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Question: 19 evaluation of proofs this type of exercise will appear...

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19. Evaluation of proofs This type of exercise will appear frequently in the book. In each case, there is a proposed proof of a proposition. However, the proposition may be true or mav be false » If a proposition is false, the proposed proof is, of course, incorrect. In this situation, you are to find the error in the proof and then provide a counterexample showing that the proposition is false. » If a proposition is true, the proposed proof may still be incorrect. In this case, you are to determine why the proof is incorrect and then write a correct proof using the writing guidelines that have been presented in this book. » If a proposition is true and the proof is correct, you are to decide if the proof is well written or not. If it is well written, then you simply must indicate that this is an excellent proof and needs no revision. On the other hand, if the proof is not well written, then you must then revise the proof by writing it according to the guidelines presented in this text.(b) Proposition. For all real numbers x and y, if xメy, x 〉 0, and Proof. Since x and y are positive real numbers, xy is positive and we can multiply both sides of the inequality by xy to obtain ー+1).xy〉2.xy By combining all terms on the left side of the inequality, we see that X2-2xy + y2 > 0 and then by factoring the left side, we obtain (x-y)2 > 0. Since x y, (x-y) proves that if x y, x > 0, and y > 0, then-+-> 2. メ0 and so (x-y)2 〉 0. This(c) Proposition. For all integers a, b, and c, ifa | (bc), then a | bora | c. Proof. We assume that a, b, and c are integers and that a divides bc. So, there exists an integer k such that bc-ka. We now factor k as k - mn, where m and n are integers. We then see that This means that b -ma or c -na and hence, a |bora|c.

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