Question

For real numbers \( x \) and \( y \), \( xRy \iff x - y + \sqrt{2} \) is an irrational number. Then the relation \( R \) is:

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. Equivalence

Hint:

A relation is reflexive if \( xRx \) holds for every element \( x \) in the set. For the given relation, \( x - x + \sqrt{2} = \sqrt{2} \) is irrational, making it reflexive.

Answer 1

The Correct Option is A

Step 1: Understanding Reflexivity.
A relation \( R \) is reflexive if for every element \( x \) in the set, the relation holds for \( xRx \). In other words, \( x - x + \sqrt{2} = \sqrt{2} \), which is irrational.

Step 2: Checking the given relation.
For the relation to be reflexive, we need to check whether \( x - x + \sqrt{2} = \sqrt{2} \) is irrational for every real number \( x \). Clearly, \( \sqrt{2} \) is irrational, so the relation is reflexive.

Step 3: Final Answer.
Since the relation is reflexive, the correct answer is option (A).