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

$\mathrm{xRy}\iff \mathrm{x}-\mathrm{y}+\sqrt{2}$ is an irrational number. 
Let $(\mathrm{x},\mathrm{x})\in \mathrm{R}$
Then, $\mathrm{x}-\mathrm{x}+\sqrt{2}=\sqrt{2}$which is an irrational number

$\therefore \mathrm{xR}\times ,\mathrm{\forall }\mathrm{x}\in \mathrm{R}$
R is an reflexive relation.
 R is not symmetric since $$\begin{gathered} \left(\sqrt{2},1\right)\in R \sin ce \sqrt{2}-1+\sqrt{2} is irrational \\ but \left(1,\sqrt{2}\right) \notin R \sin ce 1-\sqrt{2}+\sqrt{2}=1, which is rational \end{gathered}$$
 R is not transitive because  $(1,\sqrt{3})\in \mathrm{R},\text{since }(1-\sqrt{3}+\sqrt{2})$ is an irrational number

                                    and $(\sqrt{3},\sqrt{2})\in \mathrm{R},\text{since }(\sqrt{3}-\sqrt{2}+\sqrt{2})$=$\sqrt{3}$ is an irrational number. 

but  $(1,\sqrt{2})\notin \mathrm{R} \mathrm{sinc}e (1-\sqrt{2}+\sqrt{2})=1$ is a rational number.

$\therefore$R is not transitive relation.