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Sets & Relations: Real Numbers Write Xry Irrational Number Relation

JEE Maths question with a full step-by-step solution.

Question
For real numbers xx and yy, we write xRy⇔x−y+2xRy \Leftrightarrow x - y + \sqrt{2} is an irrational number. Then the relation RR is:
AReflexivecorrect
BSymmetric
CTransitive
DEquivalence
Solution
Reflexive: x−x+2=2x - x + \sqrt{2} = \sqrt{2} is irrational ⇒xRx\Rightarrow xRx holds for all xx. Symmetric: 2 R 1\sqrt{2}\,R\,1 holds since 2−1+2=22−1\sqrt{2} - 1 + \sqrt{2} = 2\sqrt{2} - 1 is irrational; but 1 R 21\,R\,\sqrt{2} fails since 1−2+2=11 - \sqrt{2} + \sqrt{2} = 1 is rational ⇒R\Rightarrow R is not symmetric. Transitive: the sum of two irrationals may be rational ⇒R\Rightarrow R is not transitive. ∴R\therefore R is reflexive only. Correct answer: (1)
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