Question

If a relation $R$ is 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

$xRy⟺x-y+\sqrt{2}$ is irrational,
$x,y\in R$

$\left(i\right)$ $R$ is reflexive relation as $x-x+\sqrt{2}=\sqrt{2}$ is irrational.

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

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