Question

Let \( U \) be the universal set and let \( A \) and \( B \) be any two subsets of \( U \). If \( n(U) = 25, n(A) = 14, n(A \cap B) = 6 \) and \( n(A \cup B) = 20 \), then \( n(B') \) is equal to:

• A. 12
• B. 13
• C. 14
• D. 15
• E. 16

Hint:

To find the complement of a set, subtract the size of the set from the size of the universal set.

Answer 1

The Correct Option is B

Step 1:Use the formula for the union of two sets.
We are given:
- \( n(U) = 25 \) (size of the universal set),
- \( n(A) = 14 \) (size of set \( A \)),
- \( n(A \cap B) = 6 \) (size of the intersection of sets \( A \) and \( B \)),
- \( n(A \cup B) = 20 \) (size of the union of sets \( A \) and \( B \)).
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 given values: \[ 20 = 14 + n(B) - 6 \] Solving for \( n(B) \): \[ n(B) = 20 - 14 + 6 = 12 \]
Step 2:Find \( n(B') \).
The number of elements in the complement of \( B \), denoted \( n(B') \), is given by: \[ n(B') = n(U) - n(B) \] Substituting the values: \[ n(B') = 25 - 12 = 13 \]
Step 3:Final Answer.
Therefore, \( n(B') = 13 \).