 4 LIMITS 102
4.1 Limits of Functions 103 4.2 Limit Theorems 111
4.3 Some Extensions of the Limit Concept 116
CHAPTER 5 CONTINUOUS FUNCTIONS 124
5.1 Continuous Functions 125
5.2 Combinations of Continuous Functions 130 5.3 Continuous Functions on Intervals 134 5.4 Uniform Continuity 141
5.5 Continuity and Gauges 149
5.6 Monotone and Inverse Functions 153
xi
CHAPTER 6 DIFFERENTIATION 161
6.1 The Derivative 162
6.2 The Mean Value Theorem 172 6.3 L’Hospital’s Rules 180 6.4 Taylor’s Theorem 188
CHAPTER 7 THE RIEMANN INTEGRAL 198
7.1 Riemann Integral 199
7.2 Riemann Integrable Functions 208 7.3 The Fundamental Theorem 216 7.4 The Darboux Integral 225 7.5 Approximate Integration 233
CHAPTER 8 SEQUENCES OF FUNCTIONS 241
8.1 Pointwise and Uniform Convergence 241 8.2 Interchange of Limits 247
8.3 The Exponential and Logarithmic Functions 253 8.4 The Trigonometric Functions 260
CHAPTER 9 INFINITE SERIES 267
9.1 Absolute Convergence 267
9.2 Tests for Absolute Convergence 270 9.3 Tests for Nonabsolute Convergence 277 9.4 Series of Functions 281
CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 288
10.1 Definition and Main Properties 289 10.2 Improper and Lebesgue Integrals 302 10.3 Infinite Intervals 308
10.4 Convergence Theorems 315
CHAPTER 11 A GLIMPSE INTO TOPOLOGY 326
11.1 Open and Closed Sets inR 326 11.2 Compact Sets 333
11.3 Continuous Functions 337 11.4 Metric Spaces 341
APPENDIX A LOGIC AND PROOFS 348
APPENDIX B FINITE AND COUNTABLE SETS 357
APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA 360 APPENDIX D APPROXIMATE INTEGRATION 364
APPENDIX E TWO EXAMPLES 367 REFERENCES 370
PHOTO CREDITS 371
HINTS FOR SELECTED EXERCISES 372 INDEX 395
CONTENTS xiii
CHAPTER 1
PRELIMINARIES
In this initial chapter we will present the background needed for the study of real analysis. Section 1.1 consists of a brief survey of set operations and functions, two vital tools for all of mathematics. In it we establish the notation and state the basic definitions and properties that will be used throughout the book. We will regard the word ‘‘set’’ as synonymous with the words ‘‘class,’’ ‘‘collection,’’ and ‘‘family,’’ and we will not define these terms or give a list of axioms for set theory. This approach, often referred to as ‘‘naive’’ set theory, is quite adequate for working with sets in the context of real analysis.
Section 1.2 is concerned with a special method of proof called Mathematical Induction. It is related to the fundamental properties of the natural number system and, though it is restricted to proving particular types of statements, it is important and used frequently. An informal discussion of the different types of proofs that are used in mathematics, such as contrapositives and proofs by contradiction, can be found in Appendix A.
In Section 1.3 we apply some of the tools presented in the first two sections of this chapter to a discussion of what it means for a set to be finite or infinite. Careful definitions are given and some basic consequences of these definitions are derived. The important result that the set of rational numbers is countably infinite is established.
In addition to introducing basic concepts and establishing terminology and notation, this chapter also provides the reader with some initial experience in working with precise definitions and writing proofs. The careful study of real analysis unavoidably entails the reading and writing of proofs, and like any skill, it is necessary to practice. This chapter is a starting point.
Section 1.1 Sets and Functions
To the reader:In this section we give a brief review of the terminology and notation that will be used in this text. We suggest that you look through it quickly and come back later when you need to recall the meaning of a term or a symbol.
If an elementxis in a setA, we write x2A
and say thatx is amember ofA, or thatxbelongstoA. IfxisnotinA, we write x2= A:
If every element of a setAalso belongs to a setB, we say thatAis asubsetofBand write AB or BA:
1
We say that a setAis aproper subsetof a setBifAB, but there is at least one element of Bthat is not in A. In this case we sometimes write
AB:
1.1.1 Definition Two sets AandBare said to beequal,and we write A¼B, if they contain the same elements.
Thus, to prove that the setsAandB are equal, we must show that
AB and BA:
A set is normally defined by either listing its elements explicitly, or by specifying a property that determines the elements of the set. IfPdenotes a property that is meaningful and unambiguous for elements of a setS, then we write
x2S:P xð Þ
f g
for the set of all elementsxinSfor which the propertyPis true. If the setSis understood from the context, then it is often omitted in this notation.
Several special sets are used throughout this book, and they are denoted by standard symbols. (We will use the symbol :¼to mean that the symbol on the left is beingdefined by the symbol on the right.)
The set ofnatural numbersN:¼f1; 2; 3;. . .g,
The set ofintegersZ:¼f0; 1; 1; 2; 2;. . .g,
The set ofrational numbers Q :¼fm=n:m; n2Zandn6¼0g,
The set ofreal numbersR.
The setRof real numbers is of fundamental importance and will be discussed at length in Chapter 2.
1.1.2 Examples (a) The set
x2N : x23xþ2¼0
consists of those natural numbers satisfying the stated equation. Since the only solutions of this quadratic equation arex¼1 andx¼2, we can denote this set more simply by {1, 2}.
(b) A natural numbernisevenif it has the formn¼2kfor somek2N. The set of even natural numbers can be written
2k : k2N
f g;
which is less cumbersome thanfn2N : n¼2k; k2Ng. Similarly, the set ofoddnatural numbers can be written
2k1 : k2N
f g: &
Set Operations
We now define the methods of obtaining new sets from given ones. Note that these set operations are based on the meaning of the words ‘‘or,’’ ‘‘and,’’ and ‘‘not.’’ For the union, it is important to be aware of the fact that the word ‘‘or’’ is used in the inclusive sense, allowing the possibility thatxmay belong to both sets. In legal terminology, this inclusive sense is sometimes indicated by ‘‘and/or.’’
1.1.3 Definition (a) Theunion of setsAandB is the set A[B:¼fx:x2Aorx2Bg:
(b) The intersectionof the setsAandBis the set
A\B:¼fx:x2Aandx2Bg:
(c) Thecomplement ofBrelative toAis the set
AnB:¼fx:x2Aandx2= Bg:
The set that has no elements is called theempty setand is denoted by the symbol;. Two setsAandBare said to bedisjointif they have no elements in common; this can be expressed by writingA\B¼ ;.
