Let $R_{1}$ and $R_{2}$ be two relations defined on $\mathbb{R}$ by $a \mathrm{R}_{1} b \Leftrightarrow a b \geq 0$ and $a R_{2} b \Leftrightarrow a \geq b$, then

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

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

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

(C) both $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ are equivalence relations

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Sets Relations and Functions

Let

R_{1}

and

R_{2}

be two relations defined on

R

a R_{1} b \Leftrightarrow ab \geq 0

and

a R_{2} b \Leftrightarrow a \geq b

, then

Answer: (D)

For

R_{1}

: The relation is reflexive since

x \times x \geq 0

is always true. It is symmetric since

x y \geq 0 \Rightarrow y x \geq 0

. However, it is not transitive. Consider

x = 2, y = 0, z = - 2

. Then

x y \geq 0

and

y z \geq 0

, but

x z = - 4 \neq \geq 0

. Therefore

R_{1}

is not an equivalence relation.

For

R_{2}

: If

a \geq b

, it does not imply

b \geq a

(not symmetric). Therefore

R_{2}

is not an equivalence relation.

Hence neither

R_{1}

nor

R_{2}

is an equivalence relation. Answer: (D)

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option