Let $n$ be a fixed positive integer.
Define a relation $R$ in $Z$ as follows $\forall a$, $b \in Z,aRb$ if and only if $a - b$ is
divisible by $n$. Show that $R$ is an equivalence relation.

It is given that,, $\forall a,b \in Z,aRb$

if and only if $a - b$ is divisible by $n$.
Now,

I. For reflexive
$aRa \Rightarrow (a - a)$ is divisible by $n$,

which is true for any integer a as 'O' is

divisible by $n$.

Hence we can say that, $R$ is reflexive.
II. For symmetric
$aRb$

$\Rightarrow$ $a - b$ is divisible by $n$.

$\Rightarrow$ $- b + a$ is divisible by $n$.

$\Rightarrow$ $- (b - a)$ is divisible by $n$.

$\Rightarrow$ $(b - a)$ is divisible by $n$.

$\Rightarrow$ $bRa$

Hence we can say that, $R$ is For symmetric.
III. For transitive
Let $aRb$ and $bRc$

$\Rightarrow$ $(a - b)$ is divisible by $n$ and $(b - c)$ is divisible by $n$

$\Rightarrow$ $(a - b) + (b - c)$ is divisibly by $n$

$\Rightarrow$ $(a - c)$ is divisible by $n$

$\Rightarrow$ $aRc$

Hence we can say that, $R$ is transitive.

So, $R$ is an equivalence relation.

Long Answer (L.A.)