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. **Understand the formula for the union of two sets**:
- The number of elements in the union of two sets A and B is given by:
\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]
where \( N(A \cap B) \) is the number of elements in the intersection of sets A and B.

3. **Calculate the maximum number of elements in \( A \cup B \)**:
- The maximum value occurs when there are no common elements between sets A and B, which means \( N(A \cap B) = 0 \).
- Therefore, the maximum number of elements in \( A \cup B \) is:
\[ N(A \cup B) = N(A) + N(B) - 0 = 3 + 6 - 0 = 9 \]

4. **Calculate the minimum number of elements in \( A \cup B \)**:
- The minimum value occurs when all elements of the smaller set (set A) are included in the larger set (set B). In this case, the maximum intersection would be the number of elements in set A, which is 3. Thus, \( N(A \cap B) = 3 \).
- Therefore, the minimum number of elements in \( A \cup B \) is:
\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) = 3 + 6 - 3 = 6 \]

5. **Conclusion**:
- The maximum number of elements in \( A \cup B \) is 9.
- The minimum number of elements in \( A \cup B \) is 6.

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