Question

For real numbers $x$ and $y$, we define $xRy$ if $x-y+\sqrt{5}$ is an irrational number. The relation $R$ is

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of these

Answer 1

The correct option is A Reflexive

$xϵR\Rightarrow x-x+\sqrt{5}=\sqrt{5}$ is an irrational number.
$\therefore (x,x)ϵR$

Hence, $R$ is reflexive
$(\sqrt{5},1)ϵR$ because $\sqrt{5}-1+\sqrt{5}=2\sqrt{5}-1$
which is an irrational number.
$\therefore (1,\sqrt{5}\text{/}ϵR$

Hence, $R$ is not a symmetric

We have, $(\sqrt{5},1)(1,2\sqrt{5})ϵR$ because
$\sqrt{5}-1+\sqrt{5}=2\sqrt{5}-1$, if
$1-2\sqrt{5}+\sqrt{5}=1-\sqrt{5}$ are irrational number.

$\therefore (\sqrt{5},2\sqrt{5})\text{/}ϵR$

Hence, $R$ is not transitive.