Question

The relation "congruence modulo m" is

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

Answer 1

The correct option is D an equivalence relation

We know that the relation ''congruence modulo $m$'', say $R$, is defined as
$xRy\iff x-y$ is divisible by $m.$

For reflexive:

Clearly $x-x$ is divisible by $m$
$⟹xRx$

So, $R$ is reflexive.

For symmetric:

Let $(x,y)\in R$
$⟹xRy\Rightarrow x-y$ is divisible by $m$
$\Rightarrow y-x$ is divisible by $m$
$\Rightarrow yRx$
So, $R$ is symmetric.

For transitive:

Let $(x,y)\in R$ and $(y,z)\in R$
$⟹xRy$ and $yRz$
$⟹x-y=k_{1}m$ and $y-z=k_{2}m$
$\therefore x-z=(k_1+k_2)m$

$⟹x-z$ is divisible by $m$

$⟹(x,z)\in R$

So, $R$ is transitive.
Hence, $R$ is an equivalence relation.