A description/proof of that composition of preimage after map of subset is identical iff it is contained in argument set
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, the composition of the preimage after the map of any subset is identical if and only if it is contained in the argument set.
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 subset, \(S_3 \subseteq S_1\), \(f^{-1} \circ f (S_3) = S_3\) if and only if \(f^{-1} \circ f (S_3) \subseteq S_3\).
2: Proof
Suppose that \(f^{-1} \circ f (S_3) \subseteq S_3\). For any \(p \in f^{-1} \circ f (S_3)\), \(p \in S_3\). For any \(p \in S_3\), \(f (p) \in f (S_3)\), which means that \(p \in f^{-1} \circ f (S_3)\) by the definition of preimage.
Suppose that \(f^{-1} \circ f (S_3) = S_3\). \(f^{-1} \circ f (S_3) \subseteq S_3\).
3: Note
It is important not to carelessly conclude that \(f^{-1} \circ f (S_3) = S_3\) without checking the condition.
