Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A ?

Options

A. {(1,1), (2,2), (3,3)}
B. {(1,1), (2,2), (3,3), (1,2), (2,1)}
C. {(1,1), (2,2), (3,3), (2,3), (3,2)}
D. {(1,1), (2,2), (2,3)}

Answer

To determine which of the given relations is not an equivalence relation on the set \( A = \{1, 2, 3\} \), we need to check each relation against the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Understand the Properties of Equivalence Relations**:
- **Reflexivity**: For every element \( a \) in set \( A \), the pair \( (a, a) \) must be in the relation. For set \( A = \{1, 2, 3\} \), the pairs must include \( (1, 1), (2, 2), (3, 3) \).
- **Symmetry**: For any pairs \( (a, b) \) in the relation, the pair \( (b, a) \) must also be included.
- **Transitivity**: If \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be included.

2. **Examine Each Option**:
- **Option 1**: \( \{(1, 1), (2, 2), (3, 3)\} \)
- Reflexive: Yes (contains \( (1, 1), (2, 2), (3, 3) \))
- Symmetric: Yes (no pairs to check)
- Transitive: Yes (no pairs to check)
- Conclusion: This is an equivalence relation.

- **Option 2**: \( \{(1, 1), (2, 2), (3, 3), (2, 1), (1, 2)\} \)
- Reflexive: Yes (contains \( (1, 1), (2, 2), (3, 3) \))
- Symmetric: Yes (contains \( (1, 2) \) and \( (2, 1) \))
- Transitive: Yes (all conditions satisfied)
- Conclusion: This is an equivalence relation.

- **Option 3**: \( \{(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)\} \)
- Reflexive: Yes (contains \( (1, 1), (2, 2), (3, 3) \))
- Symmetric: Yes (contains \( (2, 3) \) and \( (3, 2) \))
- Transitive: Yes (since \( (2, 3) \) and \( (3, 2) \) imply \( (2, 2) \))
- Conclusion: This is an equivalence relation.

- **Option 4**: \( \{(1, 1), (2, 2), (3, 3), (2, 3)\} \)
- Reflexive: No (missing \( (3, 3) \))
- Symmetric: No (missing \( (3, 2) \))
- Transitive: Not applicable since it fails reflexivity and symmetry.
- Conclusion: This is **not** an equivalence relation.

3. **Final Answer**:
- The relation that is not an equivalence relation on \( A \) is **Option 4**.