Question

Let the relation $\rho$ be defined on $R$ as $a\rho b$ iff $1+ab>0$. Determine whether the relation $\rho$ is reflexive, symmetric or transitive.

Answer 1

For reflexive:
$(a,a)\in \rho ,\forall a\in R$
Since, $1+a^2>0\forall a\in R$
$\therefore \rho$ is reflexive.

For symmetric:
$(a,b)\in \rho$, then $(b,a)\in \rho$
Since, $1+ab>0$, then $1+ba>0$
$\therefore \rho$ is symmetric.

For transitive:
$(a,b)\in \rho ,(b,c)\in \rho \Rightarrow (a,c)\in \rho$
Now, $(-2,0)\in \rho$ and $(0,1)\in \rho$
but $(-2,1)\notin \rho$
Hence, $\rho$ is not transitive.

$\therefore \rho$ is reflexive and symmetric but not transitive.