Taxicab Metric on Real Vector Space is Metric/Proof 2

From ProofWiki
Jump to navigation Jump to search

Theorem

The taxicab metric on the real vector space $\R^n$ is a metric.


Proof

The taxicab metric on $\R^n$ is:

$\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \size {x_i - y_i}$

for $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.


Proof of Metric Space Axiom $(\text M 1)$

\(\ds \map {d_1} {x, x}\) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {x_i - x_i}\) Definition of $d_1$
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n 0\)
\(\ds \) \(=\) \(\ds 0\)

So Metric Space Axiom $(\text M 1)$ holds for $d_1$.

$\Box$


Proof of Metric Space Axiom $(\text M 2)$: Triangle Inequality

\(\ds \map {d_1} {x, y} + \map {d_1} {y, z}\) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {x_i - y_i} + \sum_{i \mathop = 1}^n \size {y_i - z_i}\) Definition of $d_1$
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \paren {\size {x_i - y_i} + \size {y_i - z_i} }\)
\(\ds \) \(\ge\) \(\ds \sum_{i \mathop = 1}^n \size {x_i - z_i}\) Triangle Inequality for Real Numbers
\(\ds \) \(=\) \(\ds \map {d_1} {x, z}\) Definition of $d_1$

So Metric Space Axiom $(\text M 2)$: Triangle Inequality holds for $d_1$.

$\Box$


Proof of Metric Space Axiom $(\text M 3)$

\(\ds \map {d_1} {x, y}\) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {x_i - y_i}\)
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {y_i - x_i}\)
\(\ds \) \(=\) \(\ds \map {d_1} {y, x}\) Definition of $d_1$

So Metric Space Axiom $(\text M 3)$ holds for $d_1$.

$\Box$


Proof of Metric Space Axiom $(\text M 4)$

\(\ds x\) \(\ne\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds \exists i \in \set {1, 2, \ldots, n}: \, \) \(\ds x_i\) \(\ne\) \(\ds y_i\)
\(\ds \leadsto \ \ \) \(\ds \size {x_i - y_i}\) \(>\) \(\ds 0\) Definition of Absolute Value
\(\ds \leadsto \ \ \) \(\ds \sum_{i \mathop = 1}^n \size {x_i - y_i}\) \(>\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \map {d_1} {x, y}\) \(>\) \(\ds 0\) Definition of $d_1$

So Metric Space Axiom $(\text M 4)$ holds for $d_1$.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Taxicab_Metric_on_Real_Vector_Space_is_Metric/Proof_2&oldid=539451"