PRODUCTS

Also observe that

a

= b � ^· a _I"J b; (20)

for if

a

_{= b, then a e}

a ==:}

_{a e b}

==:}

_{a � b.} Conversely, if a � b and c e

a,

_{then c � a}

and a � b _� c _I"J b _� c e _b. Thus

a C

_b; a symmetric argument shows that li

C a

and therefore

a

= b. Next we prove:

for a,b _E ^A' either

a

_n

h

= 0 or

a

= _li. (21) If

a

_{n b} � 0, then there is an element c _E

a

_{n b.} Hence c � a ^and c � b. ^Using symmetry, transitivity and (20) we have : a � c ^and c ,-...; b => a ,-...; b

==:} a

= b.

Let A be a nonempty class and { Ai I i e I} a family of subsets of A such that:

Ai � 0, for each i e I;

U A" = A ;

iel

Ai n ^{A; = 0} for all i ^{� j} _E ^I;

then { ^{Ai I} i e I} is said to be a ^partition of A.

Theore m _4.1. If A is a nonempty set, then the assignment R � A/R defines a bijec­

tion from the set ^E(A) of all equivalence relations on ^A onto the set ^Q(A) of all parti­

tions of A.

SKETCH OF PROOF. If R is an equivalence relation on A, then the set A/ R of equivalence classes is a partition of A by (1 8), (19), and (21 ) so that R � A/ R de­

fines a function f: ^E(A) _--4 ^Q(A). Define a function g : Q(A) --4 ^E(A) as follows. If

S = { Ai I i e I} is a partition of A, let g(S) be the equivalence relation on A given by:

a � b _<=> a e ^{Ai and} b e ^Ai for some (unique) i e I. (22) Verify that g(S) is in fact an equivalence relation such that

a

= Ai for a e ^Ai. Com­

plete the proof by verifying that fg = ^{1 ocA>} and gf = 1 EcA> · Then f is bijective by (13). •

5. ^PRODUCTS

Note. In this section we deal only with sets. No proper classes are involved.

Consider the Cartesian product of two sets ^{At X} A2• An element of At ^X A2 is a pair (a1,a2) with aⁱ e ^Ai, i ⁼ 1 ,2. Thus the pair (at,a2) determines a function /: { 1 ,2}

--4 ^{A1 U A2} by : /(1) ⁼ a_.^,f⁽²⁾ ₌ a2. Conversely, every function [ : { ^{1 ,2}} _--4 ^A1 U A2 with the property that /(1) e ^A1 and /(2) e ^A2 determines an element (a�,a2) ⁼ (/(1),/(2)) of At X A2. Therefore it is not difficult to see that there is a one-to-one correspondence between the set of all functions of this kind and the set A1 ^X A2^• This fact leads us to generalize the notion of Cartesian product as follows.

Defin ition _5.1. Let { ^{Ai I i} e I} be a family of sets indexed by a (nonempty) set ^I. The

(Cartesian) product of the sets Ai is the set of all functions f : I _--4 U Ai such that f(i) E Ai for all ⁱ e ^I. It is denoted

II

Ai . ial

ial

8 P R E R EQ U I SITES AN D PRELI M I NAR I ES

If I = { 1 ,2, . . . , n}, the product IT ^A; is often denoted by A1 X A2 X · · · X An and is identified with the set of all ordered n-tuples (i£1 ^aha2^, . . . , an), where ^a; e ^A; for

i = 1,2, . . . , n just as in the case mentioned above, where I₌ ^{{1 ,2}. A similar} notation is often convenient when I is infinite. We shall sometimes denote the functionfe IT A; by ^(a;}1ei or simply ^{a;}, wheref(i) = a; e ^A; for each i e I.

If some A i£1 1 = 0, then

II

Ai ⁼ 0 since there can be no function f: ^{I --4} U Ai

i!.l

such that /U) e Ai.

If l Ai I i e I} and t Bi I i e I} are families of sets such that Bi ^c Ai for each i e I, then every function l --4 U Bi may be considered to be a function l--4 U Ai. There-

�� �I

fore we consider

II

Bi to be a subset of

II

Ai.

i,] iF.]

Let

II

Ai be a Cartesian product. For each ^k e I define a map 7rk

: II

Ai ^� Ak

iF.] iF.J

by /� f(k), or in the other notation, { ad � ak. 7rk is called the (canonical) projec- tion of the product onto its kth component (or factor). If every _Ai is nonempty, then each

1T,_ is surjective (see Exercise 7 .6).

