Question

Let \( R \) be the relation on the set \( \mathbb{R} \) of all real numbers, defined by \( aRb \) if \( |a - b| \leq 1 \). Then, \( R \) is

• A. reflexive and symmetric only
• B. reflexive and transitive only
• C. equivalence
• D. None of the above

Hint:

To check for equivalence, verify reflexivity, symmetry, and transitivity.

Answer 1

The Correct Option is A

Step 1: Understand the properties of the relation.
The relation \( R \) is reflexive because \( |a - a| = 0 \leq 1 \). It is symmetric because if \( aRb \), then \( |a - b| = |b - a| \).
Step 2: Conclusion.
The relation is reflexive and symmetric, but not transitive. Final Answer: \[ \boxed{\text{reflexive and symmetric only}} \]