To illustrate the method of proving set equalities, we will next establish one of the De Morgan lawsfor three sets. The proof of the other one is left as an exercise.
1.1.4 Theorem If A, B, C are sets, then (a) AnðB[CÞ ¼ðAnBÞ \ðAnCÞ, (b) AnðB\CÞ ¼ðAnBÞ [ðAnCÞ.
Proof. To prove (a), we will show that every element inAnðB[CÞis contained in both (AnB) and (AnC), and conversely.
Ifxis inAnðB[CÞ, thenxis inA, butxis not inB[C. Hencexis inA, butxis neither inBnor inC. Therefore,xis inAbut notB, andxis inAbut notC. Thus,x2AnBand x2AnC, which shows thatx2ðAnBÞ \ðAnCÞ.
Conversely, ifx2ðAnBÞ \ðAnCÞ, thenx2ðAnBÞandx2ðAnCÞ. Hencex2Aand bothx2= Bandx2= C. Therefore,x 2A andx2= ðB[CÞ, so thatx2AnðB[CÞ.
Since the sets ðAnBÞ \ðAnCÞ andAnðB[CÞcontain the same elements, they are
equal by Definition 1.1.1. Q.E.D.
There are times when it is desirable to form unions and intersections of more than two sets. For a finite collection of sets {A1,A2, . . . ,An}, their union is the setAconsisting of all elements that belong toat least oneof the setsAk, and their intersection consists of all elements that belong toallof the sets Ak.
This is extended to an infinite collection of sets {A1,A2, . . . ,An, . . . } as follows.
Theirunionis the set of elements that belong toat least oneof the setsAn. In this case we write
[1
n¼1
An :¼fx:x2Anfor somen2Ng:
Figure 1.1.1 (a)A[B (b)A\B (c)AnB
1.1 SETS AND FUNCTIONS 3
Similarly, theirintersectionis the set of elements that belong toallof these setsAn. In this case we write
\1
n¼1
An:¼fx:x2Anfor alln2Ng:
Functions
In order to discuss functions, we define the Cartesian product of two sets.
1.1.5 Definition IfAandBare nonempty sets, then theCartesian productABofA andBis the set of all ordered pairs (a,b) witha2Aandb2B. That is,
AB:¼fða;bÞ:a2A;b2Bg:
Thus ifA¼{1, 2, 3} andB¼{1, 5}, then the setABis the set whose elements are the ordered pairs
1;1
ð Þ; ð1;5Þ; ð2;1Þ; ð2;5Þ; ð3;1Þ; ð3;5Þ:
We may visualize the setABas the set of six points in the plane with the coordinates that we have just listed.
We often draw a diagram (such as Figure 1.1.2) to indicate the Cartesian product of two setsAandB. However, it should be realized that this diagram may be a simplification.
For example, ifA:¼fx2R : 1x2gandB:¼fy2R : 0y1 or 2y3g, then instead of a rectangle, we should have a drawing such as Figure 1.1.3.
We will now discuss the fundamental notion of a functionor amapping.
To the mathematician of the early nineteenth century, the word ‘‘function’’ meant a definite formula, such as f xð Þ:¼x2þ3x5, which associates to each real number x another number f(x). (Here, f(0) ¼ 5, f(1) ¼ 1, f(5) ¼ 35.) This understanding excluded the case of different formulas on different intervals, so that functions could not be defined ‘‘in pieces.’’
As mathematics developed, it became clear that a more general definition of
‘‘function’’ would be useful. It also became evident that it is important to make a clear distinction between the function itself and the values of the function. A revised definition might be:
Figure 1.1.2 Figure 1.1.3
A functionffrom a setAinto a setBis a rule of correspondence that assigns to each elementx inAa uniquely determined elementf(x) inB.
But however suggestive this revised definition might be, there is the difficulty of interpreting the phrase ‘‘rule of correspondence.’’ In order to clarify this, we will express the definition entirely in terms of sets; in effect, we will define a function to be itsgraph. While this has the disadvantage of being somewhat artificial, it has the advantage of being unambiguous and clearer.
1.1.6 Definition LetAandBbe sets. Then afunctionfromAtoBis a setfof ordered pairs inABsuch that for eacha2Athere exists a uniqueb2Bwith (a,b)2f. (In other words, if (a,b)2fand (a,b0)2f, thenb¼b0.)
The setAof first elements of a functionfis called thedomainoffand is often denoted byD(f). The set of all second elements inf is called therangeoffand is often denoted by R(f). Note that, althoughD(f)¼A, we only haveRðfÞ B. (See Figure 1.1.4.)
The essential condition that:
a; b
ð Þ 2f and ða; b0Þ 2f implies that b¼b0
is sometimes called thevertical line test. In geometrical terms it says every vertical line x¼awitha2Aintersects the graph offexactly once.
The notation
f :A!B
is often used to indicate thatf is a function fromAinto B. We will also say that fis a mappingofAintoB, or thatfmapsAintoB. If (a,b) is an element inf, it is customary to write
b¼f að Þ or sometimes a7!b:
Ifb ¼f(a), we often refer tob as thevalueoffata, or as theimageof aunderf. Transformations and Machines
Aside from using graphs, we can visualize a function as atransformationof the setD(f)¼ Ainto the setRðfÞ B. In this phraseology, when (a,b)2f, we think offas taking the
Figure 1.1.4 A function as a graph
1.1 SETS AND FUNCTIONS 5
element a from A and ‘‘transforming’’ or ‘‘mapping’’ it into an element b ¼ f(a) in RðfÞ B. We often draw a diagram, such as Figure 1.1.5, even when the setsAandBare not subsets of the plane.
There is another way of visualizing a function: namely, as a machinethat accepts elements ofD(f)¼Aasinputsand produces corresponding elements ofRðfÞ Basoutputs. If we take an elementx2D(f) and put it intof, then out comes the corresponding valuef(x).
If we put a different elementy2D(f) intof, then out comesf(y), which may or may not differ fromf(x). If we try to insert something that does not belong toD(f) intof, we find that it is not accepted, forfcan operate only on elements fromD(f). (See Figure 1.1.6.)
This last visualization makes clear the distinction betweenfandf(x): the first is the machine itself, and the second is the output of the machinefwhenxis the input. Whereas no one is likely to confuse a meat grinder with ground meat, enough people have confused functions with their values that it is worth distinguishing between them notationally.
Direct and Inverse Images
Letf: A!Bbe a function with domainD(f)¼Aand rangeRðfÞ B.
