If set A and B has 3 and 6 elements respecitvely. Find the maximum and minimum number of elements in `A cup B`.

Answer

To solve the problem of finding the maximum and minimum number of elements in the union of sets A and B, we can follow these steps:

### Step-by-Step Solution:

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

2. **Understanding Union of Sets**:
- The union of two sets, denoted as \( A \cup B \), includes all elements that are in either set A or set B.

3. **Finding the Minimum Number of Elements in \( A \cup B \)**:
- The minimum number of elements in \( A \cup B \) occurs when there is maximum overlap between the two sets.
- If all elements of set A are also in set B, then the union would contain only the unique elements from both sets.
- In this case, the elements in \( A \) (3 elements) are the same as 3 of the elements in \( B \) (6 elements), leading to:
\[ |A \cup B| = |B| = 6 \]
- Thus, the minimum number of elements in \( A \cup B \) is **6**.

4. **Finding the Maximum Number of Elements in \( A \cup B \)**:
- The maximum number of elements in \( A \cup B \) occurs when there is no overlap between the two sets.
- If all elements of set A are different from all elements of set B, then the total number of unique elements will be the sum of the elements in both sets:
\[ |A \cup B| = |A| + |B| = 3 + 6 = 9 \]
- Thus, the maximum number of elements in \( A \cup B \) is **9**.

### Final Answer:
- Minimum number of elements in \( A \cup B \) = 6
- Maximum number of elements in \( A \cup B \) = 9