JEE MAIN - Mathematics (2023 - 1st February Morning Shift - No. 3)
Let $$R$$ be a relation on $$\mathbb{R}$$, given by $$R=\{(a, b): 3 a-3 b+\sqrt{7}$$ is an irrational number $$\}$$. Then $$R$$ is
an equivalence relation
reflexive and symmetric but not transitive
reflexive and transitive but not symmetric
reflexive but neither symmetric nor transitive
Explanation
For reflexive :
$3 a-3 a+\sqrt{7}$ is an irrational number $\forall a \in R R$ is reflexive
For symmetric :
Let $3 a-3 b+\sqrt{7}$ is an irrational number
$\Rightarrow 3 b-3 a+\sqrt{7}$ is an irrational number
For example, Let $3 a-3 b=\sqrt{7}$
$\sqrt{7}+\sqrt{7}$ is irrational but $-\sqrt{7}+\sqrt{7}$ is not.
$\therefore R$ is not symmetric
For transitive :
Let $3 a-3 b+\sqrt{7}$ is irrational and $3 b-3 c+\sqrt{7}$ is irrational.
$\Rightarrow 3 a-3 c+\sqrt{7}$ is irrational.
For example, take $a=0, b=-\sqrt{7}, c=\frac{\sqrt{7}}{3}$
$R$ is not transitive.
$3 a-3 a+\sqrt{7}$ is an irrational number $\forall a \in R R$ is reflexive
For symmetric :
Let $3 a-3 b+\sqrt{7}$ is an irrational number
$\Rightarrow 3 b-3 a+\sqrt{7}$ is an irrational number
For example, Let $3 a-3 b=\sqrt{7}$
$\sqrt{7}+\sqrt{7}$ is irrational but $-\sqrt{7}+\sqrt{7}$ is not.
$\therefore R$ is not symmetric
For transitive :
Let $3 a-3 b+\sqrt{7}$ is irrational and $3 b-3 c+\sqrt{7}$ is irrational.
$\Rightarrow 3 a-3 c+\sqrt{7}$ is irrational.
For example, take $a=0, b=-\sqrt{7}, c=\frac{\sqrt{7}}{3}$
$R$ is not transitive.
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