x1solnfall2009 - CONCORDIA UNIVERSITY DEPARTMENT OF...

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Unformatted text preview: CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE & SOFTWARE ENGINEERING COMP 232/2 Mathematics for Computer Science FALL 2009 Examination 1 1. Let P ( x, y ) denote the statement “ y = x 2 + x + 1”, where x, y ∈ Z . What is the truth value of each of the following? Explain your answers. a) ∀ x ∃ y P ( x, y ) b) ∃ x ∀ y P ( x, y ) c) ∀ y ∃ x P ( x, y ) 2. Prove that the following expression is a tautology without using truth tables. [ ( p ∨ q ) ∧ ( ¬ p ∨ r ) ] → ( q ∨ r ) 3. Let P ( x, y ) be the statement “ x has played tennis in y ” and T ( x ) be the statement “ x watched the US Open.” Assume that the universe of discourse for x is all students in the class and for y is all cities in Canada. Express each of the following statements using the predicates P ( x, y ), T ( x ), logical operators, and quantifiers. a) Everyone in this class has played tennis in some city in Canada....
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