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THE ALGEBRAICF OUNDAT I ON SOF MATHEMATICS


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This book is in theADDISON-WESLEY SER IES I N

INTRODUCTORY COLLEGE MATHEMATICS

Consulting EditorsRICHARD. PIETERS GAILS. YOUNG


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THE A L G E B R A I CF O U N D A T I O N SOF M A T H E M A T I C S

R O S S A . B E A U M O X Tand

R I C H A R D S . P I E R C EDepartment of MathematicsUniversity of Washington

A D D I S O K - W E S L E Y P U B L I S H I N G C O M P A N Y , I N C .R E A D I N G , M A S S A CH U S E T T S P A L O A L T O L O N D O N


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Copyright @ 1963ADDISON-WESLEY PUBLISHING COMPANY, INC.

Printed in the United States U;r; ArnericaALL RIGHTS RESERVED. THIS BOOK, OR PARTS THERE-OF, MAY NOT BE REPRODUCED I N ANY FORM WITH-OUT WRITTEN PERMISSION OF THE PUBLISHERS.

Library of Congress Catalog Card No. 623-8895


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PREFACEThis book is an offspring of two beliefs which the authors have held for

many years: it is worthwhile for the average person to understand whatrnathematics is al1 about; it is impossible to learn much about mathe-matics without doing mathematics. The first of these convictions seemsto be accepted by most educated people. The second opinion is less widelyheld. Mathematicians teaching in liberal arts colleges and universities areoften under pressure from their colleagues in the humanities and socialsciences to offer short courses which will painlessly explain mathematicsto students with varying backgrounds who are seeking a broad, liberaleducation. The extent to which such courses do not exist is a credit to thegood sense of professional mathematicians. Mathematics is a big and diffi-cult subject. I t embraces a rigid method of reasoning, a concise form ofexpression, and a variety of new concepts and viewpoints which are quitedifferent from those encountered in everyday life. There is no such thingas "descriptive7' mathematics. In order to find answers to the questions"What is mathematics?" and "What do mathematicians do?", i t is neces-sary to learn something of the logic, the language, and the philosophy ofmathematics. This cannot be done by listening to a few entertaininglectures, but only by active contact with the content of real mathematics.I t is the authors' hope tha t this book will provide the means for thisnecessary contact.

For most people, the road from marketplace arithmetic to the borderof real mathematics is long and steep. I t usually takes severa1 years tomake this journey. Fortunately, because of the improving curriculum inhigh schools, many students are completing the elementary mathematicsincluded in algebra, geometry, and trigonometry before entering college,so that as college freshmen they can begin to appreciate the attractionsof sophisticated mathematical ideas. Many of these students have evenbeen exposed to the new programs for school mathematics which introducemodern mathematical ideas and methods. Too often, such students areshunted into a college algebra or elementary calculus course, where themain emphasis is on mathematical formalism and manipulation. Anyenthusiasm for creative thinking which a student may carry into collegewill quickly be blunted by such a course. It is often claimed that themanipulative skills acquired in elementary algebra and calculus are whata student needs for the application of mathematics to science and engi-neering, and indeed to the practica1 problems of life. Although notaltogether wrong, this argument overlooks the obvious fact tha t in almostany situation, the ability to use mathematical technique and reasoning ismore valuable than the ability to manipulate and calculate accurately.

v


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vi PREFACEElementary college algebra and calculus courses usually cultivate manipula-tion a t the expense of logical reasoning, and they give the student almostno idea of what mathematics is really like. It is often painfully evidentto an instructor in, say, a senior leve1 course in abstract algebra that theaverage mathematics major in his class has a very distorted idea of thenature of mathematics.

The object of this book is to present in a form suitable for student con-sumption a small but important part of real mathematics. I t is concernedwith topics related to the principal number systems of mathematics. Thebook treats those topics of algebra which are basic for advanced studiesin mathematics and of fundamental importance for al1 working mathe-maticians. This is the reason t,hat we have entitled our book "T he AlgebraicFoundations of Mathematics. " In accord with the philosophy that studentsshould be taught mathematics by exposing them to the mathematics ofprofessional mathematicians, the book should be useful not only to stu-dents majoring in mathematics, but also to adequately prepared studentsof any speciality.

Since mathematics is a logical science, it is appropriate that any bookon real mathematics should emphasize mathematical proofs. The studentwho masters the technique and acquires the habit of mathematical proofis well on his way toward understanding the nature of mathematics. Sucha mastery is hard to achieve, but it is within the reach of a large percentageof the college population.

This book is not intended to be an easy one. I t is not meant for thecollege freshman with minimum preparation from high school. An aptstudent with three years of high school mathematics should be able tostudy most parts of the book with profit, but his progress may not be rapid.Appropriate places for the use of this book include: a freshman course toreplace the standard precalculus college algebra for students who willprogress to a rigorous treatment of calculus, a terminal course for liberalarts students with a good background in mathematics, an elementaryhonors course for mathematics majors, a course to follow a traditionalcalculus course to develop maturity, and a refresher course for high schoolmathematics teachers. The book is written in such a way that the law ofdiminishing returns will not set in too quiekly. That is, enough difficultmaterial is included in most sections and chapters so that even the beststudents will be challenged. The student of more modest ability shouldkeep this in mind in order to combat discouragement.

Some sections digress from the main theme of the book. These aredesignated by a "star." For the most part, starred sections can be omittedwithout loss of continuity, although it may be necessary to refer to themfor definitions. It should be emphasized that the starred sections are notthe most difficult parts of the book. On the contrary, much of the material


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PREFACE viiin these sections is very elementary. A star has been attached to just thosesections which are not sufficiently important to be considered indispensable,but which are still too interesting to omit.

The complete book can be covered in a two semester or three quartercourse meeting three hours per week. The following table suggests howthe book can be used for shorter courses.

Above all, this book represents an effort to show college students someof the real beauty of mathernatics. The appreciation of mathematicalbeauty is not like the enjoyment of literature, music, and other ar t forms.I t requires serious effort and hard study. I t is much more difficult for amathematician to explain his triumphs and masterpieces than for any otherkind of artist or scientist. Consequently, most mathematicains do not tryto interpret their work to the general public, but only communicate with

Chapter1 (Omit 1-3, 1-5, and

starred sections)2 (Omit starred sections)4 (Omit 4-1)5 (Omit starred sections)6 (Omit 6-1, 6-4, 6-5)7 (Omit 7-1, 7-2, 7-3, 7-6,

and starred sections)89 (Omit starred section)

10 (Omit 10-4)1 through 8 (Omit starredsections)

4 (Omit 4-1, 4-3, 4-5, 4-6)5 (Omit starred sections)6 (Omit 6-1, 6-4, 6-5)89 (Omit starred section)

10 (Omit 10-3, 10-4)1 (Omit starred sections)2 (Omit starred sections)45

CourseCollege algebra

Development of theclassical numbersystemsTheory of equations

Elementary theoryof numbers

Time required1 Semester, 3 hours1 Quarter, 5 hours

1 Semester, 3 hours1 Quarter, 5 hours

1Semester, 2 hours1Quarter, 3 hours

1 Semester, 2 hours1 Quarter, 3 hours


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viii PREFACEcolleagues having similar interests. For this reason, a mathematician isoften considered to be a rather aloof person who lives partly in this worldand partly in some other mysterious realm. This is in fact a fairly accurateconception. However, the door to the world of mathematics is neverlocked, and anyone who will make the effort can enjoy the beauties of anintellectual domain which comes closer to aesthetic perfection than anyother science.Acknowledgements. Writing a textbook is not a routine chore. Withoutthe help of many people, we might never have finished this one. We areparticularly indebted to Professors C. W. Curtis, R. A. Dean, and H. S.Zuckerman, who read most of the manuscript of this book, and gave usmany valuable suggestions. Our publisher, Addison-Wesley, has watchedover our work from beginning to end with remarkable patience andbenevolence. The swift and expert typing of Mary Pierce is sincerelyappreciated. Finally, we are grateful to many friends for sincere encourage-ment during the last two years, and especially to our wives, who havelived with us through these trying times.Seattle, WashingtonJ a n u a r y 1963


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. . . . . . . . . . . . . . . . . .-1 Sets1-2 T he cardinal number of a set . . . . . . . . . . .. . . . . . .-3 T he con struction of sets from given sets. . . . . . . . . . . . .-4 T he algebra of sets . . . .-5 Fu rthe r a lgcbra of se ts. Ge nera l rules of o pera tion. . . . . . . . . . . . . .1-6 Measures on se ts . . . . . . . .1-7 Properties an d esamples of measures. . . . . . . . . . . . . .-1 Proof by induct ion . . . . . . . . . . . . .-2 T he binomial theorem . . . . . . .-3 Generalizations of the induction principlc. . . . . . . . . . .2-4 S h e technique of induction

2-5 Induct iv e proper t ies of th e natura l numbcrs . . . . . .. . . . . . . . . . . . .2-6 Inductive definit ions. . . . . . . . . . .-1 The definition of numbers . . . . . . . .-2 Operat ions with the na tura l numbers . . . . . . . .-3 Th e ordcring of t he na tura l num bers. . . . . . . . . . .-1 Con struction of th e integers. . . . . . . . . . . . . . . . .-2 Rings . . . . . . . . . .-3 General ized sums and products. . . . . . . . . . . . . .4 In tegra l domains . . . . . . . . . . .-5 T h c ordering of t he integers. . . . . . . . . . . . . .-6 Properties of order

. . . . . . . . . . . .-1 S h e div is ion a lgor i thm . . . . . . . . . . . .-2 Grea tes t common divisor . . . . . . .-3 T he fundam enta l theorem of ar i thm etic. . . . . . . . . . . . .5-4 More about pr imes . . .5-5 Applications of th e fundamen tal theorem of ari thm etic. . . . . . . . . . . . . . .-6 Congruences . . . . . . . . . . . . .-7 Linear congruences . . . . . . . . .5-8 Th e theorems of Fe rm at and E u le r. . . . . . .-1 Basic properties of the rational num bcrs. . . . . . . . . . . . . . . . .-2 Fields . . . . .-3 T h e characteristic of integral dom ains an d fields

ix


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X CONTENTS

. . . . . . . . . . . . .-4 Equivalence relations 213. . . . . . . . . . . . .-5 The construction of & 218. . . . . . . . .-1 Development of the real numbers. . . . . . . . . . . . .-2 The coordinate line. . . . . . . . . . . . . . .-3 Dedekind cuts . . . . . . . . .-4 Construction of the real numbers

7-5 The completeness of the real numbers . . . . . . . .7-6 Properties of complete ordered fields . . . . . . . .. . . . . . . . . . . . . .7-7 Infinite sequences. . . . . . . . . . . . . . .7-8 Infinite series

*7-9 Decimal representation . . . . . . . . . . . .*7-10 Applications of decimal representations . . . . . . . .

