Page 159 - HOW TO PROVE IT: A Structured Approach, Second Edition
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P1: PIG/
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                                               Proofs Involving Disjunctions           145
                            d 23. Suppose A, B, and C are sets.
                            p ∗
                                 (a) Prove that (A ∪ B)   C ⊆ (A   C) ∪ (B   C).
                                 (b) Find an example of sets A, B, and C such that (A ∪ B)   C  =
                                     (A   C) ∪ (B   C)
                             p d24. Suppose A, B, and C are sets.
                                 (a) Prove that (A   C) ∩ (B   C) ⊆ (A ∩ B)   C.
                                 (b) Is it always true that (A ∩ B)   C ⊆ (A   C) ∩ (B   C)? Give
                                     either a proof or a counterexample.
                             p d25. Suppose A, B, and C are sets. Consider the sets (A \ B)   C and
                                 (A   C) \ (B   C). Can you prove that either is a subset of the other?
                                 Justify your conclusions with either proofs or counterexamples.
                              26. Consider the following putative theorem.
                             ∗
                                 Theorem? For every real number x, if |x − 3| < 3 then 0 < x < 6.
                                 Is the following proof correct? If so, what proof strategies does it use?
                                 If not, can it be fixed? Is the theorem correct?
                                 Proof. Let x be an arbitrary real number, and suppose |x − 3| < 3. We
                                 consider two cases:
                                   Case 1. x − 3 ≥ 0. Then |x − 3|= x − 3. Plugging this into the as-
                                 sumption that |x − 3| < 3, we get x − 3 < 3, so clearly x < 6.
                                   Case 2. x − 3 < 0. Then |x − 3|= 3 − x, so the assumption |x −
                                 3| < 3 means that 3 − x < 3. Therefore 3 < 3 + x,so0 < x.
                                   Since we have proven both 0 < x and x < 6, we can conclude that
                                 0 < x < 6.
                              27. Consider the following putative theorem.

                                 Theorem? For any sets A, B, and C,if A \ B ⊆ C and A  ⊆ C then
                                  A ∩ B  =∅.

                                 Is the following proof correct? If so, what proof strategies does it use?
                                 If not, can it be fixed? Is the theorem correct?

                                 Proof. Since A  ⊆ C, we can choose some x such that x ∈ A and x /∈
                                 C. Since x /∈ C and A \ B ⊆ C, x /∈ A \ B. Therefore either x /∈ A or
                                 x ∈ B. But we already know that x ∈ A, so it follows that x ∈ B. Since
                                 x ∈ A and x ∈ B, x ∈ A ∩ B. Therefore A ∩ B  =∅.

                             *28. Consider the following putative theorem.
                                                          2
                                 Theorem? ∀x ∈ R∃y ∈ R(xy  = y − x).
                                 Is the following proof correct? If so, what proof strategies does it use?
                                 If not, can it be fixed? Is the theorem correct?
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