Question

Show that the relation 'a R b if and only if $a-b$ is an even integer defined on the Z of integers is an equivalence relation.

Answer 1

(i) Since $a-a=0$ and 0 is an even integer
$(a,a)\in R$
$\therefore$ R is reflexive.
(ii) If $(a-b)$ is even, then $(b-a)$ is also even. then, if $(a-b)\in R,(b,a)\in R$
$\therefore$ The relation is symmetric.
(iii) If $(a,b)\in R,(b,c)\in R,$ then $(a-b)$ is even, $(b-c)$ is even, then $(a-b
+b-c)=a-c$ is even.
$\therefore$ If $(a,b)\in R,(b,c)\in R$ implies $(a,c)\in R$
$\therefore$ R is transitive.
Since R is reflexive, symmetric and transitive, it is an equivalence relation.