description/proof of that map preimage of range is whole domain
Topics
About: set
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any map, the map preimage of the range is the whole domain
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:
\(S_1\): \(\in \{\text{ the sets }\}\)
\(S_2\): \(\in \{\text{ the sets }\}\)
\(f\): \(S_1 \to S_2\)
//
Statements:
\(f^{-1} (f (S_1)) = S_1\).
//
2: Natural Language Description
For any sets, \(S_1, S_2\), and any map, \(f: S_1 \to S_2\), \(f^{-1} (f (S_1)) = S_1\).
3: Proof
Obviously, \(f^{-1} (f (S_1)) \subseteq S_1\).
By the proposition that for any map, the composition of the preimage after the map of any subset is identical if and only if it is contained in the argument set, \(f^{-1} (f (S_1)) = S_1\).
