Let`A = {1, 2, 3}` Then number of relations containing `(1, 2)" and "(1, 3)`which are reflexive and symmetric but not transitive is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to find the number of relations on the set \( A = \{1, 2, 3\} \) that contain the pairs \( (1, 2) \) and \( (1, 3) \), and are reflexive and symmetric but not transitive.

### Step-by-Step Solution:

1. **Identify Required Pairs**:
- Since the relation must contain \( (1, 2) \) and \( (1, 3) \), we start with these pairs.
- Additionally, for the relation to be reflexive, we must include \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \).

**Pairs so far**: \( (1, 1), (2, 2), (3, 3), (1, 2), (1, 3) \)

2. **Add Symmetric Pairs**:
- For the relation to be symmetric, if \( (1, 2) \) is included, then \( (2, 1) \) must also be included.
- Similarly, if \( (1, 3) \) is included, then \( (3, 1) \) must also be included.

**Pairs now**: \( (1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1) \)

3. **Consider Additional Pairs**:
- The only pairs left to consider for inclusion are \( (2, 3) \) and \( (3, 2) \).
- If we include \( (2, 3) \), we must also include \( (3, 2) \) to maintain symmetry.

4. **Check for Transitivity**:
- We need to ensure that the relation is not transitive.
- If we include both \( (2, 3) \) and \( (3, 2) \), we will have the pairs \( (1, 2) \) and \( (2, 3) \), which would imply \( (1, 3) \) due to transitivity. This is not allowed as we want the relation to be not transitive.

5. **Conclusion**:
- Therefore, we cannot include \( (2, 3) \) and \( (3, 2) \) in our relation.
- The only pairs that can be included while satisfying all conditions (reflexive, symmetric, and not transitive) are:
- \( (1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1) \)
- This gives us only one valid relation.

### Final Answer:
The number of relations containing \( (1, 2) \) and \( (1, 3) \) which are reflexive and symmetric but not transitive is **1**.

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