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Equivalence Class

In the set Z ...

Question

In the set Z of all integers, which of the following relation R is not an equivalence relation?
(a) x R y : if x ≤ y
(b) x R y : if x = y
(c) x R y : if x − y is an even integer
(d) x R y : if x ≡ y (mod 3)

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Solution

(a) x R y : if x ≤ y

Clearly, R is not symmetric because x < y does not imply y < x.

Hence, (a) is not an equivalence relation.

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