If `n(U)=700,n(A)=200,n(B)=300,n(AnnB)=100`, then `n(A'capB')` is equal to

Options

A. 300
B. 350
C. 400
D. 500

Answer

To find \( n(A' \cap B') \), we can use the principle of set theory that relates the number of elements in the union and intersection of sets.

Given:
- \( n(U) = 700 \) (the total number of elements in the universal set)
- \( n(A) = 200 \) (the number of elements in set A)
- \( n(B) = 300 \) (the number of elements in set B)
- \( n(A \cap B) = 100 \) (the number of elements in the intersection of sets A and B)

We need to find \( n(A' \cap B') \), which represents the number of elements that are neither in A nor in B.

### Step 1: Calculate \( n(A \cup B) \)
We can use the formula for the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Substituting the values we have:
\[ n(A \cup B) = 200 + 300 - 100 = 400 \]

### Step 2: Calculate \( n(A' \cap B') \)
Using the relationship between the universal set and the union of sets A and B:
\[ n(A' \cap B') = n(U) - n(A \cup B) \]
Substituting the values:
\[ n(A' \cap B') = 700 - 400 = 300 \]

Thus, the final result is:
\[ n(A' \cap B') = 300 \]

### Summary of Steps:
1. Calculate \( n(A \cup B) \) using the formula \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \).
2. Use the total number of elements in the universal set to find \( n(A' \cap B') \) with the formula \( n(A' \cap B') = n(U) - n(A \cup B) \).