First slide

Relations XII

Question

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

Moderate

A

reflexive
B

symmetric
C

transitive
D

None of these

Solution

$x R y \Leftrightarrow (x - y + \sqrt{2})$ is an irrational number.
Let $(x, x) \in R .$
Then, $x - x + \sqrt{2} = \sqrt{2}$ which is an irrational number.
$∴ x R x, \forall x \in R$
$∴$ R is reflexive .
$R i s n o t s y m m e t r i c \sin c e (\sqrt{2}, 1) \in R (\sin c e \sqrt{2} - 1 + \sqrt{2} i s i r r a t i o n a l) b u t (1, \sqrt{2}) \notin R (\sin c e 1 - \sqrt{2} + \sqrt{2} i s n o t i r r a t i o n a l) s i m i l a r l y R i s n o t t r a n s t i v e \sin c e (- \sqrt{2}, \sqrt{3}) \in R, (\sqrt{3}, 1) \in R b u t (- \sqrt{2}, 1) \notin R .$

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