Let `A= {p, q, r}`. Which of the following is an equivalence relation on `A`? (a) `R_1 = {(p, q), (q, r), (p, r), (p, q)}` (b) `R_2 = {(r,q), (r, p), (r.r), (q, r)}` (c) `R_3 = {(p, p), (q,q), (r, r), (p, q)}` (d) one of these

Explanation

To determine which of the given relations 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. **Check Relation \( R_1 = \{(p, q), (q, r), (p, r), (p, q)\} \)**

- **Reflexivity**: A relation is reflexive if every element is related to itself. Here, we need \( (p, p), (q, q), (r, r) \) to be present in \( R_1 \). Since none of these pairs are present, \( R_1 \) is **not reflexive**.

- **Symmetry**: A relation is symmetric if whenever \( (a, b) \) is in the relation, \( (b, a) \) must also be in the relation. Here, \( (p, q) \) is in \( R_1 \) but \( (q, p) \) is not. Thus, \( R_1 \) is **not symmetric**.

- **Transitivity**: A relation is transitive if whenever \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be in the relation. Here, \( (p, q) \) and \( (q, r) \) are in \( R_1 \), but \( (p, r) \) is already included. However, since it fails reflexivity and symmetry, we can conclude that \( R_1 \) is **not an equivalence relation**.

2. **Check Relation \( R_2 = \{(r, q), (r, p), (r, r), (q, r)\} \)**

- **Reflexivity**: For \( R_2 \) to be reflexive, we need \( (p, p), (q, q), (r, r) \). Here, \( (r, r) \) is present, but \( (p, p) \) and \( (q, q) \) are not. Thus, \( R_2 \) is **not reflexive**.

- **Symmetry**: The pair \( (r, q) \) is in \( R_2 \) but \( (q, r) \) is also present, which satisfies symmetry for this pair. However, \( (r, p) \) is in \( R_2 \) but \( (p, r) \) is not. Thus, \( R_2 \) is **not symmetric**.

- **Transitivity**: We can check pairs, but since it fails reflexivity and symmetry, we conclude that \( R_2 \) is **not an equivalence relation**.

3. **Check Relation \( R_3 = \{(p, p), (q, q), (r, r), (p, q)\} \)**

- **Reflexivity**: All elements \( p, q, r \) are related to themselves: \( (p, p), (q, q), (r, r) \) are present. Thus, \( R_3 \) is **reflexive**.

- **Symmetry**: The pair \( (p, q) \) is in \( R_3 \), but \( (q, p) \) is not. Thus, \( R_3 \) is **not symmetric**.

- **Transitivity**: Since it is not symmetric, we do not need to check transitivity. Therefore, \( R_3 \) is **not an equivalence relation**.

4. **Conclusion**: Since none of the relations \( R_1, R_2, \) or \( R_3 \) satisfy all three properties of an equivalence relation, the answer is **(d) none of these**.