Let R be a relation defined on `A={1,2,3}` by :
`R={(1,3),(3,1),(2,2)}`. R is :

Options

A. Reflexive
B. Symmetric
C. Transitive
D. Reflexive but not Transitive

Answer

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

### Step 1: Check for Reflexivity
A relation is reflexive if every element in the set relates to itself. In other words, for every \( a \in A \), the pair \( (a, a) \) must be in \( R \).

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

Since \( (1,1) \) and \( (3,3) \) are missing, \( R \) is **not reflexive**.

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

- For \( (1,3) \): \( (3,1) \) is in \( R \).
- For \( (3,1) \): \( (1,3) \) is in \( R \).
- For \( (2,2) \): \( (2,2) \) is in \( R \).

Since all pairs satisfy the symmetry condition, \( R \) is **symmetric**.

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

- Consider \( (1,3) \) and \( (3,1) \): Here, \( a = 1 \), \( b = 3 \), and \( c = 1 \). We have \( (1,1) \) which is not in \( R \).
- Consider \( (3,1) \) and \( (1,3) \): Here, \( a = 3 \), \( b = 1 \), and \( c = 3 \). We have \( (3,3) \) which is not in \( R \).
- The pair \( (2,2) \) does not contribute to any transitive condition since it relates to itself.

Since we found instances where the transitive property fails, \( R \) is **not transitive**.

### Conclusion
The relation \( R \) is symmetric but neither reflexive nor transitive.

### Summary of Properties
- Reflexive: **No**
- Symmetric: **Yes**
- Transitive: **No**