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Homework II Key

# Homework II Key - Homework II Key Due 13-Feb-2008 Textbook...

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Homework II Key Due: 13-Feb-2008 Textbook: Mathematical Structures for Computer Science: A Modern Treatment of Discrete Structures   (5     th     Edition     ) J. Gersting, W.H. Freeman and Company: New York , NY . 2003. p.57 #7 1. hyp 2. hyp 3. 1, ei 4. 2, ui 5. 3, 4, mp 6. 5, eg p.58 #11 1. 2200 x P(x) hyp 2. 5 x Q(x) hyp 3. Q(a) 2, ei 4. P(a) 1, ui 5. P(a) Q(a) 3, 4, conn 6. ( 5 x)(P(x) Q(x)) 5, eg p.58 #17 1. ( 2200 x)(P(x) Q(x)) hyp 2. ( 2200 x)P(x) hyp 3. P(x) Q(x) 1, ui 4. P(x) 2, ui 5. Q(x) 3, 4, mp 6. ( 2200 x)Q(x) 5, ug p.92 #9 (do an informal proof) Let x = 2m, y = 2n, where m and n are integers. Then x + y = 2m + 2n = 2(m + n), where m+n is an integer, so x + y is even.

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p.93 #16 3(n 2 + 2n + 3) – 2n 2 = 3n 2 + 6n + 9 – 2n 2 = n 2 + 6n + 9 = (n + 3) 2 p.93 #27 n 3 – (n – 1) 3 = n 3 – [n 3 – 3n 2 + 3n – 1] = 3n 2 – 3n + 1 = 3n(n – 1) + 1. Then n(n-1) is even by Exercise 11, and 3n(n-1) is even by
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