The smallest reflexive relation on the set A {1, 2, 3} is

Options

A. `{(1, 1), (2, 2), (3, 3)}`
B. `{(1, 2, 3)}`
C. `{ }`
D. none of the above

Answer

To find the smallest reflexive relation on the set \( A = \{1, 2, 3\} \), we need to understand what a reflexive relation is. A relation \( R \) on a set \( A \) is called reflexive if, for every element \( a \) in \( A \), the pair \( (a, a) \) is in \( R \).

### Step-by-Step Solution:

1. **Identify the set**: We have the set \( A = \{1, 2, 3\} \).

2. **Define reflexive relation**: For a relation to be reflexive, it must include pairs where each element is related to itself. This means we need to include the pairs \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \).

3. **List the required pairs**: The smallest reflexive relation will thus include the following pairs:
- \( (1, 1) \)
- \( (2, 2) \)
- \( (3, 3) \)

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

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