. . . . . . .-1 The construction of the complesnumbers 286. . . . .-2 Comples conjugates and the absolute value in C 2918-3 The geometrical representation of complex numbers . 298. . . . . . . . . . . . .-4 Polar representation 3039-1 Algebraic equations . . . . . . . . .9-2 Polynomials . . . . . . . . . . .9-3 The division algorithm for polynomials . . . .9-4 Greatest common divisor in F[x ] . . . . . .9-5 The unique factorization theorem for polynomials9-6 Derivatives . . . . . . . . . . . .9-7 The roots of a polynomial . . . . . . .9-8 The fundamental theorem of algebra . . . .

"9-9 The solution of third- and fourth-degree equations9-10 Graphs of real polynomials . . . . . . .9-1 1 Sturm's theorem . . . . . . . . . .9-12 Polynomials with rational coefficients . . . .

10-1 Polynomialsinseveralindeterminates . . . . . . . . 94. . . . . . . . . . .0-2 Systems of linear equations 410. . . . . . . . . . . .0-3 The algebra of matrices 428

10-4 The inverse of a square matrix . . . . . . . . . . 43


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INTRODUCTION

As we explained in the preface, the purpose of this book is to exhibita small, but significant and representative, part of the world of mathe-matics. The selection of a principal subject for this project poses difficul-ties similar to those which a blind man faces when he tries to discover theshape of an elephant by means of his "sense of feel." Only a few aspectsof the subject are within reach, and i t is necessary to exercise care to besure the part examined is truly representative. We might select some im-portant unifying concept of modern mathematics, such as the notion ofa group, and explore the ramifications of this idea. Alternatively, an olderand perhaps familiar topic can be examined in depth. I t is this last moreconservative program which will be followed.

We will study the principal number systems of mathematics and someof the theories related to them. An attempt will be made to answer thequestion "what are numbers?" in a way which meets the standards oflogical precision demanded in modern mathematics. This program hascertain dangers. Familiarity with ordinary numbers hides subtle diffi-culties which must be overcome before it is even possible to give an exactdefinition of them. Checking the details in the construction of the variousnumber systems is often tedious, especially for a student who does not seethe point of this effort. On the other hand, the end products of this work,the real and complex number systems, are objects of great usefulnessand importance in mathematics. Moreover, the development of thesesystems offers an opportunity to exhibit a wide variety of mathematicaltechniques and ideas, so that the student is exposed to a representativecross section of mathematics.

I t is customary in technical books to te11 the reader what he will needto know in order to understand the text. A typical description of suchrequirements in mathematical textbooks runs as follows: "This book hasno particular prerequisites. However, the reader will need a certain amountof mathematical maturity. " Usually such a statement means that the bookis written for graduate students and seasoned mathematicians. Our pre-requisites for understanding this book are more modest. The reader shouldhave successfully completed two years of high-school algebra and a yearof geometry. The geometry, although not an absolute prerequisite, willbe very helpful. For certain topics in the chapters on the complex numbersand the theory of equations, a knowledge of the rudiments of trigonometryis assumed. We do not expect that the reader will have much "mathe-

1


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2 INTRODUCTION

matical maturity." Indeed, one of the main purposes of this book is toput the reader in touch with mature mathematics.Some of the obstacles which a beginning student of mathematics facesseem more formidable than they reafly are. With a little encouragementalmost any intelligent person can become a better mathematician than hewould imagine possible. The purpose of the remainder of this introductionis to provide some encouraging words on a variety of subjects. I t is hopedthat our discussion will smooth the reader's way throughout the book.We suggest that this material be read quickly, then referred to later asit is needed.

The number systems. There are five principal number systems in mathe-matics: the natural numbers: 1, 2, 3, 4, etc.; the integers: 0, 1, -1, 2,-2 , 3, -3, etc.; the rational numbers: 0, 1, -1, +, -+,-$, -3, 3, -3,etc.; the real numbers:O, 1, *, -*, 4,-4, , 3 - $'a,etc.;and thecomplez numbers: 0, 1, i,m, +G, + S etc.

With the possible exception of the complex numbers, each of these sys-tems should be familiar. Indeed, the study of these number systems isthe principal subject of arithmetic courses in elementary school and ofalgebra courses in high school. Of course, the names of these systems maynot be familiar. For exarnple, the integers are sometimes called wholenumbers, and the rational numbers are often referred to as fractions.

In this book the number systems will be considered a t two levels. Onthe one hand, we will assume at least a superficial knowledge of numbers,and use them in examples from the first chapter on. On the other hand,Chapters 3, 4, 6, 7, and 8 each present a critica1 study of one of thesesystems. The reader has two alt,ernatives. He can either skim the materialin these chapters, relying on the knowledge of numbers which he alreadypossesses, or he can study these chapters in detail. The latter road is longerand more tedious, but it leads to a very solid foundation for advancedcourses in mathematical analysis.

Variables. If a single event can be called the beginning of modern mathe-matics, then i t may possibly be the introduction of variables as a syste-matic notational device. This innovation, due largely to the Frenchinathematician Francois Vihta (1540-1603), occurred about 1590. With-out variables, mathematics would not have progressed very far beyondwhat we now think of as its "beginnings."

By using variables it is possible to express complicated properties ofnumbers in a very simple way. Basic laws of operation, such as

z + y = y + z and x + ( y+ z ) = ( x - k y ) i - 2 ,can be stated without using the variables x, y, and z, but the resultingstatements lack the clarity of these algebraic identities. For example,


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the statement that "the product of a number by the sum of two otherilumbers is equal to the sum of the product of the first number by thesecoild with the product of the first number by the third" is more simplyand clearly-expressed by the identity

More complicated laws would be almost impossible to stat*ewithout usingvariables. The reader who doubts this should try to express in words therelatively simple identity

The variables which are encountered in high-school algebra coursesusually range over systems of numbers; that is, it is intended that thesevariables stand for real numbers, rational numbers, or perhaps only forintegers. However, variable symbols are often useful in other contexts. Forexample, the symbols 1and m in the statement "if 1and m are two differentnonparallel lines, then 1 and m have exactly one point in common" arevariables, representing arbitrary lines in a plane. In this book variableswill be used to denote many kinds of objects. However, in al1 cases, avariable is a symbol which represents an unspecified member of some defi-nite collection of objects, such as numbers, points, or lines. The givencollection is called the range of the variable, and a particular object in therange is called a value of the variable.

The notations used for variables in mathematical literature often puzzlestudents. In the simplest cases, the letters of the alphabet are used asvariable symbols. However, some mathematical statements involve avery large number of variables, and in some cases, even infinitely many.To accommodate the need for many variables, letter symbols with sub-scripts are usually employed, for example, xl, x2, x7, y3, 215, a2, b7, etc.Sometimes double subscripts are more convenient than single ones. Thus,we find expressions such as x l , ~ , 7 , ~ ~2~,52,tc. Variable symbols areoften used to denote a subscript on a variable letter. For instance xi, y,,ak, zi,j, etc. In these cases, the variable subscript is usually assumed tostand for a natural number, or possibly an integer.

Mathematical language. One of the difficulties in learning mathematicsis the language barrier. Not only must the student master many new con-cepts and the riames of these concepts, but he must also learn numerousabbreviations and symbols for common words. Except for the use ofabbreviations, the grammar of mathematics is the same as that of thelanguage in which it is written.


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4 INTRODUCTION

A sentence in mathematical writing is any expression which is a mean-ingful assertion, either true or false. According to this definition, suchformulas as

1 + 1 = 2 , 2 . ' - - 2,1 < 3 - 2 , and 0 = 0must beexample,5 7 ~ 5 ~

counted as sentences. Sentences may contain variables. Porthe statement "There is a real number x such that xlOO-

25x7+ 500 = O" is an assertion which is either true or false,although i t is not obvious which is the case.*

There are other expressions of importance in mathematics which can-not be called sentences. These are formulas, such as

,x+ zj= 1, x2+ 2x + 1 = 0, and x > 2,and expressions which have the form of sentences, except that variablesoccur in place of the subject or object; for example, "x is an integer"and "2 divides n." Expressions such as these are called sentential func-tions. They have the property that substituting numerical values (orwhatever objects the variables represent) for the variables converts theminto sentences. For instance, by suitable substitutions for x and y, thesentential function x + y = 1 is transformed into the sentences

I t makes no sense to ask whether or not a formula such as x + y = 1 istrue. For some values of x and y it is true; for others it is not. On theother hand, the formula x + y = y + x has the property that everysubstitution of numbers for x and y leads to a true sentence. Such asentential function is usually called an identi ty. Sentential functionswhich are not formulas may also have the property of being true for al1values of the variables occurring in them. For example, the statement"either x < y, or y < x" has this property of universal validity, providedthat it is understood that x and y are variables which range over realnumbers. A sentential function which is true for al1 values of the vari-ables in it is said to be identically true or identically valid (the adjective"identically 7 is sometimes omitted).

Implications. Many beginning students of mathematics have troubleunderstanding the idea of logical implication. As many as one-half of al1statements in a mathematical proof may be implications, that is, of theform " p implies q," where p and q are sentences or sentential functions.

- -* It is true.


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I STRODUCT ION 5

1 p irnplies q 1 x pos i t ive implies th a t x i s nonnrgat iveq i s impl icd b y pIf p, then qq i f Pp only if q

x noni icgativc is impl icd b y x bcing posit iveif x is posit ive, then x is nonnegativex i s non negat ive if x is posit ivex is posit ive only if x is nonnegative1 only if q is p 1 only if x i s nonnepa t ive i s x posit ive

p i s a sufricient condition

q i s a necessary condit ion x nonnegative is a necessary condit ion forl x to be pos i tivex posit ive is a sufficicnt condition for xt o b c n o n n eg a t iv e

fo r q i t i s s u f ic i e n t t h a t p

for p i t i s necessary t h a t p 1 for x to be pos i t ive , i t i s n r i es sary tha ti z be nonncgat iveI

fo r x to be norinegative, i t is suff icientt h a t x be posit ive

Icor this reasoii, it is important to be able to recogiiize a11 implicatioii,and to undcrstand what it means.

