Question

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

• A. $S$ is transitive but $T$ is not
• B. both $S$ and $T$ are symmetric
• C. neither $S$ nor $T$ is transitive
• D. $T$ is symmetric but $S$ is not

Answer 1

Answer: (D)

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