By Patrick Suppes
This transparent and well-developed method of axiomatic set thought is geared toward upper-level undergraduates and graduate scholars. It examines the uncomplicated paradoxes and heritage of set conception and complex themes corresponding to kinfolk and services, equipollence, finite units and cardinal numbers, rational and genuine numbers, and different matters. 1960 variation.
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Extra resources for Axiomatic Set Theory
A staccato style, which consists of displaying a series of equivalences and is similar to that often used for a chain of identities, has been adopted here. Some readers may find this method of presentation clearer than the more prolix one used in the proof of Theorem 28. THEoREM 34. An (A '""'B) = A '""'B. THEOREM 35. (A r-vB)uB = AuB. THEOREM 36. _. B = A "' B. THEoREM 37. (A nB) '""'B = o. THEOREM 38. (A""' B)nB = 0. THEoREM 39. A"' (B uc) THEoREM40. A"-'(BnC) = (A"' B) n (A"' C). = (A"-'B)u(A""C).
0. Hence by sentential logic X E: 0 +-+X E: A & X fl. A, and thus by Definition 7 A ~A= 0. D. The remaining theorems of this section state facts relating the set operations of intersection, union and difference. The proofs of the theorems are easily given by employing an approach similar to that used for Theorem GENERAL DEVELOPMENTS SEc. 3 29 28. As with that theorem, the proofs depend upon exploiting formal properties of the sentential connectives analogous to the formal properties asserted in the theorems.
2 THEOREM 5. A ~ 0-+ A = 23 0. PROOF. By virtue of Definition 3 and the hypothesis of the theorem, if x E: A then x E: 0. But by Theorem l x fl. 0. Hence for every x, x fl. A, and thus by Theorem 2, A = 0. D. Transitivity of inclusion is asserted in the following theorem. THEOREM 6. A ~ B & B ~ c-+ A c c. PROOF. Consider an arbitrary element x. Since A C B if x E: A then x E: B, but B ~ C; hence if x E: B then x E: C. Thus by the transitivity of implication if x E: A then x E: C. * A typical procedure used in informal proofs is exemplified here.