Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 2 - The Logic of Compound Statements - Exercise Set 2.1 - Page 38: 54

Answer

(p ˄ (~(~p ˅ q))) ˅ (p ˄ q) ≡ (p ˄ (~(~(p ˄ ~q)))) ˅ (p ˄ q) (by De Morgan’s law) ≡ (p ˄ (p ˄ ~q)) ˅ (p ˄ q) (by Double Negative law) ≡ ((p ˄ p) ˄ ~q) ˅ (p ˄ q) (by Commutative law) ≡ (p ˄ ~q) ˅ (p ˄ q) (by Idempotent law) ≡ p ˄ (~q ˅ q) (by Distributive law) ≡ p ˄ t (by Negation law) ≡ p (by Identity law)

Work Step by Step

(p ˄ (~(~p ˅ q))) ˅ (p ˄ q) ≡ (p ˄ (~(~(p ˄ ~q)))) ˅ (p ˄ q) (by De Morgan’s law) ≡ (p ˄ (p ˄ ~q)) ˅ (p ˄ q) (by Double Negative law) ≡ ((p ˄ p) ˄ ~q) ˅ (p ˄ q) (by Commutative law) ≡ (p ˄ ~q) ˅ (p ˄ q) (by Idempotent law) ≡ p ˄ (~q ˅ q) (by Distributive law) ≡ p ˄ t (by Negation law) ≡ p (by Identity law)
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