(d) Let A, B, C be sets. Then, (AUB) nC = AU (BNC). True or False Use Venn Diagram to explain your answer. (e) Let A and B be sets. Then, A- (A-B) = An B. True or False Why? Explain your answer.
(d) Let A, B, C be sets. Then, (AUB) nC = AU (BNC). True or False Use Venn Diagram to explain your answer. (e) Let A and B be sets. Then, A- (A-B) = An B. True or False Why? Explain your answer.
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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![### Set Theory Problems with Venn Diagrams
**Problem (d)**
Let \( A, B, C \) be sets. Then,
\[ (A \cup B) \cap C = A \cup (B \cap C). \]
True or False? Use a Venn Diagram to explain your answer.
**Solution:**
To determine whether the equation is true or false, we can analyze it using a Venn Diagram:
1. **Venn Diagram for \( (A \cup B) \cap C \):**
- Start by drawing a Venn Diagram with three intersecting circles labeled \( A \), \( B \), and \( C \).
- The union \( A \cup B \) includes all elements that are in either \( A \) or \( B \) or in both.
- The intersection \( (A \cup B) \cap C \) includes all elements of \( C \) that are also in \( A \cup B \).
2. **Venn Diagram for \( A \cup (B \cap C) \):**
- The intersection \( B \cap C \) includes all elements that are in both \( B \) and \( C \).
- The union \( A \cup (B \cap C) \) includes all elements in \( A \) or in the intersection of \( B \) and \( C \).
By comparing the two diagrams, we can see if the regions shaded in both diagrams (representing the sets) are identical:
\[ \Rightarrow (A \cup B) \cap C \neq A \cup (B \cap C) \]
Therefore, the statement is **False**.
**Problem (e)**
Let \( A \) and \( B \) be sets. Then,
\[ A - (A - B) = A \cap B. \]
True or False? Why? Explain your answer.
**Solution:**
To determine whether the equation is true or false, let’s analyze it logically:
1. \( A - (A - B) \):
- \( (A - B) \) represents the set of elements that are in \( A \) but not in \( B \).
- \( A - (A - B) \) represents the set of elements that are in \( A \) but not in \( (A - B) \).
2. \( A \cap B \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47a2ecfa-9432-49fb-bf43-e148a3a9aff5%2Fd88c6b1b-6935-4ab6-ba53-787050c2b6a4%2F345t7lq_processed.png&w=3840&q=75)
Transcribed Image Text:### Set Theory Problems with Venn Diagrams
**Problem (d)**
Let \( A, B, C \) be sets. Then,
\[ (A \cup B) \cap C = A \cup (B \cap C). \]
True or False? Use a Venn Diagram to explain your answer.
**Solution:**
To determine whether the equation is true or false, we can analyze it using a Venn Diagram:
1. **Venn Diagram for \( (A \cup B) \cap C \):**
- Start by drawing a Venn Diagram with three intersecting circles labeled \( A \), \( B \), and \( C \).
- The union \( A \cup B \) includes all elements that are in either \( A \) or \( B \) or in both.
- The intersection \( (A \cup B) \cap C \) includes all elements of \( C \) that are also in \( A \cup B \).
2. **Venn Diagram for \( A \cup (B \cap C) \):**
- The intersection \( B \cap C \) includes all elements that are in both \( B \) and \( C \).
- The union \( A \cup (B \cap C) \) includes all elements in \( A \) or in the intersection of \( B \) and \( C \).
By comparing the two diagrams, we can see if the regions shaded in both diagrams (representing the sets) are identical:
\[ \Rightarrow (A \cup B) \cap C \neq A \cup (B \cap C) \]
Therefore, the statement is **False**.
**Problem (e)**
Let \( A \) and \( B \) be sets. Then,
\[ A - (A - B) = A \cap B. \]
True or False? Why? Explain your answer.
**Solution:**
To determine whether the equation is true or false, let’s analyze it logically:
1. \( A - (A - B) \):
- \( (A - B) \) represents the set of elements that are in \( A \) but not in \( B \).
- \( A - (A - B) \) represents the set of elements that are in \( A \) but not in \( (A - B) \).
2. \( A \cap B \
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