The Clowder Project

Let $X$ be a set.

1. 1.

   Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto U\cup V$ define functors

   \[ \begin{array}{ccc} U\cup -\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\cup V\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -_{1}\cup -_{2}\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \end{array} \]

   In particular, the following statements hold for each $U,V,A,B\in \mathcal{P}\webleft (X\webright )$:

   1. (a)

      If $U\subset A$, then $U\cup V\subset A\cup V$.

   2. (b)

      If $V\subset B$, then $U\cup V\subset U\cup B$.

   3. (c)

      If $U\subset A$ and $V\subset B$, then $U\cup V\subset A\cup B$.

2. 2.

   Associativity. The diagram

   
   commutes, i.e. we have an equality of sets
   \[ \webleft (U\cup V\webright )\cup W = U\cup \webleft (V\cup W\webright ) \]

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

3. 3.

   Unitality. The diagrams

   
   commute, i.e. we have equalities of sets
   \begin{align*} \text{Ø}\cup U & = U,\\ U\cup \text{Ø}& = U \end{align*}

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

4. 4.

   Commutativity. The diagram

   
   commutes, i.e. we have an equality of sets
   \[ U\cup V = V\cup U \]

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

5. 5.

   Annihilation With $X$. The diagrams

   
   commute, i.e. we have equalities of sets
   \begin{align*} U\cup X & = X,\\ X\cup V & = X \end{align*}

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

6. 6.

   Distributivity of Unions Over Intersections. The diagrams

   
   commute, i.e. we have equalities of sets
   \begin{align*} U\cup \webleft (V\cap W\webright ) & = \webleft (U\cup V\webright )\cap \webleft (U\cup W\webright ),\\ \webleft (U\cap V\webright )\cup W & = \webleft (U\cup W\webright )\cap \webleft (V\cup W\webright ) \end{align*}

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

7. 7.

   Distributivity of Intersections Over Unions. The diagrams

   
   commute, i.e. we have equalities of sets
   \begin{align*} U\cap \webleft (V\cup W\webright ) & = \webleft (U\cap V\webright )\cup \webleft (U\cap W\webright ),\\ \webleft (U\cup V\webright )\cap W & = \webleft (U\cap W\webright )\cup \webleft (V\cap W\webright ) \end{align*}

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

8. 8.

   Idempotency. The diagram

   
   commutes, i.e. we have an equality of sets
   \[ U\cup U=U \]

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

9. 9.

   Via Intersections and Symmetric Differences. The diagram

   
   commutes, i.e. we have an equality of sets
   \[ U\cup V=\webleft (U\mathbin {\triangle }V\webright )\mathbin {\triangle }\webleft (U\cap V\webright ) \]

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

10. 10.

    Interaction With Characteristic Functions I. We have

    \[ \chi _{U\cup V}=\operatorname*{\operatorname {\mathrm{max}}}\webleft (\chi _{U},\chi _{V}\webright ) \]

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