Question:
In the set Z of all integers, which of the following relation R is not an equivalence relation?
(a) $x R y$ : if $x \leq y$
(b) $x R y$ : if $x=y$
(c) $x R y:$ if $x-y$ is an even integer
(d) $x R y:$ if $x \equiv y(\bmod 3)$
Solution:
(a) $x R y:$ if $x \leq y$
Clearly, $R$ is not symmetric because $x Hence, (a) is not an equivalence relation.