Given n(U) = 20, n(A) = 12, n(B) = 9, `n(AnnB)` = 4, where U is the universal set, A and B are subsets of U, then `n((AuuB)')` equals

Options

A. 3
B. 9
C. 11
D. 17

Answer

To solve the problem, we need to find the number of elements in the complement of the union of sets A and B, denoted as \( n((A \cup B)') \).

### Step-by-Step Solution:

1. **Identify the given values**:
- \( n(U) = 20 \) (the total number of elements in the universal set)
- \( n(A) = 12 \) (the number of elements in set A)
- \( n(B) = 9 \) (the number of elements in set B)
- \( n(A \cap B) = 4 \) (the number of elements in the intersection of A and B)

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

3. **Substitute the known values into the formula**:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) = 12 + 9 - 4 \]

4. **Calculate \( n(A \cup B) \)**:
\[ n(A \cup B) = 12 + 9 - 4 = 17 \]

5. **Find the number of elements in the complement of the union**:
The complement of the union of A and B, denoted as \( (A \cup B)' \), consists of the elements in the universal set that are not in \( A \cup B \). The number of elements in the complement can be calculated as:
\[ n((A \cup B)') = n(U) - n(A \cup B) \]

6. **Substitute the values**:
\[ n((A \cup B)') = 20 - 17 \]

7. **Calculate \( n((A \cup B)') \)**:
\[ n((A \cup B)') = 3 \]

### Final Answer:
Thus, \( n((A \cup B)') = 3 \).