Question

Let $r$ be relation from $R$(set of real numbers) to $R$ defined by $r=\left\{\overline{)\left(a,b\right)}a,b\in R\text{and}a-b+\sqrt{3}\text{is an irrational number}\right\}$. then relation $r$ is

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

Answer 1

The correct option is A

An equivalence relation

Step 1: Check for reflexive relation.

Let $a\in r$

$\Rightarrow a-a+\sqrt{3}=\sqrt{3}$ is an irrational number.

Therefore, relation $r$ is reflexive.

Step 2: Check for symmetric relation.

Let $\left(a,b\right)\in r$

$\Rightarrow a-b+\sqrt{3}=b-a+\sqrt{3}$ is an irrational number.

$\Rightarrow \left(b,a\right)\in r$

Therefore, relation $r$ is symmetric.

Step 3: Check for transitive relations

Let $\left(a,b\right)\in r$ and $\left(b,c\right)\in r$

$\Rightarrow a-b+\sqrt{3}$ is an irrational number

$\Rightarrow b-c+\sqrt{3}$ is an irrational number

by adding both irrational numbers we get

$\Rightarrow a-c+2\sqrt{3}$ is also an irrational number.

$\Rightarrow \left(a,c\right)\in r$

Therefore, relation $r$ is transitive.

Since $r$ satisfies the reflexive, symmetric, transitive relation, thus it is an equivalence relation.

Hence, option A is the correct answer.