Question

The relation "congruence modulo $m$" is:

• A. reflexive only
• B. symmetric only
• C. transitive only
• D. an equivalence relation

Answer 1

The correct option is D an equivalence relation

If $R$ be the relation, then

$xRy$ $\iff x-y$ is divisible by $m$.
$xRx$ because $x-x=0$ is divisible by $m$.

So, $R$ is reflexive.

If $xRy\Rightarrow x-y$ is divisible by $m$

$y-x=-(x-y)$ is divisible by $m$

$\Rightarrow yRx$

So, $R$ is symmetric

Let $xRy$ and $yRz$

$\Rightarrow x-y=k_{1}m,y-z=k_{2}m$
$\therefore x-z=x-y+y-z=k_{1}m+k_{2}m=(k_1+k_2)m$.

So, $R$ is transitive.
As R is reflexive, symmetric and transitive, it is an equivalence relation.