The product

II

Ai and its projections are precisely what we need in order to prove : iel

Theorem _5.2. Let ^{{ Ai I i} e I } be a family of sets indexed by ^I. Then there exists a set D, together with a family of maps { 7ri : D ^{----. Ai I} i e I } with the following property: for any set C and family of maps { 'Pi ^: C ----. ^Ai I i e I l , there exists a unique map q; ^: C ----. D such that 1ri'P ⁼ 'Pi/or all ^{i E L} Furthermore, D is uniquely determined up to a bijection.

The last sentence means that if ^D' is a set and { 1r/ ^: D' ---+ A, I i e /} a family of maps, which have the same property as ^D and { 1ri } , then there is a bijection D ----. D'.

PROOF OF 5.2. (Existence) Let ^{D =}

II

Ai and let the maps 1ri be the projec­

iEI

tions onto the ith components. Given ^C and the maps 'Pi-r define ^q; : C ---+

II

Ai by

i,]

c l---+ /c, where fc(i) ⁼ <Pi(c) e Ai. It follows immediately that 1ri<P ⁼ 'Pi for all i _e I. To show that q; is unique we assume that q;' : ^{C -+}

II

Ai is another map such that 1ri'P' ⁼ 'Pi for all i e I and prove that 'P ⁼ <P'. To do this we must show that for each i£1

c e C, (f?(c), and q;'(c) are the same element of

II

^{Ai -}that is, ^<P(c) and q;'(c) agree as

it! I

functions on J: ('P(c))(i) ⁼ (q;'(c)Xi) ^{for all} i e I. But by hypothesis and the definition of 1ri we have for every i e

1:

(q;'(c))(i) ⁼ 1ri'P'(c) = <Pⁱ(c) = /c(i) ⁼ (<P(c))(i).

(Uniqueness) Suppose ^D' (with maps 1r/ ^{: D' -+} Ai) has the same property as

D ₌

II

Ai. If we apply this property (for ^D) to the family of maps { 1r/ : D' ---+ Ai } and also apply it (for ial D') to the family { 7ri

:

_{D -+} Ai l , we obtain (unique) maps

'

l

6. T H E I NTEGERS ₉

'P : D' ^--+ D and 1/1 : D ^--+ D' such that the following diagrams are commutative for each i e !:

'

Combining these gives for each i e I a commutative diagram

Thus 'Pl/1 : D --+ D is a map such that 7r"i{'{Jl/l) ^{= 1ri} for all i e !. But by the proof above, there is a unique map with this property. Since the map ^ln : D --+ D is also such that 1rilD = 1ri for all i e /, we must have 'Pl/1 ^{= 1 n} by uniqueness. ^A similar argument shows that 4''P = l n ' · Therefore, 'P is a bijection by ⁽¹³⁾ and D =

II

^Ai is uniquely determined up to a bijection. •

ie.I

Observe that the statement of Theorem 5.2 does not mention elements ; it in­

volves only sets and maps. It says, in effect, that the product

II

^Ai is characterized

iel

by a certain universal mapping property. We shall discuss this concept with more pre- cision when we deal with categories and functors below.

6. THE INTEGERS

We do not intend to give an axiomatic development of the integers. Instead we assume that the reader is thoroughly familiar with the set Z of integers, the set N = { 0,1 ,2, . . . } of nonnegative integers (or natural numbers) the set N* ^{= { 1 ,2, . . .} J of positive integers and the elementary properties of addition, multiplication, and order. In particular, for all ^a,b,c e Z:

(a + h) + _c = a + (b + c) and ^(ab)c ⁼ a(bc) (associative laws) ; (23) a + b = b + a and ah = ba (commutative laws); (24) a(b + c) = ab + ac and ^{(a + b})^{c = ac + be} (distributive laws); (25) a + 0 = a and ^{al = a} (identity elements) ; (26) for each ^a e Z there exists ^{- a} e ^Z such that a + ( - a) = 0 (additive inverse); (27) we write ^{a - h} for a + (-b).

ab = 0 <=> a = 0 or ^{b = 0;}

a < b ==:) a + c < h + _c for all _{c e} Z;

a < b =::} ad < bd for all d e N*.

