Problem 15 Use Mathematical Induction to prove De Morgan's Law. A-1 A₂ Th
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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Question
\[ \overline{\bigcup_{k=1}^{n} A_{k}} = \bigcap_{k=1}^{n} \overline{A_{k}} \]
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**Explanation:**
This problem involves using the method of mathematical induction to prove De Morgan's Law for a finite collection of sets. The law essentially states that the complement of the union of several sets is equal to the intersection of their complements.
### Steps for Mathematical Induction:
1. **Base Case:**
- Prove the statement for a small initial value of \( n \), usually \( n = 1 \).
2. **Induction Hypothesis:**
- Assume the statement holds for some arbitrary positive integer \( n = k \).
3. **Inductive Step:**
- Prove the statement for \( n = k + 1 \) using the induction hypothesis.
By applying these steps, you can demonstrate that De Morgan's Law holds for any finite number of sets. This proof method is valuable in various fields including mathematics and logic, providing a foundational tool for an array of theoretical arguments and practical applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe872339c-719c-4a81-bc7c-37653a99cb41%2F986835d2-1459-4607-b31d-f9382571be18%2F0c5u9d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 15: Use Mathematical Induction to prove De Morgan's Law.**
\[ \overline{\bigcup_{k=1}^{n} A_{k}} = \bigcap_{k=1}^{n} \overline{A_{k}} \]
---
**Explanation:**
This problem involves using the method of mathematical induction to prove De Morgan's Law for a finite collection of sets. The law essentially states that the complement of the union of several sets is equal to the intersection of their complements.
### Steps for Mathematical Induction:
1. **Base Case:**
- Prove the statement for a small initial value of \( n \), usually \( n = 1 \).
2. **Induction Hypothesis:**
- Assume the statement holds for some arbitrary positive integer \( n = k \).
3. **Inductive Step:**
- Prove the statement for \( n = k + 1 \) using the induction hypothesis.
By applying these steps, you can demonstrate that De Morgan's Law holds for any finite number of sets. This proof method is valuable in various fields including mathematics and logic, providing a foundational tool for an array of theoretical arguments and practical applications.
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