Modern abstract algebra david burton

1-2 Functions and Elementary Number Theory 13 Chapter 2 Group Theory 2-2 Certain Elementary Theorems on Groupf!. Theory 3-1 Definition and Elementary Properties of Rings 141 3-6 Boolea

This book is in the

ADDISON-WESLEY SERIES IN MATHEMATICS Ocnuultiftl/ Editor: LYNN H LoOMI8

D A V I D M BUR TON, University of New Hampshire

INTRO-DUCTION TO MOO'ERN ABSTRACT ALGEBRA

ADDISON-WESLEY PUBLISHING COMPANY

ReedIng, M8ItIaChuIItItIt Menlo PIITk, California London Amatflfdem • Don MIlls 0nfarl0 S)IdIIey

COPYRIGHT ® 1967 IIY AIlIII~{)N-Wt:~Lt:Y l'URLI~IIING COMrANY, INC ALL RIGIITS Rt:St:RV}:/) TIII~ 1I00K, on l'AltT~ Tlnau:(w, MAY NOT III': Rt:I'IUJIlUCt:1> IN ANY FOUM WITHOUT WIUT'n:N I't:nMI~HION OF Til}: l'UIILISln:n 1'liINT}:/) IN TIn: UNITt:!) STATES

OF AMt:RICA I'UIILlsllt:1> SIMULTANt:OUHLY IN CANAI>A LIBRAUY OF CONGIlESS CATALOG CAIlII NO 67-19426

, _ 0·201.00722·3

IJKLMNOPO-MA 19818

A cursory examination of the table of contents will reveal few surprises; the topics chosen for discussion in COUr8es at this level are fairly standard However, our aim has been to give a presentation which is logically developed, precise, and in keeping with the spirit of the times Thus, set notation is employed throughout, and the distinction is maintained between algebraic systems as ordered pair8 or triples and their underlying sets of elements Guided by the principII! thnt 0 st(~l"ly diet of definitions and cxullIples SOOIl 1)(~c()llIel'l unpalat-able, our eiTOl-ts are directed towards establishing the most important and fruitful results of the subject in a formal, rigorous fashion The chapter on groups, fOl' example, culminates in a proof of the classic Sylow Theorems, while ring and ideal theory are developed to the point of obtaining the Stoile Repre-sentation Theorem for Boolean rings Ell route, it is hoped that the, reader will gain an appreciation of precise mathematical thought and t.he current standards

of rigor

At the eud of each section, there will be found a collection of problems of varying degrees of difficulty; these constitute an integral part of the hook They introduce a variety of topics not treated in the main t.('xt, as well as impart much additional detail ahout material covered earlier Home, especially in the latter seet.ions, pl'Ovide slIbl'ltantial extensionl'l of t.he'theory We have, on the whol(', resist.('d t.h(' t.('mptation to lise the exercises to develop results that will

be ncedl'd HllbsequC'nt.ly; aN a 1·(,~lIlt., the reader need not work all the problems

in order t.o read the reHt of the hook Problems whose solut.ions do not appear particularly straiJ!;htforward arc accompanied by hints, Besides the general index, a glossary of !!pecial Hymhol!! iN also included

v

vi PREFACE

The text is not intended to be encyclopedic in nature; many importaut topics vie for inclusion and some choice is obviously imperative To this end, we merely followed our own taste, condensing or omitting altogether certain of the concepts found in the usual first course in modem algebra Despite these omissions, we believe the coverage will meet the needs of most students; those who are stimulated to pursue the matter further will have a finn foundation upon which to build

It is a pleasure to record our indebtedness to Lynn Loomis and Frederick Hoffman, both of whom read the original manuscript and offered valuable criticism for its correction and improvement Of our colleagues at the University

of New Hampshire, the advice of Edward Batho and Robb Jacoby proved particularly h~lpful; in this regard, special thanks are due to William Witthoft who contributed a number of incisive suggestions after reading portions of the galley proofs We also take this occasion to express our sincere appreciation to Mary Ann MacIlvaine for her excellent typing of the manuscript To my wife must go the largest debt of gratitude, not only for her generous assistance with the text at the various stages of its development, but for her constant encourage-ment and understanding

Finally, we would like to acknowledge the fine cooperation of the staff of Addison-Wesley and the usual high quality of their work

Durham, New H amp8hirc

March 1967

D.M.B

CONTENTS

Chapter 1 Preliminary Notion

1-2 Functions and Elementary Number Theory 13

Chapter 2 Group Theory

2-2 Certain Elementary Theorems on Groupf! 41

Chapter 3 RI Theory

3-1 Definition and Elementary Properties of Rings 141

3-6 Boolean Rings and Boolean Algebras 219

Chapter 4 Vector Spec

4-2 Elementary Properties of Vector Spaccli 249

CHAPTER 1

PRELIMINARY NOTIONS

1-1 THE ALGEBRA OF SETS

This ehaptcr hrieAy summarizes IIOme of the bll.llic notions concerning sets, functiollH, and number theory; it alllO serves Itl! It vehicle for establishing COIl-ventions in notation and terminology used throughout the text Inasmuch as this material is intended to serve primarily for background purposes, the reader who is already acquainted with the ideas in this chapter may prefer to cmbark directly 011 the next

Within the confines of one section, it is obviously impossible to give complete coverage to set theory or, for that matter, to achieve a logically coherent exposi-tion of such a formalil!tic diseipline The subsequent presentation should thus

be regarded simply as a summary of the fundamental aspects of the subject, and not as a sYRtematic development

Rather than attempt to list the undefined terms of set theory and the various axioms relnting them, we shall take an informal or naive approach to the sub-jeet To thil! cnd, the term set will be intuitively understood to mean II collection

of objects having some common characteristic The objects that make up a given sct arc called its elements or members Sets will generally be designated

by capital letters and their elements by small letters In particular, we shall employ the following notations: Z is the set of integers, Q the set of rational

numbers, and R' the set of real numbers The symbols Z+, Q+, and R~ will stand for the positive elements of these sets

If x is an element of the set A, it is customary to use the notation x E A

and to read the symbol E as "belongs to." On the other hand, when x fails to

be an element of the set A, we shall denote this by writing x I;l A

There arc two common methods of specifying a particular set First, we may list all of its elements within braces, as with the set {-I, 0,1, 2}, or merely list some of its elements and use three dots to indicate the fact that certain obvious clements have been omitted, as with the set {I, 2, 3,4, } When such a liHting il! not practical, we may indicate instead a characteristic property whereby we can determine whether or not a given object is an element of the set IHore specifically, if P(x) is a statement concerning x, then the set of all elementR x for whieh the J;tatcmcnt P(x) is true is denoted by {x I P(x)} For example, we might have {x I x is an odd integer greater than 2I} Clearly,

1

vi PREFACE

The text is not intended to be encyclopedic in nature; many important topics vie for inclutlioll and some choice ill obviously imperative To this end, we merely followed our own taste, condensing or omitting altogether certain of the concepts found in the usual first course in modem algebra Despite these omissions, we believe the coverage will meet the needs of most students; those who are stimulated to pursue the matter further will have a firm foundation upon which to build

It is a pleasure to record our indebtedness to Lynn Loomis and Frederick Hoffman, both of whom read the original manuscript and offered valuable criticism for its correction and improvement Of our colleagues at the University

of New Hampshire, the advice of Edward Batho and Robb Jacoby proved particularly ht'lpful; in this regard, special thanks are due to William Witthoft who contributed a number of incisive suggestions after reading portions of the galley proofs We also take this occasion to express our sincere appreciation to Mary Ann Ma.eIlvaine for her excellent typing of the manuscript To my wife must go the largest debt of gratitude, not only for her generous assistance with the text at the various stages of its development, but for her constant encourage-ment and understanding

Finally, we would like to acknowledge the tine cooperation of the staff of Addison-Wesley and the usual high quality of their work

Durham, New If ampshire

March 1967

n.M.B

CONTENTS

Chapter 1 Preliminary Notion

1-2 functions and Elementary Number Theory 13

Chapter 2 Group Thoory

2-1 Definition and Examples of Groups 27 2-2 Certain Elementary Theorems on GroupR 41

