By Patrick Suppes

This transparent and well-developed method of axiomatic set concept is geared toward upper-level undergraduates and graduate scholars. It examines the uncomplicated paradoxes and background of set idea and complex issues akin to family members and services, equipollence, finite units and cardinal numbers, rational and actual numbers, and different matters. 1960 version.

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By virtue of Theorem 12 xE: (AUB)nCrtxE:AUB&xE:C, and by Theorem 20 x E: A UB & x E: Crt (x E: A V x E: B) & x E: C, and by the distributive laws of sentential logic* (x E: A V x E: B) & x E: Crt (x E: A & x E: C) V(x E: B & x E: C). Using again Theorem 12 (x E: A & X E: C) v (x E: B & X E: C)- X E: A nc v X E: B nc, *The law in question is that from (P v 0) & R we may infer (P & R) v (0 & R) and conversely, where P, 0, and R are any formulas. GENERAL DEVELOPMENTS 28 CHAP. 2 and now using Theorem 20 again xE:AnCv xE:BnC+-+xE: (AnC)U(BnC).

By hypothesis A X B = 0. Suppose now that A~ O&B ~ 0. SEc. 8 GENERAL DEVELOPMENTS 51 Then by Theorem 2 (3y)(y E:: A) & (3z)(z E:: B), and thus by Theorem 97 (y,z) E:: A X B, which contradicts the hypothesis and proves our supposition false. [Sufficiency]. From the condition that A = 0 or B = 0 and Theorem 2, we infer (1) -(3y)(y E:: A) V -(3z)(z E:: B), and it follows from (1) that -(3y)(3z)(y E:: A & z E:: B & x = (y,z)), and thus by Theorem 96, for every x xe:AXB. Hence by Theorem 2 A X B = 0.

A nB. THEoREM 71. n IA,B} THEOREM 72. n(x,y) THEOREM 73. n n(A,B) =A. PROOF. D. = {x}. By virtue of Theorem 72 n (A,B) = {A}, and by virtue of Theorem 70 n{A} =A. D. Five general implications concerning intersections of families of sets are next. THEoREM 74. PROOF. Let we must have: X A~B & (3C)(C E: A)~ nBc nA. be an arbitrary element of X nB. Then for every c E: B, E: C, but the hypothesis of the theorem assures us that if C E: A then C E: B. Whence for every CE: A, we must have: XE: C, and thus XE: nA, the desired result.