Show that An (B U C) = (Ñ n B) u Ā - Using membership tables.
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 Proof Using Membership Tables
#### Task:
**Show that:**
\[ A \cap (B \cup C)' = (C' \cap B') \cup A' \]
**Using membership tables.**
**Using rules.**
#### Explanation:
In this task, we are asked to prove the equality between two expressions involving sets and set operations using membership tables and rules. The expressions use basic set operations such as intersection (∩), union (∪), and complement (').
##### Breakdown of Expressions:
1. **Left-hand side (LHS):**
\[ A \cap (B \cup C)' \]
- \( B \cup C \) is the union of sets B and C.
- \( (B \cup C)' \) is the complement of the union of B and C.
- The entire expression means the intersection of set A with the complement of the union of B and C.
2. **Right-hand side (RHS):**
\[ (C' \cap B') \cup A' \]
- \( C' \) is the complement of set C.
- \( B' \) is the complement of set B.
- \( (C' \cap B') \) is the intersection of the complements of C and B.
- \( A' \) is the complement of set A.
- The entire expression is the union of (C' ∩ B') and A'.
#### Proof Approach:
Using membership tables, we can verify the equality between these two sets by checking each possible membership status for elements in the universal set.
#### Rules Used:
- **De Morgan's Laws**:
1. \( (B \cup C)' = B' \cap C' \)
2. \( (B \cap C)' = B' \cup C' \)
- **Distributive Law**:
\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
#### Detailed Steps:
1. **Apply De Morgan's Law to LHS**:
\[ A \cap (B \cup C)' \]
\[ = A \cap (B' \cap C') \]
2. **Distribute the Intersection**:
\[ = (A \cap B') \cap C' \]
3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa06b614-497d-40f0-b077-f4b4fca47baf%2F08f88c23-51b0-4d5f-bbfe-5dd663c1980e%2Fyeuzoh_processed.png&w=3840&q=75)
Transcribed Image Text:### Set Theory Proof Using Membership Tables
#### Task:
**Show that:**
\[ A \cap (B \cup C)' = (C' \cap B') \cup A' \]
**Using membership tables.**
**Using rules.**
#### Explanation:
In this task, we are asked to prove the equality between two expressions involving sets and set operations using membership tables and rules. The expressions use basic set operations such as intersection (∩), union (∪), and complement (').
##### Breakdown of Expressions:
1. **Left-hand side (LHS):**
\[ A \cap (B \cup C)' \]
- \( B \cup C \) is the union of sets B and C.
- \( (B \cup C)' \) is the complement of the union of B and C.
- The entire expression means the intersection of set A with the complement of the union of B and C.
2. **Right-hand side (RHS):**
\[ (C' \cap B') \cup A' \]
- \( C' \) is the complement of set C.
- \( B' \) is the complement of set B.
- \( (C' \cap B') \) is the intersection of the complements of C and B.
- \( A' \) is the complement of set A.
- The entire expression is the union of (C' ∩ B') and A'.
#### Proof Approach:
Using membership tables, we can verify the equality between these two sets by checking each possible membership status for elements in the universal set.
#### Rules Used:
- **De Morgan's Laws**:
1. \( (B \cup C)' = B' \cap C' \)
2. \( (B \cap C)' = B' \cup C' \)
- **Distributive Law**:
\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
#### Detailed Steps:
1. **Apply De Morgan's Law to LHS**:
\[ A \cap (B \cup C)' \]
\[ = A \cap (B' \cap C') \]
2. **Distribute the Intersection**:
\[ = (A \cap B') \cap C' \]
3
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