1.1.7 Definition IfEis a subset ofA, then thedirect imageofEunderfis the subsetf(E) of Bgiven by
fðEÞ:¼ffðxÞ:x2Eg:
Figure 1.1.5 A function as a transformation
Figure 1.1.6 A function as a machine
IfHis a subset ofB, then theinverse imageofHunderfis the subsetf1(H) ofAgiven by f1ðHÞ:¼fx2A:fðxÞ 2Hg:
Remark The notationf1(H) used in this connection has its disadvantages. However, we will use it since it is the standard notation.
Thus, if we are given a setEA, then a pointy12Bis in the direct imagef(E) if and only if there exists at least one pointx12Esuch thaty1¼f(x1). Similarly, given a set HB, then a pointx2is in the inverse imagef1(H) if and only ify2:¼f(x2) belongs toH. (See Figure 1.1.7.)
1.1.8 Examples (a) Letf :R!Rbe defined byf(x) :¼x2. Then the direct image of the setE:¼fx:0x2gis the setf Eð Þ ¼fy:0y4g.
IfG:¼fy:0y4g, then the inverse image ofGis the setf1ðGÞ ¼fx:2 x2g. Thus, in this case, we see that f1ðf Eð ÞÞ 6¼E.
On the other hand, we havef f 1ð ÞG
¼G. But ifH:¼fy:1y1g, then we havef f 1ð ÞH
¼fy:0y1g 6¼H.
A sketch of the graph offmay help to visualize these sets.
(b) Let f:A!B, and let G,Hbe subsets ofB. We will show that f1ðG\HÞ f 1ð Þ \G f1ð Þ:H
For, ifx2f1ðG\HÞ, thenf xð Þ 2G\H, so thatf(x)2Gandf(x)2H. But this implies thatx2f1ð ÞG andx2f 1ð ÞH , whencex2f1ð Þ \G f1ð ÞH . Thus the stated impli- cation is proved. [The opposite inclusion is also true, so that we actually have set equality
between these sets; see Exercise 15.] &
Further facts about direct and inverse images are given in the exercises.
Special Types of Functions
The following definitions identify some very important types of functions.
1.1.9 Definition Letf :A!Bbe a function fromAtoB.
(a) The function fis said to beinjective (or to beone-one)if wheneverx16¼x2, then f xð Þ 6¼1 f xð Þ2 . Iffis an injective function, we also say that fis aninjection. (b) The functionfis said to besurjective(or to mapAontoB) iff(A)¼B; that is, if the
range R(f)¼B. Iffis a surjective function, we also say thatfis asurjection. Figure 1.1.7 Direct and inverse images
1.1 SETS AND FUNCTIONS 7
(c) Iffis both injective and surjective, thenf is said to bebijective. Iff is bijective, we also say thatfis abijection.
In order to prove that a functionfis injective, we must establish that:
for allx1;x2inA; iff xð Þ ¼1 f xð Þ;2 thenx1¼x2: To do this we assume thatf(x1)¼f(x2) and show thatx1¼x2.
[In other words, the graph offsatisfies thefirst horizontal line test:Every horizontal line y¼bwithb2Bintersects the graph finat most one point.]
To prove that a functionfis surjective, we must show that for anyb2Bthere exists at least onex 2A such thatf(x)¼b.
[In other words, the graph offsatisfies thesecond horizontal line test:Every horizontal liney¼b withb 2Bintersects the graphfinat leastone point.]
1.1.10 Example LetA:¼fx2R :x6¼1gand definef xð Þ:¼2x=ðx1Þfor allx2A. To show thatfis injective, we takex1andx2inAand assume thatf(x1)¼f(x2). Thus we have
2x1
x11¼ 2x2
x21;
which implies thatx1ðx21Þ ¼x2ðx11Þ, and hencex1¼x2. Thereforefis injective.
To determine the range off, we solve the equationy¼2x=ðx1Þforxin terms ofy. We obtain x¼y=ðy2Þ, which is meaningful fory6¼2. Thus the range of fis the set B:¼fy2R:y6¼2g. Thus,fis a bijection ofAontoB. &
Inverse Functions
Iffis a function fromAintoB, thenfis a special subset ofAB(namely, one passing the vertical line test.) The set of ordered pairs inBAobtained by interchanging the members of ordered pairs infis not generally a function. (That is, the setfmay not passbothof the horizontal line tests.) However, if f is a bijection, then this interchange does lead to a function, called the ‘‘inverse function’’ off.
1.1.11 Definition Iff:A!B is a bijection ofA ontoB, then g:¼fðb; aÞ 2BA:ða; bÞ 2fg
is a function onBintoA. This function is called theinverse functionoff, and is denoted byf1. The functionf1is also called the inverseof f.
We can also express the connection between f and its inverse f1 by noting that D(f)¼R(f1) andR(f)¼D(f1) and that
b¼fðaÞ if and only if a¼f1ðbÞ: For example, we saw in Example 1.1.10 that the function
fðxÞ:¼ 2x x1
is a bijection ofA:¼fx2R:x6¼1gonto the setB:¼fy2R:y6¼2g. Solvingy¼f(x) for xin terms ofy, we find the function inverse tofis given by
f 1ðyÞ:¼ y
y2 for y2B:
Remark We introduced the notationf1(H) in Definition 1.1.7. It makes sense even iff does not have an inverse function. However, if the inverse functionf1does exist, then f1(H) is the direct image of the setHBunderf1.
Composition of Functions
It often happens that we want to ‘‘compose’’ two functionsf,gby first findingf(x) and then applyinggto getg(f(x)); however, this is possible only whenf(x) belongs to the domain of g. In order to be able to do this forall f(x), we must assume that the range offis contained in the domain ofg. (See Figure 1.1.8.)
1.1.12 Definition If f : A ! B and g : B ! C, and if RðfÞ DðgÞ ¼B, then the composite functiong f(note the order!) is the function fromA intoCdefined by
g f
ð Þð Þx :¼g f xð ð ÞÞ for all x2A:
1.1.13 Examples (a) The order of the composition must be carefully noted. For, letf andgbe the functions whose values atx2R are given by
fðxÞ:¼2x and gðxÞ:¼3x21:
SinceDðgÞ ¼RandR fð Þ R ¼D gð Þ, then the domainD(g f) is also equal toR, and the composite functiong fis given by
g f
ð Þð Þ ¼x 3 2ð Þx 21¼12x21:
On the other hand, the domain of the composite functionf gis alsoR, but f g
ð Þð Þ ¼x 2 3 x21
¼6x22: Thus, in this case, we haveg f6¼f g.
