Question

Show that the relation $R$ in the set $Z$ of integers given by $R=\left\{\right(a,b):2\text{divides}(a-b\left)\right\}$ is equivalence relation.

Answer 1

$R=\left\{\right(a,b):2divides(a-b\left)\right\}$

for $R on (a,a)

$a-a=0$

since $2$ divides $0$,R is symmetric.

For $(a,b)onR$

Let $(a-b)=2K$

$(b,a)onR$

$(b-a)=2K$ is divided by $2$

Thus $R$ is reflexive

For $(a,b)onR$

Let $(a-b)=2K$

$(b,a)onR$

$(b,c)onR$

$(b-c)=2P$

By solving above equations. We get,

$(a-c)=2(K+P)$ is divided by $2$.

So (a,c) satisfies $R$.

Thus $R$ is transitive.

So, $R$ is equivilance relation.