Book keeping.

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Bokuan Li
2026-06-29 16:34:06 -04:00
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commit a11cfe4e04
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\section{Preimages}
\label{section:preimage}
\begin{definition}[Preimage Function]
\begin{definition}[Preimage Function*]
\label{definition:preimage-function}
Let $X, Y$ be sets and $P: 2^Y \to 2^X$, then $P$ is a \textbf{preimage function} if
\begin{enumerate}[label=(PF\arabic*)]
@@ -20,7 +20,7 @@
\label{proposition:preimage-gymnastics}
Let $X$ and $Y$ be sets, then:
\begin{enumerate}
\item For any $f: X \to Y$, the mapping $S \mapsto f^{-1}(S)$ is a total preimage function.
\item For any $f: X \to Y$, the mapping $S \mapsto f^{-1}(S)$ is a total \hyperref[preimage function]{definition:preimage-function}.
\item For any total preimage function $P: 2^Y \to 2^X$, there exists a unique $f: X \to Y$ such that $P(S) = f^{-1}(S)$ for all $S \in 2^Y$.
\end{enumerate}
\end{proposition}