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(b) Prove that Cl(AU B) = ClA U CIB.

(b) Prove that Cl(AU B) = ClA U CIB.

Advanced Engineering Mathematics
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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### Problem Set #### Topological Properties of Sets Consider the following exercises related to topology and set theory: (a) **Prove that** \( \text{Int}(A \cap B) = \text{Int}A \cap \text{Int}B \). (b) **Prove that** \( \text{Cl}(A \cup B) = \text{Cl}A \cup \text{Cl}B \). ### Explanation: - **\( \text{Int}(A \cap B) \)** refers to the interior of the intersection of sets \( A \) and \( B \). - **\( \text{Int}A \cap \text{Int}B \)** signifies the intersection of the interiors of sets \( A \) and \( B \). - **\( \text{Cl}(A \cup B) \)** denotes the closure of the union of sets \( A \) and \( B \). - **\( \text{Cl}A \cup \text{Cl}B \)** refers to the union of the closures of sets \( A \) and \( B \). ### Objective: The goal is to demonstrate the equality of these expressions by using fundamental topological principles. ### Approach: 1. **Use Definitions**: - For interiors, utilize the concept of open sets contained within a given set. - For closures, use the notion of the smallest closed set containing the given set. 2. **Set Theoretic Techniques**: - Apply properties of set operations such as unions, intersections, and distributiveness. 3. **Logical Reasoning**: - Use direct proofs, contrapositives, or contradictions where necessary to establish the desired equalities. These exercises provide insight into how the operations of intersection and union interact with topological notions like interior and closure.
Transcribed Image Text:### Problem Set #### Topological Properties of Sets Consider the following exercises related to topology and set theory: (a) **Prove that** \( \text{Int}(A \cap B) = \text{Int}A \cap \text{Int}B \). (b) **Prove that** \( \text{Cl}(A \cup B) = \text{Cl}A \cup \text{Cl}B \). ### Explanation: - **\( \text{Int}(A \cap B) \)** refers to the interior of the intersection of sets \( A \) and \( B \). - **\( \text{Int}A \cap \text{Int}B \)** signifies the intersection of the interiors of sets \( A \) and \( B \). - **\( \text{Cl}(A \cup B) \)** denotes the closure of the union of sets \( A \) and \( B \). - **\( \text{Cl}A \cup \text{Cl}B \)** refers to the union of the closures of sets \( A \) and \( B \). ### Objective: The goal is to demonstrate the equality of these expressions by using fundamental topological principles. ### Approach: 1. **Use Definitions**: - For interiors, utilize the concept of open sets contained within a given set. - For closures, use the notion of the smallest closed set containing the given set. 2. **Set Theoretic Techniques**: - Apply properties of set operations such as unions, intersections, and distributiveness. 3. **Logical Reasoning**: - Use direct proofs, contrapositives, or contradictions where necessary to establish the desired equalities. These exercises provide insight into how the operations of intersection and union interact with topological notions like interior and closure.
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