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:
Show Hint
To find the complement of a set, subtract the size of the set from the size of the universal set.
Step 1: Understanding the Concept
We use the Principle of Inclusion-Exclusion for two sets and the definition of a complement set. $B'$ (or $B^c$) represents all elements in the Universal set $U$ that are not in $B$. Step 2: Key Formula or Approach
1. $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
2. $n(B') = n(U) - n(B)$
Step 3: Detailed Calculation
1. Find $n(B)$:
- $20 = 14 + n(B) - 6$
- $20 = 8 + n(B)$
- $n(B) = 20 - 8 = 12$
2. Find $n(B')$:
- $n(B') = n(U) - n(B)$
- $n(B') = 25 - 12 = 13$ Step 4: Final Answer
The value of $n(B')$ is 13.