Question

Let $I$ be the set of integers and $R$ be a relation on $I$ defined by $R=\left\{\right(x,y):(x-y)isdivisibleby11,x,y\in I\}$. Then $R$ is

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

Answer 1

The correct option is A

An equivalence relation

Explanation of the correct Option:

The correct option is A:An equivalence relation

To verify the equivalence relation we need to show that the relation is reflexive , symmetric and transitive

Reflexive: For all $a\in I,a-a=0$, which is divisible by $11$.
Thus, $(a,a)\in R$ for all $a\in I\Rightarrow R$ is reflexive

Symmetric: Let $(a,b)\in R(a-b)$ is divisible by $11$
$\Rightarrow -(a-b)$ is divisible by $11$
$\Rightarrow (b-a)$ is divisible by $11$
$\Rightarrow (b,a)\in R$
$\Rightarrow R$ is symmetric.

Transitive : Let $(x,y)\in R$ and $(y,z)\in R$
$\Rightarrow (x-y)$ is divisible by $11$ and $(y-z)$ is divisible by $11$
$\Rightarrow (x-y)+(y-z)$ is divisible by $11$
$\Rightarrow (x-z)$ is divisible by $11$
$\Rightarrow (x,z)\in R$
$\Rightarrow R$ is transitive

Hence $R$ is an equivalence relation on $I\times I$.

Therefore, the correct option is (A)