Question

A relation “congruence modulo $\mathrm{m}$" is

• A. Reflexive only
• B. Transitive only
• C. Symmetric only
• D. An equivalence relation

Answer 1

The correct option is D

An equivalence relation

Explanation for correct option.

Step1. Relation is Reflexive

If $\mathrm{R}$ is a relation define, $xRy\iff x-y$ is divisible by $\mathrm{m}$.

‘$xRx$’ because $x-x$is divisible by $\mathrm{m}$.

It is reflexive.

Step2. Relation is Symmetric.

$xRy\iff x-y$ is divisible by $\mathrm{m}$.

$y+x$ is also divisible by $\mathrm{m}$.

$yRx$.

It is symmetric.

Step3. Relation is Transitive.

$$\begin{gathered} \mathbf{\text{'}}\mathit{x}\mathit{R}\mathit{y}\mathbf{\text{'}}\mathbf{}\mathbf{\&}\mathbf{}\mathbf{\text{'}}\mathit{y}\mathit{R}\mathit{z}\mathbf{\text{'}} \\ \Rightarrow x-y=k_{1}m \\ \Rightarrow y-z=k_{2}m \\ \Rightarrow x-z=(k_1-k_2)m \\ \mathbf{\text{'}}\mathit{x}\mathit{R}\mathit{z}\mathbf{\text{'}} \end{gathered}$$

It is transitive

Since $\mathrm{R}$ is reflexive, symmetric & transitive , it is an equivalence relation.

Hence, the correct option is $\mathbf{\left(}\mathit{D}\mathbf{\right)}$