Question

For real number x and y define a relation R, xRy 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

Answer: (A)

reflexive

Answer 2

Clearly x 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. SoR is not symmetric. Since 1 R 2 and $2R \sqrt{2}$ but 1 is not related to $\sqrt{2}$ So R is not transitive.