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 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 $(a, b) \in p$
$\Rightarrow 1+a b>0$
$\Rightarrow 1+b a>0$
$\Rightarrow (b, a) \in \rho$
$\therefore \rho$ 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.