# Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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Homework 12 - MATH 310 0101 - Summer I 2010 - Due by Thursday, June 24 1. Prove that if A is countable, then A k is countable for every k N . (Hint: Use induction. You will want to use the fact that ± ± A k +1 ± ± = ± ± A k × A ± ± ; you must prove this in your solution.) 2. Prove that a ﬁnite union of countable sets is countable. (This can be done at least two ways. One way is by induction. Another is by cleverly using a theorem from class.) 3. Prove that a countable union of sets that are either ﬁnite or countable
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Unformatted text preview: is itself either ﬁnite or countable. (Hint: The proof is similar to the proof that the countable union of countable sets is countable, but you have to ﬁgure out how to deal with the ﬁnite sets. Don’t make things too complicated; there is a very simple solution.) 4. Prove that the set of all ﬁnite subsets of N is countable....
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