The Clowder Project

Let $X$ be a set.

1. Lack of Functoriality. The assignment $\webleft (U,V\webright )\mapsto U\mathbin {\triangle }V$ does not in general define functors
   \[ \begin{array}{ccc} U\mathbin {\triangle }-\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\mathbin {\triangle }V\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -_{1}\mathbin {\triangle }-_{2}\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \end{array} \]
2. Via Unions and Intersections. We have
   \[ U\mathbin {\triangle }V=\webleft (U\cup V\webright )\setminus \webleft (U\cap V\webright ) \]

   for each $U,V\in \mathcal{P}\webleft (X\webright )$, as in the Venn diagram

   

3. Symmetric Differences of Disjoint Sets. If $U$ and $V$ are disjoint, then we have
   \[ U\mathbin {\triangle }V=U\cup V. \]
4. Associativity. The diagram
   

   commutes, i.e. we have

   \[ \webleft (U\mathbin {\triangle }V\webright )\mathbin {\triangle }W = U\mathbin {\triangle }\webleft (V\mathbin {\triangle }W\webright ) \]

   for each $U,V,W\in \mathcal{P}\webleft (X\webright )$, as in the Venn diagram

   

5. Unitality. The diagrams
   
   commute, i.e. we have
   \begin{align*} U\mathbin {\triangle }\text{Ø}& = U,\\ \text{Ø}\mathbin {\triangle }U & = U \end{align*}

   for each $U\in \mathcal{P}\webleft (X\webright )$.

6. Commutativity. The diagram
   

   commutes, i.e. we have

   \[ U\mathbin {\triangle }V = V\mathbin {\triangle }U \]

   for each $U,V\in \mathcal{P}\webleft (X\webright )$.

7. Invertibility. We have
   \[ U\mathbin {\triangle }U = \text{Ø} \]

   for each $U\in \mathcal{P}\webleft (X\webright )$.

8. 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 $U,V,W\in \mathcal{P}\webleft (X\webright )$.

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

   for each $U\in \mathcal{P}\webleft (X\webright )$.

10. 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 $U\in \mathcal{P}\webleft (X\webright )$.

11. Interaction With Complements III. The diagram
    

    commutes, i.e. we have

    \[ U^{\textsf{c}}\mathbin {\triangle }V^{\textsf{c}}=U\mathbin {\triangle }V \]

    for each $U,V\in \mathcal{P}\webleft (X\webright )$.

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

    for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

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

    for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

14. 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 $U,V,W\in \mathcal{P}\webleft (X\webright )$.

15. 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

    \[ \chi _{U\mathbin {\triangle }V}\equiv \chi _{U}+\chi _{V}\ \ (\mathrm{mod}\ 2) \]

    for each $U,V\in \mathcal{P}\webleft (X\webright )$.

16. Bijectivity. Given $U,V\in \mathcal{P}\webleft (X\webright )$, the maps
    \begin{align*} U\mathbin {\triangle }- & \colon \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ),\\ -\mathbin {\triangle }V & \colon \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \end{align*}

    are bijections with inverses given by

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

    Moreover, the map

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

17. Interaction With Powersets and Groups. Let $X$ be a set.
    1. The quadruple $\webleft (\mathcal{P}\webleft (X\webright ),\mathbin {\triangle },\text{Ø},\text{id}_{\mathcal{P}\webleft (X\webright )}\webright )$ is an abelian group.¹
    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).
18. 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 Item 17;
    • 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}}}=}}\text{Ø},\\ 1\cdot U & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U; \end{align*}

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

19. 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 18.
    2. We have
       \[ \dim \webleft (\mathcal{P}\webleft (X\webright )\webright )=\# {X}. \]

20. Interaction With Powersets and Rings. The quintuple $\webleft (\mathcal{P}\webleft (X\webright ),\mathbin {\triangle },\cap ,\text{Ø},X\webright )$ is a commutative ring.²
21. Interaction With Direct Images. We have a natural transformation
    

    with components

    \[ f_{*}\webleft (U\webright )\mathbin {\triangle }f_{*}\webleft (V\webright )\subset f_{*}\webleft (U\mathbin {\triangle }V\webright ) \]

    indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.

22. Interaction With Inverse Images. The diagram
    

    i.e. we have

    \[ f^{-1}\webleft (U\webright )\mathbin {\triangle }f^{-1}\webleft (V\webright )=f^{-1}\webleft (U\mathbin {\triangle }V\webright ) \]

    for each $U,V\in \mathcal{P}\webleft (Y\webright )$.

23. Interaction With Direct Images With Compact Support. We have a natural transformation
    

    with components

    \[ f_{!}\webleft (U\mathbin {\triangle }V\webright )\subset f_{1}\webleft (U\webright )\mathbin {\triangle }f_{!}\webleft (V\webright ) \]

    indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.

---

¹Here are some examples:

1. When $X=\text{Ø}$, we have an isomorphism of groups between $\mathcal{P}\webleft (\text{Ø}\webright )$ and the trivial group:
   \[ \webleft (\mathcal{P}\webleft (\text{Ø}\webright ),\mathbin {\triangle },\text{Ø},\text{id}_{\mathcal{P}\webleft (\text{Ø}\webright )}\webright ) \cong \text{pt}. \]

2. When $X=\text{pt}$, we have an isomorphism of groups between $\mathcal{P}\webleft (\text{pt}\webright )$ and $\mathbb {Z}_{/2}$:
   \[ \webleft (\mathcal{P}\webleft (\text{pt}\webright ),\mathbin {\triangle },\text{Ø},\text{id}_{\mathcal{P}\webleft (\text{pt}\webright )}\webright ) \cong \mathbb {Z}_{/2}. \]

3. When $X=\webleft\{ 0,1\webright\} $, we have an isomorphism of groups between $\mathcal{P}\webleft (\webleft\{ 0,1\webright\} \webright )$ and $\mathbb {Z}_{/2}\times \mathbb {Z}_{/2}$:
   \[ \webleft (\mathcal{P}\webleft (\webleft\{ 0,1\webright\} \webright ),\mathbin {\triangle },\text{Ø},\text{id}_{\mathcal{P}\webleft (\webleft\{ 0,1\webright\} \webright )}\webright ) \cong \mathbb {Z}_{/2}\times \mathbb {Z}_{/2}. \]

²Warning: The analogous statement replacing intersections by unions (i.e. that the quintuple $\webleft (\mathcal{P}\webleft (X\webright ),\mathbin {\triangle },\cup ,\text{Ø},X\webright )$ is a ring) is false, however. See [Proof Wiki, Symmetric Difference With Union Does Not Form Ring] for a proof.