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

Options

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

Answer

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

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

- The elements of \( A \) are \( 1, 2, 3 \).
- Therefore, for \( R \) to be reflexive, it must contain the pairs \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \).
- However, \( R \) only contains \( (1, 2) \).

**Conclusion**: Since \( R \) does not contain \( (1, 1) \), \( (2, 2) \), or \( (3, 3) \), it is **not 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 \( R \).

- We have the pair \( (1, 2) \) in \( R \).
- For symmetry, we need to check if \( (2, 1) \) is also in \( R \).
- Since \( (2, 1) \) is not in \( R \), it fails the symmetry condition.

**Conclusion**: Therefore, \( R \) is **not symmetric**.

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

- We have \( (1, 2) \in R \).
- For transitivity, we would need another pair \( (2, c) \) in \( R \) to check if \( (1, c) \) is also in \( R \).
- However, there are no pairs of the form \( (2, c) \) in \( R \) since \( R \) only contains \( (1, 2) \).

**Conclusion**: Since we cannot find any pairs to satisfy the transitive condition, \( R \) is **not transitive**.

### Final Conclusion
Since \( R \) is neither reflexive, nor symmetric, nor transitive, we conclude that the relation \( R \) is **none of these**.