(28) (29) (30) We write a < b and ^b > a interchangeably and write a < b if a < b or ^{a = b.} The absolute value ^Ia! of ^a e Z is defined to be ^a if ^a > 0 and ^{- a} if a < 0. Finally we assume as a basic axiom the

10 P R ER EQU I SI TES AND PRELI M I NAR I ES

Law of Wel l Ordering. Every nonempty subset S of ^N contains a least element (that is, an element b e S such that b < c for all c e S).

In particular, 0 is the least element of N.

In addition to the above we require certain facts from elementary number theory, some of which are briefly reviewed here.

Theorem 6.1. (Principle of Mathematical Induction) IfS is a subset of the set ^N of natural numbers such that 0 e S and either

or

(i) n e S ^=::} n + _J e S for all ^{n e N;}

(ii) m e S for all 0 < m < n � n e S for all n e N;

then S ^{= N.}

PROOF. If N - S � fZ), ^let n ¢ 0 be its least element. Then for every m ^< n,

we must have m * N - S and hence m ^e S. Consequently either (i) or (ii) implies

n ^e S, which is a contradiction. Therefore N - S ⁼ 0 and N = S. •

REMARK. Theorem 6. 1 also holds with 0, N replaced by c, Me ^{= { x e Z I} _x ^> c l

for any c ^{e Z.}

In order to insure that various recursive or inductive definitions and proofs in the sequel (for example, Theorems 8.8 and 111.3.7 below) are valid, we need a technical result:

Theorem 6.2. (Recursion Theorem) lfS is a set, a e S and for each n e N, fn : S ^--4 S is a function, then there is a unique function <P : N --4 S such that ^{<P(O) =} a and <P(n + 1 ) =

fn(<P(n)) for every n e N.

SKETCH OF PROOF. We shall construct a relatio� R on N X S that is the graph of a function <P : N --4 S with the desired properties. Let g be the set of all subsets Y of N X S such that

(O,a) e Y; and (n,x) ^e Y � (n ⁺ l ,.fn(x)) ^e _Y for all n ^{e N.}

Then g � f2) ^{since N X S e g. Let} R ⁼ _n Y; then R e g. Let M be the sub-

YES

set of N consisting of all those n ^{e N} for which there exists a unique Xn ^e S such that

(n,xn) ^{e R.} We shall prove M ^{= N} by induction. If 0 t M, then there exists (O,h) e R with b ¢ a and the set R - { (O,b) } C ^N X S is in g. Consequently R ⁼ n Y

YES c R ^{- {(O,bJ},} which is a contradiction. Therefore, 0 ^€ M. Suppo_{se ind}u_ct_iv_ely that n ^{E M (that} is, (n,xn) ^€ R ^{for a unique} _xn ^E S). The_n (n ^{+ l} _,fn(xn)) ^E R ^{also. If} (n + l ,c) ^{€ R with} c ¥-fn(x") ^then R - {(n + 1 ,c)} e g (verify!), which leads to _a contradiction as above. Therefore, Xn+l ⁼ [n(x11) is the unique element of S such that (n ⁺ l ,xn+t) ^e R. Therefore by induction (Theorem 6. 1 ) N = M, whence the

6. T H E I NTEGERS 11

assignment n _� Xn defines a function ^{'P :} N ^-4 S with graph R. Since (O,a) ^{e R}

we must have ffJ(O) ⁼ a. ^{For each} n ^{e N,} (n,xn) ⁼ (n,cp(n)) ^{e R} and hence

(n ⁺ l ,f.n(ffJ(n)) ^{e R} since R e g. But (n ⁺ 1 ,xn+l) ^{e R} and the uniqueness of Xn+l

imply that '{J(n ⁺ 1 ) ⁼ Xn+l = f.n(ffJ(n)). •

If ^A is a nonempty set, then a ^sequence in ^A is a function N -+ A. A sequence is usually denoted {a0 ,a1 , ^{• • •}} or {a,.};eN or {a;}, where a; e A is the image of i e N.

Similarly a function N* -+ ^A is also called a sequence and denoted {a1 ,a2 , ^{• • •}} or {a;};eN• or {a;}; this will cause no confusion in context.

Theorem 6.3. (Division Algorithm) If a,b, e Z and a r!= 0, then there exists unique integers q and r such that b = aq + r, and O ^{< r <} lal .

