First slide

Relations XII

Question

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

Easy

A

Reflexive
B

Symmetric
C

Transitive
D

None of these

Solution

For any $x \in R,$ we have $x - x + \sqrt{2} = \sqrt{2}$ an irrational number.
$\Rightarrow xRx$ for all x. So, R is reflexive.
R is not symmetric, because $\sqrt{2} R 1 but 1 R \sqrt{2,}$ R is not transitive also because $\sqrt{2} R 1 and 1 R 2 \sqrt{2} but \sqrt{2} R 2 \sqrt{2} .$

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