Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 1 - Section 1.4 - Predicates and Quantifiers - Exercises - Page 56: 49

Answer

Logical Equivalences: p→q≡¬p∨q De Morgan's Law for Qualifiers: $$\neg \exists x P(x) \equiv \forall x \neg P(x) $$$$\neg \forall x P(x) \equiv \exists x \neg P(x) $$ a) $\forall x P(x) \rightarrow A$ is logically equivalent with $\neg(\forall xP(x)) \lor A $ by above Logical Equivalence. Use De Morgan's Law for Qualifiers: $$\equiv \exists x \neg P(x) \lor A$$. Use result of exercise 46 and Logical Equivalence $$\equiv \exists x (\neg P(x) \lor A)$$. $$\equiv \exists x (P(x) \rightarrow A)$$. Thus, $\forall x P(x) \rightarrow A$ is logically equivalent with $\exists x (P(x) \rightarrow A)$ b) Use Logical Equivalence, $\exists xP(x) \rightarrow A \equiv \neg \exists xP(x) \lor A$ Use De Morgan's Law for Qualifiers: $$\equiv \forall x (\neg P(x)) \lor A$$. By previous exercise 46, this is logically equivalent to $$\equiv \forall x (\neg P(x) \lor A)$$. $$\equiv \forall x (P(x) \rightarrow A)$$. Thus, $\exists x P(x) \rightarrow A$ is logically equivalent with $\forall x (P(x) \rightarrow A)$

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