Question

Which of the following relations on the set of real numbers $ \mathbb{R} $ is an equivalence relation?

• A. \( aRb \iff |a| = |b| \)
• B. \( aRb \iff a \text{ divides } b \)
• C. \( aRb \iff a \geq b \)
• D. \( aRb \iff a<b \)

Hint:

For a relation to be an equivalence relation, it must satisfy reflexivity, symmetry, and transitivity. Pay attention to these properties while analyzing the given relations.

Answer 1

The Correct Option is A

For a relation to be an equivalence relation, it must satisfy three properties: reflexivity, symmetry, and transitivity. 
Let’s analyze the options: - Option (A): \( aRb \iff |a| = |b| \) - Reflexivity: 
For any \( a \in \mathbb{R} \), we have \( |a| = |a| \), so the relation is reflexive. 
- Symmetry: If \( |a| = |b| \), then \( |b| = |a| \), so the relation is symmetric. 
- Transitivity: If \( |a| = |b| \) and \( |b| = |c| \), then \( |a| = |c| \), so the relation is transitive. 
Thus, the relation \( aRb \iff |a| = |b| \) satisfies all three properties, and it is an equivalence relation.
 - Option (B): \( aRb \iff a \text{ divides } b \) - This relation is not reflexive because a number does not divide itself unless the number is non-zero. Thus, it is not an equivalence relation. 
- Option (C): \( aRb \iff a \geq b \) - This relation is not symmetric because if \( a \geq b \), it does not imply that \( b \geq a \). 
Thus, it is not an equivalence relation. - Option (D): \( aRb \iff a<b \) 
- This relation is neither reflexive nor transitive. It is not reflexive because \( a \not< a \). 
Thus, it is not an equivalence relation. Thus, the correct answer is (A).