A description/proof of that composition of preimage after map of subset contains argument set
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, the composition of the preimage after the map of any subset contains 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, \(S_3 \subseteq S_1\), \(S_3 \subseteq f^{-1} \circ f (S_3)\).
2: Proof
For any \(p \in S_3\), \(f (p) \in f (S_3)\), which means that \(p \in f^{-1} \circ f (S_3)\) by the definition of preimage.
