Let U be the universal set for sets A and B such that `n(A) =200 ,n(B) =300 and n(A cap B)=100.` then `n(A'cap B')` is equal to 300, provided that `n(U)` is equal to

Options

A. 600
B. 700
C. 800
D. 900

Answer

To solve the problem, we need to find the number of elements in the universal set \( n(U) \) given the following information:

- \( n(A) = 200 \)
- \( n(B) = 300 \)
- \( n(A \cap B) = 100 \)
- \( n(A' \cap B') = 300 \)

We can use the principle of inclusion-exclusion and some properties of sets to find \( n(U) \).

### Step-by-step Solution:

1. **Find \( n(A \cup B) \)**:
We use the formula for the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Substituting the values:
\[ n(A \cup B) = 200 + 300 - 100 = 400 \]

2. **Use the formula for \( n(A' \cap B') \)**:
We know that:
\[ n(A' \cap B') = n(U) - n(A \cup B) \]
Here, \( n(A') \) and \( n(B') \) represent the complements of sets \( A \) and \( B \) respectively.

3. **Substituting the known values**:
We have \( n(A' \cap B') = 300 \) and \( n(A \cup B) = 400 \). Thus:
\[ 300 = n(U) - 400 \]

4. **Solve for \( n(U) \)**:
Rearranging the equation gives:
\[ n(U) = 300 + 400 = 700 \]

### Conclusion:
Thus, the number of elements in the universal set \( n(U) \) is \( 700 \).