Chapter 3 Rlnt Theory

3-1 Definition and Elementary Properties of Rings 141

Chapter 4 Vector Spac

2 PRELIMINARY NOTIONS I-I

certain sets may be described both ways:

{O, I} = {x / x E Z and x 2 = x}

It is customary, however, to depart slightly from this notation and write

{x E A / P(x)} instead of {x / x E A and P(x)}

Definition 1-1 Two sets A and B are said to be equal, written A = B,

if and only if every clement of A is an element of B and every element of B

is an element of A That is, A = B provided A and B have the same elements Thus a set is completely determined by its elements For instance,

{I, 2, 3} = {3,I,2,2},

since each set contains only the integers 1, 2 and 3 Indeed, the order in which the elements are listed in a set is immaterial, and repetition conveys no addi-tional information shout the ad

An empty set or null set, represented by the symbol 0, is any Bet which has

no elements For instance,

o = {x E R' I x 2 < O} or 0= {X/XFX}

Any two empty sets arc equal, for in a trivial sense they both contain the same elements (namely, none) In effect, then, there is just one empty set, so that

we are free to speak of the empty 8et 0

The set whose only member is the element x is called 8ingleton x and it is denoted by {x}:

{x} = {y I y = x}

In particular, {O} F 0 sincll 0 E {O}

Definition 1-2 The set A is a BUb8et of, or is contained in, the set B, indicated

by writing A ~ B, if every element of A is also an element of B

Our notation is designed to include the possibility that A = B Whenever

A ~ B but A F B, we will write A C B and say that A is a proper BUbset of B

It will be convenient to regard all sets under consideration as being subsets

of some master set U, called the universe (universal 8et, ground set) While the universe may he diffC'rent in different contexts, it will usually be fixed throughout any given difol(~llllllion

There arc several immediate (:onscquenc(~s of the definition of sct inclusion

TheOrem 1-1 If A, B, and C are subsets of some universe U, then

a) A ~ A, 0 ~ A, A ~ cr,

b) A ~ 0 if and only if A = 0,

c) {x} ~ A if and only if x E A; that is, each clement of A determines a subset of A,

1-1 THE ALGEBRA OF SETS 3

d) if A 5;; Band B 5;; C, then A 5;; C,

e) A 5;; Band B 5;; A if and only if A = B

Observe that the result 0 5;; A follows from the logical principle that a false hypothesis implies any conclusion whatsoever Thus, the statement "if x e 0,

then x e A" is true since x e 0 is always false

The last assertion of Theorem 1-1 indicates that a proof of the equality of two specified sets A and R is generally presented in two parts One part demon-

strates that if x e A, then x e B; the other part demonstrates that if x e B,

then x e A ' An illustration of such a proof will be given shortly

We now consider several important ways in which sets may be combined with one another If A amI B are subsets of some universe U, the operations

of union, intersection, and difference arc defined as follows

Deftnition 1-3 The union of A and B, denoted by Au B, is the subset

of U defined by

A U B = {x I x e A or x e IJ} •

The intersection of A and B, denoted by A n B, is the subset of U defined by

AnB= {xlxeAandxEB}

The difference of A and B (sometimes called the relative complement of B in

A), denoted by A - B, is the subset of U defined by

A - B = {x I x E A but x ~ B}

In the definition of union, the word "or" is used in the "and/or" sense Thus

the statement "x E A or x E B" allows t.he possibility that x is in both A and B

It might also he Jlot(ld par(mthetiml.\ly that, utilizing this new notion, we could define A to be a proper subset of B provided A ~ B with B - A ~ 0

The particular difference U - B is called the (absolute) complement of B

and designated simply by -B If A and B are two nonempty sets whose

inter-section is empty, that is, A n B = 0, then they are said to be disjoint We

shall illustrate these concepts with an example

Examp.e 1-1 Let the universe be U = {O, 1,2,3,4,5, 6}, the set A = {I, 2, 4}, and the set B = {2, 3, 5} Then

A uB = {1,2,a,4,5}, A n JJ = {2}, A - B = {t,4}, and

tech-x E A or x E B n C Now, if x E A, then clearly both x E Au Band

x E A U C, so that x E (A u B) n (A u 0) On the other hand, if x E B n C,

then x E B and therefore x E A u B; also x E C and therefore x E A u C

The two conditions together imply that

x E (A U B) n (A u C)

This establishes the inclusion,

A u (B n 0) ~ (A u B) n (A u 0)

Conversely, suppose x E (A U B) n (A u 0) Thcll both x E A U Band

x E AU C Since x E A U B, either x E A or x E Bi at the same time, since

x E A U C, either x E A or x E C Together, theMel conditions mean that

x E A or x E B n C; that is, x E A u (8 n C) This proves the opposite inclusion,

(A U B) n (A u 0) ~ A u (B n C)

By part (e) of Theorem 1-1, the two inclusions are sufficient to establish the equality,

A u (B n C) = (A u 8) n (A u C)

If A, B and C are sets such that C ~ A and C ~ B, then it is clear that

C ~ A n B Thus it is possible to think of A n B as the largcst set which is a· subset of both A and B Analogously, A U B may he interpreted as the

smallcBl set which contains hoth A Rnd B

The next t1worcm relates the operation of complementation to other tions of set theory

opera-1-1 THE ALGEBRA OF SETS 5

Theorem 1-3 Let A amI B be subsets of the universe U Then

nor n Thill implicH t.lmt x E -A ami x E 0 _ 0 n, from which it follows that

x E (-A) n (-B) Thus -(A u B) ~ (-A) n (-B)

Conversely, if x E ( -A) n (-B), then x belongs to both - A and -B In other words, x ~ A and x ~ B This guarantees x ~ A U B, that is

them-frequently refer to these as families of sets One family which will prove to be

of considerable importance is the so-called power set of a given set

Definition 1-4 If A iii I1.n nrhit.mry l'Ict, thcn t1w !let WhOHC clements are nl\ the HuhRI't.H of A ill known nil the power 8(~t of A allli dmlOtcu by P(A):

peA) = {B I B ~ A}

A few remarks arc in order before considering a specific example First, since 0 ~ A and A ~ A, we always have {0, A} ~ peA) no matter what the nature of the set A (If A = 0, then of course peA) = {0}.) The next thing

to observe is that if x E A, then {x} ~ A, hence {x} E peA) From this, we infer that the power set of A has, at the very least, as many elements as the

set A Indeed, it ean be shown that whenever A is a finite set with n elements,

then peA) is itself It finite set having 2n elements For this reason, the power set of A iH oft!'11 represented hy the Hymbol 2A

Example 1-2 Huppol'll! t.he l'Id A ,~ (fl, Ii, c} Tht· POWC1' tid of A, whirh has

aI:I its c11:mclItH all the subl:lCts of fa, b, c}, is then

P(A) = {0, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, A}

6 PRELIMINARY NOTIONS 1-1

It is both desirable and possible to extend our definitions of union and section from two sets to any number of sets Assume to this end that a is a nonempty family of subsets of the universe U The union and intersection of this arbitrary family are defined by,

inter-ua = {x / x E A for some set A E a},

na = {x / x E A for every set A E a}

At times we will resort to an indexing set to define these notions To be more precise, let 1 be a set, finite or infinite, and with each i E 1 associate a set A, The fCl!ulting family of sets,

a = {Ad i E I},

is then said to be indexed by the elements of I, and the set 1 is called an index Bet for a Wh(ln t.hifl notation is employed, it is customary to denote the union and intcrHCction of the fnmily a by

and n{Ai / i E f}

If the nature of the index set 1 is clearly understood or if the emphasizing of

it is inessential for some reason, we simply write,

-and

Example 1-3 If A" = {x E R' / -lin ~ x ~ lin} for n E Z+, then

u{A / n E Z+} = {x / x E A for some n E Z+} = At,

n{A / n E Z+} = {x / x E A for every n E Z+} = {O}

In passing, we should note that by a chain of sets is meant a nonempty family e of subsets of some universe U such that if A, BEe then either As; B

or B S; A For instance, the family in Example 1-3 constitutes a chain of sets From our definition of set equality, {a, b} = {b, a}, since both sets contain

the same two elements a and b That is, no preference is given to one element over the other When we wish to distinguish one of these elements as"being the first, say a, we write (a, b) and call this an ordered pair

