Question

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

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

Answer 1

Answer: (C)

We know that,
$1+a^2>0,a\in R$
$\Rightarrow (a,a)\in \rho$
$\therefore \rho$ is reflexive
Again, let $(a,b)\in p$
$\Rightarrow 1+ab>0$
$\Rightarrow 1+ba>0$
$\Rightarrow (b,a)\in \rho$
$\therefore \rho$ is symmetric
Now, $(1,-0\cdot 1)\in \rho$
and $(-0\cdot 1,-9)\in \rho$
but $(1,-9)\in \rho$
$\therefore \rho$ is not transitive.