Suppose Set A contains 77 elements and Set B contains 59 elements. If the total number elements in either Set A or Set B is 108, how many elements do Sets A and B have in common? Answer = elements
Suppose Set A contains 77 elements and Set B contains 59 elements. If the total number elements in either Set A or Set B is 108, how many elements do Sets A and B have in common? Answer = elements
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Problem Description
Suppose Set A contains 77 elements and Set B contains 59 elements. If the total number of elements in either Set A or Set B is 108, how many elements do Sets A and B have in common?
### Solution
To find the number of elements that Sets A and B have in common, we use the formula for the union of two sets:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
where:
- \(|A \cup B|\) is the total number of elements in either Set A or Set B.
- \(|A|\) is the number of elements in Set A.
- \(|B|\) is the number of elements in Set B.
- \(|A \cap B|\) is the number of elements common to both Set A and Set B.
Given:
- \(|A| = 77\)
- \(|B| = 59\)
- \(|A \cup B| = 108\)
Substituting the known values into the formula:
\[
108 = 77 + 59 - |A \cap B|
\]
Solving for \(|A \cap B|\):
\[
|A \cap B| = 77 + 59 - 108
\]
\[
|A \cap B| = 136 - 108
\]
\[
|A \cap B| = 28
\]
### Conclusion
The number of elements that Sets A and B have in common is 28 elements.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc28801d2-5244-43a8-94e3-88d6acce7d2a%2F0d7fdb53-8e74-4a39-bc0b-f5c71be78dd3%2F8o97zw_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description
Suppose Set A contains 77 elements and Set B contains 59 elements. If the total number of elements in either Set A or Set B is 108, how many elements do Sets A and B have in common?
### Solution
To find the number of elements that Sets A and B have in common, we use the formula for the union of two sets:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
where:
- \(|A \cup B|\) is the total number of elements in either Set A or Set B.
- \(|A|\) is the number of elements in Set A.
- \(|B|\) is the number of elements in Set B.
- \(|A \cap B|\) is the number of elements common to both Set A and Set B.
Given:
- \(|A| = 77\)
- \(|B| = 59\)
- \(|A \cup B| = 108\)
Substituting the known values into the formula:
\[
108 = 77 + 59 - |A \cap B|
\]
Solving for \(|A \cap B|\):
\[
|A \cap B| = 77 + 59 - 108
\]
\[
|A \cap B| = 136 - 108
\]
\[
|A \cap B| = 28
\]
### Conclusion
The number of elements that Sets A and B have in common is 28 elements.
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