Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 9 - Section 9.1 - Relations and Their Properties - Exercises - Page 583: 57

Answer

(a, a) ∈ R$^n$ ◦ R = R$^{n+1}$ for all a ∈ A.

Work Step by Step

Use mathematical induction. The result is trivial for n = 1. Assume R$^n$ is reflexive. Then (a, a) ∈ R$^n$ for all a ∈ A and (a, a) ∈ R. Thus (a, a) ∈ R$^n$ ◦ R = R$^{n+1}$ for all a ∈ A.
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