Question

Show that the relation 'a R b' defined by $(a-b)$ is an even integer, is an equivalence relation.

Answer 1

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

Sum of two even integers is even

So, $(a-b+b-c)=(a-c)$ 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.