(b) In consideringg f, some care must be exercised to be sure that the range of fis contained in the domain ofg. For example, if
f xð Þ:¼1x2 and g xð Þ:¼ ffiffiffi px
;
then, since D gð Þ ¼fx:x0g, the composite functiong fis given by the formula g f
ð Þð Þ ¼x ffiffiffiffiffiffiffiffiffiffiffiffiffi 1x2 p
only forx2D fð Þthat satisfyf(x)0; that is, for xsatisfying1x1.
Figure 1.1.8 The composition offandg
1.1 SETS AND FUNCTIONS 9
We note that if we reverse the order, then the compositionf gis given by the formula f g
ð Þð Þ ¼x 1x;
but only for thosexin the domainD gð Þ ¼fx:x0g. &
We now give the relationship between composite functions and inverse images. The proof is left as an instructive exercise.
1.1.14 Theorem Let f:A!B and g:B!C be functions and let H be a subset of C.
Then we have
g f
ð Þ1ð Þ ¼H f 1g1ð ÞH :
Note the reversalin the order of the functions.
Restrictions of Functions
Iff:A !Bis a function and ifA1A, we can define a functionf1:A1!Bby f1ð Þx :¼f xð Þ for x2A1:
The functionf1is called therestriction offtoA1. Sometimes it is denoted byf1¼fjA1. It may seem strange to the reader that one would ever choose to throw away a part of a function, but there are some good reasons for doing so. For example, iff :R!Ris the squaring function:
f xð Þ:¼x2 for x2R;
thenfis not injective, so it cannot have an inverse function. However, if we restrictfto the set A1:¼fx:x0g, then the restriction fjA1 is a bijection of A1onto A1. Therefore, this restriction has an inverse function, which is thepositive square root function. (Sketch a graph.)
Similarly, the trigonometric functionsS(x) :¼sinxandC(x) :¼cosxare not injective on all ofR. However, by making suitable restrictions of these functions, one can obtain theinverse sineand theinverse cosinefunctions that the reader has undoubtedly already encountered.
Exercises for Section 1.1
1. LetA:¼fk:k2N;k20g;B:¼f3k1:k2Ng;andC:¼f2kþ1:k2Ng:
Determine the sets:
(a) A\B\C, (b) ðA\BÞnC, (c) ðA\CÞnB.
2. Draw diagrams to simplify and identify the following sets:
(a) An(BnA), (b) An(AnB), (c) A\ðBnAÞ.
3. IfAandBare sets, show thatABif and only ifA\B¼A.
4. Prove the second De Morgan Law [Theorem 1.1.4(b)].
5. Prove the Distributive Laws:
(a) A\ðB[CÞ ¼ðA\BÞ [ðA\CÞ, (b) A[ðB\CÞ ¼ðA[BÞ \ðA[CÞ.
6. Thesymmetric differenceof two setsAandBis the setDof all elements that belong to eitherA orBbut not both. RepresentDwith a diagram.
(a) Show thatD¼ðAnBÞ [ðBnAÞ.
(b) Show thatDis also given byD¼ðA[BÞnðA\BÞ.
7. For eachn2N, letAn¼fðnþ1Þk:k2Ng. (a) What isA1\A2?
(b) Determine the sets[fAn:n2Ngand\fAn:n2Ng.
8. Draw diagrams in the plane of the Cartesian productsABfor the given setsAandB. (a) A¼fx2R : 1x2 or 3x4g;B¼fx2R : x¼1 orx¼2g.
(b) A¼f1;2;3g;B¼fx2R : 1x3g.
9. LetA:¼B:¼fx2R :1x1gand consider the subsetC:¼ðx;yÞ:x2þy2¼1 of AB. Is this set a function? Explain.
10. Letf xð Þ:¼1=x2;x6¼0;x2R.
(a) Determine the direct imagef(E) whereE:¼fx2R : 1x2g. (b) Determine the inverse imagef1(G) whereG:¼fx2R : 1x4g.
11. Letg(x) :¼x2andf(x) :¼xþ2 forx2R, and lethbe the composite functionh:¼g f.
(a) Find the direct imageh(E) ofE:¼fx2R : 0x1g. (b) Find the inverse imageh1(G) ofG:¼fx2R : 0x4g.
12. Letf(x) :¼x2forx2R, and letE:¼fx2R :1x0gandF:¼fx2R : 0x1g. Show that E\F¼f g0 and f Eð \FÞ ¼f g, while0 f Eð Þ ¼f Fð Þ ¼fy2R : 0y1g.
Hencef Eð \FÞis a proper subset off Eð Þ \f Fð Þ. What happens if 0 is deleted from the setsE andF?
13. LetfandE,Fbe as in Exercise 12. Find the setsEnFandf(E)nf(f) and show that it isnottrue thatf EnFð Þ f Eð Þnf Fð Þ.
14. Show that if f : A ! B and E, F are subsets of A, then f Eð [FÞ ¼f Eð Þ [f Fð Þ and f Eð \FÞ f Eð Þ \f Fð Þ.
15. Show that iff:A!BandG,Hare subsets ofB, thenf1ðG[HÞ ¼f1ð Þ [G f1ð ÞH and f1ðG\HÞ ¼f1ð Þ \G f1ð Þ.H
16. Show that the function f defined by f xð Þ:¼x= ffiffiffiffiffiffiffiffiffiffiffiffiffi x2þ1
p ;x2R, is a bijection of R onto y:1<y<1
f g.
17. For a, b 2 R with a < b, find an explicit bijection of A:¼fx:a<x<bg onto B:¼fy:0<y<1g.
18. (a) Give an example of two functionsf,gonRtoRsuch thatf6¼g, but such thatf g¼g f. (b) Give an example of three functionsf,g,honRsuch thatf ðgþhÞ 6¼f gþf h.
19. (a) Show that iff:A!Bis injective andEA, thenf1ðfðEÞÞ ¼E. Give an example to show that equality need not hold iffis not injective.
(b) Show that iff:A!Bis surjective andHB, thenf f 1ðHÞ
¼H. Give an example to show that equality need not hold iffis not surjective.
20. (a) Suppose thatfis an injection. Show thatf1 f(x)¼xfor allx2D(f) and thatf f1(y)¼y for ally2R(f).
(b) Iffis a bijection ofAontoB, show thatf1is a bijection ofBontoA.
21. Prove that iff:A!Bis bijective andg:B!Cis bijective, then the compositeg f is a bijective map ofAontoC.
22. Letf:A!Bandg:B!Cbe functions.
(a) Show that ifg fis injective, thenfis injective.
(b) Show that ifg fis surjective, thengis surjective.
23. Prove Theorem 1.1.14.
24. Letf,gbe functions such that (g f)(x)¼xfor allx2D(f) and (f g)(y)¼yfor ally2D(g).
Prove thatg¼f1.
