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Discrete Mathematics with Graph Theory (3rd Edition) 12

Discrete Mathematics with Graph Theory (3rd Edition) 12 -...

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10 (c) The converse is certainly true since Ou + Ov = O. (d) The negation is true: Take a = u = b = 1 and v = -1. The contrapositive of A is false since A is false. 5. (a) There exists a countable set which is infinite. (b) For all positive integers n, 1 ~ n. Solutions to' Review Exercises 6. (a) This is true. If x is positive, x + 2 is positive. In addition, if x is odd, x + 2 is odd. (b) This is false. When x = -1, x + 2 = +1 is a positive odd integer, while x is not. 7. (a) This statement expresses a well-known property of the real numbers. It is true. (b) This is false. The conclusion would have us believe that every two real numbers are equal. 8. The desired formula is ab = {a + b)2 ~ {a - b)2 which holds because {a + b)2 - {a _ b)2 = (a 2 + 2ab + b 2 ) - (a 2 - 2ab + b 2 ) = 4ab. 9. (---+) Assume n 3 is odd and suppose, to the contrary, that n is even. Thus n = 2x for some integer x. But then n 3 = 8x 3 = 2( 4x 3 ) is even, a contradiction. This means that n must be odd. (t--) Assume n is odd. This means that n = 2x + 1 for some integer x.
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