4. Let R be a relation defined on Z as follows: For all m, n € Z, m R n iff 4 | (m² — n²). a) Prove that R is an equivalence relation. b) Describe the distinct equivalence classes of the relation R. c) Do the distinct equivalence classes form a partition of Z? Explain.
4. Let R be a relation defined on Z as follows: For all m, n € Z, m R n iff 4 | (m² — n²). a) Prove that R is an equivalence relation. b) Describe the distinct equivalence classes of the relation R. c) Do the distinct equivalence classes form a partition of Z? Explain.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 8E: In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R...
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Transcribed Image Text:4.
Let R be a relation defined on Z as follows: For all m, n € Z, m R n iff 4 | (m² — n²).
a) Prove that R is an equivalence relation.
b) Describe the distinct equivalence classes of the relation R.
c) Do the distinct equivalence classes form a partition of Z? Explain.
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