Question

If a relation $R$ defined on set of real numbers as $xRy⟺x-y+\sqrt{2}$ is an irrational number. Then the relation $R$ is

• A. a reflexive relation
• B. a symmetric relation
• C. a transitive relation
• D. both reflexive and transitive relation

Answer 1

The correct option is A a reflexive relation

$xRy⟺x-y+\sqrt{2}\text{is irrational},x,y\in R$
$\left(i\right)\text{R is reflexive relation as}x-x+\sqrt{2}=\sqrt{2}\text{is irrational}\Rightarrow xRx$

$\left(ii\right)\text{R is not symmetric relation as}(\sqrt{2},1)\in R\text{but}(1,\sqrt{2})\notin R$

$\left(iii\right)\text{R is not transitive relation as}(\sqrt{2},1)\text{and}(1,2\sqrt{2})\in R\text{but}(\sqrt{2},2\sqrt{2})\notin R$