lf a relation R on the set `{1, 2, 3}` be defined by `R ={(1,2)}`, then R is:

Options

A. reflexive
B. transitive
C. symmetric
D. None of these

Explanation

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

### Step 1: Check for Reflexivity
A relation is reflexive if every element in the set is related to itself. This means that for every element \( a \) in the set, the pair \( (a, a) \) must be in the relation \( R \).

- For the set \( \{1, 2, 3\} \), we need to check:
- \( (1, 1) \) is not in \( R \)
- \( (2, 2) \) is not in \( R \)
- \( (3, 3) \) is not in \( R \)

Since none of these pairs are in \( R \), the relation is **not reflexive**.

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

- Here, we have the pair \( (1, 2) \) in \( R \). We need to check if \( (2, 1) \) is also in \( R \).

Since \( (2, 1) \) is not in \( R \), the relation is **not symmetric**.

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

- In our case, we only have one pair \( (1, 2) \). For transitivity, we need a second pair \( (b, c) \) that starts with \( 2 \). However, there are no pairs in \( R \) that start with \( 2 \). Therefore, we cannot find any counterexample to transitivity.

Since there are no pairs that violate transitivity, we can conclude that the relation is **transitive**.

### Final Conclusion
The relation \( R \) is:
- Not reflexive
- Not symmetric
- Transitive

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

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