The relation `R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}` on a set A={1, 2, 3} is

Options

A. Reflexive but not symmetric
B. Reflexive but not transitive
C. Symmetric and transitive
D. Neither symmetric nor transitive

Explanation

To determine the properties of the relation \( R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\} \) on the set \( A = \{1, 2, 3\} \), we will check if it is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element in the set \( A \) is related to itself. This means we need to check if the pairs \( (1,1) \), \( (2,2) \), and \( (3,3) \) are present in \( R \).

- Check \( (1,1) \): Present in \( R \)
- Check \( (2,2) \): Present in \( R \)
- Check \( (3,3) \): Present in \( R \)

Since all pairs are present, the relation \( R \) is **reflexive**.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for every pair \( (a,b) \) in \( R \), the pair \( (b,a) \) is also in \( R \).

- Check \( (1,2) \): \( (2,1) \) is not in \( R \)
- Check \( (2,3) \): \( (3,2) \) is not in \( R \)
- Check \( (1,3) \): \( (3,1) \) is not in \( R \)

Since there are pairs for which the symmetric counterparts are not present, the relation \( R \) is **not symmetric**.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (a,b) \) and \( (b,c) \) are in \( R \), then \( (a,c) \) must also be in \( R \).

- Check \( (1,2) \) and \( (2,3) \): Since \( (1,2) \) and \( (2,3) \) are in \( R \), we need to check if \( (1,3) \) is in \( R \). It is present.
- Check \( (1,1) \) and \( (1,2) \): Since \( (1,1) \) and \( (1,2) \) are in \( R \), we check if \( (1,2) \) is in \( R \). It is present.
- Check \( (2,2) \) and \( (2,3) \): Since \( (2,2) \) and \( (2,3) \) are in \( R \), we check if \( (2,3) \) is in \( R \). It is present.
- Check \( (1,3) \) and \( (3,3) \): Since \( (1,3) \) and \( (3,3) \) are in \( R \), we check if \( (1,3) \) is in \( R \). It is present.

Since all necessary conditions for transitivity hold, the relation \( R \) is **transitive**.

### Conclusion
The relation \( R \) is:
- Reflexive: Yes
- Symmetric: No
- Transitive: Yes

Thus, the final answer is that the relation \( R \) is reflexive and transitive but not symmetric.

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