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90 RELATIONS AND FUNETIONS Prove (where the subscripts “A” and “R' arc dropped for brevity): = (Ny Ne (x Uy) Us d) zU(LNy) = 2 Now to go the other way, assuming for any ,4,2€ A, properties (a)-(g), define: cB'yozhNy=y Prove that 4 is a latticc relative to R”, 12. We may formally define the lambda notation for abstraction: If v and w are distincl variables and w does not occur in the term t, then the identity QUO = (um): v) holts, Tind the following sets: (a) (MAIa: 2€ A &ACB) (b) (Mix: 2E BA & ACB)) (e) Ade: «CA &A = 0) 13. Prove: (1) (MAANA) =0 (b) (MAXAUA) = 0 (o) (AAA) =0 (d) QAXA/4) = 0 (The significance of (a)—(d) is that there are no sets corresponding to the set; operations. For instance, in view of (a) we may not regard the operation of a set interscoting with itsclf as a cortain set of ordered puirs. “This result for the special case of sets intersecting themselves is easily generalized to show that Lhere is no set corresponding to the binary operation of intersection for amy two possibly distinct sets.) CHAPTER 4 EQUIPOLLENCE, FINITE SETS, AND CARDINAL NUMBERS $ 41 Equipollence. The axioms listed at the end of Chapter 2 (82.10) suffice for this section and the next, but in 44.3, on cardinal numbers, we introduce a special axiom whose use will always bc indicated by a dagger pr In 81.1 Cantor's notion of two sets having the same power, or, as we shall say, being eguipollent, was mentioned as fundamental, Tb is fun- damental because it is the basis of generalizing the notion of positive integer to that of cardinal number. Two sets are equipollent if there exists a 1-1 correspondence between them, and equipollent sets have the same cardinal number, This intuitivo notion of 1-1 correspondenec is casily made precise: such a córrespondence is just a 1-1 function. Formally we have:* Derinimon 1. (1) A=B under fifandonhyiffisal-lfunciion whose domain is A and whose range is B; (1) A=BifandontyifthereisanfsuchthatA=B under. For example, if 4, = [1,3,5) A= [1,79], then À, and À, are equipollent. Any one of several functions will establish this: fi= UL 1, (6,7), (5,0), or just as well: Fa = 1,7), 3,9); (5, 1). *In this chapter, and henceforth, we use logical symbolism only sparingly in for- tmtJating definitions and theorems, but in every case the appropriate symbolio formula- ton should be obvious. 9 92 EQUIPOLLENCE CHaP, 4 Tt is clear that two finite sets are equipollent just when they have the same number of members. (We have, of course, not yet defined the notions of :* finiteness or number within our axiomatic framework.) Ft is also clear that if one finite set is a proper subset of anothcr, then the two scts cannot have the same power, that is, they cannot be equipollent. However, the situation is entirely different for infinite sets. Consider, for instance, the set N of positivo integers (1,2,3,...) and'the set E of even numbers [2,4,6,...). Obyiously, E is a proper subset of N, but E and N are equipollent, which is easily shown by considering the doubling function f such that for each positive integer n In) = 2n. We sec at once that fis 1-1, Df = N, and Gf = E. The first three theorems show that equipollence has the characteristie three properties of an equivalence relation, Tamorem li. A=4, proor, The identity function 44 is an appropriate 1-1 function. Q.E.D. Tueonem2. IfA=BthenB=aA, TuzoREM 3. IfA=B&B=CthenAÃ=O, rroor. Let f be a 1-1 function establishing that A = B with D/=A, and let g be a corresponding 1-1 function for showing that B = € with Dg = B. Then the function go fis 1-1, D(goj) = 4, and A(g of) = €, whenee 4 = €. QE.D, “ We now state a number of theorems relating equipollence to operations and relations previously introduced. These theorems make the develop- ment of cardinal axithmetic in $4.3 very simple. The first theorem is used to justify the definition of cardinal addition. The nd is used to justify the definition of cardinal multiplication; the third is used to prove the commutativity of cardinal multiplication, and so forth. “The order of the theorems here is nearly the same as that of the corresponding theorems for cardinal numbers in 94.3, Trrorem 4. IfA=B8&0C=D&ANC=0& BND=0 then AUC = BUD, rroor. By hypothesis there are 1-1 functions f and g such that 4 = B under 7 and C = D under g. It also follows from the hypothesis that Dindg = 0 and ana =, Sec. 4.1 EQUIPOLLENCE 93 «hence by virtue of Theorem 93 of $3.4, fUg is 1-1, and if is easily seen that AUC = BUD under fUg. QED, Turortm 5. fA=B&C=DihnAXC=BXD. rroor. Let 4 = B under the funetion fand C = D under the function g. “Then the function À such that for sc À and yel ha, 1) = (Hm), 9(w)) establishes the equipollence of 4 X Cand B XD. QED. Taronim 6. 