Discrete Mathematics with Graph Theory (3rd Edition) 10

Discrete Mathematics with Graph Theory (3rd Edition) 10 - 8...

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8 32. (a) [BB] False: x = y = 0 is a counterexample. (b) False: a = 6 is a counterexample. (c) [BB] False: x = 0 is a counterexample. (d) False: a = Vi, b = -Vi is a counterexample. (e) [BB] False: a = b = Vi is a counterexample. Solutions to Exercises -b± Vb 2 -4ac (f) The roots of the polynomial ax 2 + bx + c are x = 2a . If b 2 - 4ac > 0, .../b 2 - 4ac -b+ Vb 2 -4ac is real and not 0, so the formula produces two distinct real numbers x = 2a and -b - .../b 2 - 4ac x= 2a (g) False: x = ~ is a counterexample. (h) True: If n is a positive integer, then n ~ 1, so n 2 = n(n) ~ n. 33. The result is false. A square and a rectangle (which is not a square) have equal angles but not pairwise proportional sides. 34. (a) [BB] Sincen 2 +1 is even, n 2 is odd, so n must also be odd. Writingn = 2k+l, then n 2 +1 = 2m says 4k2 + 4k + 2 = 2m, so m = 2k2 + 2k + 1 = (k + 1)2 + k 2 is the sum of two squares as required. (b) [BB] We are given thatn
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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