It is possible to give a purely set-theoretic definition of the notion of an

" ordered pair as follows:

Definition 1-5 Tbe ordered flair of el(!ments a and b, with its first nent a and lK!cond component b, denoted by (a, b), ill the set

compo-(a, b) = {{a,b}, {a}}

I-I THE ALGEBRA OF SETS 7 Note that according to this definition, a and b are not elements of (a, b) but rather components The actual elements of the set (a, b) are {a, b}, the un-

ordered pair involved, and {a}, that member of the unordered pair which has

been selected to be first This agrees with our intuition that an ordered pair should be an entity representing two elements in a given order

For a ~ b, the sets {{a, b}, {a}} and {{b, a}, {b}} are unequal, having

different elements, so that (a, b) ~ (b, a) Hence, if a and b are distinct, there are two distinct ordered pairs whose components are a and b, namely, the pairs (a, b) and (b, a) Ordered pairs thus provide a way of handling two things

as one while losing track of neither

In the next th(!Orem, a useful criterion for the equality of ordered pairs is obtained; the proof ill subtle, but simple, relying mainly on Definitions 1-1 and 1-5

Theorem 1-4 Two ordered pairs (a, b) and (e, d) are equal if and only if

a = e and b = (l

Proof, If a = e and b = d, then it is clear from Definition 1-1 that

{a} = {e} and {a, b} = {e, d},

This in tum implies {{a, b}, {a}} = {{e, d}, {e}}, whence (a, b) = (e, d)

As for the converse, suppose that {{a, b}, {a}} = {{e, d}, {en We tinguish two possible cases:

dis-1) a = b In this case, the ordered pair (a, b) reduces to a singleton, since

(a, b) = (a, a) = {{a,a}, {a}} = {{a}}

According to our hypothesis, we then have

{{an = {{e, d}, {en,

which means {a} = {e, d} = {e} From this, it follows that the four elements

a, b, e, d are all equal

2) a ~ b Here, both {a} ~ {a, b} and {e} ~ {a, b} If the latter equality

were to hold, we would have e = a and e = b, hence the contradiction a = b

Now, by virtue of the hypothesis, each member of the set (e, d) belongs to

(a, b); in particular,

{e} E {{a,b}, {a}}

This means that {e} = {a} and accordingly a = e

Ap;nill hy KIIPPOKitioll, (a, II} E f fe, (l}, fr.}} with {a, b} ~ {e} It may thus

be inferred that la, b1 = {c, d} lUlll therefore b E {e, d} As b cannot equal e (this would imply that a = b), we must conclude that b = d In either case the desired result is established

8 PRELIMINARY NOTIONS 1-1 Having faced the problem of defining ordered pairs, it is natural to consider ordered tripl('s, ordered quadruples and, for that matter, ordered n-tuples What simplifies the situation is that these notions can be formulated in terms of ordered pairs For instance, the ordered triple of a, b, and c is just an ordered pair whose first component is itself an ordered pair:

(a, b, c) = (a, b), c)

Assuming that ordered (n - I)-tuples have been defined, we shall take the

ordered n-tuple of alt a2, ,an to mean the ordered pair (a" a2, ,a,.-I), a,.),

abbreviated by (a" a2, ,an), It should come as no surpriMC that two ordered n-tuples equal whenever their corresponding components are equal; in other words,

(at a2, • , a,.) = (b" b 2, • , b n )

if and only if ak = bk for k = 1,2, ,n

Definition 1-6 The Carte8ian product of two nonempty sets A and B,

designated by A X B, is the set

A X B = {(a, b) I a E A and b E B}

Whenever we employ the Cartesian product notation, it will be with the understanding that the sets involved are nonempty, even though this may not

be explicitly stated at the time Observe that if the set A contains n elements

and B contains m elements, then A X B has nm elements, which accounts for

the use of the word "product" in CartcRian product

Example 1-4 Let A = {-I,O, I} and B = {0,2} Then,

consists of al\ points in the plane lying above the line y = x; one usually writes

3 < 4 rather than the awkward (3,4) E <

Frankly, the concept of a relation as defined is far too general for our purposes

We shall instead limit our attention to a specialized relation known as an equivalence relation

1-1 THE ALGEBRA OF SETS 9

Definition 1-7 A rl'lllt.ion Il in It liet, A iii 8nid to he 11I1 (!quivalence relation

ill A provided it Ilatislied the threc propertie8,

1) refll'xive propl'rt.y: alla, for eaeh a E A,

2) Kyn\llletril~ proJ)('rt,y: if allb for IiOmc a, b E A, thcn blla,

:i) tram!itive property: if allb and bUe for IiOllle a, b, t: E A, thcn aRc

Equivalence relations arc customarily denoted by the symbol, , (pronounced

"wiggle") With this change in notation, the conditions of Definition 1-7 may

be reHtatcd in a more familiar form:

J) 0"" 0, for I'l\I'h 0 E A,

2) a , , b implic8 b , , a,

3) both a , , band b , , r imply a , c

In tJlI~ following (lxnmpII'K, WI' h~lwc to t.ho rt'ndor t,h(l t.ILl'Ik of verifying that each rdlltion dCH(~ril)(!(1 Ilet,unIly iii lUI cfJuivulmwc relation

Example 1-5 Let A be an arbitrary nonempty set and define for a, bE A"

a , b if nnd only if a = b (a = b is tacitly interpreted to mean that a and b

are identical clements of A) This yields an equivalence relation in A

Example 1-6 Consider the set L of all lilies in a fixed plane and let a, bEL

Then, , is an cquivalenee relation in [J provided a, , b means that a is parallel

to b; let, us agrce thnt any line is parallel to itl«)lf

Example 1-7 Take Z to be the set of integers Given a, b E Z, we define

an cquivalclwc reln'tion , , in Z by requiring thnt a , , b if nnd only if a - b E Z.,

the He! of (wen int,l'gerM

Example 1-8 All a linal illustration, suppose A = Z+ X Z+ and define

(a, b) , , (e, d) to mean ad = be A simplc cldculation reveals this is an alence relation in A

equiv-One is frequently led to conclude that the reflexive property is redundant in Definition 1-7 The argumcnt goes like this: If a, , b, then the symmetric property implies b , , a; since a, , band b, , a, using the transitive property,

it follows that a , , a Thus, there appears to be no necessity for the reflexive

eondition at all The ftl~W in t.his reasoning IiI'S in t,he faet that for some clement

a E A, tllI'rl' llIay not, I'xisl any b E A Slid, I.hat, a , , h AI~I~(mlil\j!;ly, we would

1101 IlIlv(' a , , (t fol' I'VI'I'Y 1I1I'lIilu·1' of It, liS llw I'l'fI('xiv(' propl'l'1.y l'("quin'M I'(!rllltps lh(, prilwipal ,'('aSOIl for ('ollsidl'l'illK ("qllivall"If"" rclat.i()m; in a IiCt A

is that they separat.e A illt.o certain (!onvcnient suhliets To be more precise, supposp , , is a KivclI I'quivall'lIl'e relation ill A For each a E A, we let [aJ

denote the suhset of A I'onsisting of all clements which are equivalent to a:

[aJ = {x E A I x , , a}

This set [aJ is referred to as the equivalence class determined by a

10 PRELIMINARY NOTIONS 1-1 Some of the baKil: properties of equivalence classes are listed in the next theorem

Theorem 1-5 J.A:~t - he an equivalence relation in the set A Then,

1) for each a E A, [a] ¢ 0,

2) if b E [a], then raj = [I)]; that is, any clement of the equivalence class

[aJ determines the cllUlS,

3) for any a, b E A, with faJ ¢ rbI, [a] n [b] = 0,

4) U{[a]laE A} = A

Proof Clearly, a E [a], since a-a To prove (2), let bE [a], RO that b - a

Now, SIIPPOKI' ;r E [a) whil'" impIiI'R.1" - a UKinK t.he KYlllmctril~ and transitive propcrt.il's of -, it follmvH that :c - b, h('nl'c x E fbI This estnhlislwH the in-clusion [a] s;; [b] A Himilllr argument yil'lds the opposite in('lusion and thus the equality [a] = fbI

