If `n(A)=4andn(B)=7`, then the difference between maximum and minimum value of `n(AuuB)` is

Options

A. 1
B. 2
C. 3
D. 11

Answer

To solve the problem, we need to find the difference between the maximum and minimum values of \( n(A \cup B) \) given that \( n(A) = 4 \) and \( n(B) = 7 \).

### Step-by-Step Solution:

1. **Understand the formula for \( n(A \cup B) \)**:
The formula for the number of elements in the union of two sets is:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Here, \( n(A) \) is the number of elements in set A, \( n(B) \) is the number of elements in set B, and \( n(A \cap B) \) is the number of elements in the intersection of sets A and B.

2. **Substitute the known values**:
We know that \( n(A) = 4 \) and \( n(B) = 7 \). Substituting these values into the formula gives:
\[ n(A \cup B) = 4 + 7 - n(A \cap B) = 11 - n(A \cap B) \]

3. **Determine the maximum value of \( n(A \cup B) \)**:
To find the maximum value of \( n(A \cup B) \), we need to minimize \( n(A \cap B) \). The minimum value of \( n(A \cap B) \) can be 0 (when there are no common elements between A and B). Thus:
\[ \text{Maximum value of } n(A \cup B) = 11 - 0 = 11 \]

4. **Determine the minimum value of \( n(A \cup B) \)**:
To find the minimum value of \( n(A \cup B) \), we need to maximize \( n(A \cap B) \). The maximum value of \( n(A \cap B) \) cannot exceed the number of elements in the smaller set, which is \( n(A) = 4 \). Therefore:
\[ \text{Minimum value of } n(A \cup B) = 11 - 4 = 7 \]

5. **Calculate the difference between the maximum and minimum values**:
Now, we find the difference between the maximum and minimum values of \( n(A \cup B) \):
\[ \text{Difference} = \text{Maximum} - \text{Minimum} = 11 - 7 = 4 \]

### Final Answer:
The difference between the maximum and minimum value of \( n(A \cup B) \) is **4**.