If A and B are two sets such that `n(A)=7, n(B)=6and (A nnB)ne phi` Then the greatest possible value of n `(A Delta B)` is

Options

A. 11
B. 12
C. 13
D. 10

Answer

To find the greatest possible value of \( n(A \Delta B) \), we start with the definitions and given values:

1. **Definitions**:
- The symmetric difference \( A \Delta B \) is defined as:
\[ A \Delta B = (A - B) \cup (B - A) \]
- The number of elements in the symmetric difference can also be expressed as:
\[ n(A \Delta B) = n(A) + n(B) - 2n(A \cap B) \]

2. **Given Values**:
- \( n(A) = 7 \)
- \( n(B) = 6 \)
- \( A \cap B \neq \emptyset \) (meaning that the intersection of A and B is not empty)

3. **Finding Maximum \( n(A \Delta B) \)**:
- We need to minimize \( n(A \cap B) \) to maximize \( n(A \Delta B) \).
- Since \( A \cap B \) cannot be empty, the minimum value for \( n(A \cap B) \) is 1.

4. **Substituting Values**:
- Substitute \( n(A) \), \( n(B) \), and \( n(A \cap B) \) into the formula:
\[ n(A \Delta B) = n(A) + n(B) - 2n(A \cap B) \]
- With \( n(A \cap B) = 1 \):
\[ n(A \Delta B) = 7 + 6 - 2 \times 1 \]
\[ n(A \Delta B) = 13 - 2 = 11 \]

5. **Conclusion**:
- The greatest possible value of \( n(A \Delta B) \) is \( 11 \).