Which one of the following relations on R is an equivalence relation?

Options

A. `a R_(1) b hArr |a|=|b|`
B. `a R_(2) b hArr a ge b`
C. `a R_(3) b hArr a" divides " b`
D. `a R_(4) b hArr a lt b`

Explanation

To determine which of the given relations on \( \mathbb{R} \) is an equivalence relation, we need to check each relation against the three properties that define an equivalence relation: reflexivity, symmetry, and transitivity.

### Given Relations:
1. \( a R_1 b \) if \( |a| = |b| \)
2. \( a R_2 b \) if \( a \geq b \)
3. \( a R_3 b \) if \( \frac{a}{b} \)
4. \( a R_4 b \) if \( a < b \)

### Step 1: Check Relation \( R_1 \)

**Reflexivity**: For any \( a \in \mathbb{R} \), \( |a| = |a| \). Thus, \( (a, a) \in R_1 \) for all \( a \).
**Conclusion**: Reflexive.

**Symmetry**: If \( (a, b) \in R_1 \) (i.e., \( |a| = |b| \)), then \( |b| = |a| \), hence \( (b, a) \in R_1 \).
**Conclusion**: Symmetric.

**Transitivity**: If \( (a, b) \in R_1 \) (i.e., \( |a| = |b| \)) and \( (b, c) \in R_1 \) (i.e., \( |b| = |c| \)), then \( |a| = |c| \), hence \( (a, c) \in R_1 \).
**Conclusion**: Transitive.

### Result for \( R_1 \): It is an equivalence relation.

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### Step 2: Check Relation \( R_2 \)

**Reflexivity**: For any \( a \in \mathbb{R} \), \( a \geq a \). Thus, \( (a, a) \in R_2 \) for all \( a \).
**Conclusion**: Reflexive.

**Symmetry**: If \( (a, b) \in R_2 \) (i.e., \( a \geq b \)), it does not imply \( b \geq a \) unless \( a = b \).
**Conclusion**: Not symmetric.

### Result for \( R_2 \): It is not an equivalence relation.

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### Step 3: Check Relation \( R_3 \)

**Reflexivity**: For any \( a \in \mathbb{R} \), \( \frac{a}{a} = 1 \), so \( (a, a) \in R_3 \) for all \( a \neq 0 \). But \( a = 0 \) is problematic.
**Conclusion**: Not reflexive (fails for \( a = 0 \)).

### Result for \( R_3 \): It is not an equivalence relation.

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### Step 4: Check Relation \( R_4 \)

**Reflexivity**: For any \( a \in \mathbb{R} \), \( a < a \) is false.
**Conclusion**: Not reflexive.

### Result for \( R_4 \): It is not an equivalence relation.

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### Final Conclusion:
The only relation that satisfies all three properties (reflexivity, symmetry, and transitivity) is \( R_1 \). Therefore, the answer is:

**The equivalence relation is \( R_1 \): \( a R_1 b \) if \( |a| = |b| \).**

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