**Transcription for Educational Website** --- Let \[ \{C_r : r \in [0, \infty)\} \] be the family of indexed sets given by \[ C_r = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = r^2\} \] **a) Describe each of the sets \(C_r\) geometrically.** Each set \(C_r\) represents a circle in the Euclidean plane \(\mathbb{R}^2\) with radius \(r\) and centered at the origin (0,0). The equation \(x^2 + y^2 = r^2\) describes all the points \((x, y)\) that are at a distance \(r\) from the origin. **b) Prove:** \[ \bigcup_{r \in [0, \infty)} C_r = \mathbb{R}^2 \] The union of all sets \(C_r\) as \(r\) ranges from 0 to infinity is equal to the entire Euclidean plane \(\mathbb{R}^2\). Essentially, this means that any point in \(\mathbb{R}^2\) can be included within at least one circle centered at the origin with a sufficiently large radius. --- This transcription provides a mathematical description and visual understanding of circles within an infinite union covering the entire plane.
**Transcription for Educational Website** --- Let \[ \{C_r : r \in [0, \infty)\} \] be the family of indexed sets given by \[ C_r = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = r^2\} \] **a) Describe each of the sets \(C_r\) geometrically.** Each set \(C_r\) represents a circle in the Euclidean plane \(\mathbb{R}^2\) with radius \(r\) and centered at the origin (0,0). The equation \(x^2 + y^2 = r^2\) describes all the points \((x, y)\) that are at a distance \(r\) from the origin. **b) Prove:** \[ \bigcup_{r \in [0, \infty)} C_r = \mathbb{R}^2 \] The union of all sets \(C_r\) as \(r\) ranges from 0 to infinity is equal to the entire Euclidean plane \(\mathbb{R}^2\). Essentially, this means that any point in \(\mathbb{R}^2\) can be included within at least one circle centered at the origin with a sufficiently large radius. --- This transcription provides a mathematical description and visual understanding of circles within an infinite union covering the entire plane.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Transcription for Educational Website**
---
Let
\[
\{C_r : r \in [0, \infty)\}
\]
be the family of indexed sets given by
\[
C_r = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = r^2\}
\]
**a) Describe each of the sets \(C_r\) geometrically.**
Each set \(C_r\) represents a circle in the Euclidean plane \(\mathbb{R}^2\) with radius \(r\) and centered at the origin (0,0). The equation \(x^2 + y^2 = r^2\) describes all the points \((x, y)\) that are at a distance \(r\) from the origin.
**b) Prove:**
\[
\bigcup_{r \in [0, \infty)} C_r = \mathbb{R}^2
\]
The union of all sets \(C_r\) as \(r\) ranges from 0 to infinity is equal to the entire Euclidean plane \(\mathbb{R}^2\). Essentially, this means that any point in \(\mathbb{R}^2\) can be included within at least one circle centered at the origin with a sufficiently large radius.
---
This transcription provides a mathematical description and visual understanding of circles within an infinite union covering the entire plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c99a240-9de7-4947-9b54-53ff2cf6c85e%2Fb8ea9d2a-a287-46b9-96cb-fb946254bd12%2Fqa7k9o0m_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
---
Let
\[
\{C_r : r \in [0, \infty)\}
\]
be the family of indexed sets given by
\[
C_r = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = r^2\}
\]
**a) Describe each of the sets \(C_r\) geometrically.**
Each set \(C_r\) represents a circle in the Euclidean plane \(\mathbb{R}^2\) with radius \(r\) and centered at the origin (0,0). The equation \(x^2 + y^2 = r^2\) describes all the points \((x, y)\) that are at a distance \(r\) from the origin.
**b) Prove:**
\[
\bigcup_{r \in [0, \infty)} C_r = \mathbb{R}^2
\]
The union of all sets \(C_r\) as \(r\) ranges from 0 to infinity is equal to the entire Euclidean plane \(\mathbb{R}^2\). Essentially, this means that any point in \(\mathbb{R}^2\) can be included within at least one circle centered at the origin with a sufficiently large radius.
---
This transcription provides a mathematical description and visual understanding of circles within an infinite union covering the entire plane.
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