The relation R on the set A = {1, 2, 3} defined as R ={(1, 1), (1, 2), (2, 1), (3, 3)} is reflexive, symmetric and transitive.

Explanation

To determine whether the relation \( R \) on the set \( A = \{1, 2, 3\} \) defined as \( R = \{(1, 1), (1, 2), (2, 1), (3, 3)\} \) is reflexive, symmetric, and transitive, we will analyze each property step by step.

### 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 \( R \).

- For \( a = 1 \): \( (1, 1) \in R \) (True)
- For \( a = 2 \): \( (2, 2) \notin R \) (False)
- For \( a = 3 \): \( (3, 3) \in R \) (True)

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

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

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

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

- Check pairs:
- From \( (1, 2) \) and \( (2, 1) \): Since \( (1, 1) \in R \) (True)
- From \( (2, 1) \) and \( (1, 2) \): Since \( (2, 2) \notin R \) (False)
- From \( (1, 1) \) and \( (1, 2) \): Since \( (1, 2) \in R \) (True)
- From \( (3, 3) \) and any other pair: No new pairs to check.

Since \( (2, 2) \) is not in \( R \) when it should be, the relation is **not transitive**.

### Conclusion
The relation \( R \) is:
- **Not Reflexive**
- **Symmetric**
- **Not Transitive**