A relation on the set of real number R is defined as \ x y: x y>0\...

We have, $x p y: x y>0$
(i) Reflexive Suppose $x \rho x \in R$
⇒
$x^{2} > 0$
which is not true when $x=0$. Hence, relation is not reflexive.
(ii) Symmetric $x p y \in R\Rightarrow \quad x y>0$
$\Rightarrow \quad y x>0$
$\Rightarrow \quad y \rho x \in R$
If $(x, y) \in \mathbb{R},$ then, relation is symmetric.
(iii) Transitive $\mathrm{xp} \mathrm{y} \in \mathrm{R}$
$x y>0$
and $\quad y \rho z \in Ry z>0$
$$\begin{array}{lc}<br/>\Rightarrow & x y^{2} z>0 \\<br/>\Rightarrow & x z>0 \\<br/>\Rightarrow & (x, z) \in \mathbb{R} \\<br/>\text { If }(x, y) \in \mathbb{R}, \text { then }(y, z) \in \mathbb{R} \\<br/>\Rightarrow & (x, z) \in R<br/>\end{array}$$
Hence, relation is transitive.$