40: Metric

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definition of metric

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Topics

About: metric space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description

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Starting Context

• The reader knows a definition of set.

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Target Context

• The reader will have a definition of metric.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$S$: $\in \{\text{ the sets }\}$
$\ast dist$: $S\times S\to \mathbb{R}$
//

Conditions:
$\mathrm{\forall }{s}_1,s_2,s_3\in S$
(
(
1) $0\le dist(s_1,s_2)$
$\wedge$
(
$dist(s_1,s_2)=0$
$⟺$
$s_1=s_2$
)
)
$\wedge$
2) $dist(s_1,s_2)=dist(s_2,s_1)$
$\wedge$
3) $dist(s_1,s_3)\le dist(s_1,s_2)+dist(s_2,s_3)$
)
//

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2: Natural Language Description

For any set, $S$, any map, $dist:S\times S\to \mathbb{R}$, such that for any elements of $S$, $s_1,s_2,s_3\in S$, 1) $0\le dist(s_1,s_2)$ with the equality holding if and only if $s_1=s_2$; 2) $dist(s_1,s_2)=dist(s_2,s_1)$; 3) $dist(s_1,s_3)\le dist(s_1,s_2)+dist(s_2,s_3)$

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References

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