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 116: 25

Answer

a. Converse: If n + 1 is an even integer, then n is a prime number that is greater than 2. Counterexample: Let n = 15. Then n + 1 is even but n is not a prime number that is greater than 2. b. Converse: If 2m is even, then m is an odd integer. Counterexample: Let 2m=12. Then m=6 which is not an odd integer. c. Converse: If two circles do not have a common center, then they intersect in exactly two points. Counterexample: Consider the case of circles a and b. Both a and b have radius 1. The center of circle a is the origin (0,0), while the center of circle b is (2,2). Circles a and b do not have a common center and they intersect at exactly zero points.

Work Step by Step

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