definition of inverse of map
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of map.
- The reader knows a definition of composition of maps.
Target Context
- The reader will have a definition of inverse 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:
\( S_1\): \(\in \{\text{ the sets }\}\)
\( S_2\): \(\in \{\text{ the sets }\}\)
\( f\): \(: S_1 \to S_2\)
\(*f^{- 1}\): \(: S_2 \to S_1\)
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Conditions:
\(f^{- 1} \circ f = id_{S_1} \land f \circ f^{- 1} = id_{S_2}\)
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2: Note
For an \(f\), \(f^{- 1}\) may not exist.
For example, let \(S_1 = \mathbb{R}\), \(S_2 = \mathbb{R}\), and \(f = 0\), the constant map, then, \(f^{- 1}\) does not exist, because for each \(r \in \mathbb{R}\), \(f^{- 1} \circ f (r) = f^{- 1} (0)\), which cannot be \(id_{S_1} (r) = r\), because whatever \(f^{- 1}\) is, \(f^{- 1} (0)\) cannot be changed according to \(r\).
If \(f^{- 1}\) exists, it is unique, by the proposition that for any map, if an inverse exists, the inverse is the unique inverse, which is the reason why the inverse is denoted as \(f^{- 1}\) as though it is determined by \(f\).
