If `A={1,2,3}` then which of the following relations are 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. all of these

Answer

To determine which of the given relations on the set \( A = \{1, 2, 3\} \) are equivalence relations, we need to check if each relation satisfies three properties: reflexivity, symmetry, and transitivity.

### Step 1: Understand the properties of equivalence relations
1. **Reflexivity**: For every element \( x \in A \), the pair \( (x, x) \) must be in the relation.
2. **Symmetry**: For any elements \( a, b \in A \), if \( (a, b) \) is in the relation, then \( (b, a) \) must also be in the relation.
3. **Transitivity**: For any elements \( a, b, c \in A \), if \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be in the relation.

### Step 2: Check each relation

#### Relation 1: \( R_1 = \{(1, 1), (2, 2), (3, 3)\} \)
- **Reflexivity**:
- \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \) are all present.
- **Conclusion**: Reflexive.

- **Symmetry**:
- Each pair is of the form \( (x, x) \), which is symmetric.
- **Conclusion**: Symmetric.

- **Transitivity**:
- There are no pairs \( (a, b) \) and \( (b, c) \) other than \( (x, x) \), hence transitivity holds.
- **Conclusion**: Transitive.

Thus, \( R_1 \) is an equivalence relation.

#### Relation 2: \( R_2 = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\} \)
- **Reflexivity**:
- \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \) are present.
- **Conclusion**: Reflexive.

- **Symmetry**:
- \( (1, 2) \) implies \( (2, 1) \) is also present.
- **Conclusion**: Symmetric.

- **Transitivity**:
- From \( (1, 2) \) and \( (2, 1) \), we do not derive any new pairs, and since we have \( (1, 1) \) and \( (2, 2) \), transitivity holds.
- **Conclusion**: Transitive.

Thus, \( R_2 \) is also an equivalence relation.

#### Relation 3: \( R_3 = \{(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)\} \)
- **Reflexivity**:
- \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \) are present.
- **Conclusion**: Reflexive.

- **Symmetry**:
- \( (2, 3) \) implies \( (3, 2) \) is also present.
- **Conclusion**: Symmetric.

- **Transitivity**:
- From \( (2, 3) \) and \( (3, 2) \), we should have \( (2, 2) \) which is present, and from \( (2, 3) \) and \( (3, 3) \), we should have \( (2, 3) \) which is also present.
- **Conclusion**: Transitive.

Thus, \( R_3 \) is also an equivalence relation.

### Final Conclusion
All three relations \( R_1 \), \( R_2 \), and \( R_3 \) are equivalence relations on the set \( A \).

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