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CMPSC360hw5soln

# CMPSC360hw5soln - CMPSC 360 Fall 2008 Homework 5 PRACTICE...

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CMPSC 360 Fall 2008 Homework 5: PRACTICE EXAM 1 Solutions 1. (25 points, 5 each) Logic, Quantifiers, Proofs (a) i. True. For each x , we can choose y = 0, and then it is true that x + 0 = x = x - 0. ii. True. We can choose y = 0, and then the statement says for all x , x + 0 = x = x - 0, which is true. iii. False. A counterexample would be y = 1, in which case there does not exist and x such that x + 1 = x - 1. (b) Note that Q ( x, y ) is the same as ¬ P ( x, y ). ¬ ( y x P ( x, y )) y ¬∃ x P ( x, y ) y x ¬ P ( x, y ) y x Q ( x, y ) (c) Assume a and b are rational. Then a = a 1 /a 2 and b = b 1 /b 2 where a 1 , a 2 , b 1 , b 2 Z , and a 2 , b 2 = 0. Multiplying, we have ab = a 1 a 2 · b 1 b 2 = a 1 b 1 a 2 b 2 . Since a 1 b 1 Z and a 2 b 2 Z (and is not zero), ab is a rational number. (d) Proof by contraposition. Assume x 1 / 3 is rational. Then x 1 / 3 = a/b , for a, b Z , b = 0. Raising both side to the 3rd power, we get x = a 3 /b 3 . Since a 3 , b 3 Z , and b 3 = 0, x is rational.

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CMPSC360hw5soln - CMPSC 360 Fall 2008 Homework 5 PRACTICE...

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