Question

Given a metric space \((X, p)\), prove that a) \(|\rho(x, z)-\rho(y, u)| \leqslant \rho(x, y)+\rho(z, u) \quad(x, y, z, u \in X)\); b) \(|\rho(x, z)-\rho(y, z)| \leqslant \rho(x, y) \quad(x, y, z \in X)\).

Short answer

Both inequalities follow from the triangle inequality of the metric space.

Step-by-step solution

Understand the Triangle Inequality Property

In a metric space \((X, \rho)\), the function \(\rho\) (the metric) satisfies the triangle inequality: \(\rho(x, z) \leq \rho(x, y) + \rho(y, z)\). This inequality is fundamental and will be repeatedly used in proving both parts of the exercise.

Begin with Part (a)

To prove \(|\rho(x, z) - \rho(y, u)| \leq \rho(x, y) + \rho(z, u)\), start by considering \(\rho(x, z)\) and \(\rho(y, u)\) separately. By the triangle inequality, we have \(\rho(x, z) \leq \rho(x, y) + \rho(y, z)\) and \(\rho(y, u) \leq \rho(y, z) + \rho(z, u)\). Use these inequalities to bound \(\rho(x, z) - \rho(y, u)\).

Prove Part (a)

First derive \(\rho(x, z) - \rho(y, u) \leq \rho(x, y) + \rho(z, u)\) by using previously mentioned triangle inequalities: \(\rho(x, z) \leq \rho(x, y) + \rho(y, z)\) and \(-\rho(y, u) \leq \rho(y, z) + \rho(z, u)\). Combine these to show that subtracting \(\rho(y, z)\) cancels out and leaves \(\rho(x, z) - \rho(y, u) \leq \rho(x, y) + \rho(z, u)\). Similarly, consider \(\rho(y, u) - \rho(x,z) \, \text{and}\, \rho(x, u) \) and apply the same method to complete the full \(|\rho(x, z) - \rho(y, u)| \leq \rho(x, y) + \rho(z, u)\) inequality.

Relay to Part (b)

For part (b), the goal is to prove \(|\rho(x, z) - \rho(y, z)| \leq \rho(x, y)\). Use the triangle inequality \(\rho(x, z) \leq \rho(x, y) + \rho(y, z)\). Begin by showing \(\rho(x, z) - \rho(y, z) \leq \rho(x, y)\).

Prove Part (b)

To tackle \(\rho(x, z) - \rho(y, z) \leq \rho(x, y)\), notice from the triangle inequality \(\rho(x, y) + \rho(y, z) \leq \rho(x, z)\) implies that subtracting \(\rho(y, z)\) from both sides gives \(\rho(x, z) - \rho(y, z) \leq \rho(x, y)\). For the other side, \(\rho(y, z) - \rho(x, z) \leq \rho(x, y)\) is similarly shown, hence completing \(|\rho(x, z) - \rho(y, z)| \leq \rho(x, y)\) by considering absolute differences.

Key concepts

Triangle Inequality

The triangle inequality is a crucial concept in the study of metric spaces. It states that for any three points, say \(x\), \(y\), and \(z\) in a metric space, the direct path from \(x\) to \(z\) must be shorter or the same length as going through a different point \(y\). Mathematically, it's expressed as \( \rho(x, z) \leq \rho(x, y) + \rho(y, z) \). This means that the direct distance from \(x\) to \(z\) cannot exceed the sum of the distances if you detour over \(y\).

This property is intuitive when you think about physical space: taking the shortest route is always more efficient than adding extra steps. In our original exercise, the triangle inequality helps us simplify and ascertain the validity of inequalities involving multiple metric distances. It acts as a tool that restricts the possible relationships between distances in metric spaces, allowing us to derive more complex inequalities efficiently.

Metric

In mathematics, a metric is a rule or a function that defines the distance between two elements in a set, forming a metric space. This set and rule combined enable us to precisely talk about geometry and distances in potentially abstract spaces beyond simple physical distances. For instance, in a metric space \((X, \rho)\), \(X\) is a set and \(\rho\) is the metric that assigns a non-negative real number to each pair of points, ensuring it satisfies certain criteria like non-negativity, symmetry, and the triangle inequality.

• **Non-negativity:** \(\rho(x, y) \geq 0\) for all \(x, y \in X\).
• **Symmetry:** \(\rho(x, y) = \rho(y, x)\) for all \(x, y \in X\).
• **Triangle Inequality:** For all \(x, y, z \in X\), \(\rho(x, z) \leq \rho(x, y) + \rho(y, z)\).

These properties ensure the metric provides a reliable framework for analyzing distances mathematically. In your exercise, leveraging these properties, especially the triangle inequality, allows for the verification of more complex assertions about distances in \(X\). Understanding metrics turn abstract spaces into navigable domains.

Absolute Difference

The concept of the absolute difference is foundational in analyzing and comparing distances or quantities. It refers to the non-negative "pure" distance between two values, calculated as \(|x - y|\). Unlike signed differences, the absolute difference disregards direction and focuses only on magnitude, which makes it especially useful in proofs and distance calculations in metric spaces.

In the context of your initial exercise, dealing with expressions such as \(|\rho(x, z) - \rho(y, u)|\), the main task is to assess how close these metric values are to each other. By employing absolute differences, we not only handle potential negative outcomes but also simplify the complexities involved in evaluating how one distance relates to another. Specifically, these absolute differences allow us to apply the triangle inequality by converting potentially intricate expressions into simpler forms that are easier to reason about. This makes them essential tools in both proving and understanding key inequalities within metric spaces.