Question

Let \( U \) be the universal set, and \( A \) and \( B \) be the subsets of \( U \). If \( n(U) = 450 \), \( n(A) = 200 \), \( n(B) = 205 \), and \( n(A \cap B) = 15 \), then \( n(\overline{A}\cap \overline{B}) \) is equal to:

• A. 55
• B. 60
• C. 65
• D. 70

Hint:

Use the formula for the union of two sets to find \( n(A \cup B) \), then subtract it from the total number of elements in the universal set to find the complement.

Answer 1

The Correct Option is C

We need to find \( n(\overline{A} \cap \overline{B}) \), which represents the number of elements in the complement of both sets \( A \) and \( B \).
First, calculate \( n(A \cup B) \) using the formula: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) = 200 + 205 - 15 = 390 \] Next, subtract \( n(A \cup B) \) from the total number of elements in the universal set: \[ \( n(\overline{A} \cap \overline{B}) \) = n(U) - n(A \cup B) = 450 - 390 = 60 \] Thus, the correct answer is \( 60 \).