The Clowder Project

Let $X$ be a set.

1. Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto U\cap V$ define functors
   \begin{gather*} \begin{aligned} U\setminus - & \colon \webleft (\mathcal{P}\webleft (X\webright ),\supset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\setminus V & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ \end{aligned}\\ -_{1}\setminus -_{2} \colon \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \supset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ), \end{gather*}

   where $-_{1}\setminus -_{2}$ is the functor where

   • Action on Objects. For each $\webleft (U,V\webright )\in \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright )$, we have
     \[ \webleft [-_{1}\setminus -_{2}\webright ]\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U\setminus V. \]

   • Action on Morphisms. For each pair of morphisms
     \begin{align*} \iota _{A} & \colon A\hookrightarrow B,\\ \iota _{U} & \colon U\hookrightarrow V \end{align*}

     of $\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright )$, the image

     \[ \iota _{U}\setminus \iota _{V}\colon A\setminus V\hookrightarrow B\setminus U \]

     of $\webleft (\iota _{U},\iota _{V}\webright )$ by $\setminus $ is the inclusion

     \[ A\setminus V\subset B\setminus U \]

     i.e. where we have

     • If $A\subset B$ and $U\subset V$, then $A\setminus V\subset B\setminus U$.
   and where $U\setminus -$ and $-\setminus V$ are the partial functors of $-_{1}\setminus -_{2}$ at $U,V\in \mathcal{P}\webleft (X\webright )$.

2. 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 $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.

3. 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 $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

4. 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 $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

5. 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 $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

6. 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 $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

7. 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 $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

8. Interaction With Complements. We have an equality of sets
   \[ U\setminus V=U\cap V^{\textsf{c}} \]

   for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.

9. Interaction With Symmetric Differences. We have an equality of sets
   \[ U\setminus V=U\mathbin {\triangle }\webleft (U\cap V\webright ) \]

   for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.

10. Triple Differences. We have
    \[ U\setminus \webleft (V\setminus W\webright )=\webleft (U\cap W\webright )\cup \webleft (U\setminus V\webright ) \]

    for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

11. Left Annihilation. We have
    \[ \emptyset \setminus U=\emptyset \]

    for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U\in \mathcal{P}\webleft (X\webright )$.

12. Right Unitality. We have
    \[ U\setminus \emptyset =U \]

    for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U\in \mathcal{P}\webleft (X\webright )$.

13. Invertibility. We have
    \[ U\setminus U = \emptyset \]

    for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U\in \mathcal{P}\webleft (X\webright )$.

14. Interaction With Containment. The following conditions are equivalent:
    1. We have $V\setminus U\subset W$.
    2. We have $V\setminus W\subset U$.
15. Interaction With Characteristic Functions. We have
    \[ \chi _{U\setminus V}=\chi _{U}-\chi _{U\cap V} \]

    for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.