Use Venn diagrams to illustrate the given identity for subsets A, B, and C of S. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Distributive law ATTACHED IS THE IMAGE TO QUESTION
Use Venn diagrams to illustrate the given identity for subsets A, B, and C of S. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Distributive law ATTACHED IS THE IMAGE TO QUESTION
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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Use Venn diagrams to illustrate the given identity for subsets A, B, and C of S.
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Distributive law
ATTACHED IS THE IMAGE TO QUESTION
![**Understanding Set Operations through Venn Diagrams**
This section uses Venn diagrams to illustrate the set identity for subsets \(A\), \(B\), and \(C\) of \(S\).
**Identity Explained:**
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]
This is known as the *Distributive Law* in set theory.
**Diagrams Explained:**
1. **First Diagram (Left):**
- The entire rectangle represents the universal set \(S\).
- Three circles within the rectangle represent sets \(A\), \(B\), and \(C\).
- The shaded area is the intersection of \(A\) with the union of \(B\) and \(C\) (\(A \cap (B \cup C)\)).
2. **Second Diagram:**
- Again, circles represent sets \(A\), \(B\), and \(C\) within the universal set \(S\).
- The highlighted region outside the circles shows all the elements within \(S\) except those in the shaded area.
- Only the area that represents the expression \((A \cap B) \cup (A \cap C)\) is not shaded.
3. **Third Diagram:**
- Shows the intersection of \(A\) with sets \(B\) and \(C\) separately, shaded to represent \((A \cap B) \cup (A \cap C)\).
- The shaded areas depict the individual intersections, illustrating how the union of these intersections forms the desired set.
4. **Fourth Diagram (Right):**
- This diagram emphasizes all elements except those outside the union of the intersections.
- Like in the second diagram, only the region \((A \cap B) \cup (A \cap C)\) is included while the rest is shaded, illustrating the distributive property for clarity.
These diagrams collectively depict and validate the distributive law in set theory using visual representations that reinforce understanding of intersections and unions within sets.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F283588a7-4b17-47d8-a8cf-ba138f1ddbe1%2F7c3a99f4-07c3-4fb7-be6b-8aa34ab8a284%2Fa12gp0r_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Set Operations through Venn Diagrams**
This section uses Venn diagrams to illustrate the set identity for subsets \(A\), \(B\), and \(C\) of \(S\).
**Identity Explained:**
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]
This is known as the *Distributive Law* in set theory.
**Diagrams Explained:**
1. **First Diagram (Left):**
- The entire rectangle represents the universal set \(S\).
- Three circles within the rectangle represent sets \(A\), \(B\), and \(C\).
- The shaded area is the intersection of \(A\) with the union of \(B\) and \(C\) (\(A \cap (B \cup C)\)).
2. **Second Diagram:**
- Again, circles represent sets \(A\), \(B\), and \(C\) within the universal set \(S\).
- The highlighted region outside the circles shows all the elements within \(S\) except those in the shaded area.
- Only the area that represents the expression \((A \cap B) \cup (A \cap C)\) is not shaded.
3. **Third Diagram:**
- Shows the intersection of \(A\) with sets \(B\) and \(C\) separately, shaded to represent \((A \cap B) \cup (A \cap C)\).
- The shaded areas depict the individual intersections, illustrating how the union of these intersections forms the desired set.
4. **Fourth Diagram (Right):**
- This diagram emphasizes all elements except those outside the union of the intersections.
- Like in the second diagram, only the region \((A \cap B) \cup (A \cap C)\) is included while the rest is shaded, illustrating the distributive property for clarity.
These diagrams collectively depict and validate the distributive law in set theory using visual representations that reinforce understanding of intersections and unions within sets.
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