A and B are any two non-empty sets and A is proper subset of B. If `n(A)=5,` then find the minimum possible value of `n(A DeltaB)`

Options

A. is 1
B. is 5
C. cannot be determined
D. none of these

Answer

To solve the problem, we need to find the minimum possible value of \( n(A \Delta B) \) given that \( A \) is a proper subset of \( B \) and \( n(A) = 5 \).

### Step-by-Step Solution:

1. **Understanding Proper Subset**:
- Since \( A \) is a proper subset of \( B \), all elements of \( A \) are contained in \( B \), but \( B \) must have at least one element that is not in \( A \).

2. **Given Information**:
- We know \( n(A) = 5 \). This means set \( A \) has 5 elements.

3. **Finding \( n(B) \)**:
- Since \( A \) is a proper subset of \( B \), the minimum number of elements in \( B \) must be at least \( n(A) + 1 \). Therefore, the minimum possible value of \( n(B) \) is \( 5 + 1 = 6 \).

4. **Calculating \( n(A \Delta B) \)**:
- The symmetric difference \( A \Delta B \) is defined as \( (A - B) \cup (B - A) \).
- Since \( A \) is a subset of \( B \), \( A - B \) will be empty. Thus, \( n(A - B) = 0 \).
- Therefore, \( n(A \Delta B) = n(B - A) \).

5. **Finding \( n(B - A) \)**:
- The set \( B - A \) consists of elements in \( B \) that are not in \( A \).
- If \( n(B) = 6 \) (the minimum possible), then \( n(B - A) = n(B) - n(A) = 6 - 5 = 1 \).

6. **Conclusion**:
- Thus, the minimum possible value of \( n(A \Delta B) \) is \( 1 \).

### Final Answer:
The minimum possible value of \( n(A \Delta B) \) is \( 1 \).