If a set A contains 3 elements and another set B contains 6 elements, then what is the minimum number of elements that `(A cup B)` can have ?

Options

A. 3
B. 6
C. 8
D. 9

Answer

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

### Step 1: Understand the problem
We are given two sets:
- Set A has 3 elements.
- Set B has 6 elements.

We need to find the minimum number of elements in the union of these two sets, denoted as \( A \cup B \).

### Step 2: Use the formula for union of sets
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.
- \( |A \cap B| \) is the number of elements common to both sets A and B.

### Step 3: Substitute the known values
From the problem:
- \( |A| = 3 \)
- \( |B| = 6 \)

Now, substituting these values into the formula gives us:
\[ |A \cup B| = 3 + 6 - |A \cap B| \]

### Step 4: Determine the minimum value of \( |A \cap B| \)
To find the minimum number of elements in \( A \cup B \), we need to maximize \( |A \cap B| \). The maximum number of elements that can be common between the two sets is limited by the smaller set, which is set A. Therefore, the maximum value of \( |A \cap B| \) is 3 (since set A has 3 elements).

### Step 5: Calculate the minimum number of elements in \( A \cup B \)
Substituting \( |A \cap B| = 3 \) into the equation:
\[ |A \cup B| = 3 + 6 - 3 = 6 \]

### Conclusion
Thus, the minimum number of elements that \( A \cup B \) can have is **6**.