Sets A and B have 3 and 6 elaments respectively. What can be theminimum number of elements in `A uu B`

Options

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

Explanation

To find the minimum number of elements in the union of sets A and B, we can use the formula for the union of two sets:

\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]

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 common to both sets A and B.

Given:
- \(N(A) = 3\)
- \(N(B) = 6\)

### Step 1: Identify the values
We have:
- \(N(A) = 3\)
- \(N(B) = 6\)

### Step 2: Determine the maximum overlap
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 at most the number of elements in the smaller set, which is set A in this case. Therefore, the maximum overlap is:

\[ N(A \cap B) = \text{min}(N(A), N(B)) = \text{min}(3, 6) = 3 \]

### Step 3: Substitute into the formula
Now we can substitute the values into the union formula:

\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]

Substituting the values we have:

\[ N(A \cup B) = 3 + 6 - 3 \]

### Step 4: Calculate the result
Now we perform the calculation:

\[ N(A \cup B) = 9 - 3 = 6 \]

### Conclusion
Thus, the minimum number of elements in \(A \cup B\) is:

\[ \boxed{6} \]