Proving the Function d(x, y) |x - y| Defines a Metric on the Set of Real Numbers R

In this article, we will delve into the concept of a metric and demonstrate that the function d(x, y) |x - y| satisfies the axioms of a metric on the set of real numbers R. We will also discuss the importance of these axioms in defining a metric space and explore how the given function adheres to these fundamental requirements. Understanding these properties is essential for anyone working with mathematical analysis and related fields.

What is the Definition of a Metric?

A metric is a function that defines a notion of distance between elements of a set. Formally, a metric d on a set X is a function that maps pairs of elements of X to the non-negative real numbers, and it must satisfy the following four axioms:

Metric Axiom 1 (Non-negativity): For all x, y in X, we have d(x, y) geq 0, with equality if and only if x y. Metric Axiom 2 (Symmetry): For all x, y in X, we have d(x, y) d(y, x). Metric Axiom 3 (Identity of Indiscernibles): For all x, y in X, we have d(x, y) 0 if and only if x y. Metric Axiom 4 (Triangle Inequality): For all x, y, z in X, we have d(x, y) leq d(x, z) d(z, y).

In this article, we aim to prove that the function d(x, y) |x - y| on the set of real numbers R satisfies these four axioms, thereby confirming that d is a metric.

Proving the Four Axioms for d(x, y) |x - y|

Let's now go through each axiom in detail to demonstrate that d(x, y) |x - y| meets the necessary conditions.

Metric Axiom 1: Non-negativity

For all x, y in R, we need to show that d(x, y) |x - y| geq 0. The absolute value function by definition ensures that:

For any real number a, |a| geq 0.

When |x - y| 0, it implies that x - y 0, which further simplifies to x y.

Thus, the function d(x, y) |x - y| satisfies the non-negativity axiom.

Metric Axiom 2: Symmetry

For all x, y in R, we need to show that d(x, y) d(y, x). This is straightforward as the absolute value function is symmetric:

|x - y| |y - x| for any x, y in R because the absolute value of a number is the same as the absolute value of its negative.

Therefore, the function d(x, y) |x - y| is symmetric.

Metric Axiom 3: Identity of Indiscernibles

For all x, y in R, we need to show that d(x, y) 0 if and only if x y. This can be broken down as follows:

If x y, then |x - y| |x - x| 0. Therefore, d(x, y) 0.

Conversely, if d(x, y) 0, then |x - y| 0. The only real number whose absolute value is 0 is 0 itself, implying that x - y 0, which simplifies to x y.

The identity of indiscernibles is thus satisfied by the function d(x, y) |x - y|.

Metric Axiom 4: Triangle Inequality

For all x, y, z in R, we need to show that d(x, y) |x - y| leq |x - z| |z - y|. This can be proven using the properties of absolute values:

By the triangle inequality for absolute values, we know that for any real numbers a, b in R, |a b| leq |a| |b|. Applying this property with a x - z and b z - y, we get |x - y| |(x - z) (z - y)| leq |x - z| |z - y|.

Therefore, d(x, y) |x - y| leq |x - z| |z - y|, and the triangle inequality is satisfied.

Conclusion

We have rigorously proven that the function d(x, y) |x - y| defined on the set of real numbers R satisfies all four axioms of a metric. This confirms that d is indeed a metric, providing a robust and well-defined measure of distance between real numbers.

Related Resources

For a deeper understanding of metrics and metric spaces, consider exploring the following resources:

Metric Space - Wikipedia Examples of Metric Functions YouTube - Metric Spaces and Distance Functions

Understanding the properties of metrics is crucial in various fields, including geometry, topology, and functional analysis. Share this article with anyone interested in exploring more about mathematical concepts like metrics and how they apply in real-world scenarios.