We derive (3) hy assuming, to the contrary, that there is some element

C E [a] n fbI Then by statement (2), which has just been verified, [a] = [e] = rbI, an obvious contradietion Finally, since each clll.SS [a] s;; A, the inclusion U{[a] I a E A} s;; A ill apparent To obtain the reverse inclusion one need only demonstrate that each clement a in A belongs to some equivalence c1aSllj but this is evident: if a E A, then a E [a]

We next conned the idea of an equivalence relation in A with the concept

of a partition of A

Definition 1-8 A partition of a set A iH a family {Ai} of nonempty subsets

of A with the properties

1) if Ai ¢ Aj, then Ai n Aj = 0 (pairwise disjoint),

2) UAi = A

Briefly, a partition of A is a family {Ai} of nonempty subsets of A such that

every element of A belongs to one and only one member of {Ai} The integers,

for instance, have a partition ('onsisting of the sets of odd and even integers:

Z = Z u Zo, z n Zo = 0 Another partition of Z might be the sets Z+

(positive integ<'rH), Z_ (nl'Kntive integers), and {O}

Theorem 1-5 may be viewed as asserting that if - is an equivalence relation

in A, then the fnmily of nil l'(Juivalent'e classes (with respect to the relation -) forms a partition of A We now reverse the situation and show that a given partition of A indu('l's a natural equivalence relation in A

Theorem 1-6 If {Ai} is a partition of the set A, then there is an equivalence relation in A whose cquivalen('e classes are precisely the sets A j

Proof For clements a, b E A, we take a - b if and only if a and b belong to

the same subset Aj Th(' reader may check that the relation -, so defined, is

1-1 THE ALGEBRA OF SETS 11

actually an equivalence relation in A Now suppose the clement a E Ai Then

b E Ai if and only if b "'" a, that is, if and only if b, E raj This demonstrates the equality A = raj

In summary, the nhove ClitlCUHHiOIl Hhows that there is no C14scntial distinction betwccn partitions of a set and equivalence relatiolls in the set; if we start with one, we get the other

Example 1-9 Let A = R' X R' and define the relation"", by

(a, b) "'" (c, d) if and only if a - c = b - d

TIII'II - is Ilil I'Iluivll1c-lIc'c- rc-laLion ill A Thl! cquivlLlmwe dlLHH determined by the clement (a, 11) iii !limply

8 a) If 1 s:;; /I nllli 1 ~ -II, prove I.hat :I , tl

b) If A !; Hand -.1 !;; H, prove that B - ll

9 Establish the two absorption laws:

A U (A () B) = A, A n (A U B) = A

10 ASRume that :1, H, and Carll Ret.s for which

Prove t.hat B = C [Hin.t: H = B () (B U \).1

]1 Let a = {.It, h, } be a family of subsets indexed by the positive integers

Z+ Define a new family m = {BI, B2, } as follows:

Bl = "h;

Show that

Bn = An - U{.h I k = 1,2, , n - I} for n > 1

a) the member" of Hare di!ljoint !lets, b) ua = um

12 For any thrlle ~ets t, JJ and C, establish t.hat

b) aRb if and only if a - b is an odd integer,

c) IJRb if and only if ab ~ 0,

d) aRb if and only if a 2 = b 2 ,

e) aRb if and only if la - bl < I

]4 Let S be a finite set, but ot.herwise arbitrary Determine if the relations defined below are equivalenre relations in P(S):

a) J, , B means 1 s:;; H,

b) 1 , , B meanll \ and B have the same number of elements

15 How many distinct equivalence relations are there in a set of 4 elements?

16 Prove that the following relations, , are equivalence relation!! in the Cartesian product R' X R':

a) (a, b) , , (e, d) if and only if b - d = m(a - c), m a fixed real number, b) (a, b) , , (e, d) if and only if a + d = b + c,

c) (a, b) , , (e, d) if and only if a - c E Z, b = d

1-2 l"UN(''TiONS ANn ELl!lMENTAltY NUMBER THEORY 13

1-2 FUNCTIONS AND ELEMENTARY NUMBER THEORY

Let us turn next to the concept of a function, one of the most important ideas in mathematics We shall avoid the traditional view of a function as a

"rule of ('Orrl~HJI(IIlClcfl(:e or nfl(ll!;ivI' inHt,\lad II definit.ion ill U1rfllli of ordered pn.irs What t.hili Inl.t.l'l· npprllueh l:Leks in f1utumluel! 'i is lIIorc thull I:ompemmted for

by its clarity II.lId precision

Definition 1-9 A Junction (or mapping) f is a set of ordered pairs Buch that

no two distill(~t pairs have the same first component Thus (x, Yl) Ef and

(x, Y2) E J implies Yl = Y2

TIll! ",)Jl(~(:t.illfl of all lirlit ('WIIJlUfI(!IItli IIf II fuuetion f iii ealled tho domain of the fuuetioll nlld is denoted by D" while the collection of all second components

is called the ran!le of the funetion and is denoted by R, In terms of set notation,

D, = {x I (x, y) EJ for some y},

R, = {y I (x, y) EJ for some x}

If J is a function and (x, y) E J, then y is said to be the functional value or

image of f at x and is denoted by f(x) That is, the symbol f(x) represents the ullique second component of that ordered pair of f in which x is the first com-ponent We lihall oCC8.:-;ionally ohscrve the cOllvention of simply writing fx for

and we write f( -1) = 0, f(O) = 0, J(I) = 2 and J(2) = 1

It is often convenient to descrihe a function by giving a formula for its ordered pairs For instanl:e, we might have

Definition 1-10 COII;;ider a funetion J ~ X X Y If /), = X, then we say t.hat J iii a fUflct.inn fmlll X int() }', or t.hat f mall'~ X into Y; thiH Kitua.tion

is expressed symboIieally by writing f: X > 1"

14 PRELIMINARY NOTIONS 1-2

The function I is said to be onto Y, or an "onto" function, whenever I is a function from X into Y and RI = Y Thus I is onto Y if and only if for each

ye Y there exists some xeD, with (x, y) e/, so that y = I(x)

Since functions are sets, we have a ready-made definition of equality of functions: two functions I and g are equal if and only if they have the same members Accordingly, I = g if and only if D, = Dg and I(x) = g(x) for each element x in their common domain

Suppose I and g are two specific functions whose ranges are subsets of a system in which addition, subtraction, multiplication and division are per-missible (one may think of functions from R' into R') The following formulas define functions 1+ g, 1- g, I· g and I/g by specifying the value of these functions at each point of their respective domains:

where x e (D, n Dg) - {x e Dill g(x) = OJ

We term f + (I, f - g, f· g and fig, the pointwise sum, difference, product and quotient of f and g

1-2 FUN<-'TIONS AND ELEMENTARY NUMB Ell THEORY 15

Definition 1-11 The composition of two functions / and 0, denoted by

/ 0 0, is the function

/0 g = {(X,7/) I for some z, (x, z) E g and (z,7/) E/}

Written in terms of functional values, this gives

(f 0 (I)(X) = I«(/(x»), where

This last notation serves to explain the order of symbols in f • 0; the letter

g is written directly beside x, since the functional value O(x) is obtained first

It is apparent from the definition that, so long as RII n D, 'F-0,/ g is ful Also, Dlo ll ~ DII and Rlo ll ~ R I

meaning-Examp'e 1-12 IA't

I = {(x, vi) I x E R', x ~ O}, and

Olin ObRf'rVeH t.Jmj, I· (I iH dilTl'rC'lIt from (f 0 Ii indeed, mn!ly does I· (f = g 0 f

The next theorem concerns some of the basic properties of the operation of functional eompositioll Its proof is an exereise in the use of the definitions of thiR S()ctioll

Theorem 1-7 If /, g and h are functions for which the following operations

1) (f g) • Ii - 10 (y • It),

2) (f + y) • It = (f h) + (g It),

3) (f g) • h = (f 0 h) (g 0 h)

16 PRELIMINARY NOTIONS 1-2

Proof We establish here only property (3) The other parts of the theorem are obtained in a similar fashion and 80 are left as an exercise Observe first that