Thc variety of ways in which mathcmaticians say "p implies q" is ofteribcwildcring to studeiits. The expressions "x = 1 iniplics x is an iiiteger";"if x = 1, thcn x is an integer "; (E = 1 orily if x is an integcr "; ".r = 1is a siificierit conditioii for x to be an iiitcgcr "; aiid "for .-c to eyual 1, it isneccssary that x be ari iiiteger" al1 have thc samc mcaiiiilg. Such state-ments as thcsc occur rcpeatcdly in aiiy book or paper oii mathcmatics.For thc readcr's coilveiiience, TVC list in Table 1 some of the forms in which"p implies q " may be written, togcthcr with cxamples of thesc locutions.

If p and q are both sciltences, then the implicatioii "p implics q " is asentence; if cither p, or q, or both p arid q are sciitential functions, theii"p implies q" is a sentential fuiictioii. Iii case " p implies q " is a seiitence,then its truth is completely dctcrmiiicd by the truth or falsity of p aiid q.Specifically, this implicatioii is truc cither if p is false or q is true. I t isfalse only if p is true and q is false. I;or cxample, "3 = 3 implies 1 < 3"is triie, "3 = 2 implies 1 < 2" is truc, " 3 = 1 implies 1 < 1 " is truc,but "3 = 3 implics 1 < 1" is falsc.

I t may seem strange to coiisidcr a scn tcilce "p implies q" to bc true eveiithough there is iio apparent conilectioil 1)etwceii p aiid q. The idea whichthe statement "p implies q" 11s~a11y O ~ VCY S s that thc validity of the


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INTRODUCTION

TABLE2

sentence q is somehow a consequence of the truth of p. I t is hard to seehow the truth of such an implication as "3 = 1 implies 1 < 1" fits thisconception. Our convention concerning the truth of an implication be-comes more understandable when we consider how a sentence of the form"p implies q" may be obtained from a sentential function by substitutionof numerical values for the variables. For example, the implication"y + 2 = x implies y < x" is a sentential function which everyone wouldagree is identically valid. That is, it is true for al1 values of x and y.However, by substituting 1 for x and 1 for y, we obtain the sentence"3 = 1 implies 1 < 1," whose truth was previously admitted only withreluctance.

Converse and equivalence. From any statements p and q it is possible toform two different implications, "p implies q" and "q implies p." Eachof these implications is called the converse of the other.

An implication does not ordinarily have the same meaning as its con-verse. For example, the converse of the statement "if n > O, then n2 > 0 "is the implication "if n 2 > O, then n > O." These assertions obviouslyhave different meanings. In fact, the first statement is identically true,whereas the second statement is not true for al1 n ; for example, ( - I ) ~=1 > O and -1 < O. If the implication "p implies q" and its converse"q implies p" are both true, then the statements p and q are said to beequivalent. In practice, the notion of equivalence of p and q is most fre-quently applied when p and q are sentential functions. For example, ifx and y are variables which range over numbers, then the formulas x = yand x + 1 = y + 1 are equivalent, since "x = y implies x + 1 = y + 1"and "x + 1 = y + 1 implies x = y" are identically valid.

There are various ways of saying that two statements p and q areequivalent. Most of these forms are derived from the terminology forimplications. Severa1 examples are given in Table 2.

Formp is equivalent to qp if a n d only if qp is a necessary and suffi-cient condition for qp implies q, and conversely

Examplex = y is equivalent t o x + 1 = y + 1x = y if an d onl y if x + 1 = y + 1x = y is a necessary and sufficient con-dition for x + 1 = y + 1x = y implies x + 1 = y + 1, and con-versely


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INTRODUCTION

TABLE

Contrapositive and inverse. In addition to the implication "p implies q"and its converse "q implies p," two other implications can be forrned usingp and q. These are "not q implies not p" and "not p implies not q." Theimplication "not q implies not p " is called the contrapositive of "p impliesq," while "not p implies not q" is called the inverse of ('p implies q." Forexample, the contrapositive of the statement "if x = 1, then x is an integer "is the implication "if x is not an integer, then x is not equal to l ."

I t is easy to see that the contrapositive of "p implies q" is true underexactly the same circumstances that this implication is itself true. Themost convincing way to demonstrate this fact is to make a table listing al1of the possible combinations of t ruth values of any two sentences p and q,together with the corresponding truth or falsity of " p implies q" and itscontrapositive (Table 3). The entries in the fifth column of Table 3 aredetermined by the combinations of true and false in the first two columns,while the entries of the last column are determined from the combinationswhich occur in the third and fourth columns. Of course, the entries of thethird column are just the opposite of those in the first column, and a similarrelation exists between the fourth and second columns.The fact that an implication is logically the same as it s contrapositive isoften very useful in mathematical proofs. Sometimes, rather than provinga statement of the form "p implies q," it is easier to prove the contra-positive "not q implies not p." This is logically acceptable. Also, if wewish to prove that p and q are equivalent, that is, " p implies q" and "qimplies p " are valid, i t is permissible to establish that "p implies q" and"not p implies not q. " This is because "not p implies not q" is the contra-positive of "q implies p. " However, beware ; it is not correct to claim thatif p implies q and not q implies not p, then p is equivalent to q.DeJinitions. Simple mathematical proofs often consist of nothing morethan showing that the conditions of some definition are satisfied. Kever-theless, beginning students frequently find such arguments difficult tounderstand. Consider, for example, the problem of showing that 222 is

P

truetruefalsefalse

Qtruefalsetruefalse

not p

falsefalsetruetrue

not q

falsetruef alsetrue

p implies q

truefalsetruetrue

not q implies not ptruefalsetruetrue


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INTRODUCTION

From this we can infer such formulas as

The logical structure of a mathematical proof may have one of two forms.The direct proof starts from certain axioms or definitions, and proceeds byapplication of logical rules to the required conclusion. The second method,the so-called indirect proof, is perhaps less familiar, even though it is oftenused unconsciously in everyday thinking. The indirect proof begins byassuming "hat the statement to be proved is false. Then, using this assump-tion, together with the appropriate axioms and definitions, a contradictionof some kind is obtained by means of a logical argument. From this con-tradiction it is inferred that the statement originally assumed to be falsemust actually be true.

For example, let us show by an indirect proof that there is no largestnatural iiumber. This proof uses three general properties of numbers,which, for our purposes can be considered as axioms:

(a) if n is a natural iiumber, theii n + 1 is a natural iiumber;(b) n < n + 1;(c) if n < m, then n 2 m is impossible.Our indirect proof begins with the assumption that the statement to be

proved is false, that is, we assume that there is a largest natural number.Let this number be deiioted by n. To say that n is the largest naturalnumber means two things:

(i) n is a natural number;(ii) if m is a natural number, then n > m.

Applying the rule of detachment to (a) aiid (i) gives(iii) n + 1 is a natural number.

Substituting n + 1 for m in (ii), we obtain(iv) if n + 1 is a natural number, then n > n + 1.The rule of detachment can nolv be applied to (iii) and (iv) to conclude that

(v) n 2 n + 1.However, substituting n + 1 for m iii (c) gives

(vi) if n < n + 1, then n > n + 1 is impossible.This, together with (b) and the rule of detachmeiit yields

(vii) n 2 n + 1 is impossible.The statements (v) and (vii) provide the contradiction which completesthis typical indirect proof.In spite of the elementary character of the logic used by mathematicians,it is a matter of experience that understanding proofs is the most difficultaspect of mathematics. Most people, mathematiciaiis included, must workhard to follow a difficult proof. The statements follow each other relent-


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10 INTRODUCTION

lessly. Each step requires logical justification, which may not be easy tofind. The result of this labor is only the beginning. After the step-by-stepcorrectness of t,he argument has been checked, it is necessary to go on andfind the mathematical ideas behind the proof. Truly, real mathematics isnot easy.


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CHAPTER 1SET THEORY

1-1 Sets. The notion of a set enters into al1 branches of modern mathe-matics. Algebra, analysis, and geometry borrow freely from elementaryset theory and its terminology. Indeed, al1 of mathematics can be foundedon the theory of sets. As is to be expected, an idea with such a wide rangeof application is quite simple, and any intelligent person can learn enoughabout set theory for most useful applications of the subject.

The central idea of set theory is that of dealing with a collection ofobjects as an individual thing. Mathematics is not alone in using thisidea, and many occurrences of it are found in everyday experience. Thus,for example, one speaks of the Smith family, meaning the collection ofpeople consisting of John Smith, his wife Mary, and their son William.Also, if we referred to Mrs. Smith's wardrobe, we would be treating as asingle thing the collection of individual pieces of clothing belonging toMrs. Smith. The mathematical use of this device of lumping thingstogether into a single entity differs from common usage only in the fre-quency and systematic manner of its application.

DEFINITION-1.1. A set is an entity consisting of a collection of objects.*Two sets are considered to be the same if they contain exactly the sameobjects. When this is the case, we say that the sets are equal. The objectsbelonging to a set are called the elements of the set.A set is usually determined by some property which the elements of the

set have in common. In the example given above, the property of being a.piece of clothing belonging to Mrs. Smith defines the set which we cal1Mrs. Smith's wardrobe. I t should be emphasized that in thinking of acollection of objects as a set, no account is takeil of the arrangement ofthe objects or any relations between them. Thus, for example, a deck of

* This statement cannot be considered as a mathematical definition of theterni "set." In mathematics, a definition is supposed to completely identifythe object being defined. Here we have only supplied the synonym "collection"for the less familiar term "set," The problem of finding a satisfactory mathe-matical definition is far more difficult than i t might seem. The uncritical use ofsets can lead to contradictions which are avoided only by imposing restrictionson the naive concept of a set, Finding a definition of "set" which is free from con-tradictions and which satisfies al1 mathematical needs has for 75 years been acentral problem of the logical foundations of mathematics. Fortunately, thesedifficult aspects of set theory can be ignored in almost al1 mathematical applica-tions of the theory.

