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 solve the problem step by step, we need to find the greatest possible value of \( n(A \Delta B) \), given that \( n(A) = 7 \), \( n(B) = 6 \), and \( n(A \cap B) \neq \emptyset \).

### Step 1: Understand the Definitions
We know that:
- \( n(A) \) is the number of elements in set A.
- \( n(B) \) is the number of elements in set B.
- \( n(A \cap B) \) is the number of elements common to both sets A and B.
- \( n(A \Delta B) \) is the number of elements in the symmetric difference of A and B, which can be calculated using the formula:
\[ n(A \Delta B) = n(A \cup B) - n(A \cap B) \]

### Step 2: Use the Formula for Union
We also have the formula for the number of elements in the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

### Step 3: Substitute Known Values
Substituting the known values into the union formula:
- \( n(A) = 7 \)
- \( n(B) = 6 \)

Thus,
\[ n(A \cup B) = 7 + 6 - n(A \cap B) = 13 - n(A \cap B) \]

### Step 4: Find the Minimum Value of \( n(A \cap B) \)
Since it is given that \( n(A \cap B) \neq \emptyset \), the minimum value for \( n(A \cap B) \) can be 1 (it cannot be 0).

### Step 5: Substitute Minimum Value into the Union Formula
Now, substituting \( n(A \cap B) = 1 \) into the union formula:
\[ n(A \cup B) = 13 - 1 = 12 \]

### Step 6: Calculate \( n(A \Delta B) \)
Now, we can calculate \( n(A \Delta B) \):
\[ n(A \Delta B) = n(A \cup B) - n(A \cap B) = 12 - 1 = 11 \]

### Conclusion
The greatest possible value of \( n(A \Delta B) \) is \( 11 \).