Question 5 (12 points) Given the following cardinalities: \( n(U)=169, n(A)=81 \), and \( n(B)=66 \), find the following: 1. if \( n(A \cap B)=47 \), find \( n(A \cup B) \). 2. if \( n(A \cup B)=147 \), find \( n(A \cap B) \).

To solve the problems, we’ll be using the basic set theory formula involving cardinalities: 1. For the first question, we need to find \( n(A \cup B) \) using the formula: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Substituting the given values: \[ n(A \cup B) = 81 + 66 - 47 = 100 \] 2. For the second question, we can rearrange the same formula to find \( n(A \cap B) \): \[ n(A \cap B) = n(A) + n(B) - n(A \cup B) \] Substituting the known values: \[ n(A \cap B) = 81 + 66 - 147 = 0 \] So the answers are: 1. \( n(A \cup B) = 100 \) 2. \( n(A \cap B) = 0 \)