If `n(xi) = 40 and n((A uu B)) = 31`, then find `n(A' nn B')`

Answer

To solve the problem, we need to find \( n(A' \cap B') \) given that \( n(U) = 40 \) and \( n(A \cup B) = 31 \).

### Step-by-Step Solution:

1. **Understand the Notation**:
- \( n(U) \): The total number of elements in the universal set \( U \).
- \( n(A \cup B) \): The number of elements in the union of sets \( A \) and \( B \).
- \( A' \) and \( B' \) represent the complements of sets \( A \) and \( B \), respectively.

2. **Use the Complement Rule**:
- We know that \( n(A' \cap B') \) can be expressed using the formula:
\[ n(A' \cap B') = n(U) - n(A \cup B) \]
- This formula states that the number of elements in the complement of the union of \( A \) and \( B \) is equal to the total number of elements in the universal set minus the number of elements in the union of \( A \) and \( B \).

3. **Substitute the Known Values**:
- From the problem, we have:
- \( n(U) = 40 \)
- \( n(A \cup B) = 31 \)
- Now substitute these values into the formula:
\[ n(A' \cap B') = 40 - 31 \]

4. **Calculate the Result**:
- Perform the subtraction:
\[ n(A' \cap B') = 9 \]

5. **Final Answer**:
- Therefore, the value of \( n(A' \cap B') \) is \( 9 \).