3. (b) Prove part (b) of Theorem 2.3.2: Let A be a nonempty family of sets and let B be a set. If A C B for all A EAnote: this is referred to as “script font" in the following subscript), then UAEA(script font) U A C B.
3. (b) Prove part (b) of Theorem 2.3.2: Let A be a nonempty family of sets and let B be a set. If A C B for all A EAnote: this is referred to as “script font" in the following subscript), then UAEA(script font) U A C B.
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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Please see image below
![### Problem Statement
**3. (b) Prove part (b) of Theorem 2.3.2:**
Let \( \mathcal{A} \) be a nonempty family of sets and let \( B \) be a set. If \( A \subseteq B \) for all \( A \in \mathcal{A} \) (note: this is referred to as "script font" in the following subscript), then
\[ \bigcup_{A \in \mathcal{A}} A \subseteq B. \]
### Explanation
In this problem, we are given a family of sets \( \mathcal{A} \) which is nonempty, and a set \( B \). The goal is to prove that if every set \( A \) within \( \mathcal{A} \) is a subset of \( B \), then the union of all sets in \( \mathcal{A} \) is also a subset of \( B \).
The family of sets \( \mathcal{A} \) is denoted using a script font to distinguish it from other types of sets or collections. The relationship \( \bigcup_{A \in \mathcal{A}} A \subseteq B \) signifies that every element in the union of the sets in \( \mathcal{A} \) is also an element of \( B \).
To summarize, the problem is asking us to show that:
1. Given \( \forall A \in \mathcal{A}, A \subseteq B \),
2. It follows that \( \bigcup_{A \in \mathcal{A}} A \subseteq B \).
### Visualization
In order to fully grasp the concept, visualize each set \( A \) in the family \( \mathcal{A} \) as a subset within the set \( B \). The union of all these sets \( A \) will therefore form a large set, which should also reside entirely within \( B \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9381f9ec-4907-481a-8789-2e44e2a64edb%2Fb5aadc88-18fa-4439-9a55-d7f3554e7314%2F16unqn8.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**3. (b) Prove part (b) of Theorem 2.3.2:**
Let \( \mathcal{A} \) be a nonempty family of sets and let \( B \) be a set. If \( A \subseteq B \) for all \( A \in \mathcal{A} \) (note: this is referred to as "script font" in the following subscript), then
\[ \bigcup_{A \in \mathcal{A}} A \subseteq B. \]
### Explanation
In this problem, we are given a family of sets \( \mathcal{A} \) which is nonempty, and a set \( B \). The goal is to prove that if every set \( A \) within \( \mathcal{A} \) is a subset of \( B \), then the union of all sets in \( \mathcal{A} \) is also a subset of \( B \).
The family of sets \( \mathcal{A} \) is denoted using a script font to distinguish it from other types of sets or collections. The relationship \( \bigcup_{A \in \mathcal{A}} A \subseteq B \) signifies that every element in the union of the sets in \( \mathcal{A} \) is also an element of \( B \).
To summarize, the problem is asking us to show that:
1. Given \( \forall A \in \mathcal{A}, A \subseteq B \),
2. It follows that \( \bigcup_{A \in \mathcal{A}} A \subseteq B \).
### Visualization
In order to fully grasp the concept, visualize each set \( A \) in the family \( \mathcal{A} \) as a subset within the set \( B \). The union of all these sets \( A \) will therefore form a large set, which should also reside entirely within \( B \).
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