For each natural number n ≥ 4, let An = [/⁄/2, 1 + ½ ] and A = {An, n ≥ 4}.
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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![For each natural number \( n \geq 4 \), let \( A_n = \left[ \frac{1}{n}, 1 + \frac{1}{n} \right] \) and \( \mathcal{A} = \{A_n, n \geq 4\} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd49a5070-9d99-4a2f-a144-be47e7ddead5%2F95afa447-136e-4eba-bfef-bbb146493f1d%2F0v0ge9a_processed.png&w=3840&q=75)
Transcribed Image Text:For each natural number \( n \geq 4 \), let \( A_n = \left[ \frac{1}{n}, 1 + \frac{1}{n} \right] \) and \( \mathcal{A} = \{A_n, n \geq 4\} \).
![### Example Problem: Union and Intersection of Indexed Families of Sets
**Problem Statement:**
Find the union and intersection of the following indexed family of collections.
---
In this problem, we are given an indexed family of collections and we are asked to find both the union and intersection of these collections.
### Definitions:
- **Union (\(\bigcup\)):** The union of a family of sets is the set of all elements that belong to at least one set in the family.
- **Intersection (\(\bigcap\)):** The intersection of a family of sets is the set of all elements that are common to all sets in the family.
### Explanation:
To determine the union and intersection, you would need the specific sets that are part of the indexed family. However, the general approach would be:
#### Union:
\[ \bigcup_{i \in I} A_i \]
Where \( A_i \) represents each set in the family, and \( I \) represents the index set.
#### Intersection:
\[ \bigcap_{i \in I} A_i \]
Where \( A_i \) again represents each set in the family, and \( I \) is the index set.
---
**Graphical Representation (Not Provided):**
If there were a Venn diagram or similar graphical representation of the sets:
1. **Union of Sets**: This would be represented by shading all areas covered by any of the sets involved.
2. **Intersection of Sets**: This would be represented by shading only the area that is common to all sets.
By understanding the definitions and approach outlined above, you can apply these to any specific sets provided to find their union and intersection.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd49a5070-9d99-4a2f-a144-be47e7ddead5%2F95afa447-136e-4eba-bfef-bbb146493f1d%2Fwmozemhg_processed.png&w=3840&q=75)
Transcribed Image Text:### Example Problem: Union and Intersection of Indexed Families of Sets
**Problem Statement:**
Find the union and intersection of the following indexed family of collections.
---
In this problem, we are given an indexed family of collections and we are asked to find both the union and intersection of these collections.
### Definitions:
- **Union (\(\bigcup\)):** The union of a family of sets is the set of all elements that belong to at least one set in the family.
- **Intersection (\(\bigcap\)):** The intersection of a family of sets is the set of all elements that are common to all sets in the family.
### Explanation:
To determine the union and intersection, you would need the specific sets that are part of the indexed family. However, the general approach would be:
#### Union:
\[ \bigcup_{i \in I} A_i \]
Where \( A_i \) represents each set in the family, and \( I \) represents the index set.
#### Intersection:
\[ \bigcap_{i \in I} A_i \]
Where \( A_i \) again represents each set in the family, and \( I \) is the index set.
---
**Graphical Representation (Not Provided):**
If there were a Venn diagram or similar graphical representation of the sets:
1. **Union of Sets**: This would be represented by shading all areas covered by any of the sets involved.
2. **Intersection of Sets**: This would be represented by shading only the area that is common to all sets.
By understanding the definitions and approach outlined above, you can apply these to any specific sets provided to find their union and intersection.
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