If `n(U)=700, n(A)=200, n(B)=240 and n(AnnB)=100`, then `n(A^(C )uuB^(C ))` is equal to

Options

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

Answer

To solve the problem, we need to find \( n(A^C \cup B^C) \) using the information provided and De Morgan's laws.

### Step-by-Step Solution:

1. **Understand the Given Values**:
- \( n(U) = 700 \) (Total number of elements in the universal set)
- \( n(A) = 200 \) (Number of elements in set A)
- \( n(B) = 240 \) (Number of elements in set B)
- \( n(A \cap B) = 100 \) (Number of elements in the intersection of sets A and B)

2. **Apply De Morgan's Law**:
According to De Morgan's law,
\[ A^C \cup B^C = (A \cap B)^C \]
Therefore, we can express the number of elements in \( A^C \cup B^C \) as:
\[ n(A^C \cup B^C) = n((A \cap B)^C) \]

3. **Calculate \( n((A \cap B)^C) \)**:
The number of elements in the complement of \( A \cap B \) can be found using the formula:
\[ n((A \cap B)^C) = n(U) - n(A \cap B) \]
Substituting the known values:
\[ n((A \cap B)^C) = 700 - 100 \]

4. **Perform the Calculation**:
\[ n((A \cap B)^C) = 600 \]

5. **Conclusion**:
Thus, we have:
\[ n(A^C \cup B^C) = 600 \]

### Final Answer:
\[ n(A^C \cup B^C) = 600 \]