A description/proof of that when image of point is on image of subset, point is on subset if map is injective with respect to image of subset
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 between any sets, when the image of any point is on the image of any subset, the point is on the subset if the map is injective with respect to the image 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, \(S \subseteq S_1\), when \(f (p) \in f (S)\), \(p \in S\) if \(f\) is injective with respect to \(f (S)\), which means that for any \(q \in f (S)\), \(f^{-1} (q)\) is a 1 element set.
2: Proof
As \(f (p) \in f (S)\), there is a \(p' \in S\) such that \(f (p') = f(p)\). As \(f^{-1} (f (p))\) is a 1 element set, \(p' = p\), so, \(p \in S\).
3: Note
Of course, \(f\) can be wholly injective in order to satisfy the condition.
