View solution: Assign a grade of A (correct), C (partially correct), or F (failure) to

Chapter 5, Problem 22

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Assign a grade of A (correct), C (partially correct), or F (failure) to each.Justify assignments of grades other than A.(a) Claim. If A and B are finite, then is finite.Proof. If A and B are finite, then there exist suchthat Let and Thenwhich shows that Thusis finite. (b) Claim. If S is a finite, nonempty set, then is finite.Proof. Suppose S is finite and nonempty. Then for someinteger k.Case 1. Then so has k elements and isfinite.Case 2. Then Thusis finite.(c) Claim. If is finite, then A is finite.Proof. Choose any b* Then {b*}. But{b*} * Since is finite, {b*}is finite. Since A is equivalent to a finite set, A is finite.(d) Claim. The set is finite.Proof. For every n in the set is finite, because ByTheorem 5.1.7, we know is finite. Since , we see that n=1n = n=1n, n n n.A = {(a, b ): a A} A B. A B A B. A A A BS {x}x S. S {x} k {x} k 1 k+1.x S. S {x} = S, S {x}S kS {x}A BA B A B m+n. 11onto f h: m+n,h: B 11onto f: A n. 11onto A m and B n. mm, n A B242 CHAPTER 5 Cardinalityis finite.5.2 Infinite SetsIn this

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