Question

For real number $x$ and $y$, define a relation $R_x,R_y$, if and only if $x-y+\sqrt{2}$ is an irrational number. Then the relation $R$ is

• A. reflexive
• B. symmetric
• C. transitive
• D. an equivalence relation

Answer 1

The correct option is A reflexive

$R_x$ as $x-x+\sqrt{2}=\sqrt{2}$ is an irrational number.
Thus $R$ is reflexive.
Also $(\sqrt{2},1)\in R$ as $\sqrt{2}-1+\sqrt{2}=2\sqrt{2}+1$ is an irrational number but $(1,\sqrt{2})\notin R$ as $1-\sqrt{2}+\sqrt{2}=1$
is a rational number. So $R$ is not symmetric.

Since $1R2$ and $2R\sqrt{2}$ but $1$ is not related to $\sqrt{2}$.
So $R$ is not transitive.