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: 48

Answer

a. False. There is no square to the right of circle k. b. $\forall x$(Circle(x) $\rightarrow$ ($\exists y$ (Square(y) $\land$ RightOf(y,x)))) c. Formal negation: ∃x(Circle(x) ∧ (∀y(∼Square(y) ∨∼RightOf(y, x))))

Work Step by Step

Recall the negation of a for all statement: ~($\forall x$ in D, P(x)) $\equiv \exists x$ in D such that ~P(x). Recall the negation of an exists statement: ~($\exists x$ in D, P(x)) $\equiv \forall x$ in D such that ~P(x). To negate a multiply quantified statement, apply the laws in stages moving left to right along the sentence. Formal logical notation: "∀x in D, P(x)" can be written as ∀x (x in D → P(x)). "∃x in D such that P(x)" can be written as "∃x (x in D ∧ P(x))."
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