for each positive integer i. Find each of the following. (Enter your answers using interval notation.) (a) 4 = i = 1 (b) 4 i = 1 (c) Are V,, V21 V31 mutually disjoint? Why or why not? O Yes, because the union of the sets V,, V,, V3 ... is empty. O Yes, because no two of the sets V,, V,, V3, have any elements in common. O Yes, because the intersection of the sets V,, V,, V3, ... is empty. O No, because no two of the sets V,, V31 ... are disjoint. O No, because the sets V,, V2, V3. ... are disjoint.
for each positive integer i. Find each of the following. (Enter your answers using interval notation.) (a) 4 = i = 1 (b) 4 i = 1 (c) Are V,, V21 V31 mutually disjoint? Why or why not? O Yes, because the union of the sets V,, V,, V3 ... is empty. O Yes, because no two of the sets V,, V,, V3, have any elements in common. O Yes, because the intersection of the sets V,, V,, V3, ... is empty. O No, because no two of the sets V,, V31 ... are disjoint. O No, because the sets V,, V2, V3. ... are disjoint.
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
![**Concept of Sets and Intervals in Mathematics**
The image presents a mathematical problem involving sets and intervals. Let's explore the components step by step for educational purposes.
**Problem Statement:**
We define a set \( V_i \) for each positive integer \( i \) as follows:
\[ V_i = \left\{ x \in \mathbb{R} \mid -\frac{1}{i} \le x \le \frac{1}{i} \right\} = \left[ -\frac{1}{i}, \frac{1}{i} \right] \]
**Tasks:**
(a) **Union of Sets \( V_i \) from \( i = 1 \) to 4:**
\[
\bigcup_{i=1}^{4} V_i =
\]
(b) **Intersection of Sets \( V_i \) from \( i = 1 \) to 4:**
\[
\bigcap_{i=1}^{4} V_i =
\]
(c) **Mutual Disjointness of Sets \( V_1, V_2, V_3, \ldots \):**
The options provided explore whether the sets are mutually disjoint or not:
- Yes, because the union of the sets \( V_1, V_2, V_3, \ldots \) is empty.
- Yes, because no two of the sets \( V_1, V_2, V_3, \ldots \) have any elements in common.
- Yes, because the intersection of the sets \( V_1, V_2, V_3, \ldots \) is empty.
- **No, because no two of the sets \( V_1, V_2, V_3, \ldots \) are disjoint.** (Selected answer)
- No, because the sets \( V_1, V_2, V_3, \ldots \) are disjoint.
(d) **Union of Sets \( V_i \) from \( i = 1 \) to \( n \):**
\[
\bigcup_{i=1}^{n} V_i =
\]
(e) **Intersection of Sets \( V_i \) from \( i = 1 \) to \( n \):**
\[
\bigcap_{i=1}^{n} V_i =
\]
(f](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd16c46a0-d8b3-4c49-8249-7270f2b9ceb6%2Fe5cc0af6-c661-4f60-a729-c5e5a3505078%2Fwvve8ij_processed.png&w=3840&q=75)
Transcribed Image Text:**Concept of Sets and Intervals in Mathematics**
The image presents a mathematical problem involving sets and intervals. Let's explore the components step by step for educational purposes.
**Problem Statement:**
We define a set \( V_i \) for each positive integer \( i \) as follows:
\[ V_i = \left\{ x \in \mathbb{R} \mid -\frac{1}{i} \le x \le \frac{1}{i} \right\} = \left[ -\frac{1}{i}, \frac{1}{i} \right] \]
**Tasks:**
(a) **Union of Sets \( V_i \) from \( i = 1 \) to 4:**
\[
\bigcup_{i=1}^{4} V_i =
\]
(b) **Intersection of Sets \( V_i \) from \( i = 1 \) to 4:**
\[
\bigcap_{i=1}^{4} V_i =
\]
(c) **Mutual Disjointness of Sets \( V_1, V_2, V_3, \ldots \):**
The options provided explore whether the sets are mutually disjoint or not:
- Yes, because the union of the sets \( V_1, V_2, V_3, \ldots \) is empty.
- Yes, because no two of the sets \( V_1, V_2, V_3, \ldots \) have any elements in common.
- Yes, because the intersection of the sets \( V_1, V_2, V_3, \ldots \) is empty.
- **No, because no two of the sets \( V_1, V_2, V_3, \ldots \) are disjoint.** (Selected answer)
- No, because the sets \( V_1, V_2, V_3, \ldots \) are disjoint.
(d) **Union of Sets \( V_i \) from \( i = 1 \) to \( n \):**
\[
\bigcup_{i=1}^{n} V_i =
\]
(e) **Intersection of Sets \( V_i \) from \( i = 1 \) to \( n \):**
\[
\bigcap_{i=1}^{n} V_i =
\]
(f
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