A description/proof of that point is on map image of subset if preimage of point is contained in subset, but not only if
Topics
About: set
About: map
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 point is on the image of any subset if the preimage of the point is contained in the subset, but not only if.
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, \(f: S_1 \rightarrow S_2\), any point, \(p \in S_2\), and any subset, \(S_3 \subseteq S_1\), \(p \in f (S_3)\) if \(f^{-1} (p) \subseteq S_3\), but not only if.
2: Proof
Suppose that \(f^{-1} (p) \subseteq S_3\). \(f (f^{-1} (p)) = \{p\} \subseteq f (S_3)\), so, \(p \in f (S_3)\).
Suppose that \(p \in f (S_3)\). If \(f\) is not injective, there may be multiple points on \(f^{-1} (p)\), one of which may not be on \(S_3\).
3: Note
We have to be careful not to assume the reverse, by confusing with another proposition that for any map between sets, the image of any point is on any subset if and only if the point is on the preimage of the subset.
