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Let $X$ and $Y$ be sets with map $\psi: X \to Y$. If $\psi$ is bijective, then the inverse map $\psi^{-1}: Y \to X$ exists.

Denote $2^X$ and $2^Y$ as power sets of $X$ and $Y$, respectively.

For any map $\psi: X \to Y$ we can always define the map $\varphi: 2^Y \to 2^X$ via $W \mapsto \{x \in X: \psi(x) \in W\}$.

Now my question: How ist $\varphi$ called?

I have found in the literature also inverse map for $\varphi$ but this collides with $\psi^{-1}$.

Do you have suggestions? Thank you for your time and thoughts!

Moritz
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    The $\varphi$ you define above can be called inverse image mapping (for example here: https://proofwiki.org/wiki/Definition:Inverse_Image_Mapping/Mapping) – david_sap Nov 12 '22 at 11:19
  • Great! Thank you: this will do nicely. If you transfer your comment to an answer I will accept it gladly. – Moritz Nov 12 '22 at 21:45

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Given $\psi : X \to Y$, the map $\varphi : 2^Y \to 2^X$ that that sends a set in $2^Y$ to its inverse image (or preimage) in $2^X$, that is $$ \varphi : W \mapsto \{x \in X \; : \; \psi(x) \in W\}$$ is usually called the inverse image (or preimage) mapping of $\psi$ (see for example here).

david_sap
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