A description/proof of that subset is contained in map preimage of image of subset
Topics
About: set
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, any subset is contained in the preimage of the image of the subset.
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, \(S_3 \subseteq S_1\), such that \(S_4 = f (S_3)\), \(S_3 \subseteq f^{-1} (S_4)\).
2: Proof
For any \(p \in S_3\), \(f (p) \in S_4\), so, \(p \in f^{-1} (S_4)\).
3: Note
The subset does not necessarily equal the preimage, as \(f\) may not be injective and a point not on \(S_3\) may map into \(S_4\).
