Item 13 (01H4) — The Clowder Project

4.3.8 Binary Unions

Let $X$ be a set and let $U,V\in \mathcal{P}(X)$.

The union of $U$ and $V$ is the set $U\cup V$ defined by

\begin{align*} U\cup V & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcup _{z\in \left\{ U,V\right\} }z\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \text{$x\in U$ or $x\in V$}\right\} . \end{align*}

Proposition 4.3.8.1.2 ▶ Properties of Binary Unions

Let $X$ be a set.

1.
Functoriality. The assignments $U,V,(U,V)\mapsto U\cup V$ define functors

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

In particular, the following statements hold for each $U,V,A,B\in \mathcal{P}(X)$:
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.
Associativity. The diagram
commutes, i.e. we have an equality of sets

\[ (U\cup V)\cup W = U\cup (V\cup W) \]

for each $U,V,W\in \mathcal{P}(X)$.
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 $U\in \mathcal{P}(X)$.
4.
Commutativity. The diagram
commutes, i.e. we have an equality of sets

\[ U\cup V = V\cup U \]

for each $U,V\in \mathcal{P}(X)$.
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 $U,V\in \mathcal{P}(X)$.
6.
Distributivity of Unions Over Intersections. The diagrams
commute, i.e. we have equalities of sets

\begin{align*} U\cup (V\cap W) & = (U\cup V)\cap (U\cup W),\\ (U\cap V)\cup W & = (U\cup W)\cap (V\cup W) \end{align*}

for each $U,V,W\in \mathcal{P}(X)$.
7.
Distributivity of Intersections Over Unions. The diagrams
commute, i.e. we have equalities of sets

\begin{align*} U\cap (V\cup W) & = (U\cap V)\cup (U\cap W),\\ (U\cup V)\cap W & = (U\cap W)\cup (V\cap W) \end{align*}

for each $U,V,W\in \mathcal{P}(X)$.
8.
Idempotency. The diagram
commutes, i.e. we have an equality of sets

\[ U\cup U=U \]

for each $U\in \mathcal{P}(X)$.
9.
Via Intersections and Symmetric Differences. The diagram
commutes, i.e. we have an equality of sets

\[ U\cup V=(U\mathbin {\triangle }V)\mathbin {\triangle }(U\cap V) \]

for each $U,V\in \mathcal{P}(X)$.
10.
Interaction With Characteristic Functions I. We have

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

for each $U,V\in \mathcal{P}(X)$.
11.
Interaction With Characteristic Functions II. We have

\[ \chi _{U\cup V}=\chi _{U}+\chi _{V}-\chi _{U\cap V} \]

for each $U,V\in \mathcal{P}(X)$.
12.
Interaction With Direct Images. Let $f\colon X\to Y$ be a function. The diagram
commutes, i.e. we have

\[ f_{!}(U\cup V)=f_{!}(U)\cup f_{!}(V) \]

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

13.
Interaction With Inverse Images. Let $f\colon X\to Y$ be a function. The diagram
commutes, i.e. we have

\[ f^{-1}(U\cup V)=f^{-1}(U)\cup f^{-1}(V) \]

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

14.

Interaction With Codirect Images. Let $f\colon X\to Y$ be a function. We have a natural transformation

with components

\[ f_{*}(U)\cup f_{*}(V)\subset f_{*}(U\cup V) \]

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

15.

Interaction With Powersets and Semirings. The quintuple $(\mathcal{P}(X),\cup ,\cap ,\text{Ø},X)$ is an idempotent commutative semiring.

Proof of Proposition 4.3.8.1.2.

Item 1: Functoriality

See [Proof Wiki Contributors, Set Union Preserves Subsets — Proof Wiki].

Item 2: Associativity

See [Proof Wiki Contributors, Union Is Associative — Proof Wiki].

Item 3: Unitality

This follows from [Proof Wiki Contributors, Union With Empty Set — Proof Wiki] and Item 4.

Item 4: Commutativity

See [Proof Wiki Contributors, Union Is Commutative — Proof Wiki].

Item 5: Annihilation With $X$

We have

\begin{align*} U\cup X & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \text{$x\in U$ or $x\in X$}\right\} \\ & = \left\{ x\in X\ \middle |\ x\in X\right\} ,\\ & = X \end{align*}

and

\begin{align*} X\cup V & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \text{$x\in X$ or $x\in V$}\right\} \\ & = \left\{ x\in X\ \middle |\ x\in X\right\} \\ & = X. \end{align*}

This finishes the proof.

Item 6: Distributivity of Unions Over Intersections

See [Proof Wiki Contributors, Union Distributes Over Intersection — Proof Wiki].

Item 7: Distributivity of Intersections Over Unions

See [Proof Wiki Contributors, Set Intersection Distributes Over Union — Proof Wiki].

Item 8: Idempotency

See [Proof Wiki Contributors, Set Union Is Idempotent — Proof Wiki].

Item 9: Via Intersections and Symmetric Differences

See [Proof Wiki Contributors, Union as Symmetric Difference With Intersection — Proof Wiki].

Item 10: Interaction With Characteristic Functions I

See [Proof Wiki Contributors, Characteristic Function of Union — Proof Wiki].

Item 11: Interaction With Characteristic Functions II