SKETCH OF PROOF. Show that the set S = { h - ax I x e Z, b - ax > 0} is a nonempty subset of N and therefore contains a least element r = b - aq (for some

q e Z). Thus h = aq + r. Use the fact that r is the least element in S to show

0 ^< r < lal and the uniqueness of q,r. •

We say that an integer a rt= 0 ^divides an integer b (written a I b) if there is an integer k such that ak ⁼ h. If a does not divide h we write a (h.

Definition 6.4. The positive integer c ij' said to be the greatest common divisor of the integers al,a2, . . . ' an if·

(I) c I ai for I < i ^< n ;

(2) ^{d e Z} and ^{d I ai} for I < i ^< n � d I c.

Theorem 6.5. If a1,a2., ^{• • •} , an are integers, not all 0, then (a1,a2, . . . , an) exists.

Furthermore there are integers kt,k2, ^{• • •} , ^kn such that

SKETCH OF PROOF. Use the Division Algorithm to show that the least posi­

tive element of the nonempty set S = { x1a1 ^{+ x�2 +} · · · + Xnan _I Xi ^e Z,

�

Xiai > OJ

•

is the greatest common divisor of ah . . . , an. For details see Shockley [51 ,p.1 0]. •

The integers a1,a^z, . . . , an are said to be relatively prime if (a¹,a2, ^{• • •} , an) = I. A positive integer p > 1 is said to be ^prime if its only divisors are ± 1 and ±p. Thus if p

is prime and a ^e Z, either (a,p) ⁼ p (if p ^I a) or (a,p) ^{= 1} (if p .(a).

Theorem 6.6. If a and b are relatively prime integers and ^{a I} be, then a I c. If p is prime and p I a1a2 ^{· ·} · an, then p I ai for some ^{i .}

12 P R E R EQU ISITES A N D PRELI M I NA R I ES

SKETCH OF PROOF. By Theorem 6.5 1 ⁼ ra + sb, whence c ⁼ rae + sbc.

Therefore a _I c. The second statement now follows by induction on n. •

Theore m 6.7. (Fundamental Theorem of Arithmetic) Any positive integer n > ¹ may

be written uniquely in the form n = Pt t1P2t2 ^{• • •} Pk tk, where _P• < P2 < ^{· ·} · < ^Pk are primes and ti > ⁰ for all i.

The proof, which proceeds by induction, may be found in Shockley [5 1 , p.l7].

Let m _> 0 be a fixed integer. If a,b e Z and m ^I (a - h) then a is said to be ^con­

gruent to h ^{modulo m.} This is denoted by a = h (mod m).

Theore m 6.8. Let m > ⁰ be an integer and a,b,c,d e Z.

(i) Congruence modulo _\. m is an equivalence relation on the set of integers Z, which has precisely m equivalence classes.

(ii) ^If a ⁼ b (mod m) and _c = d (mod m), then a + ^c = b + d (mod m) and ac = bd (mod m).

(iii) If ab ⁼ ac (mod m) and a and m are relatively prime, then b = c (mod m).

PROOF. (i) The fact tl' at congruence modulo m is an equivalence relation is an easy consequence of the appropriate definitions. Denote the equivalence class of an integer a ^by

a

and recall property (20), which can be stated in this context as:

li = b ^� a = b (mod m). (20')

Given any a e Z, there are integers q and r, with 0 < r < m, such that a ⁼ mq + r.

Hence a - r ⁼ mq and a = r (mod m) ; therefore,

a

= r by (20'). Since a was ar­

bitrary and 0 � r < m, it follows that every equivalence class must be one of

O,i,2,3, . ^. . , (m ^-- _1). However, these m equivalence classes are distinct: for if 0 < i < j < m, then 0 < U - i) < m and m t (J - i). Thus i _f= j (mod m) and hence

I � j by (20'). Therefore, there are exactly m equivalence classes.

(ii) We are given m I a - b ^and m ^I c - d. ^Hence m ^divides (a - b) + (c - d)

= (a + c) - (h + d) and therefore a + ^c = b + d (mod m). Likewise, m divides

(a - b)c + (c - d)h and therefore divides ac - be + cb - db ⁼ ac - hd; thus

ac = bd (mod m) .

(iii) Since ab = ac (mod m), m I a(b - c). Since (m,a) ⁼ 1 , m I b - c by Theo­

rem 6.6, and thus b = c (mod m). •

7. TH E AX IOM OF CHOICE, ORDER, AND ZORN'S LEMMA

Note. In this section we deal only with sets. No proper classes are involved.

