Question

If \( n(B) = 61 \), \( n(A \cup B) = 99 \), \( n(A \cap B) = 28 \), find \( n(A') \).

Hint:

Remember the complement rule: \( n(A') = n(U) - n(A) \), where \( n(U) \) is the total number of elements in the universal set.

Answer 1

Step 1: Use the principle of inclusion-exclusion.
We are given the following information: - \( n(A \cup B) = 99 \) - \( n(A \cap B) = 28 \) From the principle of inclusion-exclusion, we know that: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Substituting the given values: \[ 99 = n(A) + 61 - 28 \] \[ n(A) = 99 - 33 = 66 \]
Step 2: Find \( n(A') \).
The complement of \( A \) is denoted as \( A' \), and by the complement rule, we know: \[ n(A') = n(U) - n(A) \] where \( n(U) \) is the total number of elements in the universal set. Since \( n(A \cup B) = 99 \) and this represents all the elements in the universal set, we have \( n(U) = 99 \). Therefore: \[ n(A') = 99 - 66 = 33 \]