Let $X$ be a set.
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The Inclusion of Characteristic Relations Associated to a Function. Let $f\colon A\to B$ be a function. We have an inclusion[1]
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Interaction With Unions I. We have
\[ \chi _{U\cup V}=\operatorname*{\text{max}}\webleft (\chi _{U},\chi _{V}\webright ) \]
for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Unions II. We have
\[ \chi _{U\cup V}=\chi _{U}+\chi _{V}-\chi _{U\cap V} \]
for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Intersections I. We have
\[ \chi _{U\cap V}=\chi _{U}\chi _{V} \]
for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Intersections II. We have
\[ \chi _{U\cap V}=\operatorname*{\text{min}}\webleft (\chi _{U},\chi _{V}\webright ) \]
for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Differences. We have
\[ \chi _{U\setminus V}=\chi _{U}-\chi _{U\cap V} \]
for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Complements. We have
\[ \chi _{U^{\textsf{c}}}=1-\chi _{U} \]
for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Symmetric Differences. We have
\[ \chi _{U\mathbin {\triangle }V}=\chi _{U}+\chi _{V}-2\chi _{U\cap V} \]
and thus, in particular, we have
\[ \chi _{U\mathbin {\triangle }V}\equiv \chi _{U}+\chi _{V}\mod {2} \]for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction Between the Characteristic Embedding and Morphisms. Let $f\colon X\to Y$ be a map of sets. The diagram commutes.
