Shade the region for the following expressions
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Question
Shade the region for the following expressions:
![### Complement of Intersections of Sets
**Expression:**
\[ A \cap (B \cup C)' \]
In the given image, we have a Venn diagram illustrating three sets: \( A \), \( B \), and \( C \).
1. **Sets Description:**
- **Set A** is represented by the left ellipse.
- **Set B** is represented by the top-right ellipse.
- **Set C** is represented by the bottom ellipse.
2. **Complement Representation:**
- The complement of a set is represented by areas that are not part of said set.
**Step-by-Step Explanation:**
1. **Union of B and C:**
- The union \( B \cup C \) includes all elements in either \( B \) or \( C \), or both.
- Graphically, this is represented by the entire area covered by both ellipses \( B \) and \( C \).
2. **Complement of Union (B ∪ C):**
- The complement \( (B \cup C)' \) contains all elements that are not in \( B \cup C \).
- This is represented by the areas outside of both ellipses \( B \) and \( C \).
3. **Intersection with A:**
- To find \( A \cap (B \cup C)' \), we look for the area within Set \( A \) that does not overlap with \( B \) or \( C \).
- Graphically, this is represented by the portion of ellipse \( A \) that lies completely outside ellipses \( B \) and \( C \).
**Visual Representation in Diagram:**
- The overall rectangle in the diagram represents the universal set \( S \).
- The overlapping areas between the ellipses represent their intersections.
- The area inside ellipse \( A \), but outside of ellipses \( B \) and \( C \), represents the subset \( A \cap (B \cup C)' \).
By analyzing this diagram, students can gain a clear understanding of how to compute and visualize the complement of the union of sets and the corresponding intersection with another set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabbd02aa-e289-47d3-b3b4-87f7eeaaef64%2Fd5fecd4d-48e1-465c-8b75-2566db10a5ba%2Fx8nadj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Complement of Intersections of Sets
**Expression:**
\[ A \cap (B \cup C)' \]
In the given image, we have a Venn diagram illustrating three sets: \( A \), \( B \), and \( C \).
1. **Sets Description:**
- **Set A** is represented by the left ellipse.
- **Set B** is represented by the top-right ellipse.
- **Set C** is represented by the bottom ellipse.
2. **Complement Representation:**
- The complement of a set is represented by areas that are not part of said set.
**Step-by-Step Explanation:**
1. **Union of B and C:**
- The union \( B \cup C \) includes all elements in either \( B \) or \( C \), or both.
- Graphically, this is represented by the entire area covered by both ellipses \( B \) and \( C \).
2. **Complement of Union (B ∪ C):**
- The complement \( (B \cup C)' \) contains all elements that are not in \( B \cup C \).
- This is represented by the areas outside of both ellipses \( B \) and \( C \).
3. **Intersection with A:**
- To find \( A \cap (B \cup C)' \), we look for the area within Set \( A \) that does not overlap with \( B \) or \( C \).
- Graphically, this is represented by the portion of ellipse \( A \) that lies completely outside ellipses \( B \) and \( C \).
**Visual Representation in Diagram:**
- The overall rectangle in the diagram represents the universal set \( S \).
- The overlapping areas between the ellipses represent their intersections.
- The area inside ellipse \( A \), but outside of ellipses \( B \) and \( C \), represents the subset \( A \cap (B \cup C)' \).
By analyzing this diagram, students can gain a clear understanding of how to compute and visualize the complement of the union of sets and the corresponding intersection with another set.
![### Set Theory Diagrams on Educational Website
#### a. \[(A' ∩ B')'\]
This diagram represents the set operation \[(A' ∩ B')'\] within the universal set \(S\). The diagram features two overlapping circles labeled \(A\) and \(B\) inside a rectangle that represents the universal set \(S\).
- **\(A'\)**: The complement of set \(A\) includes all elements in \(S\) that are not in \(A\).
- **\(B'\)**: The complement of set \(B\) includes all elements in \(S\) that are not in \(B\).
- **\(A' ∩ B'\)**: The intersection of the complements of \(A\) and \(B\), which consists of all elements in \(S\) that are neither in \(A\) nor in \(B\).
The notation \[(A' ∩ B')'\] indicates the complement of the intersection of \(A'\) and \(B'\). In other words, it includes elements that are in either \(A\) or \(B\).
#### b. \[(A ∩ B) ∪ C'\]
This diagram represents the set operation \[(A ∩ B) ∪ C'\] within the universal set \(S\). The diagram features three overlapping circles labeled \(A\), \(B\), and \(C\) inside a rectangle that represents the universal set \(S\).
- **\(A ∩ B\)**: The intersection of sets \(A\) and \(B\), which includes all elements that are in both \(A\) and \(B\).
- **\(C'\)**: The complement of set \(C\), which includes all elements in \(S\) that are not in \(C\).
The union \((A ∩ B) ∪ C'\) includes all elements that are either in the intersection of \(A\) and \(B\) or are not in \(C\).
These diagrams illustrate the visual representation of set operations, aiding in understanding the relationships between different sets within a universal set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabbd02aa-e289-47d3-b3b4-87f7eeaaef64%2Fd5fecd4d-48e1-465c-8b75-2566db10a5ba%2Fzw4ph6s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Set Theory Diagrams on Educational Website
#### a. \[(A' ∩ B')'\]
This diagram represents the set operation \[(A' ∩ B')'\] within the universal set \(S\). The diagram features two overlapping circles labeled \(A\) and \(B\) inside a rectangle that represents the universal set \(S\).
- **\(A'\)**: The complement of set \(A\) includes all elements in \(S\) that are not in \(A\).
- **\(B'\)**: The complement of set \(B\) includes all elements in \(S\) that are not in \(B\).
- **\(A' ∩ B'\)**: The intersection of the complements of \(A\) and \(B\), which consists of all elements in \(S\) that are neither in \(A\) nor in \(B\).
The notation \[(A' ∩ B')'\] indicates the complement of the intersection of \(A'\) and \(B'\). In other words, it includes elements that are in either \(A\) or \(B\).
#### b. \[(A ∩ B) ∪ C'\]
This diagram represents the set operation \[(A ∩ B) ∪ C'\] within the universal set \(S\). The diagram features three overlapping circles labeled \(A\), \(B\), and \(C\) inside a rectangle that represents the universal set \(S\).
- **\(A ∩ B\)**: The intersection of sets \(A\) and \(B\), which includes all elements that are in both \(A\) and \(B\).
- **\(C'\)**: The complement of set \(C\), which includes all elements in \(S\) that are not in \(C\).
The union \((A ∩ B) ∪ C'\) includes all elements that are either in the intersection of \(A\) and \(B\) or are not in \(C\).
These diagrams illustrate the visual representation of set operations, aiding in understanding the relationships between different sets within a universal set.
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