1.1 SETS AND FUNCTIONS 11
Section 1.2 Mathematical Induction
Mathematical Induction is a powerful method of proof that is frequently used to establish the validity of statements that are given in terms of the natural numbers. Although its utility is restricted to this rather special context, Mathematical Induction is an indispensable tool in all branches of mathematics. Since many induction proofs follow the same formal lines of argument, we will often state only that a result follows from Mathematical Induction and leave it to the reader to provide the necessary details. In this section, we will state the principle and give several examples to illustrate how inductive proofs proceed.
We shall assume familiarity with the set of natural numbers:
N¼f1; 2; 3;. . .g;
with the usual arithmetic operations of addition and multiplication, and with the meaning of a natural number being less than another one. We will also assume the following fundamental property of N.
1.2.1 Well-Ordering Property ofN Every nonempty subset ofNhas a least element.
A more detailed statement of this property is as follows: IfSis a subset ofN and if S6¼ ;, then there existsm2Ssuch thatmkfor allk2S.
On the basis of the Well-Ordering Property, we shall derive a version of the Principle of Mathematical Induction that is expressed in terms of subsets of N.
1.2.2 Principle of Mathematical Induction Let S be a subset ofNthat possesses the two properties:
(1) The number1 2S.
(2) For every k2N,if k2S,then kþ12S. Then we have S¼N.
Proof. Suppose to the contrary thatS6¼N. Then the setNnSis not empty, so by the Well- Ordering Principle it has a least elementm. Since 12Sby hypothesis (1), we know that m>1. But this implies thatm1 is also a natural number. Sincem1<mand sincemis the least element in Nsuch thatm2= S, we conclude thatm12S.
We now apply hypothesis (2) to the elementk:¼m1 inS, to infer thatkþ1¼ m1
ð Þ þ1¼mbelongs toS. But this statement contradicts the fact thatm2= S. Sincem was obtained from the assumption thatNnSis not empty, we have obtained a contradiction.
Therefore we must haveS¼N. Q.E.D.
The Principle of Mathematical Induction is often set forth in the framework of statements about natural numbers. IfP(n) is a meaningful statement aboutn2N, thenP(n) may be true for some values ofnand false for others. For example, ifP1(n) is the statement:
‘‘n2¼n,’’ thenP1(1) is true whileP1(n) is false for alln>1,n2N. On the other hand, if P2(n) is the statement: ‘‘n2>1,’’ thenP2(1) is false, whileP2(n) is true for alln>1.
In this context, the Principle of Mathematical Induction can be formulated as follows.
For each n2N , let P(n)be a statement about n. Suppose that:
(10) P(1) is true.
(20) For every k2N , if P(k)is true, then P(kþ1) is true. Then P(n)is true for all n2N.
The connection with the preceding version of Mathematical Induction, given in 1.2.2, is made by lettingS:¼fn2N : PðnÞis trueg. Then the conditions (1) and (2) of 1.2.2 correspond exactly to the conditions (10) and (20), respectively. The conclusion thatS¼N in 1.2.2 corresponds to the conclusion thatP(n) is true for alln2N.
In (20) the assumption ‘‘if P(k) is true’’ is called the induction hypothesis. In establishing (20), we are not concerned with the actual truth or falsity ofP(k), but only with the validity of the implication ‘‘if P(k), then P(k þ 1).’’ For example, if we consider the statementsP(n): ‘‘n ¼n þ5,’’ then (20) is logically correct, for we can simply add 1 to both sides ofP(k) to obtainP(kþ1). However, since the statement P(1): ‘‘1¼6’’ is false, we cannot use Mathematical Induction to conclude thatn¼nþ5 for alln2N.
It may happen that statementsP(n) are false for certain natural numbers but then are true for allnn0for some particularn0. The Principle of Mathematical Induction can be modified to deal with this situation. We will formulate the modified principle, but leave its verification as an exercise. (See Exercise 12.)
1.2.3 Principle of Mathematical Induction (second version) Let n02Nand let P(n) be a statement for each natural number nn0. Suppose that:
(1) The statement P(n0)is true.
(2) For all kn0,the truth of P(k)implies the truth of P(kþ1).
Then P(n)is true for all n n0.
Sometimes the numbern0in (1) is called thebase,since it serves as the starting point, and the implication in (2), which can be writtenP kð Þ )P kð þ1Þ, is called thebridge, since it connects the casekto the casekþ1.
The following examples illustrate how Mathematical Induction is used to prove assertions about natural numbers.
1.2.4 Examples (a) For eachn2N, the sum of the firstnnatural numbers is given by 1þ2þ þn¼12n nð þ1Þ:
To prove this formula, we letSbe the set of alln2Nfor which the formula is true.
We must verify that conditions (1) and (2) of 1.2.2 are satisfied. Ifn ¼1, then we have 1¼121ð1þ1Þso that 12S, and (1) is satisfied. Next, weassumethatk2Sand wish to infer from this assumption thatkþ12S. Indeed, ifk2S, then
1þ2þ þk¼12k kð þ1Þ:
If we addkþ1 to both sides of the assumed equality, we obtain 1þ2þ þkþðkþ1Þ¼12k kð þ1Þ þðkþ1Þ
¼12ðkþ1Þðkþ2Þ:
Since this is the stated formula forn ¼kþ1, we conclude that kþ12S. Therefore, condition (2) of 1.2.2 is satisfied. Consequently, by the Principle of Mathematical Induction, we infer thatS¼N, so the formula holds for alln2N.
(b) For each n2N, the sum of the squares of the firstnnatural numbers is given by 12þ22þ þn2¼16n nð þ1Þð2nþ1Þ:
1.2 MATHEMATICAL INDUCTION 13
To establish this formula, we note that it is true forn¼1, since 12¼16123. If we assume it is true fork, then adding (kþ1)2to both sides of the assumed formula gives
12þ22þ þk2þðkþ1Þ2 ¼16k kð þ1Þð2kþ1Þ þðkþ1Þ2
¼16ðkþ1Þ2k2þkþ6kþ6
¼16ðkþ1Þðkþ2Þð2kþ3Þ:
Consequently, the formula is valid for all n2N.
(c) Given two real numbersaandb, we will prove thatabis a factor ofanbnfor all n2N.
First we see that the statement is clearly true forn¼1. If we now assume thatabis a factor ofakbk, then
akþ1bkþ1 ¼akþ1abkþabkbkþ1
¼a a kbk
þbkðabÞ:
By the induction hypothesis,abis a factor ofa(akbk) and it is plainly a factor of bk(ab). Therefore,abis a factor ofakþ1bkþl, and it follows from Mathematical Induction thata b is a factor ofanbnfor alln2N.
