definition of metric
Topics
About: metric space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of set.
Target Context
- The reader will have a definition of metric.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( S\): \(\in \{\text{ the sets }\}\)
\(*dist\): \(S \times S \to \mathbb{R}\)
//
Conditions:
\(\forall s_1, s_2, s_3 \in S\)
(
(
1) \(0 \le dist (s_1, s_2)\)
\(\land\)
(
\(dist (s_1, s_2) = 0\)
\(\iff\)
\(s_1 = s_2\)
)
)
\(\land\)
2) \(dist (s_1, s_2) = dist (s_2, s_1)\)
\(\land\)
3) \(dist (s_1, s_3) \le dist (s_1, s_2) + dist (s_2, s_3)\)
)
//
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) \(dist (s_1, s_3) \le dist (s_1, s_2) + dist (s_2, s_3)\)