11


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12 SET THEORY [CHAP.152 cards, considered as a set, remains the same whether it is in its originalpackage or is shuffled and distributed into four bridge hands.

EXAMPLE 1. The set consisting of the numbers O and 1.EXAMPLE2. The set of numbers which are roots of the equation x2 - = 0.EXAMPLE3. The set of numbers which are roots of the equation x

x 2 = o.EXAMPLE4. The set of numbers a on the real line (Fig. 1-1) which satisfy-1 5 a 5 l .

EXAMPLE5. The set of al1 numbers x/2, where x is a real number whichsatisfies -2 5 x 5 2.EXAMPLE 6. The set of al1 points a t a distance less than one from a point

p in some plane.EXAMPLE7. The set of al1 points inside a circle of radius one with center

a t the point p in the plane of the preceding example.EXAMPLE8. The set of al1 circles with center at the point p in the plane ofExample 6.EXAMPLE 9. The set consisting of he single number O.EXAMPLE 0. The set which contains no objects whatsoever.

According to our definition of equality of sets, we see that the setsof Examples 1, 2, and 3 are the same. Although O occurs as a so-calleddouble root of the equation x 3 - 2 = O in Exaniple 3, only its presenceor absence matters when speaking of the set of roots. The sets of Exanlples4 and 5 are the same, as are those of Examples 6 arid 7. If we considera circle to be the same thing as the set of al1 of it s poiiits, then the eleineiitsof the set of Example 8 are themselves sets. Sets of sets will be studiedmore thoroughly in Section 1-5. The set described in Example 9 con-tains a single element. Such sets are quite common. I t is conventionalto regard such a set as an eiltity which is different from the eleinent whichis its only member. Even in ordinary conversation this distinctioii isoften made. If Robert Brown is a bachelor with no known rclsttions, thenwe would say that the Brown family consists of one meniber, but we wouldnot say that Mr. Brown consists of oiie nlember. The reader may feelthat the set of Example 10 does not satisfy the description given in


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1-11 SETS 13Definition 1-1.1. However, it is customary in mathcmatics to interpret theterm "collcction" i11 such a ivay that this ~iotionncludes the collection ofno objects. Actiially, thc sct containing no elements arises quite naturallyin many situations. For instante, in consideriiig thc scts of real numberswhich are roots of algebraic equatioris, it ~voilldbe awkjvard to makc aspecial rase for equations like x2 + 1 = O, ivhich has no real roots.13ecause of it s importailcc, thc set coiitaiilirig no elemcilts has a specialnamc, thc empty set, and it is rcprcscntcd by a special symbol, a. Whenit is neccssary to cal1 atteiltioii to the fuct that a set A is not the emptyset, then ve will say that A is nonempty.

Oiie reason that set theory is used in so many brailches of mathcmaticsis thc versatility of its notation. As aiiyoiie ~ vho as studied elcmentaryalgebra might expect, thc, letters of the alphabet are used to rcpresent sets.111 this book, sets will be represeiitcd by capital letters, and the elementsof sets ~ i l lsually bc represeiitcd by small letters.

Thc statemeiit tha t an object a is an elemeiit of a set A is symbolized by

We rcad thc cxpression a E A as "a is in A," or sometimes, "a in A."To give a specific example, lct, A be t,he set of roots of the equation

x2 - x = O (Example 2). TheiiIVe often wish to cxpress the fact t,hat an object is not contained in

a certaiii set. If a is not an element of t,he set A, \ve writc

and rcad this cxpressioii "a is iiot in A . " Thus if A again is the set ofExamplc 2, wc would havc 2 4 A, 3 4 A, 4 4 A, etc.

I t \vas mentioned earlier that a set is ofteri dcfincd by some propertypossessed by its elements. There is a very uscful ilotatioiial device iri settheory which gives a standard method of symbolizing thc sct of al1 objectshaving a certain property. For instaiice, the sets of Kxamples 2 and 4are respcct ivcly writtcn

{ 2 - ~ z 2 - x = 0 ] , and { a l - l 5 a < l ) .The symbolic form (* I * ] is sometimcs called the set builder. In using it,we replace ttie first asterisk by a variable elemcnt symbol (x aild a in theexamples), aiid the second astcrisk is rcplaced by a meaningful conditionwhich the object represented by thc variable miist satisfy to be an elementof the set (x2 - : = O and - 5 a 5 1 in the examplcs). Thus, the


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SET THEORY

set builder occurs in the form(xlcondition on x)

(or with some variable other than x), and this expression represents theset of al1 elements which satisfy the stated condition. Often, the totalityof possible objects for which the variable stands is evident from the con-dition required of the variable. For example, if the real roots of algebraicequations are under discussion, then in (x/x2- = O ), it is clear thatx stands for a real number, and that the set consists of the real numberswhich satisfy x2 - x = O In {al-1 5 a 5 11, it may not be clearwhat kind of numbers are allowed as values of the variable. If it is neces-sary to be more explicit, we would write

where R is the set of al1 real numbers. Similarly, in Example 6, the setbuilder notation would be

where P is the set of al1 points in some plane and d(p, q) is the distancebetween points p and q in the plane. Here, the variable q can take as itsvalue any point in the plane P, and the set in question consists of thosepoints in P which satisfy d(p, q) < l . Other forms of the set buildernotation for this example are

I t is often convenient to use general symbols or expressions in place ofthe variable element in the set builder notation. For example, (x21x E R) ,where R is the set of al1 real numbers, is the set of al1 squares of realnumbers; (x/ylx E N, y E N), where N is the set of al1 natural numbers,is the set of al1 positive fractions.A variation of the set builder notation can be used to denote sets whichcontain only a few elements. This coilsists of listing al1 of the elementsof the set between braces. For exalnple, the sets of Examples 1 and 8would be written

(0,l) and (01,respectively. I t is sometimes convenient to repeat the same element one ormore times in the notation for the set. Thus, in Example 3, we mightfirst write the set of roots of x3 - r2= x x (x - 1) = O as (O, 0, 1),since O is a double root. Of course, by the definition of equality, {OJO,1) =0 l . The notation (0, O, 1) conveys no more iiiformation about


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1-11 SETS 15xs - 2 = O th a n {O, 1) doe s. Sim ilarly, (a, b, a , a , b) represeiits th esame se t a s {a , b ) .There is aiiother good reason for allowing repetitioii of one or moree lements in the no ta t ion fo r a se t . Cons ider th e exam ple {a , b] . Wc canthin k of th is as the se t whose members ar e th e le t tcrs a and b. 111 mathe-matical applications, however, i t is often coiivenient to regard {a, b ) a sth e sct containing variable quan ti t ies a an d b. As s i ich, i t would becomea specific sct if particular values where substituted for a arid b. E'orexample, if we allow natural numbers to be substi tuted for a a n d b, t he neach choice of valucs for a an d b determ ines a se t whose memb ers are thesese lec ted na tura l r iumbers . In th i s example, a and b m ay tak e on the samcvalue . For ins ta i~c e , f a = 1 , b = 1 , t h e i ~ a , b) = (1, 1)'= (1). Ifwe did n ot al low repeti t ion of th e elements in designsting sets , th e collec-t ion of se ts (a , b) dctermin ed b y sub sti tutin g values for a an d b would beconsidcrably m ore difficult t o describe. T h is difficiilty would be increasediii mo re com plicated examples.Scts conta ining ma ny e lements can of ten be represented by l is ting someof t h e elem ents between braces aild using a sequence of d ot s to ind ica teom it ted e lemci its. l ior examplc, i t i s c lear th a t

{1, 2, 3 , . . . , 2165)represents th c set of al1 na tu ral nu mb ers from 1 t o 2165. Some infinitesets caii also he represented in this way . Fo r example,

deno tes the s et of al1 na tur al i ium bers .DE FISITIO S 1-1.2. A se t A is called a subset of th e set R (or A is in-cluded in R) if every element of A is a n e lemen t of B. I t i s c u s tom a ryto express the fac t tha t . A is a sub set of R by w riting A 2 B or B 2 A .Aily set A is a subset of itself, A A , according to this definit ion.If A 2 B, b u t A # R ( th a t i s, A is no t t h c s a m e a s t he s e t R ) , the n A iscalled a proper subset of B a n d iii th i s ca se ~ v c r it c A C R or B > A . If

A is iiot a subset of B, ~ v c ritc A g R or B 2 A .

EXAMPLEl . T h e se t of al1 even integ ers, 23 = (0, &2, &4, . . .), is a propersubsc t of th e sct Z of al1 in tcge rs .EXAMPLE2. T h e set of a11 poirits a t distance Iess th an one from a point pin a plane P is a proper subset of the set of points of P a t d is tance less tha n orequa l to o iie f r o n ~ .


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16 SET THEORYEXAMPLE3. {O, 1) C O, 1, 2, 4).EXAMPLE4. (a10 < a 5 1) C {a10 5 a 5 1 ) .EXAMPLE5. Q, E A for every set A .

The reader should carefully check to see that in each of these examplesthe condition of Definition 1-1.2 is satisfied. The fact that Q, is a subset ofevery set may seem straiige. However, it is certainly true according toour definitions: every element of is an element of A , or in other words,no element of can be found which is not in A . Since Q, has no elements,this condition is certainly satisfied.

The inclusion relation has three properties which, although direct con-sequences of our definition, are quite important. The first of these hasalready been noted.

THEOREM-1.3. For any sets A, B, and C,(a) A E A ,(b) if A c and B 5 A, then A = B,(c) if A E B and B S C, then A E C.

Proof. We will prove property (b) in detail, leaving the proof of (c) tothe reader. If A c and B E A , then every element of A is an element ofB and every element of B is an element of A. That is, A and B containexactly the same elements. Thus, by Definition 1-1.1, A = B.

Certain sets occur so frequently in mathematical work that it is con-venient to use particular symbols to designate them throughout anymathematical paper or book. An example is the practice of denoting theempty set by the symbol a. In this book, the number systems of mathe-matics, considered as sets, will occur repeatedly. We therefore adopt thef ollowing conventions :

N designates the set of al1 natural numbers : (1, 2, 3, . . .) ;Z designates the set of al1 integers: {. . . , -3, -2, -1 ,0 , 1, 2, 3, . . .);Q designates the set of al1 rational numbers: (albja E 2, b E N) ;R designates the set of al1 real numbers.

