### Understanding Venn Diagrams #### Refer to the Venn Diagram. How many elements are in the following set? \[ (A \cup B)' \] The Venn diagram is composed of two overlapping circles labeled A and B within a rectangle labeled U. Each area within the Venn diagram contains the number of elements specified: - Circle A (exclusive): 37 elements - Circle B (exclusive): 19 elements - Intersection of circles A and B: 4 elements - Area outside both circles A and B (within U but not in A or B): 40 elements To find the number of elements in \((A \cup B)'\), which is the complement of the union of sets A and B: 1. Calculate \(A \cup B\): \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] where: - \(n(A) = 37 + 4 = 41\) - \(n(B) = 19 + 4 = 23\) - \(n(A \cap B) = 4\) Therefore: \[ n(A \cup B) = 37 + 19 + 4 = 60 \] 2. Calculate \((A \cup B)'\), which represents the elements in U that are not in \(A \cup B\): \[ n((A \cup B)') = n(U) - n(A \cup B) \] Given the total number of elements in U is the sum of all individual segments within the rectangle: \[ n(U) = 37 + 19 + 4 + 40 = 100 \] Therefore: \[ n((A \cup B)') = 100 - 60 = 40 \] Thus, the number of elements in \((A \cup B)'\) is 40. #### Diagram Explanation - The Venn diagram has two overlapping circles labeled A and B within a rectangle labeled U. - The regions of the Venn diagram contain numbers representing the number of elements in each region: - Circle A (left circle): Contains 37 elements exclusive to A. - Circle B (right circle): Contains 19 elements exclusive to B. - Intersection of circles A and B:
### Understanding Venn Diagrams #### Refer to the Venn Diagram. How many elements are in the following set? \[ (A \cup B)' \] The Venn diagram is composed of two overlapping circles labeled A and B within a rectangle labeled U. Each area within the Venn diagram contains the number of elements specified: - Circle A (exclusive): 37 elements - Circle B (exclusive): 19 elements - Intersection of circles A and B: 4 elements - Area outside both circles A and B (within U but not in A or B): 40 elements To find the number of elements in \((A \cup B)'\), which is the complement of the union of sets A and B: 1. Calculate \(A \cup B\): \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] where: - \(n(A) = 37 + 4 = 41\) - \(n(B) = 19 + 4 = 23\) - \(n(A \cap B) = 4\) Therefore: \[ n(A \cup B) = 37 + 19 + 4 = 60 \] 2. Calculate \((A \cup B)'\), which represents the elements in U that are not in \(A \cup B\): \[ n((A \cup B)') = n(U) - n(A \cup B) \] Given the total number of elements in U is the sum of all individual segments within the rectangle: \[ n(U) = 37 + 19 + 4 + 40 = 100 \] Therefore: \[ n((A \cup B)') = 100 - 60 = 40 \] Thus, the number of elements in \((A \cup B)'\) is 40. #### Diagram Explanation - The Venn diagram has two overlapping circles labeled A and B within a rectangle labeled U. - The regions of the Venn diagram contain numbers representing the number of elements in each region: - Circle A (left circle): Contains 37 elements exclusive to A. - Circle B (right circle): Contains 19 elements exclusive to B. - Intersection of circles A and B:
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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![### Understanding Venn Diagrams
#### Refer to the Venn Diagram. How many elements are in the following set?
\[ (A \cup B)' \]
The Venn diagram is composed of two overlapping circles labeled A and B within a rectangle labeled U. Each area within the Venn diagram contains the number of elements specified:
- Circle A (exclusive): 37 elements
- Circle B (exclusive): 19 elements
- Intersection of circles A and B: 4 elements
- Area outside both circles A and B (within U but not in A or B): 40 elements
To find the number of elements in \((A \cup B)'\), which is the complement of the union of sets A and B:
1. Calculate \(A \cup B\):
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]
where:
- \(n(A) = 37 + 4 = 41\)
- \(n(B) = 19 + 4 = 23\)
- \(n(A \cap B) = 4\)
Therefore:
\[
n(A \cup B) = 37 + 19 + 4 = 60
\]
2. Calculate \((A \cup B)'\), which represents the elements in U that are not in \(A \cup B\):
\[
n((A \cup B)') = n(U) - n(A \cup B)
\]
Given the total number of elements in U is the sum of all individual segments within the rectangle:
\[
n(U) = 37 + 19 + 4 + 40 = 100
\]
Therefore:
\[
n((A \cup B)') = 100 - 60 = 40
\]
Thus, the number of elements in \((A \cup B)'\) is 40.
#### Diagram Explanation
- The Venn diagram has two overlapping circles labeled A and B within a rectangle labeled U.
- The regions of the Venn diagram contain numbers representing the number of elements in each region:
- Circle A (left circle): Contains 37 elements exclusive to A.
- Circle B (right circle): Contains 19 elements exclusive to B.
- Intersection of circles A and B:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0cd4ae9-86e1-45b4-8824-a362b78dd084%2Ff2f32873-c9cb-4ea2-b1e5-dd53e689ed3d%2Fh4b3nha_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Venn Diagrams
#### Refer to the Venn Diagram. How many elements are in the following set?
\[ (A \cup B)' \]
The Venn diagram is composed of two overlapping circles labeled A and B within a rectangle labeled U. Each area within the Venn diagram contains the number of elements specified:
- Circle A (exclusive): 37 elements
- Circle B (exclusive): 19 elements
- Intersection of circles A and B: 4 elements
- Area outside both circles A and B (within U but not in A or B): 40 elements
To find the number of elements in \((A \cup B)'\), which is the complement of the union of sets A and B:
1. Calculate \(A \cup B\):
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]
where:
- \(n(A) = 37 + 4 = 41\)
- \(n(B) = 19 + 4 = 23\)
- \(n(A \cap B) = 4\)
Therefore:
\[
n(A \cup B) = 37 + 19 + 4 = 60
\]
2. Calculate \((A \cup B)'\), which represents the elements in U that are not in \(A \cup B\):
\[
n((A \cup B)') = n(U) - n(A \cup B)
\]
Given the total number of elements in U is the sum of all individual segments within the rectangle:
\[
n(U) = 37 + 19 + 4 + 40 = 100
\]
Therefore:
\[
n((A \cup B)') = 100 - 60 = 40
\]
Thus, the number of elements in \((A \cup B)'\) is 40.
#### Diagram Explanation
- The Venn diagram has two overlapping circles labeled A and B within a rectangle labeled U.
- The regions of the Venn diagram contain numbers representing the number of elements in each region:
- Circle A (left circle): Contains 37 elements exclusive to A.
- Circle B (right circle): Contains 19 elements exclusive to B.
- Intersection of circles A and B:
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