Let A = {1, 2, 3} and R = {(1, 1), (2,2), (1, 2), (2, 1), (1,3)} then R is

Options

A. Symmetric
B. Transitive
C. Reflexive
D. None of these

Answer

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

Given:
- Set \( A = \{1, 2, 3\} \)
- Relation \( R = \{(1, 1), (2, 2), (1, 2), (2, 1), (1, 3)\} \)

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

- For \( 1 \in A \): \( (1, 1) \in R \) (Yes)
- For \( 2 \in A \): \( (2, 2) \in R \) (Yes)
- For \( 3 \in A \): \( (3, 3) \notin R \) (No)

Since \( (3, 3) \) is not in \( R \), the relation \( R \) is **not reflexive**.

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

- For \( (1, 1) \): \( (1, 1) \in R \) (Yes)
- For \( (2, 2) \): \( (2, 2) \in R \) (Yes)
- For \( (1, 2) \): \( (2, 1) \in R \) (Yes)
- For \( (2, 1) \): \( (1, 2) \in R \) (Yes)
- For \( (1, 3) \): \( (3, 1) \notin R \) (No)

Since \( (3, 1) \) is not in \( R \), the relation \( R \) is **not symmetric**.

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

- Consider \( (1, 2) \in R \) and \( (2, 1) \in R \):
- Here, \( (1, 1) \in R \) (Yes)
- Consider \( (1, 2) \in R \) and \( (2, 2) \in R \):
- Here, \( (1, 2) \in R \) (Yes)
- Consider \( (2, 1) \in R \) and \( (1, 3) \in R \):
- Here, \( (2, 3) \notin R \) (No)

Since \( (2, 3) \) is not in \( R \), the relation \( R \) is **not transitive**.

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