QJEEM22Ma1395: JEE Main 2022 Mathematics - Class 11 JEEM22Ma1395 Solution: Let and be two relations defined on by and Then,

Question

Verified Solution • 2025–26 Session moderate

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,

Accepted Answer

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

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

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