Question

For real numbers 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

Clearly $xRx$ 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 irrational 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