assn1 - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE...

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CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE & SOFTWARE ENGINEERING COMP 232/2 Mathematics for Computer Science FALL 2010 Assignment 1 1. (10 marks) Professor Smullyan went to a remote island where all inhabitants were either knights or knaves. Knights always tell the truth and knaves always lie. While he was visiting the island, he came across two of the inhabitants, A and B, resting under a tree. Professor Smullyan asked them, “Is either of you a knight?” and A responded to his question. The professor immediately knew what both A and B were. What were A and B? How did the professor figure it out? 2. (15 marks) Classify each of the following as a contradiction, a tautology, or a contingency. a) ( p q ) ( ¬ ( p q )) b) ( p q ) ( q r ) ∨ ¬ r p c) ( p r ) ( p ∨ ¬ r ) F 3. (15 marks) Prove the following logical equivalences without using truth tables. a) ( ¬ p F ) p T b) ( p q ) ( ¬ p r ) ( q r ) T c) ( p q ) ( ¬ q r ) (( ¬ q r ) q ) q 4. (20 marks) Let P ( x, y
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