How elements are in the set? B Many (AUB) ^ (AUB)' = A 366 19 39
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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![### How many elements are in the set?
Given the set notation:
\[ (A \cup B)' \]
we are asked to find the number of elements in the complement of the union of sets \(A\) and \(B\).
#### Venn Diagram Explanation
The Venn diagram provided shows the following details:
- **Set A** and **Set B** are two overlapping circles.
- The universal set is represented by a rectangle surrounding the two circles.
- **Set A**:
- Contains 36 elements uniquely.
- **Set B**:
- Contains 14 elements uniquely.
- **Intersection of A and B**:
- Contains 6 elements.
- Elements outside both sets in the universal set:
- Contains 39 elements.
To find the complement of \(A \cup B\), we need to calculate the elements that are not in either set \(A\) or set \(B\).
1. Find the total number of elements in \(A \cup B\):
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
\[ |A \cup B| = 36 + 14 + 6 \]
\[ |A \cup B| = 56 \]
2. The total number of elements in the universal set (represented by the rectangular boundary) is the sum of all elements inside and outside the sets \(A\) and \(B\):
\[ U = 36 + 14 + 6 + 39 = 95 \]
3. The complement of \(A \cup B\) is the total elements in the universal set minus the elements in \(A \cup B\):
\[ n((A \cup B)') = U - |A \cup B| \]
\[ n((A \cup B)') = 95 - 56 = 39 \]
Thus, the number of elements in the complement of \((A \cup B)\) is 39.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0cd4ae9-86e1-45b4-8824-a362b78dd084%2F39fbfc13-ebcf-44bf-b205-a5dd40174d60%2Fvz06wpg_processed.png&w=3840&q=75)
Transcribed Image Text:### How many elements are in the set?
Given the set notation:
\[ (A \cup B)' \]
we are asked to find the number of elements in the complement of the union of sets \(A\) and \(B\).
#### Venn Diagram Explanation
The Venn diagram provided shows the following details:
- **Set A** and **Set B** are two overlapping circles.
- The universal set is represented by a rectangle surrounding the two circles.
- **Set A**:
- Contains 36 elements uniquely.
- **Set B**:
- Contains 14 elements uniquely.
- **Intersection of A and B**:
- Contains 6 elements.
- Elements outside both sets in the universal set:
- Contains 39 elements.
To find the complement of \(A \cup B\), we need to calculate the elements that are not in either set \(A\) or set \(B\).
1. Find the total number of elements in \(A \cup B\):
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
\[ |A \cup B| = 36 + 14 + 6 \]
\[ |A \cup B| = 56 \]
2. The total number of elements in the universal set (represented by the rectangular boundary) is the sum of all elements inside and outside the sets \(A\) and \(B\):
\[ U = 36 + 14 + 6 + 39 = 95 \]
3. The complement of \(A \cup B\) is the total elements in the universal set minus the elements in \(A \cup B\):
\[ n((A \cup B)') = U - |A \cup B| \]
\[ n((A \cup B)') = 95 - 56 = 39 \]
Thus, the number of elements in the complement of \((A \cup B)\) is 39.
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