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 \( R \) defined on the set \( A = \{1, 2, 3, 4\} \) 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 \( x \) in the set \( A \) is related to itself. This means that for all \( x \in A \), the pair \( (x, x) \) must be in \( R \).

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

**Required pairs 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**:
- All required pairs \( (1, 1), (2, 2), (3, 3), (4, 4) \) are present in \( R \).

**Conclusion**: \( R \) is reflexive.

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

**Given pairs in \( R \)**:
- \( (1, 1) \) → \( (1, 1) \) (symmetric)
- \( (2, 2) \) → \( (2, 2) \) (symmetric)
- \( (3, 3) \) → \( (3, 3) \) (symmetric)
- \( (4, 4) \) → \( (4, 4) \) (symmetric)
- \( (1, 2) \) → \( (2, 1) \) (symmetric)
- \( (2, 1) \) → \( (1, 2) \) (symmetric)
- \( (3, 1) \) → \( (1, 3) \) (symmetric)
- \( (1, 3) \) → \( (3, 1) \) (not present)

**Check**:
- For \( (1, 2) \), \( (2, 1) \) is present.
- For \( (3, 1) \), \( (1, 3) \) is present.
- All pairs satisfy the symmetry condition.

**Conclusion**: \( R \) 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 \( R \).

**Check pairs for transitivity**:
1. From \( (1, 2) \) and \( (2, 1) \), we should have \( (1, 1) \) (present).
2. From \( (1, 2) \) and \( (2, 1) \), we should have \( (1, 1) \) (present).
3. From \( (1, 3) \) and \( (3, 1) \), we should have \( (1, 1) \) (present).
4. From \( (1, 2) \) and \( (2, 3) \) (not present).

**Conclusion**: Since \( (1, 2) \) and \( (2, 3) \) do not lead to \( (1, 3) \) being present, \( R \) is not transitive.

### Final Conclusion
- **Reflexive**: Yes
- **Symmetric**: Yes
- **Transitive**: No

Thus, the relation \( R \) is reflexive and symmetric but not transitive.

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