JEE MAIN - Mathematics (2022 - 27th July Morning Shift - No. 1)

Let $$R_{1}$$ and $$R_{2}$$ be two relations defined on $$\mathbb{R}$$ by

$$a \,R_{1} \,b \Leftrightarrow a b \geq 0$$ and $$a \,R_{2} \,b \Leftrightarrow a \geq b$$

Then,

$$R_{1}$$ is an equivalence relation but not $$R_{2}$$

$$R_{2}$$ is an equivalence relation but not $$R_{1}$$

both $$R_{1}$$ and $$R_{2}$$ are equivalence relations

neither $$R_{1}$$ nor $$R_{2}$$ is an equivalence relation

$$a\,{R_1}\,b \Leftrightarrow ab \ge 0$$

So, definitely $$(a,a) \in {R_1}$$ as $${a^2} \ge 0$$

If $$(a,b) \in {R_1} \Rightarrow (b,a) \in {R_1}$$

But if $$(a,b) \in {R_1},(b,c) \in {R_1}$$

$$\Rightarrow$$ Then $$(a,c)$$ may or may not belong to R₁

{Consider $$a = - 5,b = 0,c = 5$$ so $$(a,b)$$ and $$(b,c) \in {R_1}$$ but $$ac < 0$$}

So, R₁ is not equivalence relation

$$a\,{R_2}\,b \Leftrightarrow a \ge b$$

$$(a,a) \in {R_2} \Rightarrow $$ so reflexive relation

If $$(a,b) \in {R_2}$$ then $$(b,a)$$ may or may not belong to R₂

$$\Rightarrow$$ So not symmetric

Hence it is not equivalence relation