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Unformatted text preview: 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 6 = 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 6 = 0. Raising both side to the 3rd power, we get x = a 3 /b...
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This note was uploaded on 10/08/2008 for the course CMPSC 360 taught by Professor Haullgren during the Fall '08 term at Pennsylvania State University, University Park.
- Fall '08