S is a relation over the set R of all real numbers and it is given by
$(a, b) \in S \Leftrightarrow a b \geq 0$. Then, $S$ is
(a) symmetric and transitive only
(b) reflexive and symmetric only
(c) antisymmetric relation
(d) an equivalence relation
(d) an equivalence relation
Reflexivity: Let $a \in R$
Then
$a a=a^{2}>0$
$\Rightarrow(a, a) \in R \forall a \in R$
So, S is reflexive on R.
Symmetry: Let $(a, b) \in S$
Then,
$(a, b) \in S$
$\Rightarrow a b \geq 0$
$\Rightarrow b a \geq 0$
$\Rightarrow(b, a) \in S \forall a, b \in R$
So, S is symmetric on R.
Transitivity:
If $(a, b), \quad(b, c) \in \mathrm{S}$
$\Rightarrow a b \geq 0$ and $b c \geq 0$
$\Rightarrow a b \times b c \geq 0$
$\Rightarrow a c \geq 0$ $\left[\because b^{2} \geq 0\right]$
$\Rightarrow(a, c) \in S$ for all $a, b, c \in$ set $R$
Hence, S is an equivalence relation on R.
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