5) For each + ive integer i, where i is equal to three, then find the union and intersection of three set, Let (7) A₁ = {x = R| − } < x < }} = ( − }, }). a. Find A₁ U A₂ U A3 and A₁ A2 A3. b. Find UA; and A₁. i=1
5) For each + ive integer i, where i is equal to three, then find the union and intersection of three set, Let (7) A₁ = {x = R| − } < x < }} = ( − }, }). a. Find A₁ U A₂ U A3 and A₁ A2 A3. b. Find UA; and A₁. i=1
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 5: Finding Union and Intersection of Sets**
For each positive integer \( i \), where \( i \) is equal to three, we are to find the union and intersection of three sets. Let:
\[ A_i = \{ x \in \mathbb{R} \mid -\frac{1}{i} < x < \frac{1}{i} \} = \left( -\frac{1}{i}, \frac{1}{i} \right) \]
### Part (a)
Find \( A_1 \cup A_2 \cup A_3 \) and \( A_1 \cap A_2 \cap A_3 \).
### Part (b)
Find:
\[ \bigcup_{i=1}^{\infty} A_i \]
and
\[ \bigcap_{i=1}^{\infty} A_i \]
### Explanation
**General Form of the Sets \( A_i \)**:
Each set \( A_i \) is defined as an open interval centered at 0 with a radius of \(\frac{1}{i}\). This means as \( i \) increases, the interval \( \left( -\frac{1}{i}, \frac{1}{i} \right) \) becomes smaller and approaches 0.
**Union and Intersection**:
1. **Union of Sets**:
- The union \( A_1 \cup A_2 \cup A_3 \) entails finding the complete range of values covered by at least one of the sets.
- For the infinite union, \( \bigcup_{i=1}^{\infty} A_i \), consider the set of all values covered by any set \( A_i \) as \( i \) increases without bounds.
2. **Intersection of Sets**:
- The intersection \( A_1 \cap A_2 \cap A_3 \) involves finding the common values present in all three sets.
- The infinite intersection \( \bigcap_{i=1}^{\infty} A_i \) concerns the common values present in all sets \( A_i \) as \( i \to \infty \).
**Results**:
- For finite sets, each set \( A_i \) becomes narrower as \( i \) increases.
- For infinite union and intersection:
- \( \bigcup_{i=1}^{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4c11ea4-622e-4c72-b403-aa10850a497f%2Fa14d3412-b630-4cf5-82c3-fe0845f37fda%2Fm3w13fl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5: Finding Union and Intersection of Sets**
For each positive integer \( i \), where \( i \) is equal to three, we are to find the union and intersection of three sets. Let:
\[ A_i = \{ x \in \mathbb{R} \mid -\frac{1}{i} < x < \frac{1}{i} \} = \left( -\frac{1}{i}, \frac{1}{i} \right) \]
### Part (a)
Find \( A_1 \cup A_2 \cup A_3 \) and \( A_1 \cap A_2 \cap A_3 \).
### Part (b)
Find:
\[ \bigcup_{i=1}^{\infty} A_i \]
and
\[ \bigcap_{i=1}^{\infty} A_i \]
### Explanation
**General Form of the Sets \( A_i \)**:
Each set \( A_i \) is defined as an open interval centered at 0 with a radius of \(\frac{1}{i}\). This means as \( i \) increases, the interval \( \left( -\frac{1}{i}, \frac{1}{i} \right) \) becomes smaller and approaches 0.
**Union and Intersection**:
1. **Union of Sets**:
- The union \( A_1 \cup A_2 \cup A_3 \) entails finding the complete range of values covered by at least one of the sets.
- For the infinite union, \( \bigcup_{i=1}^{\infty} A_i \), consider the set of all values covered by any set \( A_i \) as \( i \) increases without bounds.
2. **Intersection of Sets**:
- The intersection \( A_1 \cap A_2 \cap A_3 \) involves finding the common values present in all three sets.
- The infinite intersection \( \bigcap_{i=1}^{\infty} A_i \) concerns the common values present in all sets \( A_i \) as \( i \to \infty \).
**Results**:
- For finite sets, each set \( A_i \) becomes narrower as \( i \) increases.
- For infinite union and intersection:
- \( \bigcup_{i=1}^{\
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