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: 9

Answer

-R is Reflexive -R is not symmetric -R is Transitive

Work Step by Step

-- R is reflexive: -R is reflexive ⇔ for all real numbers x,(x R x). By definition of R, this means that for all real numbers x,x ≥ x. In other words, for all real numbers x,x > x or x = x. But this is true. --R is not symmetric: -R is symmetric⇔for all real numbers x and y, if x R y then y R x. By definition of R, this means that for all real numbers x and y, if x ≥ y then y ≥ x. But this is false. As a counterexample, take x =1 and y =0. Then x ≥ y but y ≥ x because 1≥0 but 0 is not greater than 1. --R is transitive: -R is transitive⇔for all real numbers x, y, and z, if x R y and y R z then x R z.By definition of R, this means that for all real numbers x, y and z,if x ≥ y and y ≥ z then x ≥ z. But this is true by definition of ≥ and the transitive property of order for the real numbers.
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