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Discrete Mathematics with Graph Theory (3rd Edition) 7

Discrete Mathematics with Graph Theory (3rd Edition) 7 -...

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Section 0.2 5 (b) Converse: If n~l is not an integer, then n is an integer. This is false: n = ! is a counterexample ( n _ 1) n+1 - 3" Contrapositive: If n~l is an integer, then n is not an integer. This is false: n = 0 is a counterex- ample. Negation: There exists an integer n such that n~l is an integer. This is true: Take n = o. 9. (a) n prime ---+ 2 n - 1 prime. (b) n prime is sufficient for 2 n - 1 to be prime. (c) A is false. For example, n = 11 is prime, but 211 - 1 = 2047 = 23(89) is not. The integer n = 11 is a counterexample to A. (d) 2 n - 1 prime ---+ n prime. (e) The converse of A is true. To show this, we establish the contrapositive. Thus, we assume n is not prime. Then there exists a pair of integers a and b such that a > 1, b > 1, and n = abo Using the hint, we can factor 2 n - 1 as 2 n -1 = (2 a )b -1 = (2 a _1)[(2 a )b-1 + (2 a )b-2 + ... + 2 a + 1]. Since a> 1 and b > 1, we have 2 a -1 > 1 and (2 a )b-1 + (2 a )b-2 + ... + 2 a + 1 > 1, so 2 n -1 is the product of two integers both of which exceed one. Hence, 2 n - 1 is not prime.
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