MAS111_09_assignment_05-hints

# MAS111_09_assignment - NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 5 HINTS 1 Let p(x and q(x denote the following

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NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 5 HINTS 1. Let p ( x ) and q ( x ) denote the following predicates: p ( x ) : x 3; q ( x ) : x + 1 is odd . If the domain of x consists of all integers, what are the truth values of the following propositions: (a) q (1) (b) ¬ p (3) (c) p (7) q (7) (d) ¬ ( p ( - 5) q ( - 25)) (e) ¬ ( p ( - 5) q ( - 25)) (f) x p ( x ) (g) x q ( x ) Hints : q (1) is false, as 1 + 1 is even. p (3) is true, so ¬ p (3) is false. p (7) is false and q (7) is false, so p (7) q (7) is also false. p ( - 5) is true and q ( - 25) is false. So p ( - 5) q ( - 25) is true, and p ( - 5) q ( - 25) is false. That means, ¬ [ p ( - 5) q ( - 25)] is false, and ¬ [( p ( - 5) q ( - 25))] is true. As p (5) is false, x p ( x ) is false (5 is a counterexample). x q ( x ) is true, as q (0) is true. 2. Let p ( x,y ) and q ( x,y ) denote the following predicates: p ( x,y ) : x 2 y ; q ( x,y ) : x + 2 < y. If the domains of x and y consist of all real numbers, what are the truth values of the following propositions:

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(a) q (1 ) (b) p ( - 3 , 8) q (1 , 3) (c) p ( 1 2 , 1 3 ) ∨ ¬ q ( - 2 , - 3) (d) ¬ ( p ( - 5 , 74) q (0 . 3 , - 25)) (e) x y p ( x,y ) (f) x y q ( x,y ) Hints : q (1 ) is true, as 1 + 2 < π . p ( - 3 , 8) is false, because ( - 3) 2 = 9 6≤ 8. Hence p ( - 3 , 8) q (1 , 3) is false. p ( 1 2 , 1 3 ) is true, as ( 1 2 ) 2 < 1 3 , and hence p ( 1 2 , 1 3 ) ∨ ¬ q ( - 2 , - 3)) (we do not need to care whether ¬ q ( - 2 , - 3) is true or not as we already know that p ( 1 2 , 1 3 ) is true). p
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## This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.

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MAS111_09_assignment - NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 5 HINTS 1 Let p(x and q(x denote the following

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