Set Theory and the Structure of Arithmetic by Norman Tyson Hamilton, Joseph Landin

By Norman Tyson Hamilton, Joseph Landin

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Respectively, in the following simple Defififff'on JO". Definition IT. A fl B = A C\ B \J B = f| {x \ x C = GA {x x \ ways and x 12 and 13, E B}. A HB and x C}. xAorxB}. Definition 12'. A Definition 73'. A-B={x\xA and x {x 10', 11, : ( B}. EXERCISES In the following, use the notation "{x \ P x}" to denote the sets de scribed. 1. 2. The set of all natural numbers (to be defined in Chapter 2), which are factors of 12. What The numbers n such that if m G n then m is a (Just what a natural number is does not affect other notation can you give for this set?

B \J C. if Prove U if ACC, BCD then In elementary algebra, we are familiar with the distributive law which all real numbers, = ab ac. In the a,6,c, a(b + c) algebra of sets, we have two distributive laws, namely + states that for Theorem 4. For all sets 1. AH(UC) = 2. A u = (5 n c) A, B, C, (An)U(AnC); (A u B) n (A u c). 9 A n A n x U C) n B) u (B u o c (A u c). e A n (B U C) If x C. We make Here z case (a): n B) U (A p| B) (A n &A c) and (A p| O we show and (A n B) u re cases according as (a) x GA and n C) (A then x = re G B, hence z (Exercise 4, U C) e 65 or B.

S. citizen named "Samuel Snork" who has at least one female ancestor born outside of the U. S. The mere fact that x is alive assures us that x is less than 2,500 years of age. membership in B. Consequently, x also x satisfies all Thus from the criteria for G A follows x G B A C B. Conversely, from x G B follows x G A, and therefore B C A. ) whence From these considerations we may conclude, by the Axiom of Extensionality, that A = 2. C B. D equilateral triangles and equiangular triangles in the plane of elementary Let be the set of Euclidean plane).

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