Question

Let $S$ be the set of all real numbers. Then, the relation $R=\{(a$, $b)$ : $1+ab>0\}$ on $S$ is

• A. Reflexive and symmetric but not transitive
• B. Reflexive and transitive but not symmetric
• C. Symmetric and transitive but not reflexive
• D. Reflexive, transitive and symmetric

Answer 1

Answer: (A)

Reflexive : As $1+a\cdot a=1+a^2>0$, $a\in S$.
$\therefore (a$, $b)\in R$
$\therefore R$ is reflexive.
Symmetric : $(a$, $b)\in R$
$\Rightarrow 1+ab>0$
$\Rightarrow 1+ba>0$
$\Rightarrow (b$, $a)\in R$.
$\therefore R$ is symmetric.
Transitive : $(a$, $b)\in R$ and $(b$, $c)\in R$ need not imply $(a$, $c)\in R$.
Hence, $R$ is not transitive.