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,...
Related questions
Question
![**Venn Diagram Problem Analysis**
**Given Problem:**
- \( n(A) = 10 \)
- \( n(B) = 26 \)
- \( n(A \cup B) = 31 \)
**Objective:**
Use a Venn diagram to find \( n(A \cap B) \).
**Explanation:**
To solve the problem, we need to find the intersection of sets A and B, \( n(A \cap B) \).
**Venn Diagram Overview:**
The Venn diagram illustrated consists of two overlapping circles labeled A and B. The overlap between these circles represents the elements common to both sets A and B.
**Solution Steps:**
1. Use the formula for the union of two sets:
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]
2. Substitute the known values into the equation:
\[
31 = 10 + 26 - n(A \cap B)
\]
3. Simplify to find \( n(A \cap B) \):
\[
31 = 36 - n(A \cap B)
\]
4. Solve for \( n(A \cap B) \):
\[
n(A \cap B) = 36 - 31 = 5
\]
**Conclusion:**
The number of elements in the intersection of sets A and B, \( n(A \cap B) \), is 5.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfa31bf4-d2b8-4d05-bb93-d68a2eb13a52%2F27d0814c-4a8e-4d6a-8f6d-58bad078446e%2Fb7w7te_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Venn Diagram Problem Analysis**
**Given Problem:**
- \( n(A) = 10 \)
- \( n(B) = 26 \)
- \( n(A \cup B) = 31 \)
**Objective:**
Use a Venn diagram to find \( n(A \cap B) \).
**Explanation:**
To solve the problem, we need to find the intersection of sets A and B, \( n(A \cap B) \).
**Venn Diagram Overview:**
The Venn diagram illustrated consists of two overlapping circles labeled A and B. The overlap between these circles represents the elements common to both sets A and B.
**Solution Steps:**
1. Use the formula for the union of two sets:
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]
2. Substitute the known values into the equation:
\[
31 = 10 + 26 - n(A \cap B)
\]
3. Simplify to find \( n(A \cap B) \):
\[
31 = 36 - n(A \cap B)
\]
4. Solve for \( n(A \cap B) \):
\[
n(A \cap B) = 36 - 31 = 5
\]
**Conclusion:**
The number of elements in the intersection of sets A and B, \( n(A \cap B) \), is 5.
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