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Discrete Mathematics with Graph Theory (3rd Edition) 11

Discrete Mathematics with Graph Theory (3rd Edition) 11 -...

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Unformatted text preview: Chapter 0 9 38. We have proven in the text that y'2 is irrational. Thus, if y'2../2 is rational, we are done (with a = b = y'2). On the other hand, if y'2../2 is irrational, then let a = y'2../2 and b = y'2 in which case a b = y'22 = 2 is rational. Chapter 0 Review 1. (a) This implication is true because the hypothesis is always false: a - b > 0 and b - a > 0 give a > b and b > a, which never holds. (b) This implication is false: When a = b, the hypotheses are true while the conclusion is false. 2. (a) x is a real number and x ~ 5. (b) For every real number x, there exists an integer n such that n ~ x. (c) There exist positive integers x, y, z such that x 3 + y3 = z3. (d) There exists a graph with n vertices and n + 1 edges whose chromatic number is more than 3. (e) There exists an integer n such that for any rational number a, a =F n. (f) a =F 0 or b =F O. 3. (a) Converse: If ab is an integer, then a and b are integers....
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