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

Options

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

Answer

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

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( a \) in set \( A \), the pair \( (a, a) \) is in \( R \).

- The elements of \( A \) are \( 1, 2, 3 \).
- We need to check if \( (1,1) \), \( (2,2) \), and \( (3,3) \) are in \( R \).
- From \( R \), we see that \( (1,1) \) is present, but \( (2,2) \) and \( (3,3) \) are not.

**Conclusion:** Since not all pairs \( (a, a) \) are in \( R \), the relation \( R \) is **not reflexive**.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( (a, b) \) is in \( R \), then \( (b, a) \) must also be in \( R \).

- We check the pairs in \( R \):
- For \( (1,2) \) in \( R \), we check if \( (2,1) \) is also in \( R \). It is.
- For \( (2,1) \) in \( R \), we check if \( (1,2) \) is also in \( R \). It is.
- The pair \( (1,1) \) is symmetric by itself.

**Conclusion:** Since all pairs satisfy the symmetry condition, the relation \( R \) is **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 \( R \).

- We check the pairs:
- We have \( (1,2) \) and \( (2,1) \). Here, \( a = 1 \), \( b = 2 \), and \( c = 1 \).
- Since \( (1,2) \) and \( (2,1) \) are in \( R \), we check if \( (1,1) \) is in \( R \). It is.

- No other combinations of pairs lead to new pairs that need to be checked.

**Conclusion:** Since the transitive condition is satisfied, the relation \( R \) is **transitive**.

### Final Conclusion
The relation \( R \) is **symmetric** and **transitive**, but **not reflexive**.