Let $X$ be a set.
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Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto U\cap V$ define functors
\[ \begin{array}{ccc} U\setminus -\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\supset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\setminus V\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -_{1}\setminus -_{2}\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \supset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \end{array} \]
In particular, the following statements hold for each $U,V,A,B\in \mathcal{P}\webleft (X\webright )$:
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De Morgan’s Laws. We have equalities of sets
\begin{align*} X\setminus \webleft (U\cup V\webright ) & = \webleft (X\setminus U\webright )\cap \webleft (X\setminus V\webright ),\\ X\setminus \webleft (U\cap V\webright ) & = \webleft (X\setminus U\webright )\cup \webleft (X\setminus V\webright ) \end{align*}
for each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Unions I. We have equalities of sets
\[ U\setminus \webleft (V\cup W\webright )=\webleft (U\setminus V\webright )\cap \webleft (U\setminus W\webright ) \]
for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Unions II. We have equalities of sets
\[ \webleft (U\setminus V\webright )\cup W=\webleft (U\cup W\webright )\setminus \webleft (V\setminus W\webright ) \]
for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Unions III. We have equalities of sets
\begin{align*} U\setminus \webleft (V\cup W\webright ) & = \webleft (U\cup W\webright )\setminus \webleft (V\cup W\webright )\\ & = \webleft (U\setminus V\webright )\setminus W\\ & = \webleft (U\setminus W\webright )\setminus V \end{align*}
for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Unions IV. We have equalities of sets
\[ \webleft (U\cup V\webright )\setminus W=\webleft (U\setminus W\webright )\cup \webleft (V\setminus W\webright ) \]
for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Intersections. We have equalities of sets
\begin{align*} \webleft (U\setminus V\webright )\cap W & = \webleft (U\cap W\webright )\setminus V\\ & = U\cap \webleft (W\setminus V\webright ) \end{align*}
for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Complements. We have an equality of sets
\[ U\setminus V=U\cap V^{\textsf{c}} \]
for each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Symmetric Differences. We have an equality of sets
\[ U\setminus V=U\mathbin {\triangle }\webleft (U\cap V\webright ) \]
for each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Triple Differences. We have
\[ U\setminus \webleft (V\setminus W\webright )=\webleft (U\cap W\webright )\cup \webleft (U\setminus V\webright ) \]
for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.
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Left Annihilation. We have
\[ \text{Ø}\setminus U=\text{Ø} \]
for each $U\in \mathcal{P}\webleft (X\webright )$.
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Right Unitality. We have
\[ U\setminus \text{Ø}=U \]
for each $U\in \mathcal{P}\webleft (X\webright )$.
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Right Annihilation. We have
\[ U\setminus X=U \]
for each $U\in \mathcal{P}\webleft (X\webright )$.
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Invertibility. We have
\[ U\setminus U = \text{Ø} \]
for each $U\in \mathcal{P}\webleft (X\webright )$.
- Interaction With Containment. The following conditions are equivalent:
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Interaction With Characteristic Functions. We have
\[ \chi _{U\setminus V}=\chi _{U}-\chi _{U\cap V} \]
for each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Direct Images. We have a natural transformation
with components
\[ f_{*}\webleft (U\webright )\setminus f_{*}\webleft (V\webright )\subset f_{*}\webleft (U\setminus V\webright ) \]indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Inverse Images. The diagram
commutes, i.e. we have
\[ f^{-1}\webleft (U\setminus V\webright )=f^{-1}\webleft (U\webright )\setminus f^{-1}\webleft (V\webright ) \]for each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Direct Images With Compact Support. We have a natural transformation
with components
\[ f_{*}\webleft (U\webright )\setminus f_{*}\webleft (V\webright )\subset f_{*}\webleft (U\setminus V\webright ) \]indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.
