Modern abstract algebra david burton

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
Menlo PIITk, California

.

London

.

Amatflfdem

www.pdfgrip.com

•

ReedIng, M8ItIaChuIItItIt
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

www.pdfgrip.com


PREFACE

This book has been written with the intention of providing an introduction to
model'll aht:!t.ract algebra for mathematics major8. The r('.ader is not presumed

at thc Outllct to possess any previout:! knowledge of the concepts of modem
algebra. Accordingly, our beginuing is somewhat elementary, with the exposition
in the earlier sections proceeding at a leisurely pace; much of this early material
may be covered rapidly on a first reading. An attempt. has been made to keep
the book as self-contained as possible. To smooth the path for the unexperienced
reader, the fir8t chapter is devoted to a review of the basic facts concerning
sets, functions and number theory; it also serves as a suitable vehicle for introducing some of the notation and terminology used subsequently.
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 unpalatable, 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 Representation 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

www.pdfgrip.com


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 encouragement 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.

www.pdfgrip.com


CONTENTS

Chapter 1
1-1

Preliminary Notion.

The Algebra of Sets

1-2 Functions and Elementary Number Theory
Chapter 2

2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
Chapter 3

3-1

3-2
3-3
3-4
3-5
3-6
Chapter 4
4-1

1
13

Group Theory

Definition and Examples of Groups .
Certain Elementary Theorems on Groupf!
Two Important Groups
Subgroups
Normal Subgroups and Quotient Groups
Homomorphisms
The }<'undamental Theorems .
The Jordan-Holder Theorem .
Sylow Theorems

27
41
52
64

75
89

103
117
128

RI. . Theory

Definition and Elementary Properties of Rings
Ideals and Quotient Rings
Fields
Certain Special Ideals .
Polynomial Rings
Boolean Rings and Boolean Algebras

141
156
'172
183
196
219

Vector Spec..

The Algebra of Matrices

4-2 Elementary Properties of Vector Spaccli
4-3 HaMes a.nd Dimension
4-4 Linear Mappings

235
249

263
277

Selected Reference.

297

Inde. of Special Symbol. and Nototion. ,

299

Inde.

305
vii

www.pdfgrip.com


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 COIlventions 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 exposition 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 subjeet. 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

www.pdfgrip.com


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 encouragement 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.
..n.M.B.

Durham, New Ifampshire
March 1967

www.pdfgrip.com

CONTENTS

Chapter 1

Preliminary Notion.

1-1 The Algebra of Sets
1-2 functions and Elementary Number Theory
Chapter 2

2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
Chapter 3

Group Thoory

Definition and Examples of Groups .
Certain Elementary Theorems on GroupR
Two Important Groups
Subgroups
Normal Subgroups and Quotient Groups
Homomorphisms
The l!'undamental Theorems .

The Jordan-Holder Theorem .
Sylow Theorems

Definition and Elementary Properties of Rings
Ideals and Quotient Rings
Fields
Certain Special Ideals.
Polynomial Rings .
3-6 Boolean Rings and Boolean Algebras

4-1

27
41
52
64

75
89
103
117
128

Rlnt Theory

3-1
3-2
3-3
3-4
3-5

Chapter 4

1
13

141
156
'172
183
196
219

Vector Spac..

The Algebra of MatriceR

4-2 l!~lementary Propertie.'l of Vector Spaces
4-3 U8l!C1I and Dimension
4-4 Linear Mappings

235
249
263
277

Selected Reference.

297

Indox of Spoeial Symbol. and Notation. .

299

Index

305

vii

www.pdfgrip.com


2

I-I

PRELIMINARY NOTIONS

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 additional 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
~ 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.

