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

$x R y\iff \left(\mathrm{x}-\mathrm{y}+\sqrt{2}\right)$ is an irrational number.
Let $\left(\mathrm{x},\mathrm{x}\right)\in \mathrm{R}.$
Then, $\mathrm{x}-\mathrm{x}+\sqrt{2}=\sqrt{2}$ which is an  irrational number.
$\therefore \mathrm{x} \mathrm{R} \mathrm{x},\forall \mathrm{x} \in \mathrm{R}$
$\therefore$R is reflexive .
 $$\begin{gathered} R is not symmetric \sin ce \left(\sqrt{2},1\right)\in R \left(\sin ce \sqrt{2}-1+\sqrt{2} is irrational\right) \\ but \left(1,\sqrt{2}\right)\notin R \left(\sin ce1- \sqrt{2}+\sqrt{2} is not irrational\right) \\ similarly R is not transtive \sin ce \left(-\sqrt{2},\sqrt{3}\right)\in R ,\left(\sqrt{3},1\right)\in R but \\ \left(-\sqrt{2},1\right)\notin R . \end{gathered}$$