A variety of divisibility results can be derived from this fact. For example, since 117¼4, we see that 11n7nis divisible by 4 for all n2N.
(d) The inequality 2n>2nþ1 is false forn¼1, 2, but it is true forn¼3. If we assume that 2k>2kþ1, then multiplication by 2 gives, when 2kþ2>3, the inequality
2kþ1>2 2ð kþ1Þ ¼4kþ2¼2kþð2kþ2Þ>2kþ3¼2ðkþ1Þ þ1: Since 2kþ2>3 for allk1, the bridge is valid for allk1 (even though the statement is false fork¼1, 2). Hence, with the basen0¼3, we can apply Mathematical Induction to conclude that the inequality holds for all n3.
(e) The inequality 2n(nþ1)! can be established by Mathematical Induction.
We first observe that it is true forn ¼1, since 21¼2 ¼1 þ1. If we assume that 2k(kþ1)!, it follows from the fact that 2kþ2 that
2kþ1¼22k2ðkþ1Þ!ðkþ2Þðkþ1Þ!¼ðkþ2Þ!:
Thus, if the inequality holds fork, then it also holds forkþ1. Therefore, Mathematical Induction implies that the inequality is true for alln2N.
(f) Ifr2R,r6¼1, andn2N, then
1þrþr2þ þrn¼1rnþ1 1r :
This is the formula for the sum of the terms in a ‘‘geometric progression.’’ It can be established using Mathematical Induction as follows. First, if n ¼1, then 1 þr ¼ (1r2)=(1r). If we assume the truth of the formula forn¼kand add the termrkþ1 to both sides, we get (after a little algebra)
1þrþrkþ þrkþ1 ¼1rkþ1
1r þrkþ1¼1rkþ2 1r ;
which is the formula forn¼kþ1. Therefore, Mathematical Induction implies the validity of the formula for alln2N.
[This result can also be proved without using Mathematical Induction. If we let sn:¼1þrþr2þ þrn, then rsn¼rþr2þ þrnþ1, so that
1r
ð Þsn¼snrsn¼1rnþ1: If we divide by 1r, we obtain the stated formula.]
(g) Careless use of the Principle of Mathematical Induction can lead to obviously absurd conclusions. The reader is invited to find the error in the ‘‘proof’’ of the following assertion.
Claim: Ifn2Nand if the maximum of the natural numberspandqisn, thenp¼q.
‘‘Proof.’’ LetSbe the subset ofNfor which the claim is true. Evidently, 12Ssince if p,q2Nand their maximum is 1, then both equal 1 and sop¼q. Now assume thatk2Sand that the maximum ofpandqiskþ1. Then the maximum ofp1 andq1 isk. But since k2S, thenp1¼q1 and thereforep¼q. Thus,kþ12S, and we conclude that the assertion is true for alln2N.
(h) There are statements that are true formanynatural numbers but that are not true for allof them.
For example, the formulap(n) :¼n2nþ41 gives a prime number forn¼1, 2, . . . , 40. However,p(41) is obviously divisible by 41, so it is not a prime number.
Another version of the Principle of Mathematical Induction is sometimes quite useful.
It is called the ‘‘Principle of Strong Induction,’’ even though it is in fact equivalent to 1.2.2.
1.2.5 Principle of Strong Induction Let S be a subset ofN such that (100) 12S.
(200) For every k2N,iff1;2; . . .; kg S,then kþ12S. Then S¼N.
We will leave it to the reader to establish the equivalence of 1.2.2 and 1.2.5.
Exercises for Section 1.2
1. Prove that 1=12þ1=23þ þ1=n nð þ1Þ ¼n=ðnþ1Þfor alln2N. 2. Prove that 13þ23þ þn3¼12n nð þ1Þ2
for alln2N. 3. Prove that 3þ11þ þð8n5Þ ¼4n2nfor alln2N.
4. Prove that 12þ32þ þð2n1Þ2¼ð4n3nÞ=3 for alln2N.
5. Prove that 1222þ32þ þ 1ð Þnþ1n2¼ 1ð Þnþ1n nð þ1Þ=2 for alln2N.
6. Prove thatn3þ5nis divisible by 6 for alln2N. 7. Prove that 52n1 is divisible by 8 for alln2N.
8. Prove that 5n4n1 is divisible by 16 for alln2N.
9. Prove thatn3þ(nþ1)3þ(nþ2)3is divisible by 9 for alln2N.
10. Conjecture a formula for the sum 1=13þ1=35þ þ1=ð2n1Þð2nþ1Þ, and prove your conjecture by using Mathematical Induction.
11. Conjecture a formula for the sum of the firstnodd natural numbers 1þ3þ þð2n1Þ, and prove your formula by using Mathematical Induction.
12. Prove the Principle of Mathematical Induction 1.2.3 (second version).
1.2 MATHEMATICAL INDUCTION 15
13. Prove thatn<2nfor alln2N. 14. Prove that 2n<n! for alln4,n2N.
15. Prove that 2n32n2for alln5,n2N.
16. Find all natural numbersnsuch thatn2<2n. Prove your assertion.
17. Find the largest natural numbermsuch thatn3nis divisible bymfor alln2N. Prove your assertion.
18. Prove that 1= ffiffiffi p1
þ1= ffiffiffi p2
þ þ1=pffiffiffin>pffiffiffin
for alln2N,n>1.
19. LetSbe a subset ofNsuch that (a) 2k2Sfor allk2N, and (b) ifk2Sandk2, then k12S. Prove thatS¼N.
20. Let the numbersxnbe defined as follows:x1:¼1,x2:¼2, andxnþ2:¼12ðxnþ1þxnÞfor all n2N. Use the Principle of Strong Induction (1.2.5) to show that 1xn2 for alln2N.
Section 1.3 Finite and Infinite Sets
When we count the elements in a set, we say ‘‘one, two, three, . . . ,’’ stopping when we have exhausted the set. From a mathematical perspective, what we are doing is defining a bijective mapping between the set and a portion of the set of natural numbers. If the set is such that the counting does not terminate, such as the set of natural numbers itself, then we describe the set as being infinite.
The notions of ‘‘finite’’ and ‘‘infinite’’ are extremely primitive, and it is very likely that the reader has never examined these notions very carefully. In this section we will define these terms precisely and establish a few basic results and state some other important results that seem obvious but whose proofs are a bit tricky. These proofs can be found in Appendix B and can be read later.
1.3.1 Definition (a) The empty set; is said to have 0elements.
(b) If n2N, a set Sis said to have n elementsif there exists a bijection from the set Nn :¼f1; 2;. . .; ng ontoS.
