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 1: Understand the given information
We know that:
- Set A has 3 elements.
- Set B has 6 elements.

### Step 2: Use the formula for union of sets
The formula for the number of elements in the union of two sets is given by:
\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]
Where:
- \( N(A) \) is the number of elements in set A.
- \( N(B) \) is the number of elements in set B.
- \( N(A \cap B) \) is the number of elements in the intersection of sets A and B.

### Step 3: Calculate the maximum number of elements in \( A \cup B \)
To find the maximum number of elements in \( A \cup B \), we need to minimize \( N(A \cap B) \). The minimum value of \( N(A \cap B) \) is 0 (when there are no common elements between A and B).

Using the values:
- \( N(A) = 3 \)
- \( N(B) = 6 \)
- \( N(A \cap B) = 0 \)

Substituting these values into the formula:
\[ N(A \cup B) = 3 + 6 - 0 = 9 \]

### Step 4: Calculate the minimum number of elements in \( A \cup B \)
To find the minimum number of elements in \( A \cup B \), we need to maximize \( N(A \cap B) \). The maximum value of \( N(A \cap B) \) can be the total number of elements in the smaller set, which is 3 (since set A has 3 elements).

Using the values:
- \( N(A) = 3 \)
- \( N(B) = 6 \)
- \( N(A \cap B) = 3 \)

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

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