DU'II)% = {x E D/o I hex) E DI"}

= {x E D/o I hex) E DI n D II}

= {x E D" I hex) ED,} n {x E D/o I hex) E D II}

= D,o/a n Vll0/o = DUo/a)'(II0/a) Now, for x E D("II)o/a, we have

Once again, consider an arbitrary function f: X ~ Y While no element of

X can be mapped under / onto more than one element of Y, it is clearly possible that several (perhaps, even all) elements may map onto the same element of Y

When we wish to avoid this Hituation, the notion of a one-to-one function is

-Definition 1-12 A function / is termed one-ta-one if and only if XI, X2 E D"

with XI ' " X2, impli(!s/(xl) '" /(X2)' That is, distinct clements in the domain have distinct functional values

When establiilhing one-f,o-oncness, it will often prove to be morc convenient

to use the contrapollitive of Definition 1-12:

In terms of ordered pairs, It function I ill onc-to-onll if and only if no two distinct ordcred pairs of / have the same second component Thus the collection

of ordered pairs obtained by interchanging the components of the pairs of /

is also a funct.ion This oh8(·rvat.ion indicates the importance of such functions More specifically, the inverse of a one-to-one function /, symbolized by /-1,

is the set of ordered pllirs,

/-1 = fey, x) I (x, y) Ef}

The function /-1 has the properties

(f-I • I)(x) ~ x for xED"

(fo/-I)(y) = yforYEDrt = RI

1-2 FUNCTIONS AND ELEMENTARY NUMBER THEORY 17

To state this result a little more concisely, let us introduce some special terminology

Definition 1-13 Given a nonempty set X, the function ix: X -+ X defined

by ix(x) = x for each x E X is called the identity [unction on Xj that is to say, ix merely maps caeh element of X onto itself

ExpreRRed in terms of the identity funetion, what was just seen is that for

any function j: X -+ Y which is both one-to-one and onto Y,

and [0[-1 = iy

It migftt also be mentioned at this point that the identity function ix is itself

a one-to-one mapping onto the set X such that iXI = ix

Example 1-13 The fUlwtioll [ = {(x,3x - 2) I x E nil is on(,> to-one, for

ax - 2 ~ ax:.! !! impliml x = X2' Cont:I(Jqucmtly, the inVllftj(J of j exists and

is the set of ordered pail~[-I = {(3x - 2, x) I x E R'} It is preferable, ever, to have[-t defined in terms of its domain and the image at each point of the domain Observing that

how-{(3x-2,x)lxER'} = {(x,!(x+2»lxER'},

we choose to write

r 1 = {(x,!(x + 2) I x E R'}

In terms of functional vl\lueM, [-I(X) = !(x + 2) for each x E R'

An important situation ariscs when we consider the behavior of a function

on a subset of its domain For example, it it! frequently advantageous to limit the domain 80 I,hnt t.he fUlldion beeomcM one-to-one Suppose, in general,

that j: X -+ Y it! an nrbitrary function and the subset A ~ X The tion j 0 i : A -+ Y is known as the restriction of [ to the set A and is, by estab-lished cllstom, denoted by f I A; dually, the funetion [ is referred to as an

composi-extension of f I A to all of X For the reader who prllfers the ordered pair approach,

In any event, if the clement x E A, then (f I A)(x) = [(x) so that both [and

[ I A coineide on the set A It is well worth noting that while there is only one restriction of the given function [ to the subsct A, [ is not necessarily uniquely determined by [I A The particular restrietion ix I A = iA , when viewed as a funetion from A into X, is termed the inclusion or injection map from A to X

Thl! lwxt ddinitinn 1'lIIilodil'fl lL fn~CJUlmtly employed uotational device Observe that despite the use of the symbol j-1, the function [ is not required

to be one-to-one

18 PRELIMINARY NOTIONS 1-2

Definition 1-14 Conllider a (unction f: X -+ Y If A !;; X, then the direct

image of A, denoted by f(A), is the subset of Y defined by

f(A) = {f(z} I z e A}

On the ot.her hand, if B !;; Y, then the inverse image of B, denoted by f-l(B},

is the subset of X defined by

f-l(B) = {z /f(z) e B}

It shall be our convention to omit unnecessary parentheses whenever possible

In regard t.o RinglctonR, for iIlRt,allce, we Rhl\lI write dirc(~t l\n<1 inverse images

lUI f(:z:) ami J-I (x), fnUlI'r UIILII J( :r; ) nncl r I ( (x:) TIII~ HI,II(hmt who worriuH about notlltion nmy fcelllOnll'what UIICIJ.'W about this double usc of the symbol

f-1• The abulIC of notation Ilhould not cauRe any confusion, howev<lr, for in any given eontext it should be perfectly clear how f-I ill to be interpreted Certain properties of the function f may be conveniently characterized in terms of inverR<l illlllg(~R To he more (~xJllidl., J iR 1\ onc-t.o-QII(! fun<~tion if and only if the invcrHC imagc! of each clement of III ill a Ringleton Whereas j maps onto Y if and only if the inverse image of eaeh nonempty subset of Y is non-empty

We shall now preRC'nt two short theorems which will establish the re\.ationship between direct and inverse images; these results are rather trivial but will be essential for later study In the two theorems, supposc f to be an arbitrary function from the sct X into the sct Y

Theorem 1-8 For each subset B !;; Y,

J(rl(B») !;; B

this, it follows that f(a) e B, and conscquclntly b e B

Corollory If, in addition, j maps onto the sct Y, then

Proof In view of the indusion proved in the theorem, we need only establish that, under the existing hypothesis, B !;;f(j-I(B») For this, let be B; then,

as j is by supposition an onto function, b = f(a) for some choice of a in X

Since a ej-l(B),

80 that

1-2 FUNCTIONS AND ELEMENTARY NUMBEU THEORY 19 Theorem 1-9 For ea.ch subset A k X,

Proof The proof is almost obvious, for, if a e A, thenf(a) ef(A); hence,

Corollary If, in addition, f is a one-to-one function, then

Proof H Jllninly HllfIlI'I'H 1.0 I'HI.nhliflh t.hn 0111' il",hlllicm , - I <t(A») !;;; A To

HLILI'L with, 11'1 II Ef I (j(A») We then hlWt! f(a) ef(A) ILUU thUl"f(a) = f(a')

for some a' in A Since' is assumed to be one-to-one, a = a' and 80 a e A

Before terminating this section, it may be well to review, quickly, some of the facts from number theory which we shall require later Most of these results depend on the HC)-(~nlll!(l Wcll-Ordnring Prirwiplc:

Well.Orderlng Principle Every nonempty subset B of nonnegative integers

contains a smallest element; that is, there exist ~ some (unique) element

a e S with a ::; b for all b E S

Let us start with the following result

Theorem 1-10 If a, be Z, with b > 0, then there exist unique integers

q and r such that

a= qb+ r, 0::; r < b

Proof We begin by proving that the set,

S = {a - xb I x e Z; a - xb ~ O},

is not empty Since b ~ 1, lalb ~ lal, and

a - (-Ial)b = a + lalb ~ a + lal ~ O

Hence, for x = -Ial, a - xb E B By the Well-Ordering Principle, S contains

a smallest integer, say r In ot.her words, there is somc q e Z for which

Corollary (Division Algorithm) If 0, b E Z, with b ¢ 0, then there exist

unique integers q and r such that

Let us now make the following definition

Deftnitlon 1-15 Let a, b E Z, with 0 ¢ O Thc integer a is said to divide b,

or 0 is a divi81w of h, in JoIymhols a I h, provid<.'<.i there exists HOme c E Z such that b = ac If a does not divide b, then we write arb

When the notation a I h is employed, it is to be understood (even if not plicitly mentioned) that a ¢ O

ex-Some immediate consequences of this definition are noted below; the reader

is asked to verify each of them

Theorem 1-11 Let a, b, c E Z Then

1) 0 10, 1 I a, a I a,

2) a I ±1 if and only if a = ±1,

3) if a I b, then ac I be,

1-2 21

4) if a I band b I c, then a I c,

5) a I band b I a if and only if a = ±b,

6) if c I a and c I b, then c I (ax + by) for every x, y E Z

From (1) above, we see that every integer a ¢ 0 is divisible by 1 and a,

diviflOrH whidl ar(! fn'IJIlI'IIUy f(·f(·rn,d t.o II.H illlpropl'r lliviH()fH An int.eger

a > 1 having no divisol'H other than the improper ones is said to be a prime

number; all integer a > 1 that is not prime is termed composite Thus, according

to our definition, 1 is neither prime nor composite In particular, an integer

a > 1 is composite if and only if there exist integers b, c with a = be, 1 < b < a, l<c<a

