If A and B are two non - empty equivalent sets, then a possible value of `n(AxxB)` is ___________.

Options

A. 1
B. 2
C. 6
D. 3

Answer

To solve the problem, we need to determine the possible value of \( n(A \times B) \) given that \( A \) and \( B \) are two non-empty equivalent sets.

### Step-by-Step Solution:

1. **Understanding Equivalent Sets**:
- Two sets \( A \) and \( B \) are said to be equivalent if they have the same number of elements. This means \( n(A) = n(B) \).

2. **Let the Number of Elements be \( x \)**:
- Since \( A \) and \( B \) are equivalent and non-empty, we can denote the number of elements in both sets as \( x \). Thus, \( n(A) = n(B) = x \).

3. **Finding the Number of Elements in the Cartesian Product**:
- The Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element is from set \( A \) and the second element is from set \( B \).
- The formula for the number of elements in the Cartesian product is given by:
\[ n(A \times B) = n(A) \times n(B) = x \times x = x^2 \]

4. **Determining Possible Values of \( x^2 \)**:
- Since \( A \) and \( B \) are non-empty sets, \( x \) must be at least 1 (i.e., \( x \geq 1 \)).
- The possible values for \( x \) can be 1, 2, 3, etc. Therefore, the possible values for \( x^2 \) will be:
- If \( x = 1 \), then \( x^2 = 1^2 = 1 \)
- If \( x = 2 \), then \( x^2 = 2^2 = 4 \)
- If \( x = 3 \), then \( x^2 = 3^2 = 9 \)
- And so on.

5. **Conclusion**:
- The possible values of \( n(A \times B) \) are perfect squares: \( 1, 4, 9, 16, \ldots \)
- The question asks for a possible value, and since we are looking for at least one value, we can conclude that a possible value of \( n(A \times B) \) is \( 1 \).

### Final Answer:
A possible value of \( n(A \times B) \) is **1**.

---