- - 1. For any x, y € R², define d₁ (x, y) = max{|x1 — Y₁|, |x2 — Y2|}. Prove that d₁ is a metrics on R². 2. For any x, y ER², define d₂ (x, y) = |x1 − Y1| + |x2 − y2|. Prove that d2 is a metrics on R².
- - 1. For any x, y € R², define d₁ (x, y) = max{|x1 — Y₁|, |x2 — Y2|}. Prove that d₁ is a metrics on R². 2. For any x, y ER², define d₂ (x, y) = |x1 − Y1| + |x2 − y2|. Prove that d2 is a metrics on R².
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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![## Metrics on \( \mathbb{R}^2 \)
1. **Metric Definition Using Maximum Function**
For any \( x, y \in \mathbb{R}^2 \), define the distance function \( d_1(x, y) \) as follows:
\[
d_1(x, y) = \max\{|x_1 - y_1|, |x_2 - y_2|\}
\]
**Objective:** Prove that \( d_1 \) is a metric on \( \mathbb{R}^2 \).
2. **Metric Definition Using Absolute Values**
For any \( x, y \in \mathbb{R}^2 \), define the distance function \( d_2(x, y) \) as follows:
\[
d_2(x, y) = |x_1 - y_1| + |x_2 - y_2|
\]
**Objective:** Prove that \( d_2 \) is a metric on \( \mathbb{R}^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d5e88e6-b1af-4aea-9b08-2dadd85f5e2c%2Fbaa51e82-1354-46e5-be85-324a6d33565f%2Frl7icrm_processed.png&w=3840&q=75)
Transcribed Image Text:## Metrics on \( \mathbb{R}^2 \)
1. **Metric Definition Using Maximum Function**
For any \( x, y \in \mathbb{R}^2 \), define the distance function \( d_1(x, y) \) as follows:
\[
d_1(x, y) = \max\{|x_1 - y_1|, |x_2 - y_2|\}
\]
**Objective:** Prove that \( d_1 \) is a metric on \( \mathbb{R}^2 \).
2. **Metric Definition Using Absolute Values**
For any \( x, y \in \mathbb{R}^2 \), define the distance function \( d_2(x, y) \) as follows:
\[
d_2(x, y) = |x_1 - y_1| + |x_2 - y_2|
\]
**Objective:** Prove that \( d_2 \) is a metric on \( \mathbb{R}^2 \).
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