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Math · Advanced Math
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Consider the set Zt of positive integers. a) Prove that Zt can be expressed as a countably infinite union of disjoint countably infinite sets. b) Use part (a) to give a different proof of Theorem 1.25THEOREM 1.25 A countable union of countable sets is countable Proof. It is sufficient to prove that a countably infinite union of disjoint countably infinite sets is countably infinite (see the exercises). In order to have a countably infinite number of sets, there must be one set corresponding to each positive integer n. Let (An i n e Z) be a countably infinite collection of sets. Suppose that each An is a countably infinite set and that none of the sets have any elements in common. We must prove that the set A- U An is countably infinite. Since each An is countably infinite, its elements can be put into a one-to-one correspondence with the set of positive integers. For each n, let An = {Xnk-k = 1, 2, . . .}, that is, А,-(x11, Х12, X13,X14, .. .], A2 (x2l+ X22, X23,K24, . . .], and so on. By the Fundamental Theorem of Arithmetic, which states that the factorization of positive integers into products of primes is unique (see Appendix C), the pairing xnk-2 3k is a one-to-one correspondence between A and a subset of the positive integers. By Corollary 1.24, the set A is countably infinite.

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