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Math 220 Assignment 9 Due November 25th 1. Let A and B be finite sets. Prove that |A × B| = |A| × |B|. Hint: try induction on |A|. 2. Let A and B be finite sets, and let AB denote the set of functions from B to A. Prove that |AB | = |A||B| . Hint: Induction on |B|, and note that you might need to prove a lemma: given b ∈ B, there is a bijection between AB and A × (AB\{b} ). 3. For any real number a 6= 1, and a positive integer n, prove that 1 + a + n a2 + · · · + an−1 = 1−a 1−a . 4. Let n be a natural number such that n ≥ 12. Prove that there exists numbers a, b, c ∈ N such that n = 4a + 6b + 9c. Hint: there will be cases. Interesting fact: this result states that you can order n ≥ 12 chicken mcnuggets from McDonald’s by ordering a 4-piece, b 6-piece, and c 9piece containers. You are allowed to use strong induction if you prefer. p 5. Define f : N → R by f (0) = 1 and f (n + 1) = 2 + f (n). Prove that, for all n, f (n) < 2. Also, prove that, for all n, f (n) < f (n + 1), using induction. Fact: a result from advanced calculus implies then that limn→∞ f (n) exists, and is at most 2. In that course, you will gain the techniques to show that the limit is 2. 6. Bonus extra credit: Let A and B be finite sets. How many injective functions are there from B to A? Prove your result by induction. 1