Let $X$ be a set.
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Functoriality. The assignment $U\mapsto U^{\textsf{c}}$ defines a functor
\[ \webleft (-\webright )^{\textsf{c}}\colon \mathcal{P}\webleft (X\webright )^{\mathsf{op}}\to \mathcal{P}\webleft (X\webright ), \]
where
- Action on Objects. For each $U\in \mathcal{P}\webleft (X\webright )$, we have
\[ \webleft [\webleft (-\webright )^{\textsf{c}}\webright ]\webleft (U\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}. \]
- Action on Morphisms. For each morphism $\iota _{U}\colon U\hookrightarrow V$ of $\mathcal{P}\webleft (X\webright )$, the image
\[ \iota ^{\textsf{c}}_{U}\colon V^{\textsf{c}}\hookrightarrow U^{\textsf{c}} \]
of $\iota _{U}$ by $\webleft (-\webright )^{\textsf{c}}$ is the inclusion
\[ V^{\textsf{c}}\subset U^{\textsf{c}} \]
i.e. where we have
- If $U\subset V$, then $V^{\textsf{c}}\subset U^{\textsf{c}}$.
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De Morgan’s Laws. We have equalities of sets
\begin{align*} \webleft (U\cup V\webright )^{\textsf{c}} & = U^{\textsf{c}}\cap V^{\textsf{c}},\\ \webleft (U\cap V\webright )^{\textsf{c}} & = U^{\textsf{c}}\cup V^{\textsf{c}} \end{align*}
for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Involutority. We have
\[ \webleft (U^{\textsf{c}}\webright )^{\textsf{c}}=U \]
for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Characteristic Functions. We have
\[ \chi _{U^{\textsf{c}}}=1-\chi _{U} \]
for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U\in \mathcal{P}\webleft (X\webright )$.