2023-10-22

394: Composition of Preimage After Map of Subset Contains Argument Set

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A description/proof of that composition of preimage after map of subset contains argument set

Topics


About: set

The table of contents of this article


Starting Context



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 contains 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, \(S_3 \subseteq S_1\), \(S_3 \subseteq f^{-1} \circ f (S_3)\).


2: Proof


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.


References


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