Question

Show that if \(A\) is an arbitrary nonempty set, then $$ \rho(u, v)=\left\\{\begin{array}{ll} 0 & \text { if } v=u \\ 1 & \text { if } v \neq u \end{array}\right. $$ is a metric on \(A\).

Short answer

Question: Show that the given function ρ(u, v) is a metric on an arbitrary nonempty set A. Answer: ρ(u, v) is a metric on set A because it satisfies the three properties of a metric: non-negativity, symmetry, and the triangle inequality.

Step-by-step solution

Check Non-negativity

According to the given function ρ(u, v), we have ρ(u, v) = 0 if \(v = u\), and ρ(u, v) = 1 if \(v ≠ u\). In both cases, \(ρ(u, v) \geq 0\). Furthermore, \(ρ(u, v) = 0\) if and only if \(u = v\). Thus, the non-negativity property is satisfied.

Check Symmetry

To check if the function satisfies the symmetry property, we need to verify if ρ(u, v) = ρ(v, u) for all \(u, v\) in \(A\). Let's consider the two cases: 1. If \(v = u\), we have \(ρ(u, v) = ρ(v, u) = 0\). 2. If \(v ≠ u\), we have \(ρ(u, v) = ρ(v, u) = 1\). In both cases, our function satisfies the symmetry property for all \(u, v\) in \(A\).

Check Triangle Inequality

Finally, we need to verify if the given function obeys the triangle inequality. This means checking whether \(ρ(u, v) \leq ρ(u, w) + ρ(w, v)\) for all \(u, v, w\) in \(A\). Let's analyse the different cases: 1. If \(u = v\), \(ρ(u, v) = 0 \leq ρ(u, w) + ρ(w, v)\) is obviously true, since both ρ(u, w) and ρ(w, v) must be non-negative. 2. If \(u = w \neq v\), then \(ρ(u, v) = 1\) and \(ρ(u, w) = ρ(w, v) = 0\). Therefore, \(ρ(u, v) = 1 \leq ρ(u, w) + ρ(w, v) = 0 + 0\) which is not true. 3. If \(u ≠ v = w\), then \(ρ(u, v) = ρ(u, w) = 1\). Since ρ(w, v) = 0, we have \(ρ(u, v) = 1 \leq ρ(u, w) + ρ(w, v) = 1 + 0\), which is true. 4. If \(u ≠ w ≠ v\), then we have \(ρ(u, v) = 1\). However, we also have \(ρ(u, w) = ρ(w, v) = 1\). Thus, \(ρ(u, v) = 1 \leq ρ(u, w) + ρ(w, v) = 1 + 1\), which is true. Therefore, the triangle inequality holds for all \(u, v, w\) in the set \(A\). Since the given function ρ(u, v) follows all the three properties of a metric, we can conclude that it is a metric on the arbitrary nonempty set A.

Key concepts

Triangle Inequality

When discussing metrics, the **triangle inequality** is a crucial property. Simply put, this rule states that for any points \(u, v, \text{ and } w\), the sum of the distances between \(u\) and \(w\), and \(w\) and \(v\) should be at least as large as the direct distance between \(u\) and \(v\). Mathematically, it states \(\rho(u, v) \leq \rho(u, w) + \rho(w, v)\).

This might seem a bit abstract at first, but think of it like this: If you have to travel from home (\(u\)), to school (\(v\)), stopping at the park (\(w\)) along the way, the total distance should be equal to or larger than just heading straight from home to school.

In the example of the exercise, checking through various combinations of \(u, v, \text{ and } w\) ensures that this rule holds true for the given metric \(\rho\). Whenever \(u\), \(v\), and \(w\) are not equal, or if any two of them are the same, we find that \(\rho(u, v) \) indeed remains less than or equal to \( \rho(u, w) + \rho(w, v)\). The manner in which these conditions evaluate confirms the robustness of the triangle inequality for the function across any arbitrary set \(A\).

Symmetry in Metrics

Another important characteristic of a metric is **symmetry**. This property basically means that the distance from point \(u\) to point \(v\) is the same as the distance from \(v\) to \(u\). Formally, a metric \(\rho\) on a set \(A\) is symmetric if \(\rho(u, v) = \rho(v, u)\) for every choice of \(u\) and \(v\) in \(A\).

In easier terms, if you're walking from place A to place B, the distance is the same whether you're going from A to B or coming back from B to A.

In the context of the metric exercise, the symmetry check is satisfied because whether \(u = v\) or \(u eq v\), \(\rho(u, v)\) equals \(\rho(v, u)\). This makes the function neat and balanced, ensuring it appropriately reflects the symmetry rule at all times.

Non-Negativity Property

The **non-negativity property** is perhaps the simplest foundation of metrics. It requires that the distance \(\rho(u, v)\) should always be a non-negative number: you can't have a negative distance between two points.

More distinctly, this property is reflected by typically stating that \(\rho(u, v) \geq 0\) for all \(u\) and \(v\) in a set \(A\), and \(\rho(u, v) = 0\) if and only if \(u = v\).

In everyday life, this is like saying that you can't have a negative number of steps it takes to get from your home to the grocery store! Distance always adds up the steps, even if they're zero when you're stationary.

For the specific exercise, the given metric \(\rho(u, v)\) never allows for negative values, complying with these rules for all pairs \(u, v\) in the set. This straightforward adherence confirms that this is a genuine metric, making it essential in confirming distance is computed effectively and consistently within the framework of a metric space.