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,

Question

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

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