4AXB=BXA. »roor. The function f such thatforze AandycB TG, 1) = (2) is appropriato to establish the desired equipollence. Q.E.D. Trrorem 7. AX(BXC) =(AXB)xC. Turorem8. AXful=4A&[z)xA=A, proor. For the first half of the theorem the function f defined on AX fx) such that for ye À TU =) = 3 is appropriate. A similar function is suitable for the second half. Q.E.D. Temonum 9. There are sets Cand D such that A=C &B=D &CnD=0, eroor. Define C=AX 10) D=Bx ftol). “Then by the preceding thcorem 4 = C and B = D, and elearly CND = 0. QD, “Lhe next theorem is used to justify the definition of cardinal exponenti- ation, Tueornm 10. IA =B&C=Dthen AC=BP proor. By hypothesis there arc 1-1 functions f and q such that 4 = B under f and € = D under g. If he Aº then (65) fohe Bº and from (1) we infer fohogle B?, 96 EQUIPOLLENCE Crrap, 4 Since every CE D is a subset of UD, we conclude from (1) and (2) H CeDthen (3) CCA-94B-fUD). By Thecorem 63 of Chapter 2 we may infer from (3) that (4) UDCA-94B=fUD). Now let (5 F=AG4B-fUD). Then by virtue of (1), (4), and (5) Av GB fUD) GA gB o fe), that is, FEA-qUB fm, whence we conclude FeD, that is (6) ASGBSfUD) E UD. From (4) and (6), we have UD = Am g4B=f“UD), which for K = UD is equivalent to: GBP) = ASK, the desired conclusion. Q.ED. We shall have oscasion to use tho Schroder-Bernstein theorem in proofs of several subsequent theorems, For the moment we complete our list of theorems on the relation <. Mainly, we have the following monotonicity result: Tuzorem 19. IA
eln + D) then (Vela). In the next chapter we prove this principle for the integers. 'Tho eor- responding principle of induction for a finite set adds the hypothesis that the set is finite and replaces (ii) by (VV Ba e A & o(B) > (BU ta), thc idea being that if the set A is finite we start with the empty set and add elements of 4, one at a time, until we have exhausted 4. The prof of the theorem about induction for finite sets is facilitated by having available the already defined notion of maximal element of a family of subsets. Proofs of Lhe two thcorems about maximal elements we leave as exercises, TuHeorEm 30. Every non-emply family of subseis of. a finite sei has q maximal element, The second thcorem is the converso of Theorem 30 and the two together thus show that a finite set may be defined in terms of every non-empty family of its subscts having a maximal element. Tuzorem 31. If every non-empty family of subscis of a set A has q maximal element then A is finito. We are now ready for the first induction thcorem. Tunorem ScHema 32. If (1) A is fimite, GD a(o), Gi) (Vo(VBZEA&BCA & AB) = BUÍzx) then (A). proor. Supposc (1) and (ii) hold. We define 1) K=ÍB: BCA&g(B)). Sec, 4.2 FINIVE SETS 103 :. The set K is not empty sinec0C 4 and q(0), and thus 0€ K. Whenco by virtuc of Theorem 29 and (1), K has a maximal element, say B. We want to show that B — A, and thus q(4). Suppose it were the case that BxA. On the basis of (1), BC A and thus AvB=O. Letz€C AB, Then BUÍz)C 4 and by virtucol (ii) o(BU [)), whence BUlv) CK, srhich contradicts B being a maxima! element of K and proves our sup- position false. Q.E.D. By taking o(B) as 'B € K”, we immediately infer from Theorem 32 the following “set” formulation of induetion for finit: Tt is worth noting, that the converso holds; namely, Theorem 81 follows from Theorem 32 if west K=|B: BE GA &a(B)). TeroREM 33. 1f () A is finite, (ii) 0€K, Gi) (VolVBxeA&GBCA&BEKSBUÍZ)CK) then AEK. That this inductivc property of finite sets may be used to characterize them is established by the next thcorem. Turorgu 34. 4 is finite if and only if A belongs to every set K satisfy- ing (ii) and (ii) of Theorem 32. »roor. The necessity follows from Theorem 33. To prove suficicney, assume that 4 belongs to every set K satisfying (ii) and (ii). Let K, be the family of all finite subsets of 4. By virtue of Theorem 24, 0€ K,. Moreover if BC K, and 2€ A then by Theorem 29 BUÍz| E K,, whenco by hypothesis 4 € K, and is thus finite. QE.D., The definition of Sierpinski [1918], as modified slightly by Tarski, is given in the following thcorem whose proof is similar to the preceding one. THrorEM 35. A is finite if and onty if A belongs to every set K such that () 0€K, ) fecA, them (o) CK, Gi) FBEKECEK then BUCEK. The definition of Sicrpinski is modified in Kuratowski [1920] to yield the following thcorem, whose proof wc leave as an exercise. 