A

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,

www.pdfgrip.com


1-1

THE ALGEBRA OF SETS

d) if A 5;; Band B 5;; C, then A 5;; C,
e) A 5;; Band B 5;; A if and only if A

=

3

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 demonstrates 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 intersection 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

and the set B

=

= {O, 1,2,3,4,5, 6}, the set A = {I, 2, 4},

{2, 3, 5}. Then

A uB = {1,2,a,4,5},

A

n JJ

=

A - B

{2},

and

B - A

=

{3,5}.

Also,
-A = {O, 3, 5, 6},

-B

=

{O, 1,4, 6}.

Observe that A - B !lnd B - A are disjoint.

www.pdfgrip.com

=

{t,4},

4

1-1

PRELIMINARY NOTIONS

In the following theorem, some simple consequences of the definitions of
union, intersection, and complementation are listed.
Theorem 1-2. If A, B, and C are subsets of some universe U, then
a) A U A = A,
AnA = A,
b) A uB = B u A,
A nB = B nA,
e) A U (B u 0) = (A u B) u
A n (B nO) = (A n B) n C,

c,

d) A u (B nO) = (A u B) n (A
A n (B u C) = (A n B) u (A
e) A U 0 = A, A n 0 = 0,
f) A u U = U, A n U = A.

u C),

nO),

We shall verify the first equality of (d), since its proof illustrates a technique mentioned previously. Suppose that x E Au (B n C). Then, either

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
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 operations of set theory.

C

~

www.pdfgrip.com


1-1

THE ALGEBRA OF SETS

5

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

a) -(A U B) = (-A) n (-B),
b) -(A n B) = (-A) U (-B),
c) if A ~ B, then (-B) ~ (-A),
d) -(-A) = A, _o~ = U, -V = 0,
e) A u (-A) = V, A n (-A) = 0.

To give the reader a little more familiarity with set-theoretic argument, we
shall establish the first of the above assertions. For the proof, let x be an
arbitrary (!Icmcnt of -(A U B). Then x ~ A U B. Hence x is in neither A
nor n. Thill implicH t.lmt x E - A ami x E 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
0_0

xE-(AUB).
We consequently have the inclusion (-A)
desired equality holds.

n (-B)

~

-(A

u B)

and the

The first two parts of the above theorem are commonly known as DeMorgan'8
rules.
There will be occasions when we wish to consider sets whose elements themselves are sets; in order to avoid the awkward repetition "set of sets," we shall
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 2 n 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·
aI:I

POWC1'

tid of A, whirh has

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}.

www.pdfgrip.com


6

1-1

PRELIMINARY NOTIONS

It is both desirable and possible to extend our definitions of union and intersection 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,

ua =
na =

{x / x E A for some set A E a},
{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+}

=

n{A .. / n E Z+}

=

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

At,

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 component a and lK!cond component b, denoted by (a, b), ill the set

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

www.pdfgrip.com


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 unordered 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
= e and b = (l.

a

Proof, If a

=

e and b

=

{a}

d, then it is clear from Definition 1-1 that

=

{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 distinguish two possible cases:
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.

www.pdfgrip.com


8

1-1

PRELIMINARY NOTIONS

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" b2, .•. , bn )

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 XB

=

{(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,

A X B = {(-I, 0), (-1,2), (0,0), (0, 2), (1,0), (1, 2)},

while
B XA

=

{CO, -1), (0,0), (0, 1), (2, -1), (2,0), (2, I)}.

Clearly the sets A X Band B X A are not identical. In fact, A X B = B X A
if and only if A = B.

By a (binary) rrlatinn in a nOllf'lIIpt.y Ret A if:! meant a subset R of the CartcRian
A X A. H t.h(~ e1elllent (a, b) E R, wc KUY thllt a iH related to b with
respect to the relat.ion R and write aRb. For instance, the relation < in R'
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.
prod\l(~t

www.pdfgrip.com


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 equivalence relation in A.

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.

www.pdfgrip.com


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 inclusion [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:
n Zo = 0. Another partition of Z might be the sets Z+
Z = Z. u Zo,
(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.

z.

