Question

On the set $R$ of all real numbers, a relation $R$ is defined by $R=(a,b):1+ab>0$.Then $R$ is

• A. Reflexive and symmetric only
• B. Reflexive and Transitive only
• C. Symmetric and Transitive only
• D. An equivalence relation

Answer 1

The correct option is A

Reflexive and symmetric only

Explanation for correct answer:

The correct option is A: Reflexive and symmetric only

Relations on R

For reflexive:- $(a,a)\in R\Rightarrow 1+a\times a=1+a^2>0\forall$ real numbers $a$

$\Rightarrow R$ is reflexive.

For symmetric:$(a,b)\in R\Rightarrow 1+ab>0\Rightarrow 1+ba>0\Rightarrow (b,a)\in R$

$\Rightarrow R$ is symmetric.

For transitivity: since $\left(2,\frac{1}{2}\right)\in R$ and $\left(\frac{1}{2},-1\right)\in R$ but $\left(2,-1\right)\notin R$as $1+2\times \left(-1\right)=-1$which is not greater than $0$

$\Rightarrow R$is not transitive.

Therefore the option (A) is correct