104 FINITE SETS CHar. 4 Tmrorem 36. A is finite if and only if the power set PA is the only set K which satisfies the conditions: (1) KC GA, (ii) 0€K, (ii) feeAthen (rick, (iv) YBEK&ECER then BUCEK. Our next objective is to prove some facts concerning finiteness about the power set and sum sct of a given set. We first prove a preliminary theorem asserting that if we can map 4 onto B and 4 is finite then B is finite, TuzoreM 37. If A is finite and f is a function whose domain is A and range is B then B is finile. proor. We define: K=[0: COCA &fC is finite). We want to prove by induêtion (using 'Lheorem 33) that 4 € K, whence B às finite, since fºA = B. First we observe immediately that 0 € K because 0CA and fº0 = 0, Assume, for the second part of the induction, that ze A and CEk. We nedto show that CUfalc K. Obviously, CUÍz]C A. Since fis a function, fS fx) is a unit set and thus finite (The- orem 25), and since C € K, fºC is finite. Therefore, in view of Theorem 28, (ºC) USStx) is finite. But by virtue of Theorem 35 of Chapter 3 Sou tr) = (POUf ta), and f(CU [2)) is finite. We conclude that CuÍziek, which compictes the proof. Q.E.D. Twrorey 38. Ifa set is finite then its power set is finite. proor. As in the preceding proof, we define a certain set and then prove by induction that 4 is a member cf it. We define: K= |B: BCA& Bis fnite). Again àt is obvious that 0 € K. Assume, as usual, for the second part of the induction that BC Kandz CA. fz € B the induction is immediate, so we may suppose that x 7 B. Clemly, BU[z)C 4. H remains to show that the power set (BU [z)) is finito, given that PB is Ênite, in order to prove that BU(z) € K. Let us definc: + (D) f=tc Cutz): Ce oB). Seo. 4,2 FINTIE SETS 105 We show that f is a function whose range is (BU Íz)) — EB. (Obviously the domain of fis €B.) That f is a function is evident from (1). The problem is to show that its range is the desired set. If C € GB, then culz) € ABuUfz) — 6B gince zg B. On the other hand, if De eBUÍz) =» 6B then D- tz) € OB, whenco Dt, Des. Wo conclude that G(BU (xl) » GB is the range of f. Applying now the precoding theorem and using the inductive hypothesis that OB is finite, we infer that G(BU fz)) — OB is finite. We note that (BU [x)) = (o(BU[2)) » 6B)00B and since by virtue of 'Lheorem 28 the union of two finito sets is finite, we obtain that G(BU [2)) is finite, whence BUfalcX and the induction is completed. Q.E.D. We lcave as an exercise the inductive proof of the following theorem. Tasonux 39. 17 A is finito and every set which is q member of A is Jinite, then UA is finito. “he last two theorems may be uscd to prove each other's converse, that is, Theorem 39 is useful in proving the converse of Theorem 88, and vice versa. We leave thesc two converses as exercises. Tenorey 40. If GA is finite then A is finite, Turorem 41. If A és a family of sets and UA is finite then is À finite and every set which is a member of À is finite. Notice that Theorem 41 is not the exact converse of Theorem 39, for the additional hypothesis is required that 4 be a family of sets. Obviously UA could be finite and À infinito as long as 4 has only à finite number of sets as members, since individuals in A do not contribute to UA. We now consider some thcorems about equipollenco of finite sets. The proof of the first one follows at once ftom Theorem 37. 108 TINTIE SETS Cnar. 4 Tuzoney 48. I/A, Band C are finiteandif A 1-2), In case the reader is puzaled by tho use of the phrases 'finite cardinaP, “infinite cardinal, “transfinito cardinal, it may be said that the topios connoted by these phrases will be explained subsequentiy. In the next section we discuss the finite cardinals, and in the latter part of the next chapter we distinguish between infinito cardinals“and transfinite cardinals; mis an infinite cardinal if there exists an infinite set A such that (4) = m; mis a transfinite cardinal if there exists a Dedekind infinite set À such that (4) = m. As is obvious from remarks in the section on finite sots every known proof requires the axiom of choice to show that a cardinal is infinite % and enly if it is transfinite. EXERCISES Prove Theorem 56 in detail. Prove Thcorem 57. Prove Theorem 62. Prove Theorem 68. Prove Theorems 64-66. Prove Theorems 67-69. Prove Theorem 70. Prove Theorem 73. di go pp Prove Theorem 76. Sne. 4.4 CARDINAL NUMBERS 121 10. For each of the following, either prove it holds or give, a counterexample: (9) MAB) < RA), . (DD) R(A-B) 
