If n(U) = 700, n(A) = 200, n(B) = 240, n`(A cap B)` = 100, then n(A' `cup` B') equals

Options

A. 260
B. 560
C. 360
D. 600

Answer

To solve the problem, we need to find \( n(A' \cup B') \), where:

- \( n(U) = 700 \) (the total number of elements in the universal set)
- \( n(A) = 200 \) (the number of elements in set A)
- \( n(B) = 240 \) (the number of elements in set B)
- \( n(A \cap B) = 100 \) (the number of elements in the intersection of sets A and B)

### Step-by-Step Solution:

1. **Understanding the Complement**:
- The complement of set A, denoted \( A' \), includes all elements in the universal set that are not in A.
- Similarly, the complement of set B, denoted \( B' \), includes all elements in the universal set that are not in B.

2. **Using the Formula for Union of Complements**:
- We can use the formula:
\[ n(A' \cup B') = n(U) - n(A \cap B) \]
- This formula states that the number of elements in the union of the complements of A and B is equal to the total number of elements in the universal set minus the number of elements in the intersection of A and B.

3. **Calculating \( n(A \cap B) \)**:
- From the problem, we know \( n(A \cap B) = 100 \).

4. **Substituting Values into the Formula**:
- Now, substituting the known values into the formula:
\[ n(A' \cup B') = n(U) - n(A \cap B) \]
\[ n(A' \cup B') = 700 - 100 \]

5. **Final Calculation**:
- Performing the subtraction:
\[ n(A' \cup B') = 600 \]

### Conclusion:
Thus, the value of \( n(A' \cup B') \) is \( 600 \).