Question

If \( R \) be a relation defined as \( a \, R \, b \) iff \( |a - b|>0 \), \( a, b \in \mathbb{R} \), then \( R \) is :

• A. reflexive
• B. symmetric
• C. transitive
• D. symmetric and transitive

Hint:

To check if a relation is symmetric, verify if \( a \, R \, b \implies b \, R \, a \).

Answer 1

The Correct Option is B

For reflexivity, the condition \( a \, R \, a \) must hold. This is not satisfied as \( |a - a| = 0 \), whereas the relation requires \( |a - b|>0 \). For symmetry, if \( a \, R \, b \), then \( b \, R \, a \) must be true. Given that \( |a - b| = |b - a| \), the relation is indeed symmetric.
Therefore, option (B) is the correct choice.