Scaled Euclidean Metric is Metric
Jump to navigation
Jump to search
Theorem
Let $\R_{>0}$ be the set of (strictly) positive integers.
Let $\delta: \R_{>0} \times \R_{>0} \to \R$ be the metric on $\R_{>0}$ defined as:
- $\forall x, y \in \R_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Then $\delta$ is a metric.
Proof
Proof of Metric Space Axiom $(\text M 1)$
| \(\ds \map \delta {x, x}\) | \(=\) | \(\ds \dfrac {\size {x - x} } {x^2}\) | Definition of $\delta$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) | as $\size {x - x} = 0$ |
So Metric Space Axiom $(\text M 1)$ holds for $\delta$.
$\Box$
Proof of Metric Space Axiom $(\text M 2)$: Triangle Inequality
| \(\ds \map \delta {x, y} + \map \delta {y, z}\) | \(=\) | \(\ds \frac {\size {x - y} } {x y} + \dfrac {\size {y - z} } {y z}\) | Definition of $\delta$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {z \size {x - y} + x \size {y - z} } {x y z}\) | Sum of Quotients of Real Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\size {x z - y z} + \size {x y - x z} } {x y z}\) | Valid, as $x, z > 0$ | |||||||||||
| \(\ds \) | \(\ge\) | \(\ds \frac {\size {x z - y z + x y - x z} } {x y z}\) | Triangle Inequality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\size {x y - y z} } {x y z}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\size {x - z} } {x z}\) | simplifying further | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \delta {x, z}\) | Definition of $\delta$ |
So Metric Space Axiom $(\text M 2)$: Triangle Inequality holds for $\delta$.
$\Box$
Proof of Metric Space Axiom $(\text M 3)$
| \(\ds \map \delta {x, y}\) | \(=\) | \(\ds \frac {\size {x - y} } {x y}\) | Definition of $\delta$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\size {y - x} } {y x}\) | Definition of Absolute Value and Real Multiplication is Commutative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \delta {y, x}\) | Definition of $\delta$ |
So Metric Space Axiom $(\text M 3)$ holds for $\delta$.
$\Box$
Proof of Metric Space Axiom $(\text M 4)$
| \(\ds x\) | \(\ne\) | \(\ds y\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \size {x - y}\) | \(>\) | \(\ds 0\) | Definition of Absolute Value | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\size {x - y} } {x y}\) | \(>\) | \(\ds 0\) | as $x y > 0$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \delta {x, y}\) | \(>\) | \(\ds 0\) | Definition of $\delta$ |
So Metric Space Axiom $(\text M 4)$ holds for $\delta$.
$\blacksquare$