Which of one of the following relations on `R` is equivalence relation

Options

A. `xR_(1)yhArrx^(2)=y^(2)`
B. `xR_(2)yhArrxgey`
C. `xR_(3)yhArrx|y(x "divides" y)`
D. `xR_(4)yhArrxlty`

Explanation

To determine which of the given relations on \( \mathbb{R} \) is an equivalence relation, we need to check each relation for three properties: reflexivity, symmetry, and transitivity. Let's analyze each option step by step.

### Step 1: Analyze Option 1: \( x^2 = y^2 \)

1. **Reflexivity**: For any \( x \in \mathbb{R} \), we check if \( x^2 = x^2 \).
- This is true for all \( x \), so the relation is reflexive.

2. **Symmetry**: If \( x^2 = y^2 \), then we need to check if \( y^2 = x^2 \).
- This is also true, hence the relation is symmetric.

3. **Transitivity**: If \( x^2 = y^2 \) and \( y^2 = z^2 \), we need to check if \( x^2 = z^2 \).
- This holds true, so the relation is transitive.

Since all three properties are satisfied, **Option 1 is an equivalence relation**.

### Step 2: Analyze Option 2: \( x \geq y \)

1. **Reflexivity**: For any \( x \in \mathbb{R} \), check if \( x \geq x \).
- This is true, so the relation is reflexive.

2. **Symmetry**: If \( x \geq y \), does it imply \( y \geq x \)?
- This is not true in general (e.g., \( 2 \geq 1 \) does not imply \( 1 \geq 2 \)), so the relation is not symmetric.

Since symmetry fails, **Option 2 is not an equivalence relation**.

### Step 3: Analyze Option 3: \( x \text{ divides } y \)

1. **Reflexivity**: For any \( x \in \mathbb{R} \), check if \( x \) divides \( x \).
- This is true, so the relation is reflexive.

2. **Symmetry**: If \( x \) divides \( y \), does \( y \) divide \( x \)?
- This is not true in general (e.g., \( 2 \) divides \( 4 \) but \( 4 \) does not divide \( 2 \)), so the relation is not symmetric.

Since symmetry fails, **Option 3 is not an equivalence relation**.

### Step 4: Analyze Option 4: \( y > x \)

1. **Reflexivity**: For any \( x \in \mathbb{R} \), check if \( x > x \).
- This is false, so the relation is not reflexive.

Since reflexivity fails, **Option 4 is not an equivalence relation**.

### Conclusion
The only relation that satisfies reflexivity, symmetry, and transitivity is **Option 1: \( x^2 = y^2 \)**. Therefore, this is the equivalence relation.

---