A description/proof of that preimage under surjection is saturated w.r.t. surjection
Topics
About: set
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any surjection, the preimage of any codomain subset is saturated with respect to the surjection.
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 surjection, \(f: S_1 \rightarrow S_2\), and any subset, \(S_3 \subseteq S_2\), the preimage, \(f^{-1} (S_3)\), is saturated with respect to \(f\), which means \(f^{-1} \circ f (f^{-1} (S_3)) = f^{-1} (S_3)\).
2: Proof
\(f (f^{-1} (S_3)) = S_3\), because \(f\) is a surjection, by the proposition that for any map, the composition of the map after any preimage is identical if and only if the argument set is a subset of the map image. So, \(f^{-1} \circ f (f^{-1} (S_3)) = f^{-1} (S_3)\).
