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