If the numbers of different reflexive relations on a set A is equal to the number of different symmetric relations on set A, then the number of elements in A is

Options

A. 1
B. 3
C. 1 and 3
D. 3 and 7

Explanation

To solve the problem step by step, we need to find the number of elements in set A such that the number of different reflexive relations on set A is equal to the number of different symmetric relations on set A.

### Step 1: Understand the formulas for reflexive and symmetric relations
- The number of different reflexive relations on a set A with n elements is given by:
\[ 2^{n^2 - n} \]
- The number of different symmetric relations on a set A with n elements is given by:
\[ 2^{\frac{n(n+1)}{2}} \]

### Step 2: Set up the equation
According to the problem, we have:
\[ 2^{n^2 - n} = 2^{\frac{n(n+1)}{2}} \]

### Step 3: Equate the exponents
Since the bases are the same (both are powers of 2), we can equate the exponents:
\[ n^2 - n = \frac{n(n+1)}{2} \]

### Step 4: Clear the fraction
To eliminate the fraction, multiply both sides by 2:
\[ 2(n^2 - n) = n(n + 1) \]
This simplifies to:
\[ 2n^2 - 2n = n^2 + n \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 2n^2 - 2n - n^2 - n = 0 \]
This simplifies to:
\[ n^2 - 3n = 0 \]

### Step 6: Factor the equation
Factoring out n gives:
\[ n(n - 3) = 0 \]

### Step 7: Solve for n
Setting each factor to zero gives us:
1. \( n = 0 \)
2. \( n - 3 = 0 \) which implies \( n = 3 \)

### Step 8: Conclusion
The possible values for n are 0 and 3. Since a set cannot have a negative number of elements, the only valid solution is:
\[ \text{The number of elements in } A \text{ is } 3. \]