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

Answer: (A)

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