`A` and `B` two sets such that `n(A)=3` and `n(B)=6`, then

Options

A. minimum value of `n(AuuB)=6`
B. minimum value of `n(AuuB)=9`
C. maximum value of `n(AuuB)=6`
D. maximum value of `n(AuuB)=9`

Answer

To solve the problem, we need to find the minimum and maximum values of the union of two sets \( A \) and \( B \) given that \( n(A) = 3 \) and \( n(B) = 6 \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We are given two sets \( A \) and \( B \).
- The number of elements in set \( A \) is \( n(A) = 3 \).
- The number of elements in set \( B \) is \( n(B) = 6 \).

2. **Finding the Minimum Value of \( n(A \cup B) \)**:
- The minimum value of the union \( n(A \cup B) \) occurs when the two sets share the maximum number of elements.
- If all elements of \( A \) are also in \( B \), then the total number of unique elements in \( A \cup B \) will be equal to the number of elements in \( B \).
- Therefore, the minimum value of \( n(A \cup B) \) is:
\[ n(A \cup B)_{\text{min}} = n(B) = 6 \]

3. **Finding the Maximum Value of \( n(A \cup B) \)**:
- The maximum value of the union \( n(A \cup B) \) occurs when the two sets have no elements in common (i.e., they are disjoint).
- In this case, the total number of unique elements in \( A \cup B \) will be the sum of the number of elements in both sets.
- Therefore, the maximum value of \( n(A \cup B) \) is:
\[ n(A \cup B)_{\text{max}} = n(A) + n(B) = 3 + 6 = 9 \]

4. **Conclusion**:
- The minimum value of \( n(A \cup B) \) is \( 6 \).
- The maximum value of \( n(A \cup B) \) is \( 9 \).

### Final Answer:
- Minimum value of \( n(A \cup B) = 6 \)
- Maximum value of \( n(A \cup B) = 9 \)