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 A j . Th(' reader may check that the relation -, so defined, is

www.pdfgrip.com


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
[(a, b)J

=

{(c, d) I a - c

=

b -

d}.

This set may be represented geometrically as a straight line with slope 1 passing
through the point (a, b). Therefore, the relation"", partitions A into a family
of parallellinel!.

PROBLEMS

In the following excrciscl! :I, B, and C are subset!! of some universe U.
I. Provo t.hat. A n B ~ .. I U B.
2. Suppose ,1 ~ B. Hhow that
\ a) A n C ~ B n C,

b) A U C ~ B U C.
3. Prove that A - B = A n (-B), and use this result to verify each of the following identities:

0 = :I, 0 - A = 0, ..t - .1 = 0,
b) A - B = A - ( ..1 n B) = (11 U B) - B,
c) (11 - B) n (B - A) = 0.

a) A -

4. Simplify the following expressions to one of the symbols A, B, II U B, An B,
A -B:
a) A n (A U B),
b) A - (11 - B),
c)
n B) U (-.1».

-«..I

5. Prove that A n (B U C) = (A n B) U (A n C).
6. Establish the following reKults on differences:
a) (A - B) - C = A - (B U C),
b) .1 - (B - C) ... (II - B) U (A n C),
c) A U (B - C) = (:I U B) - (C - A),
d) .t n (B - C) = (A n B) - (.1 n C).

www.pdfgrip.com


12

1-1

PRELIMINAlty NOTIONS

7. The notion of set indw;ion may be expressed either in terms of union or intersection. To see this, prove that
a) AS:;; B if and only if .1 U B = B,
b) ..l s:;; R if and only if .1 n R = :1.
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,

n (A

A

U B) = A.

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

and

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

a = {.It, .h, ...} be a family of subsets indexed by the positive integers
Z+. Define a new family m = {BI, B2, ... } as follows:

]1. Let

Bl = "h;

Bn

=

An - U{.h I k

1,2, ... , n - I} for n

=

>

1.

Show that
a) the member" of Hare di!ljoint. !lets,

b)

ua

=

um.

12. For any thrlle ~ets .. t, JJ and C, establish t.hat
a) .1 X (HUe) = (.1 X H) U (:1 X C),

b) 11 X (B n C) = (.1 X B) n (.t X C),
r) A X (B - e) = (ot x B) - (A X C),
d) A X B = U{A X {b} I bE B}.

13. Classify earh of the following relations R in the set Z of integers as to whether
they do or do not have the properties of being reflexive, symmetric, and transitive:
a) aRb if and only if a < b,
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 = b2 ,
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.

+

+

www.pdfgrip.com


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.

f is a set of ordered pairs Buch that
pairs have the same first component. Thus (x, Yl) Ef and
(x, Y2) E J implies Yl = Y2.

Definition 1-9. A Junction (or mapping)

no two

distill(~t

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,

=
R, =

D,

{x I (x, y)

EJ for some y},

{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 component. We lihall oCC8.:-;ionally ohscrve the cOllvention of simply writing fx for
f(x).
Example 1-10. If the function f is the finite set of ordered pairs,

f = {(-1,0), (0,0), (1, 2), (2, I)},
then
D, = {-1,0, 1, 2},
and we write f( -1)

R,

=

{O, 1, 2},

= 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

J=

{(x,x 2 +2) IXER'}.

