Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 8 - Relations - Exercise Set 8.2 - Page 458: 15

Answer

-D is Reflexive -D is not symmetric -D is transitive

Work Step by Step

--D is reflexive: -[We must show that for all positive integers m,m D m.]Suppose m is any positive integer. Since m = m·1, by definition of divisibility m | m. Hence m D m by definition of D. -- D is not symmetric: - For D to be symmetric would mean that for all positive integers m and n, if m D n then n D m . By definition of divisibility, this would mean that for all positive integers m and n, if m|n then n|m. But this is false. As a counterexample, take m =2 and n =4. Then m|n because 2|4 but n (not|)m because 4(not|2). -- D is transitive: - To prove transitivity of D, we must show that for all positive integers m,n, and p, if m D n and n D p then m D p. By definition of D, this means that for all positive integers m,n, and p, if m|n and n | p then m | p.
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