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hw3-sol - CS 173 Discrete Structures Fall 2010 Homework 3...

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CS 173: Discrete Structures, Fall 2010 Homework 3 Solutions This homework contains 4 problems worth a total of 38 points. It is due on Friday, September 17th at 4pm. When a problem specifies a particular proof technique, you must use that technique in your solution, even if it’s not the only reasonable approach to the mathematical problem. This is because the main point of these problems is to learn how to use the various different proof techniques. 1. [10 points] Proof using divides Consider the following claim: For any integers a , b , and c , if a 2 b | cb , then a | c . (a) This claim is slightly buggy. Give a concrete counter-example showing why it fails. (b) Adding one extra (simple) condition to the hypotheses will make this claim true. State the revised claim. (c) Prove the revised claim, directly from the definition of “divides.” (That is, don’t use any lemmas about divides from the text or lecture.) Solution: (a) This fails for any integers a and c where b = 0 and a does not divide c . For example: a = 2, b = 0, c = 3. In this, (2 2 )(0) | (3)(0) is logically equivalent to 0 | 0, which yields the equation 0 = 0 j for any integer j. By definition of “divides”, this holds. However, consider that 2 | 3 yields the equation 3 = 2 k for some integer k. Dividing both sides by 2, we find 3 2 = k . But this cannot happen, since k was defined as an integer. Thus, the inputs above yield a counter-example.
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