Question

Consider a relation $R$ defined as $aRb$ if $2+ab>0$ where $a,b$ are real numbers. Then, the relation $R$ is

• A. reflexive and symmetric
• B. symmetric and transitive
• C. transitive and reflexive
• D. None of these

Answer 1

Answer: (A)

Relation $R$ is reflexive, since $2+a\times a>0,\forall $ real number $a$ .
It is symmetric, since $2+ab>0\Rightarrow 2+ba>0.$
However, $R$ is not transitive.
Consider three real numbers $2,-\frac{1}{6}$ and $-2$ . We have
$\text{2}+2\times \left(- \frac{1}{6}\right)=\frac{5}{3}>0$
and $\text{2}+\left(- \frac{1}{6}\right)\left(- 2\right)=\frac{7}{3}>0$
So, $2R\left(- \frac{1}{6}\right)$ and $\left(- \frac{1}{6}\right)R\left(- 2\right)$