Question

Let a relation R in the set R of real numbers be defined as (a, b) $ϵ$ R if and only if 1 + ab > 0 for all a, b $ϵ$ R. The
relation R is

• A. Reflexive and symmetric
• B. symmetric and transitive
• C. an equivalence relation
• D. None of these

Answer 1

The correct option is A Reflexive and symmetric

We have, R = {(a, b) : 1 + ab > 0, ab $ϵ$ R}.
Let a $ϵ$ R. $\therefore a^2\ge$ 0 or 1+ $a^2$ > 0 or (a,a) $ϵ$ R
$\therefore$ R is reflexive.
Let (a,b) $ϵ$ R $\Rightarrow$ 1 + ab > 0 $\Rightarrow$ 1 + ba > 0
$\Rightarrow$ (b,a) $ϵ$ R
$\therefore$ R is symmetric.
$(2,\frac{1}{3})ϵ$ R because 1 + 2 $\left(\frac{1}{3}\right)=\frac{5}{3}$ > 0
$(\frac{1}{3},-1)ϵ$ R because 1+$\frac{1}{3}$ (-1) = $\frac{2}{3}$ > 0
Now, (2, -1) $ϵ$ R if 1 + 2 (- 1) = - 1 < 0, which in not true.
$\therefore$ R is not transitive.