Question

Show that the relation R in the set of integers given by $R=\{(a,b):5divides(a-b)\}$ is symmetric and transitive.

Answer 1

Reflexive

Since $a-a=0$ and $5$ divides $0$

So, $5$ divides $a-a$

So, it is reflexive

Symmetric

$5$ divides $a-b$ then $5$ also divide $-(a-b)$ that is $(b-a)$

$(a,b)$ belongs to $R$, so $(b,a)$ belongs to $R$

So, it is symmetric.

Transitive

If $5$ divides $(a-b)$ and $5$ divides $(b-c)$ then $5$ divides $(a-b)+(b-c)=(a-c)$

So it is Transitive also.