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CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE & SOFTWARE ENGINEERING COMP 232/2 Mathematics for Computer Science FALL 2010 Assignment 2 In each of the problems below it is especially important that your proof (or counter example) is correct, clear, complete, concise, and carefully presented, using proper mathematical notation. Points will be deducted if your presentation does not satisfy these requirements. 1. If the following equivalence is valid then give a proof. If the equivalence is invalid then give a counter example. x P ( x ) ∧ ∃ x Q ( x ) ≡ ∀ x y P ( x ) Q ( y ) 2. Prove that ( p q ) ( p s ) ( q t ) = s t using a direct proof (with cases). 3. Prove that ( p q ) ( p s ) ( q t ) = s t using a proof by contradiction. 4. Prove that ( p q ) ( p s ) ( q t ) = s t by proving the contrapositive. 5. Rewrite the three propositions “Tomorrow it rains or we go to the beach” and “Tomorrow it does not rain or we go see a movie” and “Tomorrow we
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