Question

(a) Suppose that \((A, \rho)\) is a metric space, and define $$ \rho_{1}(u, v)=\frac{\rho(u, v)}{1+\rho(u, v)} $$ Show that \(\left(A, \rho_{1}\right)\) is a metric space. (b) Show that infinitely many metrics can be defined on any set \(A\) with more than one member.

Short answer

In summary, the given distance function \(\rho_1(u, v) = \frac{\rho(u, v)}{1+\rho(u, v)}\) defines a valid metric on the set A, since it satisfies all the required metric properties, and we can define infinitely many metrics on any set A with more than one element using suitable functions. This demonstrates the flexibility in defining various metrics on a given set, based on specific requirements and applications.

Step-by-step solution

Part (a): Showing \((A, \rho_1)\) is a metric space

Since \((A, \rho)\) is a metric space, the distance function \(\rho\) satisfies the four properties of a metric for all \(u, v, w \in A\). Now, we will show \(\rho_1\) satisfies these properties as well: 1. Non-negativity: Since \(\rho(u, v) \ge 0\), it follows that: $$ \rho_1(u, v)=\frac{\rho(u, v)}{1+\rho(u, v)} \ge 0 $$ 2. Identity of indiscernibles: If \(u = v\), then \(\rho(u, v) = 0\). Therefore, \(\rho_1(u, v) = \frac{\rho(u, v)}{1+\rho(u, v)} = 0\). Conversely, if \(\rho_1(u, v) = 0\), we have: $$ 0 = \frac{\rho(u, v)}{1+\rho(u, v)} \Rightarrow \rho(u, v) = 0 $$ This implies that \(u = v\). 3. Symmetry: Since \(\rho(u, v) = \rho(v, u)\), it follows that: $$ \rho_1(u, v)=\frac{\rho(u, v)}{1+\rho(u, v)} = \frac{\rho(v, u)}{1+\rho(v, u)} = \rho_1(v, u) $$ 4. Triangle inequality: We need to show that: $$ \rho_1(u, v) + \rho_1(v, w) \ge \rho_1(u, w) $$ Let \(a = \rho(u, v)\), \(b = \rho(v, w)\), and \(c = \rho(u, w)\). Since \((A, \rho)\) is a metric space, we have, by the triangle inequality: $$ a + b \ge c \Rightarrow 1 - \frac{1}{a+1} + 1 - \frac{1}{b+1} \ge 1 - \frac{1}{c+1} $$ Now, we want to show that: $$ \frac{a}{a+1} + \frac{b}{b+1} \ge \frac{c}{c+1} $$ This inequality is equivalent to the previous one. Thus, the triangle inequality is satisfied for \(\rho_1\). Therefore, \((A, \rho_1)\) is a metric space.

Part (b): Infinitely many metrics on set A

Let's consider the set \(A\) with more than one element. We can define infinitely many metrics on \(A\) using the following approach: Given any non-negative function \(f:\mathbb{R}\rightarrow\mathbb{R}\) with \(f(0)=0\), and continuous with increasing behavior, we can define a new metric \(d_f(u, v)=f(\rho(u, v))\). If we choose infinitely many such functions \(f_i\), we will have infinitely many metrics \(d_{f_i}(u, v)\) on the set \(A\). For example, we can consider the family of functions \(f_k(x)=\frac{x^k}{1+x^k}\), where \(k\in\mathbb{N}\). Each \(f_k\) is non-negative, has \(f_k(0)=0\), is continuous, and increasing, so we can use them to define infinitely many metrics \(d_{f_k}(u, v)\) on the set \(A\). Thus, we have demonstrated that infinitely many metrics can be defined on any set \(A\) with more than one element.

Key concepts

Distance Function

Understanding the distance function in a metric space is foundational for comprehending the nature of the space itself. A distance function, often represented as \( \rho(u, v) \) for elements \( u \) and \( v \) within the set \( A \), provides a way to measure the 'distance' between these two points. This measurement isn't just a physical one but an abstract representation that fulfills certain conditions.

The conditions that \( \rho(u, v) \) must satisfy to be a distance function in a metric space include non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. These establish the structure and cohesion within the space, ensuring that the concept of distance is stable, consistent, and mathematical meaningful—allowing us to navigate through and analyze the metric space with these reliable measurements.

Non-Negativity

The non-negativity property of a metric space's distance function ensures that the distance between any two points is always greater than or equal to zero. This concept is intuitive as in the real world; it is not possible for distance to be negative. Mathematically, for all points \( u, v \) in a set \( A \) with a distance function \( \rho \), it is confirmed that \( \rho(u, v) \geq 0 \).

This characteristic is fundamental since it establishes the baseline that no distance can exist in a deficit. When presented with a new function \( \rho_1(u, v) \) derived from \( \rho(u, v) \) in an exercise, verifying non-negativity is crucial to ensure that it could serve as a potential metric for the space.

Identity of Indiscernibles

The identity of indiscernibles is a pivotal aspect of a metric space that upholds the principle that if two points are indistinguishable under the distance function—meaning the distance between them is zero—then they must be the same point, confirming \( \rho(u, v) = 0 \) if and only if \( u = v \).

This property of the distance function underscores a strict identification rule within our metric space: no two distinct points can be located 'zero distance' apart. When we adapt a given metric, like changing \( \rho \) to \( \rho_1 \) in a problem, demonstrating that this property holds true under the new function fortifies our understanding that we have a proper metric.

Symmetry

Symmetry in a metric space implies that the distance measured from point \( u \) to point \( v \) is the same as the distance measured from point \( v \) to point \( u \), forming an equation \( \rho(u, v) = \rho(v, u) \).

This is akin to understanding that the path between two destinations can be traversed both ways with the same effort or distance, a relateable concept even outside of mathematics. When assessing a new metric, such as \( \rho_1 \) from an exercise, ensuring that this two-way street of distance holds true is vital in validating the function's candidacy as a true metric for that space.

Triangle Inequality

The triangle inequality is a non-negotiable cornerstone of a metric space, ensuring that the distance between two points taken directly is always less than or equal to the distance when taking a detour through a third point. In other words, for all points \( u, v, w \) in \( A \) with a distance function \( \rho \), this inequality \( \rho(u, w) \leq \rho(u, v) + \rho(v, w) \) explains that the direct route is the shortest path.

This principle aligns well with our intuitive understanding of distance in the physical world—taking a detour usually means adding mileage. In a mathematical context, illustrating this property for a modified distance function like \( \rho_1 \) is essential to confirm it reflects a true metric, capable of mapping the space appropriately.