Question

If \( n(X) = 700 \), \( n(A) = 200 \), \( n(B) = 300 \), \( n(A \cap B) = 100 \), where \( X \) is the universal set and \( A \) and \( B \) are subsets of \( X \), then \( n(A' \cap B') = \)

• A. 300
• B. 400
• C. 340
• D. 240

Hint:

When calculating the number of elements in the complement of the union of two sets, use the formula \( n(A' \cap B') = n(X) - n(A \cup B) \), and remember to apply the principle of inclusion and exclusion for the union.

Answer 1

The Correct Option is D

Step 1: Use the formula for the complement of the union.
We are asked to find \( n(A' \cap B') \), the number of elements in the complement of \( A \) and \( B \). This can be calculated using the formula: \[ n(A' \cap B') = n(X) - n(A \cup B) \] where \( n(A \cup B) \) is the number of elements in the union of \( A \) and \( B \).
Step 2: Calculate \( n(A \cup B) \).
Using the formula for the union of two sets: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Substitute the given values: \[ n(A \cup B) = 200 + 300 - 100 = 400 \]
Step 3: Calculate \( n(A' \cap B') \).
Now, using the formula for \( n(A' \cap B') \): \[ n(A' \cap B') = 700 - 400 = 240 \]
Step 4: Conclusion.
Thus, \( n(A' \cap B') = 240 \).