Let `A={1,2,3,4}` and `R` be a relation in A given by `R={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(3,1),(1,3)}`. Then show that `R` is reflexive and symmetric but not transitive.

Answer

To determine whether the relation \( R \) on the set \( A = \{1, 2, 3, 4\} \) is reflexive, symmetric, and transitive, we will analyze each property step by step.

### Step 1: Check if \( R \) is Reflexive
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 \( R \).

**Elements of \( A \)**:
- 1
- 2
- 3
- 4

**Pairs needed for reflexivity**:
- \( (1, 1) \)
- \( (2, 2) \)
- \( (3, 3) \)
- \( (4, 4) \)

**Given relation \( R \)**:
\[ R = \{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (3,1), (1,3)\} \]

**Check for reflexivity**:
- \( (1, 1) \) is in \( R \)
- \( (2, 2) \) is in \( R \)
- \( (3, 3) \) is in \( R \)
- \( (4, 4) \) is in \( R \)

Since all required pairs are present, we conclude that \( R \) is reflexive.

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

**Given relation \( R \)**:
\[ R = \{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (3,1), (1,3)\} \]

**Check for symmetry**:
- For \( (1, 2) \), \( (2, 1) \) is in \( R \)
- For \( (2, 1) \), \( (1, 2) \) is in \( R \)
- For \( (1, 3) \), \( (3, 1) \) is not in \( R \) (this is incorrect, as \( (3, 1) \) is in \( R \))

Since for every \( (a, b) \) in \( R \), \( (b, a) \) is also in \( R \), we conclude that \( R \) is 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 \( R \).

**Check for transitivity**:
- Consider \( (1, 2) \) and \( (2, 1) \). We need to check if \( (1, 1) \) is in \( R \) (which it is).
- Now consider \( (1, 2) \) and \( (2, 3) \). We need to check if \( (1, 3) \) is in \( R \) (which it is).
- Now consider \( (1, 3) \) and \( (3, 1) \). We need to check if \( (1, 1) \) is in \( R \) (which it is).
- However, consider \( (1, 2) \) and \( (2, 3) \). We need to check if \( (1, 3) \) is in \( R \) (which it is).

However, if we take \( (2, 1) \) and \( (1, 3) \), we need to check if \( (2, 3) \) is in \( R \) (which it is not).

Since there exists at least one instance where the transitive property does not hold, we conclude that \( R \) is not transitive.

### Conclusion
- The relation \( R \) is **reflexive**.
- The relation \( R \) is **symmetric**.
- The relation \( R \) is **not transitive**.