This notation, though not universal, would be recognized by most modernmathematicians. Throughout this book, the letters N, 2 , Q, and R willnot be used to denote any set other than the corresponding ones listedabove.

In mathematical literature, a considerable amount of variation in nota-tion can be found. The terminology and symbolism introduced in thissection will be used in the remainder of this book, but i t is by no means


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1-11 SETS 17universal. Fo r th e reader's coiivenience, \ve list some common alternat ivetermii~ology

Set : class, ensemble, aggregate, collect ion.Element of a set : member of a se t, poiiit of a se t.Empt y set : void set, vaciious set, null set, zero set.CP: o, A.a ~ A : A 3 a .a 4 A : ~ E ' A , ~ E A , A ~ ~ .(*]*) : (* : *), [*)*],[* : *l.

l . Using the set builder form, write expressions for the follon-ing sets.(a) The set of al1 even integers.(b) The set of al1 integers which are divisible by five.(c) The set of al1 integers which leave a remainder of one mhen divided

by five.(d) The set of al1 rational numbers greater than five.(e) The set of al1 points in space which are inside a sphere with center a t

the point p and radius r .(f) The set of solutions of the equation x3 - x2 - + 2 = 0.2. Te11 in words what sets are represented by the following expressions.(a) {x E NIx > 10)(b) {x E QIX- 3 E N)( 4 ( 5 , 6 , 7, . . -1(4 {a, b, , Y, 2 )(e) { X ~ X y2 + z2, y E R, z E R, x2 - 2 = (x - ) (x + y))

3. Describe the following sets by listing their elements.(a) {xIx2 = 1)(b) {x1x2- 2x = 0)(c) {x1x2- x + 1 = O)

4. List the following collections of sets.(a) The sets {a, b, c), where a, b, and c are natural numbers less than or

equal to 3.(b) The sets {a2 + a + 1), where a is a natural number less than or

equal to 5 .(c) The three element sets (a, b, c), where a, b, and c are integers between

-2 and 4.5. State al1 inclusion relations which exist between the folloing sets: N, 2,

Q, R, the set of al1 even integers, {n(n = m2, m E Z), {zjx = y2, y E Q),{ X ~ X = y2 - , y E R, n E 2 ) .

6. Prove Theorem 1-1.3(c).7. Prove that if A B, B C C, then A C C, and if A C B and B G C,then A c C.


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18 SET THEORY [CHAP. 11-2 The cardinal number of a set. The simplest and most important

classification of sets is given by the distinction between finite and infinitesets. Returning to the examples considered in Section 1-1, the set ofExamples 1, 2, and 3, and the sets of Examples 8 and 9 are finite, while thesets of Examples 4 through 7 are infinite. Note that the empty set @ isconsidered to be finite. I t is not altogether easy to explain the differencebetween a finite and an infinite set, although almost everyone with someexperieiice learns to distinguish finite from infinite sets. Roughly speaking,a finite set is either the empty set, or a set in which we can designate afirst element, a second element, a third element, and so on, until at somestage we reach an nth element and find that there are no more left. Ofcourse, the number n of elements in the set may be one, two, three, four,. . . ,a million, or any natural number whatsoever.A set is said to be infinite if it is not finite, that is, if its elements cannotbe counted. Examples of infinite sets are the set N of al1 natural numbers,the set Z of al1 integers, the set Q of al1 rational numbers, the set R of al1real numbers, etc. These examples show that some of the most importantsets encountered in mathematics are infinite.If A is a finite set, it is meaningful to speak of the number of elementsof A. This number is called the cardinal number (or cardinality) of the setA. We will use the notation iA to designate the cardinal number of A .As examples,

0 , a = l(O,1,4)1=3.I t was remarked above that the empty set 4> is regarded as being finite:Since @ has no elements, it is natural to say that the cardinality of (P iszero. Thus, in symbols, ( @ l = 0.

There are many synonyms for the expression "cardinal number ofA." Besides the term "cardinality of A," which we have already men-tioned, one finds such expressions as "power of A " and "potency of A. "The notation IA( for the cardinal number of the set A is not universaleither. The symbolism A s perhaps even more common (but difficultto print and type), and such expressions as card A or N(A) can alsobe found.

These descriptions of finite and infinite sets, and of the cardinal numberof a finite set are too vague to be called mathematical definitions. More-over, we have not said anything about the cardinal numbers of sets whichare not finite. The first man to systematically study the cardinal numberconcept for arbitrary sets (both finite and infinite) was Georg Cantor(1845-1918). His researches have had a profound influence on al1 aspectsof modern mathematics. In the remainder of this section, we will examineone of Cantor's most important ideas, and see in particular how it enablesas to explain the concept of a finite set in more exact terms.


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1-21 THE CARDINAL KUMBER OF A SET 19DEFINITION-2.1. A pairing between the elements of two sets A and Bsuch th at each element of A is matched with exactly one element of B,and each element of B is matched with exactly one element of A , iscalled a one-to-one correspondence between A and B.The reader should study the following examples to be sure that he fully

understands the meaning of the fundamental concept defined in Defini-tion 1-2.1.

EXAMPLE . Let A = (1, 2, 31, and B = (a, b, e] . Then there are six pos-sible one-to-one correspondences between -4 and B:

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 35 5 5 5 5 5 5 5 5 5 5 5 5 5a b c b c a c a b b a c a c b c b a

EXAMPLE . It is impossible to obtain a one-to-one correspondence betweenthe set A = (1, 2, 3) and the set B = (1, 2). No matter how we tr y to pairoff the elements of ,4 and B, we find that more than one element of A mustcorrespond to a single element of B. If 1t, 1 and 2 t, 2, then 3 must correspondto 1 or 2 in B. In the correspondence 1 t, 1, 2 t, 2, 3 1, the element 1 E Bis paired with both 1 E A and 3 E A, so that the correspondence is not one-to-one. A similar situation occurs in al1 possible correspondences between A and B.The more general fact that there is no one-to-one correspondence between(1, 2, 3, . . . ,m) and (1, 2, 3, . . . ,n) if m < n is also true. This can be provedusing the properties of the natural numbers, which will be discussed in the nexttwo chapters.

EXAMPLE. There is a one-to-one correspondence bet.c~-een he set Z =(. . . , -3, -2, -1, 0, 1, 2, 3, . . .) of al1 integers and the set N = (1, 2, 3, . . )of al1 natural numbers. The elements of Z and N can be paired off as follows:

Note that in order to construct a one-to-one correspondence between Z and N,no t al1 of the numbers of N can be paired with themselves. Otherwise, we woulduse up al1 of N and have nothing left to associate with O, -1, -2, . . . .

Usiiig Defiiiitioii 1-2.1, we caii clarify the notioil of a fiiiite set.DEFINITION-2.2. Let A be a set and let n be a natural number. Thenthe cardinal iliimber of A is n if there is a one-to-oiie correspondencebetween A and the set (1, 2, 3, . . . ,n), consisting of the first n natural


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20 SET THEORY [CHAP. 1numbers. A set A is Jinite if A = @, or there is a natural number nsuch that the cardinal number of A is n. Otherwise A is called in$nite.This definition is no more than a careful restatement of the informaldescriptions of finite and infinite sets, and their cardinal numbers, which

were given at the beginning of this chapter.The usual practice of writing a finite set in the form

{al, az, a3, ,a,) (without repetition)exhibits the one-to-one correspondence between A and (1, 2, 3, . . . ,n),namely, a l a2 a3 . . . a ,

Cantor observed that it is possible to say when two sets A and B havethe same number of elements, without referring to the exact number ofelements in A and B. This idea is illustrated in the following example.Suppose that in a certain mathematics class, every chair in the room isoccupied and no students are standing. Then without counting the numberof students and the number of chairs, it can be asserted that the numberof students in the class is the same as the number of chairs in the room.The reason is obvious; there is a one-to-one correspondence between theset of al1 students in the class and the set of al1 chairs in the room.

DEFINITION-3.3. Two sets A and B are said to have the same cardinalnumber, or the same cardinality, or to be equivalent if there exists aone-to-one correspondence between A and B.By Example 1, the two sets {1,2,3) and {a, b, c ) have the same cardi-

nality. By Example 3, so do the sets N and 2. However, according toExample 2, the sets {1,2, 3) and (1,2) do not have the same cardinalnumber .

In accordance with Definition 1-2.3, the existence of any one-to-onecorrespondence between A and B is enough to guarantee that A and Bhave the same cardinal number. As in Example 1, there may be manyone-to-one correspondences between A and B.

EXAMPLE. Every set *4 is equivalent to itself, since a ++ a for a E A isobviously a one-to-one correspondence of A with itself. If A contains more thanone element, then there are other ways of defining a one-to-one correspondenceof A with itself. For example, let L4 = (1, 2). Then there are two one-to-onecorrespondences of A with itself: 1* , 2 t,2 and 1* , 2o 1. Any one-

, to-one correspondence of a set with itself is called a permutation of the set.


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1-21 THE CARDINAL KUMBER OF A SET 21

If A = {al, a2, . . . ,a,) and B = {bl, b2, . . . , b,) are finite sets whichboth have the cardinal number n, then there is a one-to-one correspondencebetween A and B:

so that A and B are equivalent in the sense of Definition 1-2.3. That is,if A and B are finite sets, then A and B are equivalent if 1A = 1BI.

The important fact to observe about Definition 1-2.3 is that it applies toinfinite as well as finite sets. One of Cantor's most remarkable discoverieswas that infinite sets can have different magnitudes, that is, in some sense,certain infinite sets are "bigger than" others. To appreciate this factrequires some work. Example 3 has already illustrated the fact that in-finite sets which seem to have different magnitudes may in fact have thesame cardinality. An even more striking example of this phenomenon isthe following one.

