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

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

$R_{1}$ 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

answer is A.

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Detailed Solution

$\begin{array}{l} (a, b) \in R_{1} \Leftrightarrow ab \geq 0, Clearly, a^{2} \geq 0 \forall a \in R \Rightarrow (a, a) \in R, \forall a \in R \\ (a, b) \in R_{1} \Rightarrow ab \geq 0 \\ \Rightarrow ba \geq 0 \\ \Rightarrow (b, a) \in R_{1} \\ (a, b) \in R_{1}, (b, c) \in R_{1} \Rightarrow (a, c) \in R_{1} \end{array}$
$∴ R_{1}$ is an equivalence relation.

Clearly,
$(∵ a = a \forall a \in R)$
$a \geq a \forall a \in R \Rightarrow (a, a) \in R_{2} (a, b) \in R_{2} \Rightarrow a \geq b but (b, a) \notin R_{2} as b \leq a So R_{2} is not symmetric relation$
$∴ R_{2}$ is not an equivalence