(c) A setSis said to befiniteif it is either empty or it hasnelements for somen2N. (d) A set Sis said to beinfiniteif it is not finite.
Since the inverse of a bijection is a bijection, it is easy to see that a set S has n elements if and only if there is a bijection from S onto the set {1, 2, . . . ,n}. Also, since the composition of two bijections is a bijection, we see that a set S1 has n elements if and only if there is a bijection from S1 onto another set S2 that has n elements. Further, a set T1 is finite if and only if there is a bijection from T1 onto another set T2 that is finite.
It is now necessary to establish some basic properties of finite sets to be sure that the definitions do not lead to conclusions that conflict with our experience of counting. From the definitions, it is not entirely clear that a finite set might not havenelements formore than onevalue ofn. Also it is conceivably possible that the setN:¼ f1;2;3;. . .gmight be a finite set according to this definition. The reader will be relieved that these possibilities do not occur, as the next two theorems state. The proofs of these assertions, which use the fundamental properties ofN described in Section 1.2, are given in Appendix B.
1.3.2 Uniqueness Theorem If S is a finite set, then the number of elements in S is a unique number in N.
1.3.3 Theorem The setN of natural numbers is an infinite set.
The next result gives some elementary properties of finite and infinite sets.
1.3.4 Theorem (a) If A is a set with m elements and B is a set with n elements and if A\B¼ ;, then A[B has m þn elements.
(b) If A is a set with m2Nelements and CA is a set with1element, then AnC is a set with m1elements.
(c) If C is an infinite set and B is a finite set, then CnB is an infinite set.
Proof. (a) Letfbe a bijection ofNmontoA, and letgbe a bijection ofNn ontoB. We define h on Nmþn by h(i) :¼ f(i) for i¼1;. . .; m and h(i) :¼ g(i m) for
i¼mþ1;. . .;mþn. We leave it as an exercise to show that h is a bijection from
Nmþn ontoA[B.
The proofs of parts (b) and (c) are left to the reader, see Exercise 2. Q.E.D.
It may seem ‘‘obvious’’ that a subset of a finite set is also finite, but the assertion must be deduced from the definitions. This and the corresponding statement for infinite sets are established next.
1.3.5 Theorem Suppose that S and T are sets and that TS. (a) If S is a finite set, then T is a finite set.
(b) If T is an infinite set, then S is an infinite set.
Proof. (a) IfT ¼ ;, we already know thatTis a finite set. Thus we may suppose that T 6¼ ;. The proof is by induction on the number of elements inS.
IfShas 1 element, then the only nonempty subsetTofSmust coincide withS, soTis a finite set.
Suppose that every nonempty subset of a set withkelements is finite. Now letSbe a set havingkþ1 elements (so there exists a bijectionfofNkþ1 ontoS), and letT S. If f kð þ1Þ2= T, we can consider T to be a subset of S1:¼Snff kð þ1Þg, which has k elements by Theorem 1.3.4(b). Hence, by the induction hypothesis,Tis a finite set.
On the other hand, iff kð þ1Þ 2T, thenT1 :¼Tnff kð þ1Þgis a subset ofS1. Since S1haskelements, the induction hypothesis implies thatT1is a finite set. But this implies thatT¼T1[ff kð þ1Þg is also a finite set.
(b) This assertion is the contrapositive of the assertion in (a). (See Appendix A for a
discussion of the contrapositive.) Q.E.D.
Countable Sets
We now introduce an important type of infinite set.
1.3.6 Definition (a) A setSis said to bedenumerable(orcountably infinite) if there exists a bijection ofN ontoS.
(b) A set Sis said to becountable if it is either finite or denumerable.
(c) A setS is said to beuncountableif it is not countable.
From the properties of bijections, it is clear thatSis denumerable if and only if there exists a bijection ofSontoN. Also a setS1is denumerable if and only if there exists a 1.3 FINITE AND INFINITE SETS 17
bijection fromS1onto a setS2that is denumerable. Further, a setT1is countable if and only if there exists a bijection from T1 onto a set T2that is countable. Finally, an infinite countable set is denumerable.
1.3.7 Examples (a) The setE:¼f2n:n2Ngofevennatural numbers is denumerable, since the mappingf :N!Edefined byf(n) :¼2nforn2Nis a bijection ofNontoE. Similarly, the setO:¼f2n1:n2Ngofoddnatural numbers is denumerable.
(b) The set Zof allintegers is denumerable.
To construct a bijection ofNontoZ, we map 1 onto 0, we map the set of even natural numbers onto the setNof positive integers, and we map the set of odd natural numbers onto the negative integers. This mapping can be displayed by the enumeration:
Z¼f0;1; 1;2; 2; 3; 3;. . .g:
(c) The union of two disjoint denumerable sets is denumerable.
Indeed, if A¼fa1; a2; a3;. . .g and B¼fb1;b2; b3;. . .g, we can enumerate the elements ofA[Bas:
a1; b1; a2; b2; a3; b3;. . .: &
1.3.8 Theorem The setNN is denumerable.
Informal Proof. Recall thatNNconsists of all ordered pairs (m,n), wherem,n2N. We can enumerate these pairs as:
1; 1
ð Þ; ð1; 2Þ; ð2; 1Þ; ð1;3Þ; ð2;2Þ; ð3;1Þ; ð1;4Þ;. . .;
according to increasing summþn, and increasingm. (See Figure 1.3.1.) Q.E.D.
The enumeration just described is an instance of a ‘‘diagonal procedure,’’ since we move along diagonals that each contain finitely many terms as illustrated in Figure 1.3.1.
The bijection indicated by the diagram can be derived as follows. We first notice that the first diagonal has one point, the second diagonal has two points, and so on, withkpoints in thekth diagonal. Applying the formula in Example 1.2.4(a), we see that the total number of points in diagonals 1 through kis given by
cð Þ ¼k 1þ2þ þk¼12k kð þ1Þ
Figure 1.3.1 The setNN
The point (m,n) lies in thekth diagonal whenk¼mþn1, and it is themth point in that diagonal as we move downward from left to right. (For example, the point (3, 2) lies in the 4th diagonal since 3þ21¼4, and it is the 3rd point in that diagonal.) Therefore, in the counting scheme displayed by Figure 1.3.1, we count the point (m, n) by first counting the points in the firstk1¼mþn2 diagonals and then addingm. Therefore, the counting functionh:NN!N is given by
h m;ð nÞ :¼cðmþn2Þ þm
¼12ðmþn2Þðmþn1Þ þm:
For example, the point (3, 2) is counted as number hð3;2Þ ¼1234þ3¼9, as shown by Figure 1.3.1. Similarly, the point (17, 25) is counted as numberh(17, 25)¼c(40) þ17¼837.