If I ^� 0 and { ^{Ai I} i e /} is a family of sets such that ^Ai � 0 for all i e I, then we would like to know that

II

^{Ai � 0.} It has been proved that this apparently in- nocuous conclusion cannot be deduced from the usual axioms of set theory (al-id

though it is not inconsistent with them - see P. _J. Cohen [59]). Consequently we shall assume

I

7. T H E AX I OM OF C H OI C E, ORDER, AND ZOR N 'S LEMMA 13

The Axiom of Choice. The product of a family ofnonempty sets indexed by a non­

empty set is nonempty.

See Exercise 4 for another version of the Axiom of Choice. There are two propo­

sitions equivalent to the Axiom of Choice that are essential in the proofs of a number of important theorems. In order to state these equivalent propositions we must introduce some additional concepts.

A partially ordered set is a nonempty set A together with a relation R on A X A

(called a partial ordering of A) which is reflexive and transitive (see (15), (17) in section 4) and

antisymmetric : (a,b) _e R and (h,a) e ^R � a = b. (31)

If R is a partial ordering of A , then we usually write a < h in place of (ap) e R. In this notation the conditions (1 5), (1 7), and (3 1 ) become (for all a,b,c e ^{A) :}

a < a· - '

a < b and b < c =:}

a < b and b < a _�

We write a < b if a < b and a ¢ b.

a < c· - '

a = b.

Elements a,b _e A are said to be comparable, provided a < b or b < a. However, two given elements of a partially ordered set need not be comparable. A partial ordering of a set A such that any two elements are comparable is called a linear

(or ^total or simple) ordering.

EXAMPLE. Let A be the power set (set of all subsets) of { l ,2,3,4,5 } . Define

C < D if and only if ^C C D. Then A is partially ordered, but not linearly ordered (for example, { ₁ ^,2} and { 3,4 } are not comparable).

Let (A, < ) be a partially ordered set. An element a _e A is maximal in A if for every

c e A which is comparable to a, c < a; in other words, for all c _e A, a < c � a = c.

Note that if a is maximal, it need not be the case that c < a for all c e A (there may exist ^c e A that are not comparable to a). Furthermore, a given set may have many maximal elements (Exercise 5) or none at all (for example, Z with its usual ordering).

An upper bound of a nonempty subset B of A is an element d e A such that b < d for

every b e B. A nonempty subset B of ^A that is linearly ordered by < is called a chain

in A .

Zorn's Le m ma . If A is a nonempty partially ordered set such that every chain in A

has an upper hound in A� then A contains a maximal element.

Assuming that all the other usual axioms of set theory hold, it can be proved that Zorn's Lemma is true if and only if the Axiom of Choice holds ; that is, the two are equivalent - see E. Hewitt and K. Stromberg [57 ; p. 14]. Zorn's Lemma is a power­

ful tool and will be used frequently in the sequel.

Let B be a nonempty subset of a partially ordered set (A, < ). An element c e B is a

least (or minimum) element of B provided c < b for every b e B. If every nonempty subset of A has a least element, then A is said to be well ordered. Every well-ordered set is linearly ordered (but not vice versa) since for all a,h e A the subset { a,b } must

14 P R ER EQ U I S I TES AN D PRELI M I NAR I ES

have a least element ; that is, a < b or b < a. Here is another statement that can be proved to be equivalent to the Axiom of Choice (see E. Hewitt and K. Stromberg

[57; p.l4]).

The Wel l Ordering Princi ple. If A is a nonempty set, then there exists a linear ordering < of A such that (A, < ) is well ordered.

EXAMPLFS. We have already assumed (Section ₆₎ that the set N of natural numbers is well ordered. The set Z of all integers with the usual ordering by magni­

tude is linearly ordered but not well ordered (for example, the subset of negative integers has no least element). However, each of the following is a well ordering of Z

(where by definition a < b � a is to the left of b):

(i) 0,1 , - 1 ,2,- 2,3,- 3, ^{. . .} , n, -n, . . . ;

(ii) 0,1 ,3,5,7, . . . , 2,4,6,8, . . . ' - 1 , - 2 , - 3, - 4, . . ^:;

(iii) 0,3,4,5,6, . ^{. . t} - 1 , - 2,-3,-4, . . . ' 1 ,2.

