Let A = {p, q, r}. Which of the following is an equivalence relation on A?

Options

A. `R_(1) = {(p, q), (q, r), (p, r), (p, p)}`
B. `R_(2) = {(r, q), (r, p), (r, r), (q, q)}`
C. `R_(3) = {(p, p), (q, q), (r, r), (p, q)}`
D. None of the above

Explanation

To determine which of the relations R1, R2, or R3 is an equivalence relation on the set A = {p, q, r}, we need to check each relation for three properties: reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Understanding Equivalence Relations**:
An equivalence relation must satisfy three properties:
- **Reflexive**: For every element \( a \) in set A, the pair \( (a, a) \) must be in the relation.
- **Symmetric**: For any \( (a, b) \) in the relation, the pair \( (b, a) \) must also be in the relation.
- **Transitive**: If \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be in the relation.

2. **Analyzing R1**:
- Given R1 = {(p, p), (p, q), (q, r), (p, r)}.
- **Reflexivity**: Check if (p, p), (q, q), and (r, r) are present.
- (q, q) and (r, r) are missing. Hence, R1 is **not reflexive**.
- Since R1 is not reflexive, it cannot be an equivalence relation.

3. **Analyzing R2**:
- Given R2 = {(r, r), (r, p), (r, r), (q, q)}.
- **Reflexivity**: Check if (p, p), (q, q), and (r, r) are present.
- (p, p) is missing. Hence, R2 is **not reflexive**.
- Since R2 is not reflexive, it cannot be an equivalence relation.

4. **Analyzing R3**:
- Given R3 = {(p, p), (q, q), (r, r), (p, q), (q, p)}.
- **Reflexivity**: Check if (p, p), (q, q), and (r, r) are present.
- All pairs (p, p), (q, q), and (r, r) are present. Hence, R3 is **reflexive**.
- **Symmetry**: Check if for every (a, b) in R3, (b, a) is also in R3.
- (p, q) is present and (q, p) is also present. Hence, R3 is **symmetric**.
- **Transitivity**: Check if for any (a, b) and (b, c) in R3, (a, c) is also in R3.
- Since we have (p, q) and (q, p), we can check combinations, and they hold. Hence, R3 is **transitive**.
- Since R3 satisfies reflexivity, symmetry, and transitivity, it is an **equivalence relation**.

5. **Conclusion**:
- R1 is not an equivalence relation.
- R2 is not an equivalence relation.
- R3 is an equivalence relation.

### Final Answer:
R3 is an equivalence relation on A.