Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Problem Statement:**
Suppose \( n(A) = 10 \), \( n(B) = 26 \), and \( n(A \cup B) = 31 \). Use a Venn diagram to find \( n(A \cap B) \).
**Solution:**
- The Venn diagram on the bottom right displays two intersecting circles within a rectangle. Circle A represents set A, and Circle B represents set B.
- The overlap between circles A and B indicates the intersection \( A \cap B \), the elements common to both sets.
**Calculation:**
To find \( n(A \cap B) \), use the formula for the union of two sets:
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]
Given:
- \( n(A \cup B) = 31 \)
- \( n(A) = 10 \)
- \( n(B) = 26 \)
Substitute into the formula:
\[
31 = 10 + 26 - n(A \cap B)
\]
Simplify to find \( n(A \cap B) \):
\[
31 = 36 - n(A \cap B)
\]
\[
n(A \cap B) = 36 - 31
\]
\[
n(A \cap B) = 5
\]
**Conclusion:**
The number of elements in the intersection of sets A and B is \( n(A \cap B) = 5 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfa31bf4-d2b8-4d05-bb93-d68a2eb13a52%2F776d87f1-a363-415c-9e65-4aebad28f5b8%2Fmnlcn6k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Suppose \( n(A) = 10 \), \( n(B) = 26 \), and \( n(A \cup B) = 31 \). Use a Venn diagram to find \( n(A \cap B) \).
**Solution:**
- The Venn diagram on the bottom right displays two intersecting circles within a rectangle. Circle A represents set A, and Circle B represents set B.
- The overlap between circles A and B indicates the intersection \( A \cap B \), the elements common to both sets.
**Calculation:**
To find \( n(A \cap B) \), use the formula for the union of two sets:
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]
Given:
- \( n(A \cup B) = 31 \)
- \( n(A) = 10 \)
- \( n(B) = 26 \)
Substitute into the formula:
\[
31 = 10 + 26 - n(A \cap B)
\]
Simplify to find \( n(A \cap B) \):
\[
31 = 36 - n(A \cap B)
\]
\[
n(A \cap B) = 36 - 31
\]
\[
n(A \cap B) = 5
\]
**Conclusion:**
The number of elements in the intersection of sets A and B is \( n(A \cap B) = 5 \).
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