definition of map
Topics
About: collection
The table of contents of this article
Starting Context
- The reader knows a definition of collection.
Target Context
- The reader will have a definition of map.
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:
\( C_1\): \(\in \{\text{ the collections }\}\)
\( C_2\): \(\in \{\text{ the collections }\}\)
\(*f: C_1 \to C_2\): \(\in \{\text{ the functions }\}\)
//
Conditions:
\(Dom (f) = C_1 \land Ran (f) \subseteq C_2\)
//
2: Note
As for the distinction between 'collection' and 'set', refer to Are You Talking About Sets or About Collections?.
\(C_2\) is called "codomain" of \(f\), which is different from the range of \(f\) in general.
'map' is a 'function' with a codomain.
Having the codomain is sometimes crucial, for example, a map from a \(C^\infty\) manifold into a \(C^\infty\) manifold can be talked about being \(C^\infty\) only because the map has the codomain: the definition of map being \(C^\infty\) requires the codomain as a \(C^\infty\) manifold (possibly with boundary).