EXAMPLE. The set iV of natural numbers has the same cardinality as theset

F = (m/nlm E N, n E N)of al1 positive rational numbers. This can be seen with the aid of a dia.gramas in Fig. 1-2. By following the indicated path, each fraction will eventuallybe passed. If we number the fractions in the order tha t they are encountered,skipping fractions like S, 2, S, and 2, hich are equal to numbers which have


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22 SET THEORY [CHAP. 1been ~ re v io u s ly assed, 1%-e et th e desired one-to-one correspondence betw eenN and F:

Cantor showed that many important collections of numbers have the samecardinali ty as th e set N of al1 natural numbers. For example, this is true forthe se t d of al1 real algebraic num bers , th a t is, real nu m be rs r which are solu-tions of an equation of the form

where no, ni, . . . , nk-1, n k are an y integers. Th e set A includes al1 ratio nalnumbers, since m/n is a root of the equation nx - m = O; A also includesnum bers l ike d2,%)4 3 ,a , . . .

Judging from these examples, one might guess that al1 infinite setshave the same cardinality. But this is not the case. Cantor proved that i tis impossible to give a pairing between N and the set R of al1 real numbers.Later, we will be able to present Cantor's proof that these sets do not havethe same cardinal numbers. The fact that the set A of real algebraicnumbers has the same cardinality as N and the result that R and N donot have the same cardinal numbers together imply that R # A . Thatis, there are real numbers which are not solutions of any equation

with no, nl , . . . , nk-l, nk integers. This interesting fact is by no meansevident. I t is fairly hard to exhibit such a real number, but Cantor'sresults immediately imply that they do exist.

Although Cantor's work on the theory of sets was highly successful inmany ways, it raised numerous new and difficult problems. One of theseranks among the three most famous unsolved problems in mathematics(the other two: the Fermat conjecture, which we will describe in Chapter 5 ,and the Riemann hypothesis, urhich is too technical t o explain in thisbook.) Cantor posed the problem of urhether or not there is some set Sof real numbers whose cardinality is different from both the cardinalityof N and the cardinality of R. The conjecture that no such set S exists


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1-21 THE CARDINAL NUMBER OF A SET 23is known as the continuum hypothesis. I t was first suggested in 1878, andto date, it has been neither proved nor disproved.

An infinite set is called denumerable if i t has the same cardinality as theset N of al1 natural numbers. If S is denumerable, then it is possible topair off the elements of S with the numbers 1 ,2 ,3 , . . . . Thus, the elementscan be labeled a l , a2, a3, . . . , where a, is the symbol which stands for theelement corresponding to the number n. Hence, if S is denumerable, thenS can be written {al, a2 ,a3, . . .),with the elements of S listed in the formof a sequence. The converse statement is also true. That is, a set whichcan be designat,ed {al, a2, a3, . . .) is denumerable (or possibly finite,since distinct symbols might represent the same element of S). As we haveshown in this section, the set of al1 integers and the set of al1 positive ra-tional numbers are examples of denumerable sets.

We conclude this section by listing for future reference the followingimportant propert'ies of the equivalence of sets.

(1-2.4). Let A, B, and C be arbitrary sets. Then(a) A is equivalent to A ;(b) if A is equivalent to B, then B is equivalent to A; and(c) if A is equivalent to B and B is equivalent to C, then A is equiva-

lent to C.I t has already been noted in Example 4 that (a) is satisfied. Property

(b) follows from the fact that the definition of a one-to-one correspondenceis symmetric. That is, if A and B are interchanged in Definition 1-2.1, thedefinition says the same thing as before. Thus, a one-to-one correspondencebetween A and B is a one-to-one correspondence between B and A. Theproof of (c) is left as an exercise for the reader (see Problem 8).

l . S ta te which of th e following sets are finite.(a) {(x, Y,z>lxE (0, 1, 2)) Y E (3, 41, 2 E (0, 2,411(b) ( X ~ XZ, x < 5 )(e) {xlx E N, x2- = 0)(d) (x1x E Q , O < x < 1)

2. W ha t is th e cardinal num ber of t h e following finite se ts?(a) {nln E N, n < 1000) (b) {nln E 2,n2 5 86)(e ) (n21n E Z, n2 5 36) (d) (nln E N, n3 5 27)(e) {n31n.E N, n3 < 27)

3. Let A = (1, 2, 3, 4) and B = (a, b, c, d). List al1 one-to-one correspond -ences between -4 a n d B.


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24 SET THEORY [CHAP. 14. Using the method by which we proved tha t the positive rational numbers

have the same cardinality as the set N of natural numbers, indicate how toprove that N has the same cardinal number as the set of al1 pairs (m, n,) ofnatural numbers. List the pairs which correspond to al1 numbers up to 21.5. Prove that the set of al1 rational numbers Q has the same cardinality asthe set of al1 natural numbers N.

6. Let A be the set of al1 positive real numbers x, and let B be the set of al1real numbers y satisfying O < y < l . Show that the pairing x ++ y, wherey = 1/(1 + x) is a one-to-one correspondence between A and B.7. Let A be a denumerable set, and let B be a finite set. Show th at the setS = ((2, y)lx E A, y E B) is denumerable.

8. Suppose that sets A and B have the same cardinality, and that sets Band C have the same cardinality. Show tha t A and C have the same cardinality.

1-3 The construction of sets from given sets. In this section, we willdiscuss two important methods of constructing sets from given sets. Thefirst process combines two sets X and Y to obtain a set called the productof X and Y. The second construction leads from a single set X to anotherset called the power set of X. There are severa1 other methods of buildingsets from given sets, but they will not be considered in this book.

The definition of the product of two sets is based on the concept of anordered pair of elements. Suppose that a and b denote any objects what-soever. If the elements a and b are grouped together in a definite order(a, b) , where a is the first elernent and b is the second element, then theresulting object (a, b) is called an ordered pair of elements. Two orderedpairs are the same if and only if they have the same first element and thesame second element. Thus we arrive a t the following definition."

DEFINITION-3.1. (a, b) = (e, d) if and only if a = c and b = d.

EXAMPLE. Let A = (1, 2, 3). Then the following distinct ordered pairs ofelements of A can be fornied: (1, l), (1, 2)) (1, 3)) (2, l), (2, 2), (2, 3), (3, l),(3, 2), (3, 3). Note that (a, b) = (b, a) only if a = b. By Definition 1-3.1, thisis true in general.

EXAMPLE. A man has tulo pairs of shoes, one brown pair and one blackpair. If he dresses in th e dark, what are the possible combinations of shoes* There is a simple way to define an ordered pair in the framework of set theory,

namely, for objects a and b let (a, b) = ({a, b), a) . An ordered pair is then adefinite object, and i t is possible t o prove t ha t (a, b) = (e, d) if and only ifa = c and b = d. However, we will use the informal description given in thetext and regard this property of ordered pairs as a definition.


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1-31 THE CONSTRUCTION OF SETS FROM GIVEN SETS 25which he can put on? Let X = {left brown shoe, left black shoe) and Y ={right brown shoe, right black shoe). Then the set of al1 possible combinationswhich the man might wear is the set of al1 ordered pairs with the first elementtaken from the set X and the second element taken from the set Y, tha t is, theset of al1 pairs (left brown shoe, right brown shoe), (left brown shoe, right blackshoe), (left black shoe, right brown shoe), (left black shoe, right black shoe).

EXAMPLE . The set of al1 ordered pairs of natural numbers is the setS = ((n, m)(n E N, m E N ) .

Thus,

DEFINITION-3.2. Let X and Y be sets. Then the product of the sets Xand Y is t,he set of al1 ordered pairs (x, y), where x E X and y E Y.The product of X and Y is denoted by X X Y. Thus, in symbols

X x Y = {(x, y)lx E X ) y E Y).EXAMPLE. The ordered pairs listed in Examples 1, 2, and 3 are exactly the

elements of the products A X A, X X Y, and N X N, respectively.EXAMPLE. Let U = (1, 2, 3), V = (1, 31, and PV = (2, 3). Then

u x V = i(1, l) , (1, 3)) (2? 1)) (2, 3), (3, l) , (3, 3)), and V X U = ((1, l),(1, 2), (1, 3), (3, l ), (3, 2), (3, 3)). I t follows tha t U X V f V X U . Thus, informing the product of two sets, the order in which the sets are taken is significant.We also have( U x x = {(O, 0,2), ((1) 3), 2), ((2) 1), 2), ((2, 3), 2)) ((3, 1), 2),((3, 3)) 2)) ((1) l ) , 3), ((1) 3), 3), ((2, 1), 3)) ((2, 3), 3),

((3, l ) , 3), ((3) 3), 3)),

Note that the elements of ( U X V )X TV are different from al1 of the elements ofU X (V X TV). In fact, the elements of ( U X V )X TV are ordered pairs whosefirst element is an ordered pair of numbers, and the second element is a number.In U X ( V X TV) i t is just the other way around: the elements are ordered pairsin which the first element is a number and the second element is an orderedpair of numbers. The reader must be careful to make a distinction between((2, l ) , 3) and (2, (1, 3)), for example.


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26 SET THEORY [CHAP. 1EXAMPLE. If X is any set, then X X @ = cP X X = cP. Indeed, since theempty set contains no element, there cannot be any ordered pair whose firstor second element belongs to the empty set.

Even though U X V # V X U and (U x V) X W # U X (V x W)in Example 5, it is true that U X V is equivalent to V X U and ( U X V)x W is equivalent to U X (V X W), as we see by counting the elementsin each of these sets. It is easy to prove tha t these results hold in general.

THEOREM-3.3. Let X, Y, and Z be sets. Then(a) X X Y is equivalent to Y X X, and(b) (X X Y) X Z is equivalent to X X (Y X 2 ) .

Proof. We will prove (a) and leave the proof of (b) as an exercise forthe reader. According to Definition 1-2.3, we must show that there is aone-to-one correspondence between X X Y and Y X X. Every element ofX X Y is an ordered pair (x, y), with x E X y E Y; every element ofY X X is an ordered pair (z, w), with z E Y and w E X. If (x, y) EX X Y, then (y, x) E Y X X, so tha t (x, y) can be matched with (y, x).The pairing (x, y) - y, x) is the desired one-to-one correspondencebetween X X Y and Y x X.

The definition of the product of two sets can be generalized to a finitecollection of sets X1, X B , . . ,Xn. The product of these sets, deiloted byX1 x Xz X . X X,, is the set of al1 ordered strings of elements

- (xl, x2, . . . ,x,), where xi E Xi for i = 1, 2, . . . ,n. For example, ifn = 3, then

EXAMPLE. Let U, V, and TV be the sets defined in Example 5 , that isU = (1, 2, 31, V = (1, 3), and TV = (2, 3). Then

I t is possible to generalize Theorem 1-3.3 to products of finite collec-tions of sets (see Problems 6, 7, and 8).

