Write the smallest reflexive relation on set A = {1, 2, 3, 4, 5}.

Answer

To find the smallest reflexive relation on the set \( A = \{1, 2, 3, 4, 5\} \), we need to understand what a reflexive relation is. A relation \( R \) on a set \( A \) is called reflexive if every element in \( A \) is related to itself. This means that for every element \( a \in A \), the pair \( (a, a) \) must be included in the relation \( R \).

### Step-by-step Solution:

1. **Identify the Set**: The given set is \( A = \{1, 2, 3, 4, 5\} \).

2. **Define Reflexive Relation**: For a relation to be reflexive, it must include all pairs of the form \( (a, a) \) for each element \( a \) in the set \( A \).

3. **List the Required Pairs**:
- For element \( 1 \): The pair is \( (1, 1) \)
- For element \( 2 \): The pair is \( (2, 2) \)
- For element \( 3 \): The pair is \( (3, 3) \)
- For element \( 4 \): The pair is \( (4, 4) \)
- For element \( 5 \): The pair is \( (5, 5) \)

4. **Construct the Relation**: The smallest reflexive relation on set \( A \) will include all these pairs. Therefore, the smallest reflexive relation \( R \) can be written as:
\[ R = \{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\} \]

5. **Conclusion**: The smallest reflexive relation on the set \( A \) is:
\[ R = \{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\} \]