If a, bE Z, we sayan integer d ill a common divisor of a and b if d I a and

d I b Also,

Definition 1-16 Let a and b be integers, not both of which are zero The

{JreairHt r()mmon tiiviR01' of a I\lId b, dell()u,d by Ked (a, b), is t.he positive teger tl Ilueh t.hnt

positive integers d and d' whieh satisfy the conditions of Definition 1-16 Then

by (2), we must have did' as well as d' I d, whence d = ±d' [Theorem 1-11(5)]

Since tl and d' are hoth positive integers, it follows that d = d' Thus, the

greatest COlIllllOII diviHor of a and b is unique, when it exists The following theorem will prove that any two integenl, which are not both zero, actually

do have a greatest common divisor

Theorem 1-12 If a, b are integers, not both of which are zero, then ged (a, b) exists; in fnet, t.here ('xiRt integerR x II.l\d y such that

gcd (a, b) = ax + by

Proof First, define t.he set S by

S = {au -I-bv I u, v E Z; au + bv > O}

This set S is not empty For example, if a ¢ 0, the integer lal = au + bO

will lie in S, where we ehoose u = lor -1 according as a is positive or tive By the Well-Ordering Principle, S must contain a smallest element d > 0;

nega-that is to say, there exist x, y E Z for which d = ax + by We assert that

d = gcd (a, b)

22 PRELIMINARY NOTIONS 1-2

From the Division Algorithm, one can obtain integers q and r such that

a = qd + r, ° ~ r < d But then r will be of the form

r = a - qd = a - q(ax + by)

= a(1 - qx) + b( -qy)

Were r> 0, this representation would imply rES, and contradict the fact that d is the least integer in S Thus r = 0, so that d I a A similar argument establishes that d I b, making d a common divisor of a and b On the other hand, if e I a and e I d, then by Theorem 1-11(6), e I (ax + by), or rather,

c I d From these two statements, we conclude that d is the greateflt common divisor of a and b

It may be well to record the fact that the integers x a.nd y in the tion gcd (a, b) = ax + by are by no means unique More concretely, if a = 90 and b = 252, then

representa-ged (90, 252) = 18 = (3)90 + (-1)252

Among other possibilities, we also have

18 = (3 + 252)90 + (-1 - 90)252 = (255)90 + (-91)252

There is a special case of Theorem 1-12 which will play an important role

in the future; while it is, in effect, a corollary of the foregoing result, we shall single it out as a theorem But first, a definition: two integers a and b, not both of which are zero, are said to be relatively prime (or prime to each other)

if and only if gcd (a, b) = 1 For instance, the integers 8 and 15 are relatively prime, although neither is itself a prime

The.Nm 1-13 Let a, ,) E Z, lIot both zero Th(m a Ilnd b are relatively prime if and only if there exiflt integers x and 11 such that

1 = ax + b y ) Proof If a and b are relatively prime, 80 that gcd (a, b) = 1, Theorem 1-12

guarantees the existence of x and y satisfying 1 = ax + by Conversely, suppose 1 = ax + by for suitable x, y E Z and that d = gcd (a, b) Since

d I a, d I b, Theorem 1-11(6) implies d I (ax + by), or rather d 11 Because d is positive, this forces d = 1 [Theorem 1-11(2»), as desired

In light of Theorem 1-13, one may easily prove

Theorem 1-14 (Euclid's Lemma) If a I be, with a and b relatively prime,

Proof Since gcd (a, b) = I, there exist integers x and y for which 1 = ax + by

MUltiplying bye, wc obtain

e = (ax + by)e = a(cx) + (be)y

1-2 FUNL'TIONS ANI) ELI!:MENTARY NUMB.m TIIEOItY 23

Now a I a trivinlly ILlld a I be by hypothcNi8, 80 that a IlIU8t divide thc sum

acx + bey; hence a I e, as asserted

Corollary If p i8 a prime and pi (ala2' a,,), then p I at for some k,

1 :::; k:::; n

Proof OUf proof is by induction on n For n = I, the result obviously holds Supposc, al:! the induction hypothesis, that n > 1 and that whenever p divides

a product of less than n factors, then it divides at least one of the factors of

this product Now, let pi (ala2' an) If p divides at, there is nothing to prove In thll contmry I!MC, p and al arc rdativcly prime; hence, by the theorem, p I (a2' an)· Since the product a2' • an contains n - 1 factors, the induction hypothe8is implies p I at for some k with 2 :::; k :::; n

Having developed the machinery, it might be of interest to give a proof of the Fundamental Theorem of Arithmetic

Theorem 1-15 (Fundamental Theorem of Arithmetic) Every positive integer

a > I I~nll hll (lxpreHllCd u '\ a P1"lltluct of prime!! i thi!! rcpretIClltu.tioll i8 unique, apart from the order in which the factors occur

Proof The first part of the proof-the existence of a prime factorization-is proved by induction on the values of a The statement of the theorem is trivi-ally true for the integer 2, since 2 is itself a prime Assume the result holds

for all positive integers 2 :::; b < a If a is already a prime, we are through; otherwise, 4 = be for suitable integers b, e with 1 < b < a, 1 < e < 4 By the induction hypothesis,

b = PIP2" • p"

with P., p, nil prime!! Hut theil,

a = be = PI' PTP~ .• p!

is a product of prinles

To establish uniqueness, let us supposc the integer a can be represented as a

product of primes ill two ways, say

where the Pi and qi are primes The argument proceeds by induction on the integer n In the case n = 1, we have a = PI = ql(q2' • q ) Since PI is prime, it possesses no proper factorization, so that m = 1 and PI = ql

Next, assume n > 1 and that whenever a can be expressed as a product of less than n factors, this representation is unique, except for the order of the factors

From the equality PIP2 Pn = qlq2' • q , it follows that PI I (qlq2' •• q )

Thus, by the preceding corollary, there is some prime qk, 1 :::; k :::; m, for which

PI I qki relabeling, if Iwccssary, we may suppose, PI I ql But then PI = qt,

24 PRELIMINARY NOTIONS 1-2

for ql has no divisors other than 1 and itself Canceling this common factor,

we conclude 1'2 • PA = q2 • qm' According to the induction hypothesis, a product of n - 1 primes can be factored in essentially onc way Therefore, the primes Q2, •• , qm are simply a rearrangement of the primes 1'2,'" , PA'

The two prime factorization!'! of a are thus identical, (~ompleting the induction step

An immediate consequence of this theorem is the following:

Proof Assume the Rtatement is false; that is, assume there are only a finite

number of primes PIr 112, ,1'" Consider the positive integer

a = (1'11'2' PA) + 1

None of the primeR Pi divides a If a were divisible by PIr for instance, we would then have p, 1(0 - P,1'a'" 1'A) by Theorem 1-11(6), or PI 11; this is im-possible by part (2) of the same theorem But, since a > 1, the Fundamental Theorem asserts it must have a prime factor Accordingly, a is divisible by a prime which ifol not amonK our list of all primes ThiR lugument shows there

is no finite listing of till! prillle int<~gerH

This comJlI(!t<~H our Hurvc~y of HOmel of till! fUluJu.numtlLl notion!!! c:onceming sets, functions, and arithml'tie in thll int,egers Although the treatment was purposely sketchy, it is hoped that the reader did not find it too superficial In the subsequent chapters of the text, we shall utilize the foregoing concepts by applying these ideas to certain specific! situations

PROBLEMS

1 Let I: X ~ }' be an arbitrary function Define a relat-ion in the Ret X as follows: for any two elementR x, y E X, x,." y if and only if I(x) = I(Y) Verify that ,."

