Assignment 12

Assignment 12 - Homework #12 Section 5.4 _ A means the...

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Homework #12 Section 5.4 _ A means the cardinal number of the set A. ___ 1) Let n be a natural number. n = N_n which is a subset of N. So ___ _ n = N_n <= N = aleph not < c. So n < c. ___ _ ___ 2) P(N) = c = R < P(R) last inequality is by Cantor's Thm. _ _ _ _ _ _ 3) Prove that if A <= B and B = C, then A <= C. Proof: _ _ Since A <= B, there exists a 1-1 map f: A->B. _ _ Since B = C, there exists a bijection g:B->C. g o f is a map from A to C. By the Thm 4.11, g o f is 1-1. By the defintion of <=, _ _ A <= C. _ _ 5a) False Let A={1,2,3}, B={3,4,5,6}. A < B, but B is not a subset of A. b) True Use Thm 5.30 e) and that A intersect B is a subset of B. _ _ c) True A <= B means there exists a 1-1 fctn f:A->B. The induced fctn f:P(A)->P(B) would also be 1-1. 6b) Let f:A->B be one-to-one and onto and let g:B->C be one-to-one and onto. Then g o f: A->C is one-to-one and onto. 7) Prove that there is no largest cardinal number.

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This note was uploaded on 03/18/2012 for the course MAT 108 taught by Professor Staff during the Winter '10 term at UC Davis.

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Assignment 12 - Homework #12 Section 5.4 _ A means the...

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