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Assignment1-solutions

# Assignment1-solutions - MACM 101 Discrete Mathematics I...

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MACM 101 — Discrete Mathematics I Exercises on Propositional Logic. Due: Tuesday, Septem- ber 29th (at the beginning of the class) SOLUTIONS 1. Construct a truth table for the following compound proposition: ( p q ) ( p q ) Solution p q ( p q ) ( p q ) 0 0 0 0 1 1 1 0 1 1 1 1 2. Which of the following statements are tautologies? ( p q ) ( ¬ p q ) ( p q ) ( ¬ ( p ∧ ¬ q )) ( p q ) ( p q ) Solution Method 1: Use truth tables. Method 2: Reason about it: 1) Yes, each of ( p q ) and ( ¬ p q ) is false if and only if p is true and q is false. 2) Yes, it follows from 1), De Morgan’s law and double negation law: ( p q ) ( ¬ p q ) ( p q ) ( ¬¬ ( ¬ p q )) ( p q ) ( ¬ ( ¬¬ p ∧ ¬ q )) ( p q ) ( ¬ ( p ∧ ¬ q )) 3) No, if we set p = 0 , q = 1 then the formula turns into 0. 3. Show that ( p r ) ( q r ) and ( p q ) r are logically equivalent. Method 1: Use truth table. 1

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Method 2: Use the laws of logic. ( p r ) ( q r ) (definition of ) ( ¬ p r ) ( ¬ q r ) (distributivity) ( ¬ p ∧ ¬ q ) r (De Morgan) ( ¬ ( p q )) r (definition of ) ( p q ) r 4. Show that ( a b c d e ) f and ( a b c d e ) f are not logically equivalent Solution If we set a = b = c = d = 0 , e = 1 , f = 0 then we have the first statement to be 1 and the second to be 0.
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