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 3.2 - Page 117: 27

Answer

Contrapositive: $\forall$ integers d, if $d \neq 3$, then 6/d is not an integer. Converse: $\forall$ integers d, if d=3, then 6/d is an integer. Inverse: $\forall$ integers d, if 6/d is not an integer, then $d \neq 3$. The original statement and the contrapositive are logically equivalent and they are both false. Counterexample: Let x=1. Then 6/1 = 6 which is an integer. The converse and inverse are contrapositives of each other, so they are logically equivalent to each other. Both are true.

Work Step by Step

A statement of the form: $\forall x \in D$, if P(x) then Q(x), has as its contrapositive statement: $\forall x \in D$, if ~Q(x) then ~P(x), as its converse statement: $\forall x \in D$, if Q(x) then P(x), and as its inverse statement: $\forall x \in D$, if ~P(x) then ~Q(x).
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