is an equivalence relation in X and describe its equivalence c19.!l8e8

2 Show that the ordered triples (a, b, c) and (a', b', c') are equal if and only if a = a',

1-2 FUNCTlO!IIS AND ELEMENTARY NUMnER THEORY 25

6 Let I, g and h hE' fundionl! from R' into R' with Rio ~ D, n /)/1' PrO'!e that

a) Show t.hat /I 0 /'.0 = lab

b) For a ;o! 0, prove that I"" ill both one-to-one and onto

c) For a ;o! 0, obtain 1 1•

9 Using fllnl'tions/: R' R', give an example of a function which ill

a) one-to-one but not onto,

b) onto but not one-to-one

10 For functionR g: X -+ }' and I: Y -+ Z, flhow that the following statements are true:

a) If I 0 ( / iH all nnto funl!tioll, then I ill 8\1111

b) If log i, 8 (1ll1 t~l-Illl(l fllnl't.ion, th(\n g ill aillo

n) If I nlld 0 urI' hnth nnn-tn-nlle fUllnt.jllllll, t.Iwn log iH I~IHCI nnu-tn-one and (jo g)-I = g-I 0/-1

11 Establish the following characterizations for any function I: x Y:

a) I ill onto Y if and only if for all functions g, h: Y -+ Z, g I = hoi implies that g = h

b) I is one-to-one if and only if for all functions g, h: Z -+ X, log = , 0 h implies that g = h

12 If I: X Y, g: Y - Z and A ~ X, prove that

26 PRELIMINARY NOTIONS 1-2

15 If I: X -+ Y, prove that I is a one-to-one function if and only if

I(A n B) = I(A) nf(B)

for all set.'! .1, B ~ X

16 Given intt'.j/;ers a and b, which arc not hoth zero, establish the following facts concerning gcd (a, b):

a) p;cd (a, -b) = gcd ( -a, b) == ged (-a, -b),

b) whenever a ;o! 0, p;ed (a, 0) == lal,

c) p;ed (a, b) = lal if and only if a I b,

d) p;ed (ea, eb) = lei ged (a, b), provided c ;o! 0,

e) ged (a, b) = ged (a, b + ea), for every c E Z

17 Prove that if a, b, e arc integers, no two of which arc zero, then

11:1'" (gl'" (a, b), c) g('d (a, p;1:d (b, c»

= gcd (I/:cd (a, c), b)

18 Prove the two 8I!.'lCrtions below:

a) If lI;ed (a, b) = 1I;"d (a, c) == 1, then gcd (a, be) 1

b) If gcd (a, b) = 1, a I c and b I e, then ab I c

]9 Let a and b be intep;erR, not both zero The least common multiple of a and b,

denoted by lem (a, b), iH the positive integer e such that

1) a I e and b I e; that iH, e is a multiJlle of both a and b,

2) if a I e and b I c, tht'n e I c

Show that the least common multiple of a and b is related to the greatest common divisor of a and b by

(11:m (a, b) )(gl:d (a, b» == labl

20 Let a, b, c E Z, with a and b not hoth zero, and let d = ged (a, b) Verify that there exiMt intcgcrM x and 1/ Hueh that

ax + by = c,

if and only if die

CHAPTER 2

GROUP THEORY

2-1 DEFINlnON AND EXAMPLES OF GROUPS

In this I!hapter, nlld t.hroughout the renllLind(~r of thc text" we shall deal with mathemntielll HYHtemK whidl ar(~ d(~fined hy a preserihcd list of properties Emphasis will be on deriving theoremll that follow logically from t.he postulates and which, at the same time, help to describe the algebraic structure of the particular system under consideration This axiomatic approal!h not only penn its us to concentrate on essential ideas, but also unifies the prescntation

by showing the basic similarities of many diverse and apparent.1y unrelated examples

We first confine our attention to systems involving just one operation, since they are amenable to the simplest fornlal description Despite this simplidty, the axioms permit the construction of a profuse and elegant theory in which one encounters many of the fundamental notions common to all algebraic systems

Before beginning, however, it iH necessary to arrive at some understanding conceming the UKe of the equivalClwc relation = We will henceforth take the equality Hign t.o mean, intuitively, "is the same as." In other words, the Hymhol = aHK(~rt.K thnt the two pllrtieular expreHllions involved arc merely differcnt nameH for, or descriptions of, one and the HaOle objcct; jUHt onc objed

is being conllidered, and it ill named twice To indicate that a and b arc not the same object we shall, naturally enough, write a F- b

As a fimt steJl in our program, we introduce the concept of a binary operation This idea is the cornerstone of all that follows

Definition 2-1 Given a nonempty set S, any function from the Cartesian

produet S X S int.o S is "ailed 1\ binary operation on S

A biliary op('rntion Oil S thuH 1l 'i."IigIlK to ('ad I ordl'J'('(\ pair of clements of S

a uniquely determined third element of the same sct S For instllnee, if P(A)

denotes the power set of a fixed set A, then hoth U Ilnd n are binllry operations

on P(A) In praetice, we shall generally use the symbol * to represent a binllry openlt.ioll lind write a * II, inst.ead of *(a, IJ)), for its vlLlue nt the ordered pair (a, b) E S X S While this eonvcntion is at varimlee with the functional notation developed in the previous ehapter, its use in the prescnt

27

28 W({JUl' TH~;OHY 2-J situation is dietated by long-standing mathematical tradition At the very least, it has t.he Ildvantage of avoiding some rather clumsy notation

From tim<.' to timl~, we shall permit ourselves to make such informal ments us "('ombine a with b" or "form a * b." In a precise sense, what is really meant of ('ours!' is to upply the fUlwtion * to the ordered pair (a, b) The most uscfulaspcet of a binary opcmtion is that, having once formed the clement

state-a * b, we may ill turn combine it with other members of S; the result of all sueh ealeulatiolls again lies in S

Needless to :;ay, the particular notation used for the abstract product of two clement:; is of no great importance On occasions ROme other symbol, as equally nOlICollllllittal as *, will he employed Specifically, we will frequently

('hooSt· to write a 0 IJ in phwe of a * b (in this context, the symhol 0 is not tended to hav(' allY Sll('pial (·onlleet.ion with fUlwtiQlml eomposition) In gelleral,

in-a and IJ will havl' 110 llulllI'J'ieal valul' hut will simply Iw arbitrary element.!! in our lIIulpriyilll!; SP! S, whatever this set may be, while * may well be some law of ('ompo:;itioll whid! bear:; no resemblance to the usual operations of

!'1('lnl'lltllry :dp;I'I>rIt

Closely aliiI'd to the notion of It binary opl'ration is the so-called closure condition For It formal statement of this property, suppose that * is a binary operation 011 the set S nnd A ~ S; the subset A is sllid to be closed under the

opcmtion • Jlmvided a • b E A whenever a and b are in A The desirable feature here is that when A is closed under the operation , the restriction of • to the

subset A is Il binary operation on A as well as S

Example 2-1 Ordinary slIbtra!'tion is dearly a binary operution on the set

Z of illti!g!'l'H; the SUhH!'t Z+ of pOtlitive integer!!, however, is not closed under

Conversely, such a table could equally well serve to define a binary operation

on 8, for the re8ult, of ('ombining any pair of elements of S would be displayed somewhere in tllP tahle

Example 2-2 A binal'Y ·operation • may be defined on the three-element set S = (1,2, 3} by means of the operation table below:

1 1 2 3

2 3 1 2

3 2 3 1

2-1 In:FINITION AND ~:XAMJ>I :I'\ OF <moups 29

According to the t.able, the product 2 • 3, for inst.ance, iR equal to the clement 2, located at the interRection of the row marked 2 and t.he column marked 3 Given an arbitrary binary operation , there iH ('ertninly no ren.son to expect that a • b will be the Rllllle IlS 11 • a for nil a Ilnd b In faet, it can be seen in the

above example that 1 • 2 = 2, whereas 2 1 = a One must consequently take care to refer to a • b as the product of a and b and to b • a as the product

of b and a; the distinction is quite importallt We should also point out that it is obviously possible to combine an element with itself Thnt is to say, a • a

can be defined

Deflnition 2-2 By a mathematiml .~y.~lcm (01' malhrmatical structure), we shldl mean Il IlOTwmpty s!'t of (·Ienwnls t.og('ther wilh olle or more biliary operat.ions d!'fined on t.his set

