Explanation Given information: Define a relation on Z by a R b if a 2 = b 2 . Proof: To show R is an equivalence relation, we need to show that R is reflexive, symmetric and transitive. Let R be a relation on Z by a R b if a 2 = b 2 or a 2 − b 2 = 0 . (i) Reflexive: Let a ∈ Z , then a 2 = a 2 or a 2 − a 2 = 0 . a R a So, R is reflexive. (ii) Symmetry: Let a , b ∈ Z , then a 2 − b 2 ∈ Z or a 2 − b 2 = m , m ∈ Z . Then b 2 − a 2 = − ( a 2 − b 2 ) = − m and − m ∈ Z Thus, a R b ⇒ b R a and therefore R is symmetric To determine (b) Describe the equivalence classes of R.

3rd Edition

ISBN: 9781284056242

Author: David J. Hunter

Publisher: Jones & Bartlett Learning

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Chapter 2.4, Problem 9E

To determine

(a)

To prove:

R is an equivalence relation.

To determine

(b)

Describe the equivalence classes of R.

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Chapter 2.4, Problem 8E

Chapter 2.4, Problem 10E

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(a) To prove: R is an equivalence relation.