The taxicab metric p on the plane R² is defined by the rule: p((x1, y1), (x2, Y2)) = |x1 – x2| + |y1 - y2| Verify that the taxicab metric satisfies the criteria for being a metric.
The taxicab metric p on the plane R² is defined by the rule: p((x1, y1), (x2, Y2)) = |x1 – x2| + |y1 - y2| Verify that the taxicab metric satisfies the criteria for being a metric.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Taxicab Metric on \( \mathbb{R}^2 \)
The taxicab metric \( \rho \) on the plane \( \mathbb{R}^2 \) is defined by the following rule:
\[ \rho \left( (x_1, y_1), (x_2, y_2) \right) = |x_1 - x_2| + |y_1 - y_2| \]
This formula calculates the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a grid-like path, much like how a taxicab would travel through city streets, making only horizontal and vertical moves.
To verify that the taxicab metric satisfies the criteria for being a metric, we need to check the following properties:
1. **Non-negativity (Positivity):**
\[ \rho \left( (x_1, y_1), (x_2, y_2) \right) \geq 0 \]
2. **Identity of Indiscernibles:**
\[ \rho \left( (x_1, y_1), (x_2, y_2) \right) = 0 \iff (x_1, y_1) = (x_2, y_2) \]
3. **Symmetry:**
\[ \rho \left( (x_1, y_1), (x_2, y_2) \right) = \rho \left( (x_2, y_2), (x_1, y_1) \right) \]
4. **Triangle Inequality:**
\[ \rho \left( (x_1, y_1), (x_3, y_3) \right) \leq \rho \left( (x_1, y_1), (x_2, y_2) \right) + \rho \left( (x_2, y_2), (x_3, y_3) \right) \]
By ensuring that \( \rho \) meets these criteria, we can confirm that the taxicab metric is indeed a valid metric on \( \mathbb{R}^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb449e86c-4325-4629-81be-ddccdcc6fdc4%2F1e0b6d00-1626-403f-811c-6ba7c0507bd1%2F7l8kinc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Taxicab Metric on \( \mathbb{R}^2 \)
The taxicab metric \( \rho \) on the plane \( \mathbb{R}^2 \) is defined by the following rule:
\[ \rho \left( (x_1, y_1), (x_2, y_2) \right) = |x_1 - x_2| + |y_1 - y_2| \]
This formula calculates the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a grid-like path, much like how a taxicab would travel through city streets, making only horizontal and vertical moves.
To verify that the taxicab metric satisfies the criteria for being a metric, we need to check the following properties:
1. **Non-negativity (Positivity):**
\[ \rho \left( (x_1, y_1), (x_2, y_2) \right) \geq 0 \]
2. **Identity of Indiscernibles:**
\[ \rho \left( (x_1, y_1), (x_2, y_2) \right) = 0 \iff (x_1, y_1) = (x_2, y_2) \]
3. **Symmetry:**
\[ \rho \left( (x_1, y_1), (x_2, y_2) \right) = \rho \left( (x_2, y_2), (x_1, y_1) \right) \]
4. **Triangle Inequality:**
\[ \rho \left( (x_1, y_1), (x_3, y_3) \right) \leq \rho \left( (x_1, y_1), (x_2, y_2) \right) + \rho \left( (x_2, y_2), (x_3, y_3) \right) \]
By ensuring that \( \rho \) meets these criteria, we can confirm that the taxicab metric is indeed a valid metric on \( \mathbb{R}^2 \).
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