S is a relation over the set R of all real numbers and it is given by

Solution:

(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.