### Sets And Relations Question 6

#### Question 6 - 31 January - Shift 2

Among the relations

$S={(a, b): a, b \in \mathbb{R}-{0}, 2+\frac{a}{b}>0}$ and $T={(a, b): a, b \in \mathbb{R}, a^{2}-b^{2} \in Z}$,

(1) $S$ is transitive but $T$ is not

(2) $T$ is symmetric but $S$ is not

(3) Neither $S$ nor $T$ is transitive

(4) Both $S$ and $T$ are symmetric

## Show Answer

#### Answer: (2)

#### Solution:

#### Formula: Symmetric relation (v), Transitive relation (vi)

For relation $T=a^{2}-b^{2}=-I$

Then, $(b, a)$ on relation $R$

$\Rightarrow b^{2}-a^{2}=-I$

$\therefore T$ is symmetric

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

$2+\frac{a}{b}>0 \Rightarrow \frac{a}{b}>-2, \Rightarrow \frac{b}{a}<\frac{-1}{2}$

If $(b, a) \in S$ then

$2+\frac{b}{a}$ not necessarily positive

a

$\therefore S$ is not symmetric