Problem 3 - Basic Analysis: Introduction to Real Analysis

Question

Let \(X:=\\{a, b\\}\) be a set. Can you make it into two distinct metric spaces? (define two distinct metrics on it)

Step-by-Step Solution

Answer

Yes, use metrics \(d_1(x, y) = 0\) if \(x = y\) and 1 if \(x \neq y\); \(d_2(x, y) = 0\) if \(x = y\) and 2 if \(x \neq y\).

Step 1: Understanding Metric Spaces

A metric space is a set \(X\) together with a function \(d: X \times X \rightarrow \mathbb{R}\) called a metric or distance function, which satisfies the following conditions for all \(x, y, z \in X\):1. \(d(x, y) \geq 0\) (Non-negativity)2. \(d(x, y) = 0\) if and only if \(x = y\) (Identity of indiscernibles)3. \(d(x, y) = d(y, x)\) (Symmetry)4. \(d(x, z) \leq d(x, y) + d(y, z)\) (Triangle inequality).For creating two distinct metric spaces on set \(X=\{a, b\}\), we need two different metrics that satisfy these properties.

Step 2: Define the First Metric

Let \(d_1\) be a metric defined on \(X\) by the formula:\[d_1(x, y) = \begin{cases} 0, & \text{if } x = y \1, & \text{if } x eq y\end{cases}\]This metric satisfies all the required properties for a metric: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.

Step 3: Define the Second Metric

Let \(d_2\) be another metric defined on \(X\) by the formula:\[d_2(x, y) = \begin{cases} 0, & \text{if } x = y \2, & \text{if } x eq y\end{cases}\]This different set of values also satisfies the properties of a metric space: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Note that since \(d_2(a, b) = 2\), it is distinct from \(d_1(a, b) = 1\).

Step 4: Verification of Distinct Metrics

To verify that both \(d_1\) and \(d_2\) are distinct, observe:- \(d_1(a, b) = 1\), whereas \(d_2(a, b) = 2\).- Both metrics have the same values for \( x = y \) (i.e., 0), but different values for \( x eq y \).Hence, the two metrics are distinct since they assign different distances between any two different elements.

Key Concepts

Set Theory

In mathematics, set theory is a fundamental concept that deals with collections of objects, known as sets. Sets can contain anything, including numbers, people, or even other sets. In this context, our set is \( X = \{a, b\} \). Set theory provides the basis for defining many mathematical concepts, including metric spaces.

- **Set:** In our example, it's \( \{a, b\} \).- **Elements:** The objects within a set, like \( a \) and \( b \).

By understanding set theory, you gain a foundation to explore other mathematical concepts, such as metric spaces, and how elements within sets can relate to each other through various functions.

Distance Function

The distance function, or metric, is a way to define the "distance" between two elements within a set. For our set \( X = \{a, b\} \), we define such a function to satisfy particular properties that make it a valid metric.

A metric \( d(x, y) \) is a function that assigns a non-negative real number to every pair of elements \( x \) and \( y \) in set \( X \). The concept of a metric helps us quantify how far apart two elements are, even in abstract spaces.

For example, in our solution, two distinct metrics were defined:

• **First metric** \( d_1 \) assigns a distance of 1 between different elements \( a \) and \( b \).
• **Second metric** \( d_2 \) assigns a distance of 2 between \( a \) and \( b \).

By defining different metrics, you can create distinct metric spaces, giving unique ways to measure "distance" depending on the context of the problem.

Triangle Inequality

The triangle inequality is a pivotal property of metric spaces. It ensures that the shortest distance between any two points is a straight line, metaphorically speaking. This rule states that for any three points \( x, y, z \) in a set \( X \), the inequality \( d(x, z) \leq d(x, y) + d(y, z) \) must hold.

- Think of the triangle inequality as a way of preventing detours when measuring distances. The direct path between two points should never be longer than taking a roundabout way through a third point.

In practice, this means that if you define a metric space correctly, no sequence of measurements should violate this principle. For our metrics \( d_1 \) and \( d_2 \), both satisfy the triangle inequality:

• In \( d_1 \), the distances are 0 when points are the same and 1 when they're different. The inequality naturally holds.
• In \( d_2 \), you similarly find that direct paths are the shortest, respecting the triangle principle.

The triangle inequality is crucial for defining meaningful and consistent measurements in metric spaces.

Symmetry in Metrics

Symmetry in metrics refers to the idea that the distance from one point to another should be the same in both directions. In mathematical terms, for any points \( x \) and \( y \) in a set, symmetry is expressed as \( d(x, y) = d(y, x) \).

This property holds true in both metrics we defined:

• With \( d_1 \), whether you measure from \( a \) to \( b \) or from \( b \) to \( a \), the result is the same: 1.
• In \( d_2 \), whether in reverse or forward, from \( a \) to \( b \), the distance remains 2.

Symmetry ensures that a metric is consistent and logical. If going from point A to B is a certain distance, returning from B to A should not differ. This fundamental idea in defining a metric reinforces consistency and predictability in your mathematical models.