Book keeping.
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\section{Preimages}
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\label{section:preimage}
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\begin{definition}[Preimage Function]
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\begin{definition}[Preimage Function*]
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\label{definition:preimage-function}
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Let $X, Y$ be sets and $P: 2^Y \to 2^X$, then $P$ is a \textbf{preimage function} if
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\begin{enumerate}[label=(PF\arabic*)]
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@@ -20,7 +20,7 @@
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\label{proposition:preimage-gymnastics}
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Let $X$ and $Y$ be sets, then:
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\begin{enumerate}
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\item For any $f: X \to Y$, the mapping $S \mapsto f^{-1}(S)$ is a total preimage function.
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\item For any $f: X \to Y$, the mapping $S \mapsto f^{-1}(S)$ is a total \hyperref[preimage function]{definition:preimage-function}.
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\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 $S \in 2^Y$.
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\end{enumerate}
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\end{proposition}
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