Question

On $R,$ the set of real numbers, a relation $\rho$ is defined as '$a\rho b$ if and only if $1+ab>0$'. Then

• A. $\rho$ is an equivalence relation
• B. $\rho$ is reflexive and transitive but not symmetric
• C. $\rho$ is reflexive and symmetric but not transitive
• D. $\rho$ is only symmetric

Answer 1

The correct option is C $\rho$ is reflexive and symmetric but not transitive

For every $a\in R$,
$1+a^2>0$
$\therefore \rho$ is reflexive.

Let $(a,b)\in \rho$
$\Rightarrow 1+ab>0$
$\Rightarrow 1+ba>0$
$\Rightarrow (b,a)\in \rho$
$\therefore \rho$ is symmetric.

Let $a=-1,b=\frac{1}{2},c=1$
Then $1+ab>0$ and $1+bc>0$
But $1+ac\ngtr 0$
Hence, $\rho$ is not transitive.