midsol

# midsol - CS123 Spring 2010 Solution to Midterm Problem 1(25...

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CS123 Spring 2010 Solution to Midterm Problem 1 (25 points) Let p , q be two propositions. We define the following operation, denoted , as: p q = ( p q ) ( p q ) a) Show the truth table of . Solution : p q p q ( p q ) p q 0 0 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 1 0 b) Prove that p F = p and p T = p where F and T are “false” and “true,” respectively. Solution : p F = ( p F ) ( p F ) by the definition of operation = ( p T ) p by identity law = T p by domination law = p by identity law p T = ( p T ) ( p T ) by the definition of operation = ( p F ) T by domination law = p T by identity law = p by identity law c) Prove that p q = p q . (Note: you can do (b) and (c) by truth tables, but they can be done in a faster way.) Solution : p q = ( p q ) ( p q ) by the definition of operation = ( p q ) ( p q ) by double negation law = ( p q ) ( p q ) by commutative law = ( p q ) ( p q ) by double negation law = p q by the definition of operation Problem 2 (25 points) a) Let x n = 9 x n - 1 + 4 for n 1, and x 0 = 1. Prove by induction on n that x

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