Give an function which is semi metric space

A **semi-metric space** is a generalization of a metric space where some of the conditions of a metric are relaxed. Specifically, a semi-metric only requires the **non-negativity**, **identity of indiscernibles**, and **triangle inequality**, but not necessarily **symmetry**. This means that the distance between two points can be different depending on the order.

Here’s an example of a **semi-metric**:

Define \( d : \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) by:

\[
d(x, y) = |x - y|^2
\]

### Reason why this is a semi-metric but not a metric:
1. **Non-negativity**: \( d(x, y) = |x - y|^2 \geq 0 \) for all \( x, y \), so it satisfies non-negativity.
2. **Identity of indiscernibles**: If \( x = y \), then \( d(x, x) = |x - x|^2 = 0 \). If \( x \neq y \), \( d(x, y) > 0 \), so it satisfies the identity of indiscernibles.
3. **Triangle inequality**: This can be proven since \( |x - y|^2 \) still respects the triangle inequality property.
4. **Symmetry**: However, symmetry is not required in a semi-metric, but in this case, it holds true for \( d(x, y) = |x - y|^2 \). 

### A less trivial example without symmetry:
Define another function \( d : \mathbb{R} \times \mathbb{R} \to \mathbb{R} \):

\[
d(x, y) = x^2 - y^2 \text{ for } x \geq y \quad \text{and} \quad d(x, y) = 0 \text{ for } x < y
\]

This is a semi-metric because:
1. **Non-negativity**: \( d(x, y) \geq 0 \) for all \( x \geq y \), as \( x^2 - y^2 \geq 0 \), and \( d(x, y) = 0 \) for \( x < y \).
2. **Identity of indiscernibles**: \( d(x, x) = 0 \), and \( d(x, y) > 0 \) if \( x > y \), so it satisfies this condition.
3. **Triangle inequality**: This can be verified for specific cases.
4. **Symmetry**: Symmetry is not satisfied because \( d(x, y) \neq d(y, x) \) in general.

Thus, this is a semi-metric space, but not a metric.

Would you like more details or have any questions?

Here are 5 related questions:
1. What are the exact differences between a semi-metric space and a metric space?
2. Can a semi-metric space still be used in practical applications like a metric space?
3. How can we prove the triangle inequality in a semi-metric space?
4. What other generalizations of metric spaces exist besides semi-metrics?
5. How does the relaxation of symmetry affect the properties of the space?

**Tip:** In a semi-metric, distances can depend on the order of points, so always test the symmetry condition to distinguish it from a metric space!

Explore semi-metric spaces, a generalization of metric spaces where symmetry is not necessarily required. Learn about key concepts and examples like d(x, y) = |x - y|^2 and d(x, y) = x^2 - y^2. Discover the implications of relaxation in symmetry and practical applications.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation