Question
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 \( n(A \cap B) \).
Ask by Weston Donnelly. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
1. \( n(A \cup B) = 100 \)
2. \( n(A \cap B) = 0 \)
Solution
Let's solve each part step-by-step using the principles of set theory.
### **1. Finding \( n(A \cup B) \) when \( n(A \cap B) = 47 \)**
The formula for the union of two sets is:
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]
Plugging in the given values:
\[
n(A \cup B) = 81 + 66 - 47 = 147 - 47 = 100
\]
**Answer:** \( n(A \cup B) = 100 \)
---
### **2. Finding \( n(A \cap B) \) when \( n(A \cup B) = 147 \)**
Using the same formula:
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]
\[
147 = 81 + 66 - n(A \cap B)
\]
\[
147 = 147 - n(A \cap B)
\]
\[
n(A \cap B) = 0
\]
**Answer:** \( n(A \cap B) = 0 \)
This means that sets \( A \) and \( B \) are **disjoint**, having no elements in common.
---
### **Summary:**
1. \( n(A \cup B) = 100 \)
2. \( n(A \cap B) = 0 \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
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. \( n(A \cap B) = 0 \)
