Question

For real numbers x and y, we write $xRy\iff x-y+\sqrt{2}$ is an irrational number. Then the relation R is

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of these

Answer 1

The correct option is A

Reflexive

For any $x\in R$ we have $x-x+\sqrt{2}=\sqrt{2}$ an irrational number.
$\Rightarrow xRx$ for all x. So, R is reflexive.
R is not symmetric, because $\sqrt{2}R1$ but $1R\sqrt{2},R$ is not transitive also because $\sqrt{2}R1$ and $1R2\sqrt{2}$ but $\sqrt{2}R2\sqrt{2}$.