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: 51

Answer

a. False. There is no object that has a different color from every other object. b. Formal version: ∃y(∀x(x = y → ∼SameColor(x, y))) c. Formal negation: ∀y(∃x(x = y ∧ SameColor(x, y)))

Work Step by Step

c. ~(∃y(∀x(x = y → ∼SameColor(x, y)))) $\equiv$ $\forall$y ~(∀x(x = y → ∼SameColor(x, y))) (by the law of negating a $\exists$ statement) $\equiv$ $\forall$y ($\exists$x ~(x = y → ∼SameColor(x, y))) (by the law of negating a $\forall$ statement) $\equiv$ $\forall$y ($\exists$x (x = y $\land$ SameColor(x, y))) (by the law of negating an if-then statement)
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