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 on the set A = {1, 2, 3} is not an equivalence relation, we need to check each relation against the three properties that define an equivalence relation: reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Understand the Properties of Equivalence Relations**:
- **Reflexive**: For every element a in A, the pair (a, a) must be in the relation.
- **Symmetric**: For any elements a and b in A, if (a, b) is in the relation, then (b, a) must also be in the relation.
- **Transitive**: For any elements a, b, and c in A, if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.

2. **Evaluate Each Relation**:
- **Relation 1**: {(1, 1), (2, 2), (3, 3)}
- Reflexive: Yes, all (a, a) are present.
- Symmetric: Yes, since it only contains pairs of the form (a, a).
- Transitive: Yes, since there are no pairs that violate transitivity.
- **Conclusion**: This is an equivalence relation.

- **Relation 2**: {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}
- Reflexive: Yes, all (a, a) are present.
- Symmetric: Yes, (1, 2) and (2, 1) are present.
- Transitive: Yes, since (1, 2) and (2, 1) imply (1, 1) and (2, 2) which are present.
- **Conclusion**: This is an equivalence relation.

- **Relation 3**: {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}
- Reflexive: Yes, all (a, a) are present.
- Symmetric: Yes, (2, 3) and (3, 2) are present.
- Transitive: Yes, since (2, 3) and (3, 2) imply (2, 2) and (3, 3) which are present.
- **Conclusion**: This is an equivalence relation.

- **Relation 4**: {(1, 1), (2, 2), (2, 3)}
- Reflexive: No, (3, 3) is missing.
- Symmetric: Not applicable since it is not reflexive.
- Transitive: Not applicable since it is not reflexive.
- **Conclusion**: This is **not** an equivalence relation.

3. **Final Answer**: The relation that is not an equivalence relation is the fourth one.