JEE Main 2022 — Sets And Relations Question with Solution

$$a{R}_{1}b\iff ab\ge 0$$

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

$$a{R}_{2}b\iff a\ge b$$

$$(a,a)\in R_2\Rightarrow$$ 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