Discrete Mathematics with Applications 4th Edition

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

Chapter 3 - The Logic of Quantified Statements - Exercise Set 3.3 - Page 131: 52

Answer

a. False. Every circle is not to the right of every triangle. For instance, circle b is not to the right of triangle c. b. Formal: (($\forall x$ Circle(x) $\land \forall y$ Triangle(y))$\rightarrow$ RightOf(x, y)) c. Negation: ((($\exists x$ Circle(x) $\lor \exists y$ Triangle(y)) $\land$ ~RightOf(x, y)))
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Work Step by Step

c. ~((($\forall x$ Circle(x) $\land \forall y$ Triangle(y))$\rightarrow$ RightOf(x, y))) $\equiv$ ((($\exists x$ Circle(x) $\lor \exists y$ Triangle(y)) ~$(\rightarrow$ RightOf(x, y)))) (by De Morgan's Law and the law of negating $\forall$ statement) $\equiv$ ((($\exists x$ Circle(x) $\lor \exists y$ Triangle(y)) $\land$ ~RightOf(x, y))) (by the law of negating an if-then statement)
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