a1s - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE...

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CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING COMP232 MATHEMATICS FOR COMPUTER SCIENCE ASSIGNMENT 1 SOLUTIONS FALL 2011 1. Use truth tables to determine whether each of the following statements is a tautology, a contradiction, or a contingency , that is, neither a tautology nor a contradiction. SOLUTION (omitting details): (a) p ( q r ) ( p q ) ( p r ) TAUTOLOGY (b) p ∧ ¬ ( q p ) CONTRADICTION (c) ( ¬ p ( p q )) q TAUTOLOGY (d) (( p r ) ( q r )) (( p q ) r ) CONTINGENCY 2. Prove the following logical equivalences without using truth tables. (a) ( ¬ p ) ( q r ) q ( p r ) (b) ( p r ) ( q r ) (( p q ) r ) (c) ( p q ) ( p r ) ( p ( q r )) (d) (( p q ) ( ¬ p r )) ( q r ) T SOLUTION: (a) ( ¬ p ) ( q r ) p ( q r ) p ( ¬ q r ) ≡ ¬ q ( p r ) q ( p r ) (b) ( p r ) ( q r ) ( ¬ p r ) ( ¬ q r ) ≡ ¬ p ∨¬ q r ≡ ¬ ( p q ) r ( p q ) r (c) ( p q ) ( p r ) ( ¬ p q ) ( ¬ p r ) ≡ ¬ p ( q r ) p ( q r ) (d) (( p q ) ( ¬ p r )) ( q r ) ≡ ¬ ( p q ) ∨ ¬ ( ¬ p r ) q r ( ¬ p ∧ ¬ q ) ( p ∧ ¬ r ) q r q ( ¬ p ∧ ¬ q ) r ( p ∧ ¬ r ) ± ( q ∨ ¬ p ) T ² ± ( r p ) T ² ( q ∨ ¬ p ) ( r p ) ( p ∨ ¬ p ) q r T q r T 1
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3. Write down the negations of each of the following statements in their simplest form ( i.e. , do not simply state ”It is not the case that. ..”). Below, x denotes an integer, i.e. , x Z . (a) The train is late or my watch is fast. (b) 0 < x 7 (c) If x is non-negative, then x is positive or x is 0. (d) Doing homework regularly is a necessary condition for John to pass the course.
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a1s - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE...

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