A description/proof of that map image of point is on subset iff point is on preimage of subset
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, the image of any point is on any subset if and only if the point is on the preimage 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, \(f: S_1 \rightarrow S_2\), any point, \(p \in S_1\), and any subset, \(S_3 \subseteq S_2\), \(f (p) \in S_3\) if and only if \(p \in f^{-1} (S_3)\).
2: Proof
Suppose that \(p \in f^{-1} (S_3)\). By the definition of preimage, \(f (p) \in S_3 \).
Suppose that \(f (p) \in S_3\). By the definition of preimage, \(p \in f^{-1} (S_3)\).
3: Note
This proposition is of course obvious, but I tend to have to have a second thought in order not to confuse with another 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.