Using the functional value not.ation, onll would then wriw f(x) = x 2 + 2 for
each x E R'. Needles!! to say, there are funetions whose ordered pairs would
be difficult-if IIOt impoliliible-to express in terms of a formula. The diseerning
reader is advised to keep in mind the distinction between a function and its
values or, as the case may be, its formula; although the notation sometimes
leads to confusion, these concepts are obviously quite different.
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".

www.pdfgrip.com


14

1-2

PRELIMINARY NOTIONS

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 permissible (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:

+

(f gHx) = I(x) + ,,(x), )
(f - g)(x) = f(x) - g(x),
(I· g)(x) = f(x)(j(x),
(f/g)(x) = f(x)/g(x) ,

where xeD, n Dg
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.
Examp'e 1-11. SUppOtlC

f =

{(x,

V4 -

x2)

I -2 5

x

5 2}

and

g

=

((x,~) I R' ~ {O}},

so that
f(x)

Then for x e D, n Da

=

= "D,

-

v4 -

2
g(x) - -.
x

X2,

{OJ,

(f + g)(x)

=

v4 -

x2

2
+ x-,

(/ - g)(x)

=

v4 -

x2

-

_IT.r 2 )

(J. (J)(.r) -- (v <1
(f/g)(.r)

=

v4-x 2
2/x

2
-,
x

2
x'
x_~

= 2 v4 -

;1:2.

The function operations just eonsidered plainly depend on the algebraic
properties on the range; the domain merely furnishes the points for these
pointwise operations. The most important operation involving functions,

functional composition, is independent of such algebraic structure and relies
only on the underlying set.s.

www.pdfgrip.com


1-2

/

15

FUN<-'TIONS AND ELEMENTARY NUMB Ell THEORY

Definition 1-11. The composition of two functions / and 0, denoted by
0, is the function
0

/0 g =

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

Written in terms of functional values, this gives

I«(/(x»),

(f (I)(X) =
0

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 meaningful. Also, Dlo ll ~ DII and Rlo ll ~ RI .

Examp'e 1-12. IA't

I =

{(x,

vi) I x

E

R', x

~ O},

and

u=
so that/ex) =

vi, o(x)
(fog)(x)

= 2x

{(x, 2x

+ 3.

+ 3) I x E R'} ,

Then,

= /(g(x») =

+ 3) =

f(2x

v2x

+ a,

where
[x E DII I g(x) ED,}

D"II =

=

{x E R' I 2x

=

{x

I 2x + 3

+ 3 ED,}

~ O}.

On the other hand,
(0 o/)(x)

=

o(f(x»)

=

g(v'x)

=

2v'x + 3,

where
D llo,

Olin ObRf'rVeH t.Jmj,

=

{x E D, I/(x) E D II }

I· (I

iH dilTl'rC'lIt from

=

{x ~ 0 I v'x E R'}

= {xlx~O}.
(f Ii indeed, mn!ly
0

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
are defined, then
.
1)
2)

3)

(f. g) • Ii - 1 (y • It),
(f
y) • It = (f. h) + (g. It),
(f. g) • h = (f 0 h) . (g 0 h).
0

+

www.pdfgrip.com


16

1-2

PRELIMINARY NOTIONS

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

=

D,o/a

{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 }

n

Vll0/o

=

DUo/a)'(II0/a)

Now, for x E D("II)o/a, we have
[(f. g)

0

h](x) = (f. g)(h(x»

= /(h(x» . g(h(x»

=

(f h)(x) . (g h)(x)

=

[(f h) . (g h)](x),

0

0

0

0

whidl, Recording t.o UlP d(!finit,ion of equality of functions, Hhows that
(f. g)

0

h

=

(f 0 h) . (g

0

h).

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
useful. The formal definition follows.
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.

www.pdfgrip.com

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
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, however, to have[-t defined in terms of its domain and the image at each point of
the domain. Observing that

ax. -

{(3x-2,x)lxER'}
we choose to write

r

1

=

{(x,!(x+2»lxER'},

= {(x,!(x + 2) I x

In terms of functional vl\lueM, [-I(X)

E R'}.

= !(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 composition j i ... : A -+ Y is known as the restriction of [ to the set A and is, by established cllstom, denoted by f I A; dually, the funetion [ is referred to as an
extension of f I A to all of X. For the reader who prllfers the ordered pair
approach,
flA= {(x,y)l(x,Y)E[andxEA}.
0

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

[ IA

www.pdfgrip.com