Let A be a set containing n elements. If the number of reflexive relations that can be defined on A is 64, then n is equal to

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem, we need to determine the number of elements \( n \) in set \( A \) given that the number of reflexive relations on \( A \) is 64.

### Step-by-Step Solution:

1. **Understand the formula for reflexive relations**:
The number of reflexive relations on a set with \( n \) elements is given by the formula:
\[ 2^{n^2 - n} \]
This formula accounts for the fact that a reflexive relation must include all pairs \( (a, a) \) for each element \( a \) in the set.

2. **Set up the equation**:
We know from the problem statement that the number of reflexive relations is 64. Therefore, we can set up the equation:
\[ 2^{n^2 - n} = 64 \]

3. **Express 64 as a power of 2**:
We can express 64 as a power of 2:
\[ 64 = 2^6 \]
Thus, we can rewrite our equation as:
\[ 2^{n^2 - n} = 2^6 \]

4. **Equate the exponents**:
Since the bases are the same, we can equate the exponents:
\[ n^2 - n = 6 \]

5. **Rearrange the equation**:
Rearranging the equation gives us:
\[ n^2 - n - 6 = 0 \]

6. **Factor the quadratic equation**:
We can factor the quadratic equation:
\[ (n - 3)(n + 2) = 0 \]

7. **Solve for \( n \)**:
Setting each factor to zero gives us:
\[ n - 3 = 0 \quad \text{or} \quad n + 2 = 0 \]
This results in:
\[ n = 3 \quad \text{or} \quad n = -2 \]
Since \( n \) must be a non-negative integer (as it represents the number of elements in a set), we take:
\[ n = 3 \]

### Conclusion:
Thus, the number of elements \( n \) in set \( A \) is \( 3 \).