Question

Let $A$ and $B$ be two subsets of a universal set $U$. If $n(U) = 116$, $n(A \cup B) = 99$, $n(B) = 61$ and $n(A \cap B) = 28$, then $n(A')$ is equal to

• A. 47
• B. 45
• C. 53
• D. 48
• E. 50

Hint:

Always identify the universal set first. The complement of a set is simply the 'remaining' portion of the universal set. Finding the base set's cardinality is the key intermediate step.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem involves the fundamental principles of set theory, specifically the relationship between the number of elements in the union and intersection of two sets, and the definition of the complement of a set.
The cardinality of the union of two sets is given by the inclusion-exclusion principle.
The complement of a set consists of all elements in the universal set that are not in the given set.
Key Formula or Approach:
The formula for the union of two sets $A$ and $B$ is:
\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
The number of elements in the complement of $A$ is:
\( n(A') = n(U) - n(A) \)

Step 2: Detailed Explanation:
From the question, we have the following data:
\( n(U) = 116 \)
\( n(A \cup B) = 99 \)
\( n(B) = 61 \)
\( n(A \cap B) = 28 \)
First, we find the number of elements in set $A$ using the union formula:
\[ 99 = n(A) + 61 - 28 \]
\[ 99 = n(A) + 33 \]
Subtracting 33 from both sides:
\[ n(A) = 99 - 33 = 66 \]
Next, we calculate the number of elements in the complement of $A$ (denoted $A'$):
\[ n(A') = n(U) - n(A) \]
\[ n(A') = 116 - 66 \]
\[ n(A') = 50 \]

Step 3: Final Answer:
The value of $n(A')$ is 50.