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 2.4 - Page 77: 33

Answer

DEFINITIONS De Morgan’s law: $∼(p∧q)≡∼p∨∼q$ $∼(p∨q)≡∼p∧∼q$ Double negative law: $∼(∼p)≡p$ Idempotent laws: $p∨p≡p$ $p∧p≡p$

Work Step by Step

(a) By the definition of the Sheffer Stroker: $P∣Q≡∼(P∧Q)$ $(P∣Q)∣(P∣Q) ​ ≡∼((P∣Q)∧(P∣Q))$ $≡∼(∼(P∧Q)∧∼(P∧Q))$ $≡∼(∼(P∧Q))∨∼(∼(P∧Q))$ $≡(P∧Q)∨∼(∼(P∧Q))$ $≡(P∧Q)∨(P∧Q)$ $≡P∧Q$ We have derived that (P∣Q)∣(P∣Q) is a logically equivalent with P∧Q. $P∧Q≡(P∣Q)∣(P∣Q)$ ​(b): $P∧(∼Q∨R)≡P∧(∼Q∨∼(∼R))$ $≡P∧∼(Q∧(∼R))$ $≡P∧∼(Q∧(∼(R∧R)))$ $≡P∧∼(Q∧(R∣R))$ $≡P∧(Q∣(R∣R))$ We have derived that $(P∣(Q∣(R∣R)))∣(P∣(Q∣(R∣R)))$ is a logically equivalent with $P∧(∼Q∨R)$, while $(Q∣(R∣R)))∣(P∣(Q∣(R∣R)))$ contains only Sheffer strokes. RESULT: (a) $P∧Q≡(P∣Q)∣(P∣Q)$ (b) $(P∣(Q∣(R∣R)))∣(P∣(Q∣(R∣R)))$ ​
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