Question

Prove that if \(\left(X, \rho^{\prime}\right)\) and \(\left(Y, \rho^{\prime \prime}\right)\) are metric spaces, then a metric \(\rho\) for the set \(X \times Y\) is obtained by setting, for \(x_{1}, x_{2} \in X\) and \(y_{1}, y_{2} \in Y\), (i) \(\rho\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=\max \left\\{\rho^{\prime}\left(x_{1}, x_{2}\right), \rho^{\prime \prime}\left(y_{1}, y_{2}\right)\right\\}\); or (ii) \(\rho\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=\sqrt{\rho^{\prime}\left(x_{1}, x_{2}\right)^{2}+\rho^{\prime \prime}\left(y_{1}, y_{2}\right)^{2}}\)

Short answer

Both equations define valid metrics on the product space.

Step-by-step solution

Understanding the Problem

We're tasked with proving that two specified equations define valid metrics on the Cartesian product of two metric spaces, \((X, \rho')\) and \((Y, \rho'')\). A metric must satisfy the properties: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.

Non-negativity and Identity of Indiscernibles for Equation (i)

For equation (i), \(\rho((x_1, y_1), (x_2, y_2)) = \max\{\rho'(x_1, x_2), \rho''(y_1, y_2)\}\): Non-negativity is ensured since the maximum of two non-negative values is non-negative. Identity of indiscernibles holds because if \(\rho((x_1, y_1), (x_2, y_2)) = 0\), then both \(\rho'(x_1, x_2)\) and \(\rho''(y_1, y_2)\) must be 0, implying \(x_1 = x_2\) and \(y_1 = y_2\).

Symmetry and Triangle Inequality for Equation (i)

Symmetry holds because the maximum function is symmetric. Triangle inequality is shown as follows: \(\rho'(x_1, x_3) \leq \rho'(x_1, x_2) + \rho'(x_2, x_3)\) and \(\rho''(y_1, y_3) \leq \rho''(y_1, y_2) + \rho''(y_2, y_3)\), hence the maximum value will satisfy: \(\max\{\rho'(x_1, x_3), \rho''(y_1, y_3)\} \leq \max\{\rho'(x_1, x_2), \rho''(y_1, y_2)\} + \max\{\rho'(x_2, x_3), \rho''(y_2, y_3)\}\).

Non-negativity and Identity of Indiscernibles for Equation (ii)

For equation (ii), \(\rho((x_1, y_1), (x_2, y_2)) = \sqrt{\rho'(x_1, x_2)^2 + \rho''(y_1, y_2)^2}\): Non-negativity holds since the square root of a sum of non-negative numbers is non-negative. If the metric is zero, then both squares must be zero, implying \(x_1 = x_2\) and \(y_1 = y_2\).

Symmetry and Triangle Inequality for Equation (ii)

Symmetry is evident because both \(\rho'(x_1, x_2)\) and \(\rho''(y_1, y_2)\) are symmetric. For triangle inequality, use Minkowski's inequality for \(L^2\) norm: \ \sqrt{\rho'(x_1, x_3)^2 + \rho''(y_1, y_3)^2} \leq \sqrt{(\rho'(x_1, x_2)+\rho'(x_2, x_3))^2 + (\rho''(y_1, y_2)+\rho''(y_2, y_3))^2} \ which holds because both terms respect individual triangle inequalities.

Key concepts

Cartesian Product

The Cartesian product is a foundational operation in mathematics, especially important when discussing metric spaces. The Cartesian product of two sets, say set \(X\) and set \(Y\), is denoted by \(X \times Y\) and refers to the set of all possible ordered pairs \((x, y)\) where \(x\) is an element of \(X\) and \(y\) is an element of \(Y\). This is significant because it allows us to construct new spaces from existing ones, enhancing our ability to analyze relationships between complex structures.
In metric spaces, the Cartesian product is particularly useful because it allows us to define a new metric space on \(X \times Y\) by using the metrics on \(X\) and \(Y\). This metric characterizes how 'far apart' two pairs \((x_1, y_1)\) and \((x_2, y_2)\) are in the combined space. Ultimately, understanding Cartesian products enables us to extend familiar concepts in a standardized way across different mathematical contexts.

Minkowski's Inequality

Minkowski's inequality is a generalization of the triangle inequality and plays a crucial role in proving the properties of metrics especially when dealing with the Euclidean space or spaces like it. When we talk about the Minkowski inequality, we are often dealing with the \(L^p\) spaces.
For our specific purposes in examining metric spaces, Minkowski's inequality helps to establish the triangle inequality for certain norms. It states, for instance, that for sequences \(x\) and \(y\) in a finite-dimensional real or complex vector space:

• \(\|x + y\|_p \leq \|x\|_p + \|y\|_p\)

Here, \(\|\cdot\|_p\) denotes the \(p\)-norm. In the context of the Euclidean distance \((p=2)\), this inequality helps in proving that the sum of the distances cannot exceed the distance obtained by treating them independently and then combining. It underpins much of the analysis required in metric spaces to see that when we have two sets, the distance metric obtained abides by the metric properties.

Triangle Inequality

The triangle inequality is one of the key properties that any metric metric must satisfy. It describes how the distance between two points through an intermediate point will never be greater than the direct distance. In a mathematical form, if \(x\), \(y\), and \(z\) are points in a metric space with metric \(\rho\), this inequality is expressed as:

• \(\rho(x, z) \leq \rho(x, y) + \rho(y, z)\)

This principle is not just theoretical; it is intuitive and apparent in real-world scenarios. For example, in a literal sense, if you travel from your home (x) to a store (y) and then to a friend's house (z), the total distance you travel will always be at least as long as the straight line distance from your home to the friend's house. This property ensures that all points in a metric space are logically and internally consistent regarding their distances.

Identity of Indiscernibles

The identity of indiscernibles is another fundamental property of metric spaces, asserting that two points are considered the same if the distance between them is zero. Formally, a metric \(\rho\) on a set \(X\) satisfies the identity of indiscernibles if for any \(x, y \in X\):

• \(\rho(x, y) = 0 \text{ if and only if } x = y\)

This axiom is important because it ensures that our definition of 'distance' is meaningful. For if two distinct points could have a distance of zero, it would undermine the concept of distance itself. Hence, this property delineates distinct elements sharply within the metric space. This is why during the formation of metrics for Cartesian products, ensuring this property holds establishes that the combined space diligently observes that only identical pairs seem 'indiscernibly' close.