We turn now to a second method of obtaining a new set from a givenset X.

DEFIXITION -3.4. Let X be any set. The set of al1 subsets of Xis called the power set of X, and is denoted by P(X).


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1-31 THE CONSTRUCTION OF SETS FROM GIVEN SETS 27Thus, the elements of P(X) are precisely the subsets of X. In particular,E P ( X) a n d X E P(X) .

EXAMPLE. If X = a , then P(X) = (a).EXAMPLE9. If X = (a), then P(X) = (a, (a}).EXAMPLE0. If X = (a, b), then P(X) = (a, (a), (b), (a, b)) .If X is an infinite set, then it has infinitely many distinct subsets.

That is, P(X) is infinite if X is infinite. In fact, if x E X, then {x) E P(X),so that P(X ) contains a t least as many elements as X.

Suppose that X is a finite set. Let X1 be a set which is obtained byadjoining to X a new element a which is not in X. Tha t is, the elementsof X1 are al1 of the elements of X, together with the new element a. Thenevery subset of X1 either does not contain a and is therefore a subsetA of X, or else it contains a and is therefore obtained from a subset A ofX by adjoining the element a to A. Thus, every subset A of X gives riseto two distinct subsets of X1, the set A itself and the set A l obtained byadjoining a to A. Note that al1 of the sets so constructed are different.That is, A # Al, and if A # B, then A # Bl, Al # B, and Al Z B1.Therefore, there are just twice as many subsets of X1 as there are subsetsof X. That is, 1P(X1) = 21P(X) . Starting with the empty set @ (for whichIP(X)I = 1), it is possible to add elements one by one, doubling thecardinality of the resulting power set each time an element is added,until a set X containing n elements is obtained. Our reasoning shows thatthe power set of X will contain 2" elements.

THEOREM-3.5. If 1x1= n, then IP(X)I = 2".There is another way to prove this theorem which is worth examining,

since i t gives additional information about the number of subsets of afinite set. Let X consist of the distinct elements al, a,, . . . , a,. Witheach ak in X , associate a symbol xk, and consider the formal product

If this expression is multiplied out, the result is a sum of distinct productsof x'a (except the first term, which is 1). There is exactly one such prod-uct for every subset {a,,, a,,, . . . ,a,,) of {al, a,, . . . ,a,), namelyx,,x,, . . .x,,. The empty set corresponds to l . For instance, if


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28 SET THEORY

Kow replace each x k by the symbol t. Then the product becomes(1 + t)(l + t) . . . (1 + t), while in its expansion, al1 products correspond-ing to sets containing the same number j of elements become t'. In theexample, X = {ai, a2, as), we obtain (1+ t)3= 1 + + t 4- +t2 + 2 + 2 + 3 = 1 + 3t + 3t2+ t3. As in this example, al1 of theterms t j can be collected into a single expression of the form Nj,,tj, whereNj,, is precisely the number of subsets of X which have cardinality j.Therefore

We can specialize even more by letting t have the value l . Then theidentity (1-1) becomes

The sum 011 the right-hand side of this identity represents the number ofsubsets containing no elements of X (the empty set), plus the number ofsubsets containing one element of X, plus the number of subsets containingtwo elements of X, and so on, until we reach N,,,, the number of subsetscontaining n elements of X. Clearly, this sum is just the total number ofsubset,s of X, since every subset contains some number of elements of Xbetween zero and n. Thus, we have arrived a t the same conclusion asbefore: there are exactly 2" subsets of a set X with n elements.

By using the binomial theorem of algebra (see Section 2-2) to expand(1 + t),, it is possible to squeeze more information from identity (1-1).We getThe coefficient of t j is *

n(n - ) . . . (n - + 1)-- n!j! j!(n - ) !* An exclamation mark (!) following a natural number n denotes the numberobtained by multiplying together al1 the numbers from 1 to n. For example,

l ! = 1, 2! = 1 . 2 = 2, 3! = 1 . 2 . 3 = 6, 4! = 1 - 2 . 3 - 4= 24. I t i s a l s ocustomary to define O! to be 1. With this convention, the formulas for thebinomial coefficients are correct in the cases j = O and j = n.


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1-31 TH E CONSTRUCTION OF SETS FROM GIVEX SETS 29Comparing the identities (1-1) and (1-3) we see that

for al1 numbers t. This leads us to expect that the coefficients of the samepowers of t on each side of the equation must be equal. That is, No,, =1, Ni, , = n, Nz,, = n(n - 1)/2, . . . ,N,,, = 1, and in generalLater we will be able to prove that if

for al1 values of t , then a. = bo, a l = bl , . . . , a , = b,. This will justify(1-4). Thus, our somewhat longer proof of Theorem 1-3.5 yields the in-teresting fact that in a set X containing n distinct elements, there aren!/j!(n - ) !different subsets containing exactly j elements. For example,in a set containing ten elements, there are 10!/4!6! = 210 subsets ofcardinality 4.

1. List the elements of the sets A X B, B X A, ( A X B) X C, and A X(B X C), where A = (x, , 2, w ), B = (1, 2)) and C = ( a ) .2. Prove Theorem 1-3.3(b).3. Let U = (1, 2). Prove tha t U X N is equivalent to N, where N is the

set of al1 natural numbers.4. Prove that if U is a finite nonempty set and V is a denumerable set,

then U X V is equivalent to V.5 . Prove that if U and V are denumerable sets, then the following sets areequivalent: U, V, U X V, V X U.

6. State the generalization of Theorem 1-3.3 for a finite collection of setsX1) X2) ,Xn.7. Prove that U X V X W is equivalent to ( U X V) X TV, where U, V ,and W are arbitrary sets.

8. Prove that the following sets are equivalent: U X V X W, U X W X V ,V X U X W , V X W X u , w x U X V , W X V X U.9. For any set X, define

n factors7-7xn= X X X X . . . X X ,where n is a natural number. Show th at if X = (1, 2), and Y = (1, 2, . . . ,n),then IXnJ = J P ( Y ) I .


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30 SET THEORY [CHAP. 110. List the elements of P (X ), where X = {1,2, 3, 4).11. Let X be a set with 7 objects.(a) How many subsets of X of cardinality a t most three are there?

(b) How ma ny subsets of X of cardin ality a t least three are there ?12. (a) Let t = -1 in equa tion (1-1) an d inte rpre t the meaning of the result-ing ident i ty .(b) W h at is th e numb er of subse ts of ev en card ina lity of a set containingn elements?

13. Show that i f the sets A and B have the same cardinal i ty , then so doP ( A ) a n d P ( B ) .

14. Can tor proved t h a t if X is an infinite set, the n X and P (X ) do no t haveth e sam e cardinal number, th a t is, i t is impossible to give a one-to-one cor-respondence between th e elements of X an d th e elements of P ( X ) . P rove th i sfact . [Hint: Suppose th a t a o A is such a correspondence. Le t

B = (aja E X , a - A a n d a G? A ) .Show th at if b ++ B, then both b E B and b 4 B.]

1-4 The algebra of sets. The ordinary number systems satisfy severa1important laws of operation, such as a + b = b + a , a ( b . c ) = ( a .b ) cand a (b + c ) = a b + a c. There are also natural operations of com-bining sets which satisfy rules analogous to these identities. Moderiialgebra is largely eoncerned with systems which satisfy various laws ofoperation, so it is natural that the algebra of sets should be a part of thissubject. Our objective in this section is to study the principal operatingrules for sets. The first two basic operations of set theory are analogousto addition and multiplication of numbers. They are binary operations,that is, they are performed on a pair of sets to obtain a new set.

DEFIXITION-4.1. Let A and B be sets. TheilA U B = (xlx E A o r x E B ) ,A n B = ( X I X E A a n d x E B ) .

The set A U B is called the u n i o n (or join or set sum) of A and B. Theset A n B is called the intersection (or meet or set product) of A and B.As we pointed out in the Introductioii, the word "or" in mathematics

is interpreted in the inclusive sense, so that the statement "x E A orx E B" includes the case where z is in both A and B. Thus, the unionof A and B contains those elements which are in A, or in B, or in bothA and B. The intersectioii of ,4 and B contaii~s hose elements whichare in both A and B.


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1-41 THE ALGEBRA OF SETS 31These sets can be illustrated by means of simple pictures called Venn

diagrams. The elements of the sets are represented by the points inside aclosed curve in the plane. It should be emphasized that these diagrams areonly symbolic, and that the elements of the sets which they represent arenot necessarily points in the plane, but can be any objects whatsoever.In Fig. 1-3, the total shaded area is A U B and the doubly shaded areais A n B.

A n B

EXAMPLE. ( 1 , 3, 4 , 5 , 7 ) U ( 2 , 3 , 6 ) = ( 1 , 2 , 3 , 4 , 5 , 6 , 7 ) .EXAMPLE. ( 1 , 3, 4 , 5 , 7 ) n ( 2 , 3, 6 ) = (3).EXAMPLE. ( 1 , 3 , 4 , 5 , 7 ) n ( 2 , 6 ) = @.EXAMPLE. (a10 < a < 1 ) U {O, 1 ) = (a10 < a 2 1 ) .EXAMPLE. (a10 < a < 1) n (al$ < a < 2 ) = {al$ < a < 1).EXAMPLE. If *4 is the set of al1 points of a line through the point p and if

B is the set of al1 points of a second (different) line through p, then A l B = ( p ) .

In most mathematical applications of set theory, al1 of the sets underconsideration will be subsets of some particular set X. This set, called theuniversal set, may be different for different problems, but it will usuallybe fixed throughout any discussion. For the purposes of developing thealgebra of sets, we will fix a universal set X once and for all. Al1 of the setsunder consideration are assumed to be subsets of X.

The third basic operation of set theory is analogous to forming thenegative of a number. I t is a unary operation, that is, i t is performed ona single set to obtain a new set.

DEFINITION-4.2. Let A be a subset of the set X. ThenAc = ( X ~ X X , x g A) .

The set A" is called the complement of A in X (or simply the comple-ment of A if i t is understood that A is being considered as a subset ofthe universal set X) .


