A description/proof of that map preimages of disjoint subsets are disjoint
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 the preimages of any disjoint subsets under any map are disjoint.
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 disjoint subsets, \(S_{21}, S_{22} \subseteq S_2\), such that \(S_{21} \cap S_{22} = \emptyset\), \(f^{-1} (S_{21}) \cap f^{-1} (S_{22}) = \emptyset\).
2: Proof
Suppose that there was a common element, \(p \in f^{-1} (S_{21})\) and \(p \in f^{-1} (S_{22})\). \(f (p) \in S_{21}\) and \(f (p) \in S_{22}\), a contradiction.