A maf,Ju'mal iC'al syslpm ('onsist.ing of Uw set S ILnd It single binary operation will be d('llot\'d hy tlw ordered pair (8, ); analogously, a system eonsi:4ing of

the Ret S lllld two operat.ions • awl will hI' I'l~pr<'Hcnl,('(l by the ordered triple

(8, , 0)

Example 2-3 The pair (8, ), where the set S = {I, -1, i, -i} and the operation is that of mdinary mult.iplication, is a mltth('nmtieall'!ystem provided one defines i 2 = -1

Example 2-4 If Ze and Zo denote the even and odd int<'gers, respectively, then (Z., +, ) ('onstit.utes a mathematical system, while (Z., .+, ) cloes not

In the latter caRe, Ow set Z ill not ('Ios(>d undN addition, sin('l' t.he sum of two odd integerl'! is lweessarily even

The systemH to be studied Hlloscqtwntly arc dall8ificd aecording to the ties they possess or, to put it anot.her way, accordillg to the axioms they satisfy Our object will be to present a sequent.ial development of the principal mathe-matical Hystems of modern algebra, beginning with t.hose involving relatively few axioms and progressing to systellls sat.isfying more detailed hypotheses The axioms whieh form the starting point of the abstraet theory can be, by nature, ratlwr varied The growing t.endeney of modern mathematies is to isolate almost any convenient set of propl'rtie~ from its original context, to

proper-define a parti(,ular system, llnd to develop the eorrcspollding Ilbstract theory through logical d(~dul'tion ROllle of tlH'se forlllal :txiolllat ie theorie~, sueh as the notion of It group, have It fundamentlll irhportalH'(' to the whole of mathe-matics and have been instrullll'ntal in IInifyillg various apPllrently unrelated branches; other I heories, while sat.isfying the esHwt i(' and inquisitive needs of the mathemati('ian, are limited in the extent of their applicability We do not mean to create the impression that it, is the usmil praetiec for one to define a new system by arbitrarily (apart from logical (,oll~iderntions) writing down axioms Although thpr(' is no part.icular np('('ssity for the model to precede the

thcorct~al development, in most cases the axioms nrc the abstract realization

general, ill poor in (~ont<mt Certain rcQ.HOnahlll limitations must be impo8(~d

on the opcmt,jon if one itJ to obtain utlCful rctJulttJ In the following pllragrnphH,

!lOme of the mOM) bailie requirements arc named and brieRy examin(.'<i FOI' conciseness, we shall hereafter omit the word "binary" inasmuch as every operation to be considered will necessarily be binary

Given a mathematical system (8, *), the symbol a * b * e is at the moment completely meaningless, since the operation • has only been defined for paim

of clementI! of S If, however, we make t1w IItipulntion that whenever quantiti(!H are enclotJed in pnrenthctJCs these are to be evaluated first, then both the expres-sions a * (b * c) and (a • b) • e make tlCn8(~ Namely, a * (b • e) is to be inter-preted as: eombilw a with what rCfllllttJ fmm comhining b with e; while (a * b) * e

is to be interpreted atJ: timt combine a with b and then combine the result with e Of course, the resulting elements a * (b * e) and (a * b) • e will not necessarily be the same

Definition 2-3 The operation * defined on the set 8 is said to be

lJ88Oeia-tive if

a * (b • c) = (a * b) * e (associative law),_

for every triple, distinet or not, of elements a, b, and e of 8

Example 2-5 The operation of subtraction on the set R' of real numbers is not associative, since in general

a - (b - e) F (a - b) - e

Example 2-6 An associative operation * may be defined on Z, the set of integers, by taking a * b = a + b + abo (We shall frequently delete the dot and write the product of a and b under ordinary multiplication simply as ab.)

= (a + b + ab) -I-e + (a + b + able

The equality of these two expressions follows in part from the fact that addition and multiplicat.ion Ilrc themselves associative in Z

When del~ling with a Hystem who8() operation is defined by a mUltiplication table rather than II formula, it is generally quite tedious to establish the associa-

2-1 1l1<:1<'INITlON ANI> I<:XAMI'LES 01<' OROUPS 31

tivity of the operation, for one must compute all possible threefold products

On the other hand, it may be fur easier to show thltt the operation is not a.'lROciativc, as all we need do in thi::; ea.'lC is find three pnrticular clements from the underlying Ret fOf whieh the nS!lO(~iative law fails

Example 2-7 Conl'lid(!f the I)JI(!MLI.iuli • t11!fiw,t1 on the set S = {l, 2,' 3} by the operntion tlLllle:

• 1 2 3

1 1 2 3

2 3 1 2

a 2 :J From t.his I.allll!, we !j(!(' that 2 (1 • :J) = 2.:J = 2, whereas (2 1) • a =

3 * 3 = ]; I.hnt, is,

2 • (1 • a) ~ (2 1) 3

The associative law thus fails to hold in the system (8, )

The mathematical system which we shall use to build up more complieated nlgehmil! llt.rudufCS ill known 11."1 a semigroup

Definition 2-4 A Bemi(JTOUl1 ill a paif (8, ) (!onllillting of a nOnCml)ty scI 8

together with an associative (binary) operation defined on S

Let us stress t.hat it is an abuse of lnnguage to say a certain sct alone is a semigroup without also specifying the operation involved, as it may be quite possible to equip the same set with several associative operations For this reason, wc have utilized the ordered pair notation to indicate both the operation and the underlying set of elements

Observe that since any three elements from the set of a semigroup always associate, there is no particular reason for parentheses Consequently, when dealing with sueh It system, the I!ymbol a • ~ • c has meaning in the sense that

we are frL'C to interpret it either as a • (b • cj or as (a * b) • c More generally, the notlttion al • a2 • • • am is unambiguous, for it can be shown that all ways of inserting parentheses 80 as to give this expression a value yield the same felmlt, (Theorem 2-4) An opemtion whieh il! not as80ciative has the dedded disadvantage thtLt the notation for multiple-factored products can be(!ome quite unwieldy 11." a result of the eonstltnt need for parentheses

In order to ROlidify the notion of a semigroup, we present several examples

Example 2-8 There Itrl! severnl semigroups wit.h whi('h t,he reader is already familiar If, for insta.nce, Z+ denotes t.he set of ull positive integers, then both

the pairs (Z +, +) and (Z +, ) form semigroups Similar statements hold for the sets Z, Q, and R'

so that (R', *) satiRficR the requirempnts of a scmigroup •

Example 2-10 For IlIlY XC!! X, eueh of UIIl HYH!.mnH (I'(X), u) Ilnd (I'(X), n)

cOIlt!titutCR It 1-!!'luigrouJl (Tlworem 1-2)

Example 2-11 1,<'1 X bl! It lIollcmpty R(~f ulld S he t,l1P (!olledioll of all funetion8

f: X -+ X If· dl'lIot.I'1-! flllU'tiollal 1'()1IIJ10I-!itioll, t.hell the pair (8,0) provides another iIIustrat.ion of It semigroup (Problem !i, Seetion 1-2)

As we Rhall Rubs<''1uentiy SI'I', the relevance of the semigroup concept lies in the fact that many important systems contain the semigroup structure as a subsystem

We have already indicated that the order in which elements occur in a product

is quite essential If it is pos -;ible to interchange the order of eombining any two elenwnts from our set without affecting the result, then the operation is termed commutativ('

Definition 2-5 The operation * defined on the set S is called commutative

if

(commutative law),

for every pair of clements a, b E S

Examples 2-8, 2-9 and 2-10 arc of commutative semigToups (semigroups whose operation is commutative), while in Example 2-11 functional composition

is not, in grneral, a commutlttive operation Although the commutative law may fail to hold throughout an entire system, it may still be valid for particular pairs of elemrnts; accordingly, it will be convenient to make the following drtinition

Definition 2-6 Two clt'mrllts a and b are said to commute or permute

(with each other) provided a * b = b * a

Employing this terminology, we observe that the opemtion of the system

(8, *) is commutative if and only if every pair of clements of S commute Once an operation has been defined on a set, one finds that certain elements play special rol(,l-!; t1H'rl' may ('xist identity clements and inverse elements

Definition 2-7 The system (S, *) is said to have a (two-sided) identity elelllrnt for the oppmf.ioll * if there exist~ an elenwnt e ill S such that

a*e=e*a=a