A description/proof of that map preimage of whole range is whole domain
Topics
About: set
About: map
The table of contents of this article
Starting Context
- The reader knows a definition of map.
Target Context
- The reader will have a description and a proof of the proposition that the preimage of the whole range of any map is the whole domain.
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, \(S_1\) and \(S_2\), and any map, \(f: S_1 \rightarrow S_2\), the preimage of the whole range is the whole domain, which is \(f^{-1} (S_2) = S_1\).
2: Proof
Suppose \(p \in f^{-1} (S_2)\). \(p \in S_1\). Suppose \(p \in S_1\). \(f (p) \in S_2\), so \(p \in f^{-1} (S_2)\).
3: Note
The point is that the map does not have to be surjective.