These orderings are quite different from one another. Every nonzero element a in ordering (i) has an immediate predecessor (that is an element c such that a is the least element in the subset { x I c < x } ). But the elements - 1 and 2 in ordering (ii) and -1

and 1 in ordering (iii) have no immediate predecessors. There are no maximal ele­

ments in orderings (i) and (ii), but 2 is a maximal element in ordering (iii). The element 0 is the least element in all three orderings.

The chief advantage of the well-ordering principle is that it enables us to extend the principle of mathematical induction for positive integers (Theorem 6. 1 ) to any well ordered set.

Theore m 7 .1. (Principle of Transfinite Induction) If B is a subset of a well-ordered set (A, < ) such that for every a s A,

then B = A.

{ c s A I c < a J C ^{B =}} a s B,

PROOF. If A ^- B _rf 0, then there is a least element a s A - B. By the defini­

tions of least element and A ^- B we must have { c s A ^I c < a J _C B. By hypothesis then, a s B so that a s B _n (A - B) = 0, which is a contradiction. Therefore,

A ^- B ₌ 0 and A ⁼ B. •

EX E R C I S E S

1 . Let (A, < ) be a partially ordered set and B a nonempty subset. A lower bound of B

is an element d s A such that d < b for every b e B. A greatest lower bound (g.l.b.)

of B is a lower bound do of B such that d < do for every other lower bound d of B.

A least upper bound (l.u.b.) of B is an upper bound to of B such that to < t for every other upper bound t of B. (A, < ) is a lattice if for all a,b e A the set { a,h } has both a greatest lower bound and a least upper bound.

8. CARD I NAL N U M BERS 15

(a) If S =F- 0, then the power set P(S) ordered by SPt-theoretic inclusion is a lattice, which has a unique maximal element.

(b) Give an example of a partially ordered set which is not a lattice.

(c) Give an example of a lattice w.ith no maximal element and an example of a partially ordered set with two maximal elements.

2.

A

^lattice (A , < ) (see Exercise l) is said to be ^complete if every nonempty subset of

A has both a least upper bound and a greatest lower bound. A map of partially ordered sets f: A � B is said to preserve order if a < a' in A implies /(a) < f(a')

in B. Prove that an order-preserving map f of a complete lattice ^A into itself has at least one fixed element (that is, an a s A such that [(a) = a).

3 . Exhibit a well ordering of the set Q of rational numbers.

4. Let S be a set. A choice function for S is a function f from the set of all nonempty subsets of S to S such that f(A) s A for all A =F- 0, A C S. Show that the Axiom of Choice is equivalent to the statement that every set S has a choice function.

5. Let S be the set of all points (x,y) in the plane with y < 0 . Define an ordering by (xi,YI) < (x2,y.,) � x1 ⁼ x2 and Y1 < Y2- Show that this is a partial ordering of S, and that S has infinitely many maximal elements.

6. Prove that if all the sets in the family { ^Ai I i e I _=F- 0 J are nonempty, then each of the projections ^7rk :

II

^{Ai __, Ak} is surjective.

i£1

1. Let (A , < ) be a linearly ordered set. The immediate successor of a s ^A (if it exists) is the least element in the set { ^x s ^{A I} a < x } . Prove that if A is well ordered by

< , then at most one element of A has no immediate successor. Give an example of a linearly ordered set in which precisely two elements have no immediate successor.

8. CARDI NAL NUM B ERS

The definition and elementary properties of cardinal numbers wilJ be needed fre­

quently in the sequel. The remainder of this section (beginning with Theorem 8.5), however, will be used only occasionally (Theorems 11. 1 .2 and IV.2.6; Lemma V.3.5;

Theorems V.3.6 and VI.1 .9). It may be omitted for the present, if desired.

Two sets, A and B, are said to be equipollent, if there exists a bijective map A � B;

in this case we write A ^{� B .}

Theorem 8.1. Equipollence is an equivalence relation on the class S of all sets.

PROOF. Exercise ; note that 0 � 0 ^since 0 C 0 X 0 is a relation that is (vacuously) a bijective function. 3 •

Let ^{Io =} 0 and for each n E N* let ^{In =} { 1 ,2,3, ^. . . , n } . It is not difficult to prove that lm ^and In are equipollent if and only if m = n (Exercise ^1). To say that a set A

3See page ^6.