This geometric argument leading to the counting formula has been suggestive and convincing, but it remains to be proved that his, in fact, a bijection of NN ontoN. A detailed proof is given in Appendix B.
The construction of an explicit bijection between sets is often complicated. The next two results are useful in establishing the countability of sets, since they do not involve showing that certain mappings are bijections. The first result may seem intuitively clear, but its proof is rather technical; it will be given in Appendix B.
1.3.9 Theorem Suppose that S and T are sets and that TS. (a) If S is a countable set, then T is a countable set.
(b) If T is an uncountable set, then S is an uncountable set. 1.3.10 Theorem The following statements are equivalent:
(a) S is a countable set.
(b) There exists a surjection of Nonto S. (c) There exists an injection of S intoN.
Proof. (a))(b) IfSis finite, there exists a bijectionhof some setNn ontoSand we defineHonN by
HðkÞ:¼ hðkÞ for k¼1;. . .; n;
hðnÞ for k>n:
ThenHis a surjection ofN ontoS.
IfSis denumerable, there exists a bijectionHofNontoS, which is also a surjection of N ontoS.
(b))(c) IfHis a surjection ofNontoS, we defineH1:S!Nby lettingH1(s) be the least element in the setH1ðsÞ:¼fn2N:HðnÞ ¼sg. To see thatH1is an injection ofS into N, note that if s,t2S andnst:¼H1ðsÞ ¼H1ðtÞ, thens¼H(nst)¼t.
(c))(a) IfH1is an injection ofSintoN, then it is a bijection ofSontoH1ðSÞ N. By Theorem 1.3.9(a),H1(S) is countable, whence the setSis countable. Q.E.D.
1.3.11 Theorem The setQ of all rational numbers is denumerable.
Proof. The idea of the proof is to observe that the setQþof positive rational numbers is contained in the enumeration:
1
1; 12; 21; 13; 22; 31; 14;. . .;
1.3 FINITE AND INFINITE SETS 19
which is another ‘‘diagonal mapping’’ (see Figure 1.3.2). However, this mapping is not an injection, since the different fractions12and24represent the same rational number.
To proceed more formally, note that sinceNNis countable (by Theorem 1.3.8), it follows from Theorem 1.3.10(b) that there exists a surjection fof N onto NN. Ifg: NN!Qþ is the mapping that sends the ordered pair (m,n) into the rational number having a representationm=n, thengis a surjection ontoQþ. Therefore, the compositiong f is a surjection ofNontoQþ, and Theorem 1.3.10 implies thatQþis a countable set.
Similarly, the setQ of all negative rational numbers is countable. It follows as in Example 1.3.7(b) that the setQ ¼Q[ f0g [Qþ is countable. SinceQ containsN, it
must be a denumerable set. Q.E.D.
The next result is concerned with unions of sets. In view of Theorem 1.3.10, we need not be worried about possible overlapping of the sets. Also, we do not have to construct a bijection.
1.3.12 Theorem If Amis a countable set for each m2N, then the union A:¼S1
m¼1Am
is countable.
Proof. For eachm2N, letwmbe a surjection ofNontoAm. We defineb:NN!Aby bðm;nÞ:¼wmðnÞ:
We claim thatbis a surjection. Indeed, ifa2A, then there exists a leastm2Nsuch that a2Am, whence there exists a leastn2Nsuch thata¼wmðnÞ. Therefore,a¼b(m,n).
SinceNNis countable, it follows from Theorem 1.3.10 that there exists a surjection f :N!NNwhenceb fis a surjection ofNontoA. Now apply Theorem 1.3.10 again
to conclude that Ais countable. Q.E.D.
Remark A less formal (but more intuitive) way to see the truth of Theorem 1.3.12 is to enumerate the elements ofAm,m2N, as:
A1 ¼fa11;a12;a13;. . .g;
A2 ¼fa21;a22;a23;. . .g;
A3 ¼fa31;a32;a33;. . .g;
: Figure 1.3.2 The setQþ
We then enumerate this array using the ‘‘diagonal procedure’’:
a11;a12;a21; a13; a22;a31;a14;. . .; as was displayed in Figure 1.3.1.
Georg Cantor
Georg Cantor (1845–1918) was born in St. Petersburg, Russia. His father, a Danish businessman working in Russia, moved the family to Germany several years later. Cantor studied briefly at Zurich, then went to the University of Berlin, the best in mathematics at the time. He received his doctorate in 1869, and accepted a position at the University of Halle, where he worked alone on his research, but would occasionally travel the seventy miles to Berlin to visit colleagues.
Cantor is known as the founder of modern set theory and he was the first to study the concept of infinite set in rigorous detail. In 1874 he proved that Q is countable and, in contrast, thatRis uncountable (see Section 2.5), exhibiting two kinds of infinity. In a series of papers he developed a general theory of infinite sets, including some surprising results. In 1877 he proved that the two-dimensional unit square in the plane could be put into one-one correspondence with the unit interval on the line, a result he sent in a letter to his colleague Richard Dedekind in Berlin, writing ‘‘I see it, but I do not believe it.’’ Cantor’s Theorem on sets of subsets shows there are many different orders of infinity and this led him to create a theory of ‘‘transfinite’’ numbers that he published in 1895 and 1897. His work generated considerable controversy among mathematicians of that era, but in 1904, London’s Royal Society awarded Cantor the Sylvester Medal, its highest honor.
Beginning in 1884, he suffered from episodes of depression that increased in severity as the years passed. He was hospitalized several times for nervous breakdowns in the Halle Nervenklinik and spent the last seven months of his life there.
We close this section with one of Cantor’s more remarkable theorems.
1.3.13 Cantor’s Theorem If A is any set, then there is no surjection of A onto the set PðAÞof all subsets of A.
Proof. Suppose thatw:A! PðAÞis a surjection. Sincew(a) is a subset ofA, eithera belongs tow(a) or it does not belong to this set. We let
D:¼fa2A:a2= wðaÞg:
SinceDis a subset of A, ifwis a surjection, then D¼wð Þa0 for somea02A.
We must have eithera02Dora02= D. Ifa02D, then sinceD¼wð Þa0 , we must have a02wð Þa0 , contrary to the definition ofD. Similarly, ifa02= D, thena02= wð Þa0 so that a02D, which is also a contradiction.
Therefore,wcannot be a surjection. Q.E.D.
Cantor’s Theorem implies that there is an unending progression of larger and larger sets. In particular, it implies that the collectionPðNÞof all subsets of the natural numbers N is uncountable.
