The Clowder Project

Let $X$ be a set.

1. Lack of Functoriality. The assignment $\webleft (U,V\webright )\mapsto U\mathbin {\triangle }V$ need not define functors
   \begin{gather*} \begin{aligned} U\mathbin {\triangle }- & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\mathbin {\triangle }V & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ \end{aligned}\\ -_{1}\mathbin {\triangle }-_{2} \colon \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \subset \webright )\to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ).\end{gather*}
2. Via Unions and Intersections. We have^[1]
   \[ U\mathbin {\triangle }V = \webleft (U\cup V\webright )\setminus \webleft (U\cap V\webright ) \]

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

3. Associativity. We have^[2]
   \[ \webleft (U\mathbin {\triangle }V\webright )\mathbin {\triangle }W = U\mathbin {\triangle }\webleft (V\mathbin {\triangle }W\webright ) \]

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

4. Commutativity. We have
   \[ U\mathbin {\triangle }V = V\mathbin {\triangle }U \]

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

5. Unitality. We have
   \begin{align*} U\mathbin {\triangle }\emptyset & = U,\\ \emptyset \mathbin {\triangle }U & = U \end{align*}

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

6. Invertibility. We have
   \[ U\mathbin {\triangle }U = \emptyset \]

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

7. Interaction With Unions. We have
   \[ \webleft (U\mathbin {\triangle }V\webright )\cup \webleft (V\mathbin {\triangle }T\webright ) = \webleft (U\cup V\cup W\webright )\setminus \webleft (U\cap V\cap W\webright ) \]

   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 I. We have
   \[ U\mathbin {\triangle }U^{\textsf{c}}= X \]

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

9. Interaction With Complements II. We have
   \begin{align*} U\mathbin {\triangle }X & = U^{\textsf{c}},\\ X\mathbin {\triangle }U & = U^{\textsf{c}}\end{align*}

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

10. Interaction With Complements III. We have
    \[ U^{\textsf{c}}\mathbin {\triangle }V^{\textsf{c}}=U\mathbin {\triangle }V \]

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

11. “Transitivity”. We have
    \[ \webleft (U\mathbin {\triangle }V\webright )\mathbin {\triangle }\webleft (V\mathbin {\triangle }W\webright )=U\mathbin {\triangle }W \]

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

12. The Triangle Inequality for Symmetric Differences. We have
    \[ U\mathbin {\triangle }W\subset U\mathbin {\triangle }V\cup V\mathbin {\triangle }W \]

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

13. Distributivity Over Intersections. We have
    \begin{align*} U\cap \webleft (V\mathbin {\triangle }W\webright ) & = \webleft (U\cap V\webright )\mathbin {\triangle }\webleft (U\cap W\webright ),\\ \webleft (U\mathbin {\triangle }V\webright )\cap W & = \webleft (U\cap W\webright )\mathbin {\triangle }\webleft (V\cap W\webright ) \end{align*}

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

14. Interaction With Characteristic Functions. 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 $U,V\in \mathcal{P}\webleft (X\webright )$.

15. Bijectivity. Given $A,B\subset \mathcal{P}\webleft (X\webright )$, the maps
    \begin{align*} A\mathbin {\triangle }- & \colon \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ),\\ -\mathbin {\triangle }B & \colon \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \end{align*}

    are bijections with inverses given by

    \begin{align*} \webleft (A\mathbin {\triangle }-\webright )^{-1} & = -\cup \webleft (A\cap -\webright ),\\ \webleft (-\mathbin {\triangle }B\webright )^{-1} & = -\cup \webleft (B\cap -\webright ). \end{align*}

    Moreover, the map

    \[ C\mapsto C\mathbin {\triangle }\webleft (A\mathbin {\triangle }B\webright ) \]

    is a bijection of $\mathcal{P}\webleft (X\webright )$ onto itself sending $A$ to $B$ and $B$ to $A$.

16. Interaction With Powersets and Groups. Let $X$ be a set.
    1. The quadruple $\webleft (\mathcal{P}\webleft (X\webright ),\mathbin {\triangle },\emptyset ,\text{id}_{\mathcal{P}\webleft (X\webright )}\webright )$ is an abelian group.^[3]
    2. Every element of $\mathcal{P}\webleft (X\webright )$ has order $2$ with respect to $\mathbin {\triangle }$, and thus $\mathcal{P}\webleft (X\webright )$ is a Boolean group (i.e. an abelian $2$-group).
17. Interaction With Powersets and Vector Spaces I. The pair $\webleft (\mathcal{P}\webleft (X\webright ),\alpha _{\mathcal{P}\webleft (X\webright )}\webright )$ consisting of
    • The group $\mathcal{P}\webleft (X\webright )$ of ;
    • The map $\alpha _{\mathcal{P}\webleft (X\webright )}\colon \mathbb {F}_{2}\times \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (X\webright )$ defined by
      \begin{align*} 0\cdot U & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\emptyset ,\\ 1\cdot U & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U; \end{align*}

    is an $\mathbb {F}_{2}$-vector space.

18. Interaction With Powersets and Vector Spaces II. If $X$ is finite, then:
    1. The set of singletons sets on the elements of $X$ forms a basis for the $\mathbb {F}_{2}$-vector space $\webleft (\mathcal{P}\webleft (X\webright ),\alpha _{\mathcal{P}\webleft (X\webright )}\webright )$ of Item 17.
    2. We have
       \[ \dim \webleft (\mathcal{P}\webleft (X\webright )\webright )=\# \mathcal{P}\webleft (X\webright ). \]

19. Interaction With Powersets and Rings. The quintuple $\webleft (\mathcal{P}\webleft (X\webright ),\mathbin {\triangle },\cap ,\emptyset ,X\webright )$ is a commutative ring.^[4]