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 \( B \), it means that all elements of \( A \) are in \( B \), and \( B \) contains at least one element that is not in \( A \).

2. **Using the Definition of Symmetric Difference**:
The symmetric difference \( A \Delta B \) is defined as:
\[ A \Delta B = (A \cup B) - (A \cap B) \]
Since \( A \) is a proper subset of \( B \), we can simplify this to:
\[ A \Delta B = B - A \]

3. **Finding the Number of Elements**:
Let \( n(B) \) be the number of elements in set \( B \). Since \( A \) has 5 elements and \( B \) must have at least one additional element (because \( A \) is a proper subset), we can express this as:
\[ n(B) \geq n(A) + 1 = 5 + 1 = 6 \]

4. **Calculating \( n(A \Delta B) \)**:
The number of elements in the symmetric difference \( n(A \Delta B) \) can be calculated as:
\[ n(A \Delta B) = n(B) - n(A) \]
Substituting the values we have:
\[ n(A \Delta B) = n(B) - 5 \]

5. **Finding the Minimum Value**:
Since the minimum value of \( n(B) \) is 6 (from step 3), we substitute this into our equation:
\[ n(A \Delta B) = 6 - 5 = 1 \]

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