Among the relations

$\mathrm{S}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R}-\{0\}, 2+\frac{\mathrm{a}}{\mathrm{b}}>0\right\}$

and $\mathrm{T}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R}, \mathrm{a}^{2}-\mathrm{b}^{2} \in \mathbb{Z}\right\}$,

For relation $\mathrm{T}=\mathrm{a}^{2}-\mathrm{b}^{2}=-\mathrm{I}$ <br/><br/>Then, $(\mathrm{b}, \mathrm{a})$ on relation $\mathrm{R}$ <br/><br/>$\Rightarrow \mathrm{b}^{2}-\mathrm{a}^{2}=-\mathrm{I}$ <br/><br/>$\therefore \mathrm{T}$ is symmetric <br/><br/>$\mathrm{S}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathrm{R}-\{0\}, 2+\frac{\mathrm{a}}{\mathrm{b}}>0\right\}$ <br/><br/>$2+\frac{\mathrm{a}}{\mathrm{b}}>0 \Rightarrow \frac{\mathrm{a}}{\mathrm{b}}>-2, \Rightarrow \frac{\mathrm{b}}{\mathrm{a}}<\frac{-1}{2}$ <br/><br/>If $(b, a) \in \mathbf{S}$ then <br/><br/>$2+\frac{\mathrm{b}}{\mathrm{a}}$ not necessarily positive <br/><br/>$\therefore \mathrm{S}$ is not symmetric