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32 SET THEORY [CHAP.1T hu s, A" consists of those elements of X which a re ilot elem ents of A .There are many different notations in mathematical l i terature for thecom plement of a set A . Some which th e reader ma y encounter are A',

A, C( A ), an d c(A ). I n Fig. 1-4, th e shaded area represents AC .

EXAMPLE . L et X = (1, 2, 3, 4, 5). Then (1, 3)" = (2, 4, 5).E X A M P L E . Let X = (1, 2, 3 ) . Then (1, 3) " = (2).EXAMPLE . L et X be the se t of al1 real num bers. Th en {ala < O)" =

(ala2 o>

T H E O R E M-4.3. Let A , B, a n d C be subsets of X. The n thefollowingidentities ar e satisfied:(a) A u B = B u A , A n B = B n A ;(b) A u (B u C) = ( A u B ) u C, A n (B n C ) = (A n B) n C ;(c) A u A = A , A n A = A ;(d) A n (B u C ) = (A n B) u ( A n C ) ,A u ( B n C ) = ( A u B ) n ( A u C ) ;(e) A U A " = X , A n A " = i P ;(f) A u X = X , A n @ = @ ;(g) A u @ = A , A n X = A .

Proof. Al1 of these id ent itie s are sim ple consequences of th e definitionof union, intersection, and complement. We will illustrate this assertionby giving th e detailed proofs of (b) and ( d). T h e remaining identitie s areleft for the read er t o check.Suppose that x E A U (B U C). Then according to Definition 1-4.1,either x E A or x E B U C. Suppose th a t x E A . Th en by Definition 1-4.1again, x E A U B . Again, by 1-4.1, x E (A U B ) U C. On the other han d,if x E B U C, then ei ther x E B or x E C. If x E B, then x E A U B,and consequeiitly x E (A U B ) U C. If x E C, then we conclude im-mediately th a t x E (A U B ) U C. T hu s, in every case, if x E A U (B U C),


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1-41 THE ALGEBRA O F SETS 33theii x E (A U B) U C. By Definitioii 1-1.2, this means that A U (B U C)E (A U B) U C. A similar argument shows that (A U B) U C 2A u (B u C). Hence, A U (B U C) = (A U B) U C. This proves thefirst half of (b).If x E A n (B n C), then by Definition 1-4.1, x E A and x E B n C.Thus, x E A, x E B, and x E C. Consequently, x E A n B and x E C.Therefore x E (&4n B) n C. Hence, A n (B n C) G (A n B) n C.Similarly, (A n B) n C G A n (B n C). This shows that A n (B n C) =(A n B) n C.To prove the first equality of (d), suppose that x E A n (B U C).Then x E A and x E B U C. Hence, either x E A and x E B or x E A

and x E C. That is, either x E A n B or x E A n C. Consequently,x E (A n B) U (A nC). WehaveshownthatA n (B uC ) E ( A n B ) u(A n C). On the other hand, suppose tha t x E (A n B) U (A n C).Then either x E A n B or x E A n C. If x E A n B, then x E A and~ E B ,o that X E A and ~ E B u C . Therefore X E A n ( B u C ) .Similarly, if x E A n C, then x E A n (B U C). Hence, in any case,x E A n (B U C). We have shown that

This inclusion relatioii, combined with the one obtained above, yields

Let us illust'rate by a Venn diagram the identity (d) which we have justproved. The heavily outliiied region in Fig. 1-5 represents either side ofthe identity. The reader should illust,rate the other identities of Theorem1-4.3 by Venn diagrams.The identities (a) through (g) in Theorem 1-4.3 are the basic rulesof operation in the algebra of sets. By algebraic manipulations alone,it is possible to derive from these numerous other laws of operation.


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34 SET THEORY [CHAP. 1EXAMPLE0. Let A, B , and C be subsets of a universal set X.(a) (A U B) n C = C n A U B ) = ( C n A ) U ( C n B) =(A n C) U (B n C), and similarly,

(b) A U (B n A ) = (A U B) n ( A U A ) = ( AU B) n A = A n (B U A ) =( - ~ ~ ( B u A ) ) u @ = A ~ ( B u A ) ) u ( A ~ A ~ ) =n ( B u A ) u A ' ) =A n (B U ( A U AC ) ) = A n ( B U X) = A n X = A , t h a t is,A U ( B n A ) = A sncl A n B U A ) = A -

(e) If A n B = A n C and A U B = A U C, the n B = C. Indeed,B = B U ( A ~ B ) B U ( A ~ C ) ( B U A ) ~ B U C ) = ( A U B ) ~( B u C ) = (14 u C ) n ( B u C ) = ( A n B ) U C = ( A n C ) U C =c U n c ) = C.

Identities such as those of Example 10 can of course always be obtaineddirectly from the definitions of the set operations, as y e did for the proofof Theorem 1-4.3. However, identities which involve severa1 sets canusually be derived more easily by algebraic manipulations.

THEOREM-4.4. Let A, B, and C be sets.(a) A c A u B , B c A u B ; A = , A n B , B A n B .(b) If A 2 C and B C, then A U B E C; if A 2 C and B 2 C ,then A n B 2 C.(c) If A S B, then A U C B U C and A n C 2 B n C.(d) A E B if and only if A n B = A ; A 2 B if and only ifA u B = A .

The proofs of the various statements in Theorem 1-4.4 are again simpleapplications of the definitions. For exahple, let us prove the first partof (d). If A c , then x E A implies z E B, and hence x E A n B.Thus A A n B. If x E A n B, then in particular, x E A , so thatA n B c . Therefore, A = A n B . Conversely, if A = A n B , thenevery element of A is in A n B and, in particular, in B . Therefore A 2 B .

THEOREM-4.5. Let A and B be subsets of the set X.(a) If A C B, then A c 2 B c.(b) (A U B )"= A c n B c ; ( A n B)"= u B".(c) ( A C ) C A .(d) ac= X; X C= @.


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1-41 THE ALGEBRA OF SETS 35The statements (a), (e), and (d) of Theorem 1-4.5 should be clear.

Let us examine (b). To say that x E (A U B)" is the same as sayingx A U B, which in turn amounts to x A and x B. That is, x E ACand x E Bc, which means x E Ac n Bc. Thus (A U B)" and A" n Bccontain exactly the same elements, so they are equal. The proof that(A n B)"= A" U Bc is similar.

We illustrate the identity (A n B)" = A" U BC by a Venn diagram.In Fig. 1-6, the region outside of the doubly shaded region representseach of the sets (A n B)" and A" U Bc.

1. If th e universal s et is the collection N of al1 na tu ra l num bers, d eterm ine,4 U B, A n B, an d Ac in t he following cases.(a) ,4 = (nln is even) ,B = {nln < 10)(b) A = {n1n2 > 2n - ) , B = {nln2 = 2n + 3)(e) A = {nl(n + 1) /2 E N ) , B = {nln/2 E N )

2. Prove Theorem 1-4.3(a), (c), (e), ( f ) , and (g) .3. Ju stify each ste p of th e com putations in Examp le 10, using th e resultsof T heorem 1-4.3 where th ey are needed.4. Prove the following identities by algebraic manipulations, using theresults of Theorem 1-4.3 and Example 10.

(a) A U ( A c n B) = A U B, A n A C uB) = A n B(b ) A U (B n (A U C )) = A U ( B n C)(c) ((n n B) U (B n c>) (C n A)= ( ( A U B) n ( B u C ) ) n ( C U A)

5. Illus trate Theorem 1-4.4(c) b y a Venn diagram .6. Show th a t if A E C, then A U (B n C) = (A U B) 7 C.7. Prove Theo rem 1-4.5(a), (e), an d (d).8. Using Theo rem 1-4.3(d), (e), (g) an d Theo rem 1-4.4(d), show th a t ifA U B = X, t h e n A c B. Also, show t h a t if A n B =


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36 SET THEORY [CHAP. 110. Rlake Venn diagrams to i l lustrate the following identit ies.

(a) A n ( B U (C U D)) = (.(A n B) U (A n c)) U A n D)(b) ( A U (B n c ) ) ~= n ( B ~ Cc)11. If A an d B are a ny sets, then the diflerence between A an d B is definedto be A - = ( a E Ala B ). I n particular, if A an d B are subsets of someuniversal set X, then A - = A n Be. Show th at th e following are t rue .

(a) (,4 - ) - C = A - BU C )(b) A - B - A) = A(c) A - A - B) = A n B

12. Define ,4/B = A c n Bc. Prove th at t he following are true.(a) A/A = 14c (b) (A /A ) / (B /B) = A ri B(c) (A/B)/(A4/B) = U B

Th e binary operat ion (*/*) is called t h e S chefler stroke ope ration .13. Tra nsla te th e identities of Theo rem 1-4.3(d) an d The orem 1-4.4(b) in torules involving only th e Scheff er s trok e operation .14. Define A O B = (A n Be) U ( A cn B). Prove th e following.

(a) A O A = @ , A @ @ =(b) (A O B) O C = A O (B O C)(c) A n ( B o C) = ( A n B) O ( A n C )

1-5 Further algebra of sets. General rules of operation. It is possibleto extend many of the identities in the previous section to theorems con-cerning operations on any number of sets.

DEFINITION-5.1. Let S be a set whose elemeiits are sets. Thenu(S) = { x l x E A for some A E S), n(S) = ( ~1 . 2 : A for al1 A E S).As in the case of two sets, u(S) is called the un i o n of the sets of S andn( S) is called the intersection of the sets in S.

Thus, u(S) contains those elemeizts which are in any one or more of thesets in S, and n(S) contains those elemeizts which are in every set in S.For these definitions, S need not be a finite collectioiz of sets (see Example3 below).

In Fig. 1-7, S = {A , B, C, D); u(S) is the total shaded area andn(S) is the most heavily shaded area, inside the heavy outline.

EX A M PLE . Le t S = ((1, 2), {l., 3, 5:- , (2, 5, 6)). Then U(S) = (1, 2, 3, 5, 6)a n d n ( S ) = @.E X A M P L E . If S = {A ,B) , then U(S) = A U B and n (S ) = *4 l B .


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1-51 FURTHER ALGEBRA OF SETS. GENERAL RULES OF OPERATION 37

EXAMPLE. Let C be a circle in soine plane P. Let S consist of al1 sets Awhich satisfy the followin

beaumont - conjuntos

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