A collection of nonempty subsets of a nonempty set S is called a cover of S if every

Chapter 3, Problem 3.36

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A collection of nonempty subsets of a nonempty set S is called a cover of S if every element of S belongs toat least one of the subsets. (A cover is a partition of S if every element of S belongs to exactly one of thesubsets.) Consider the following.Result Let a, b Z. If a is even or b is even, then ab is even.Proof Assume that a is even or b is even. We consider the following cases.Case 1. a is even. Then a = 2k, where k Z. Thus ab = (2k)b = 2(kb). Since kb Z, it follows that ab iseven.Case 2. b is even. Then b = 2, where Z. Thus ab = a(2) = 2(a). Since a Z, it follows that ab iseven.Since the domain is Z for both a and b, we might think of Z Z being the domain of (a, b). Consider thefollowing subsets of Z Z:S1 = {(a, b) Z Z : a and b are odd}S2 = {(a, b) Z Z : a is even}S3 = {(a, b) Z Z : b is even}.(a) Why is {S1, S2, S3} a cover of Z Z and not a partition of Z Z?(b) Why does the set S1 not appear in the proof above?(c) Give a proof by cases of the result above where the cases are determined by a partition and not acover.