MHB Set equality with a function and its inverse.

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The discussion centers on the relationship between a set E and the image of E under a function F and its inverse F^{-1}. It is established that E is always a subset of F^{-1}(F(E)). The main question posed is whether the intersection of F^{-1}(F(E)) and E equals E itself. Participants agree that E is indeed a subset of F^{-1}(F(E)), reinforcing the concept of set equality in this context. The conclusion emphasizes the importance of understanding the properties of functions and their inverses in set theory.
OhMyMarkov
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Is $F^{-1}(F(E))\cap E=E$?

Thanks!
 
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OhMyMarkov said:
Is $F^{-1}(F(E))\cap E=E$?

It is always the case that $E\subseteq F^{-1}(F(E))$
 
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Plato said:
It is always the case that $E\subseteq F^{-1}(F(E))$

Yes...
 

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