A description/proof of that composition of map after preimage is contained in argument set
Topics
About: set
About: map
The table of contents of this article
Starting Context
- The reader knows a definition of set.
- The reader knows a definition of map.
Target Context
- The reader will have a description and a proof of the proposition that for any map between sets, the composition of the map after any preimage is contained in the argument set.
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: Description
For any sets, \(S_1, S_2\), any map, \(f: S_1 \rightarrow S_2\), and any subset, \(S_3 \subseteq S_2\), \(f \circ f^{-1} (S_3) \subseteq S_3\).
2: Proof
For any \(p \in f \circ f^{-1} (S_3)\), \(p = f \circ f^{-1} (p')\) for a point, \(p' \in S_3\), but \(p = f \circ f^{-1} (p') = p'\), so, \(p \in S_3\).
3: Note
It is not necessarily \(f \circ f^{-1} (S_3) = S_3\), as for which, there is another proposition.
