If A has 3 elements and B has 6 elements, then the minimum number of elements in `AuuB` is :

Options

A. 3
B. 6
C. 9
D. 18

Answer

To solve the problem, we need to find the minimum number of elements in the union of two sets A and B, given that set A has 3 elements and set B has 6 elements.

### Step-by-Step Solution:

1. **Understand the Union of Sets**:
The union of two sets A and B, denoted as \( A \cup B \), includes all elements from both sets without any repetition.

2. **Identify the Number of Elements**:
- Set A has 3 elements.
- Set B has 6 elements.

3. **Calculate the Maximum Overlap**:
To find the minimum number of elements in \( A \cup B \), we need to consider the case where there is maximum overlap between the two sets. The maximum number of elements that can be common between A and B is limited by the smaller set, which is set A in this case.

4. **Determine the Maximum Common Elements**:
Since set A has 3 elements, the maximum number of elements that can be common (repeated) between sets A and B is 3.

5. **Apply the Formula for Union**:
The formula for the number of elements in the union of two sets is:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Where \( |A| \) is the number of elements in set A, \( |B| \) is the number of elements in set B, and \( |A \cap B| \) is the number of common elements.

6. **Substitute the Values**:
- \( |A| = 3 \)
- \( |B| = 6 \)
- \( |A \cap B| = 3 \) (maximum overlap)

Substituting these values into the formula:
\[ |A \cup B| = 3 + 6 - 3 = 6 \]

7. **Conclusion**:
Therefore, the minimum number of elements in \( A \cup B \) is 6.

### Final Answer:
The minimum number of elements in \( A \cup B \) is **6**.