Item 1 (021R) — The Clowder Project

4.4.7 The Internal Hom of a Powerset

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

Definition 4.4.7.1.1 ▶ The Internal Hom of a Powerset

The internal Hom of $\mathcal{P}\webleft (X\webright )$ from $U$ to $V$ is the subset $\webleft [U,V\webright ]_{X}$¹ of $X$ defined by

\begin{align*} \webleft [U,V\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}\cup V\\ & = \webleft (U\setminus V\webright )^{\textsf{c}}\end{align*}

where $U^{\textsf{c}}$ is the complement of $U$ of Definition 4.3.11.1.1.

¹Further Notation: Also written $\mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,V\webright )$.

Proof of Definition 4.4.7.1.1.

We have

\begin{align*} \webleft (U\setminus V\webright )^{\textsf{c}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X\setminus \webleft (U\setminus V\webright )\\ & = \webleft (X\cap V\webright )\cup \webleft (X\setminus U\webright )\\ & = V\cup \webleft (X\setminus U\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}V\cup U^{\textsf{c}}\\ & = U^{\textsf{c}}\cup V,\end{align*}

where we have used:

1.
Item 10 of Proposition 4.3.10.1.2 for the second equality.
2.
Item 4 of Proposition 4.3.9.1.2 for the third equality.
3.
Item 4 of Proposition 4.3.8.1.2 for the last equality.

This finishes the proof.

Remark 4.4.7.1.2 ▶ Intuition for the Internal Hom of $\mathcal{P}\webleft (X\webright )$

Henning Makholm suggests the following heuristic intuition for the internal Hom of $\mathcal{P}\webleft (X\webright )$ from $U$ to $V$ ([B., Show that the powerset partial order is a cartesian closed category]):

1.
Since products in $\mathcal{P}\webleft (X\webright )$ are given by binary intersections (Item 1 of Proposition 4.4.1.1.4), the right adjoint $\mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,-\webright )$ of $U\cap -$ may be thought of as a function type $\webleft [U,V\webright ]$.
2.
Under the Curry–Howard correspondence (), the function type $\webleft [U,V\webright ]$ corresponds to implication $U\Rightarrow V$.
3.
Implication $U\Rightarrow V$ is logically equivalent to $\neg U\vee V$.
4.
The expression $\neg U\vee V$ then corresponds to the set $U^{\textsf{c}}\cup V$ in $\mathcal{P}\webleft (X\webright )$.
5.
The set $U^{\textsf{c}}\vee V$ turns out to indeed be the internal Hom of $\mathcal{P}\webleft (X\webright )$.

Proposition 4.4.7.1.3 ▶ Properties of Internal Homs of Powersets

Let $X$ be a set.

1.
Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}$ define functors

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

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 $\webleft [A,V\webright ]_{X}\subset \webleft [U,V\webright ]_{X}$.
2. (b)
  If $V\subset B$, then $\webleft [U,V\webright ]_{X}\subset \webleft [U,B\webright ]_{X}$.
3. (c)
  If $U\subset A$ and $V\subset B$, then $\webleft [A,V\webright ]_{X}\subset \webleft [U,B\webright ]_{X}$.
2.
Adjointness. We have adjunctions
witnessed by bijections

\begin{align*} \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U,\webleft [V,W\webright ]_{X}\webright ),\\ \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (V,\webleft [U,W\webright ]_{X}\webright ). \end{align*}

In particular, the following statements hold for each $U,V,W\in \mathcal{P}\webleft (X\webright )$:
1. (a)
  The following conditions are equivalent:
  1. (i)
    We have $U\cap V\subset W$.
  2. (ii)
    We have $U\subset \webleft [V,W\webright ]_{X}$.
2. (b)
  The following conditions are equivalent:
  1. (i)
    We have $U\cap V\subset W$.
  2. (ii)
    We have $V\subset \webleft [U,W\webright ]_{X}$.
3.
Interaction With the Empty Set I. We have

\begin{align*} \webleft [U,\text{Ø}\webright ]_{X} & = U^{\textsf{c}},\\ \webleft [\text{Ø},V\webright ]_{X} & = X, \end{align*}

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
4.
Interaction With $X$. We have

\begin{align*} \webleft [U,X\webright ]_{X} & = X,\\ \webleft [X,V\webright ]_{X} & = V, \end{align*}

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
5.
Interaction With the Empty Set II. The functor

\[ D_{X} \colon \mathcal{P}\webleft (X\webright )^{\mathsf{op}}\to \mathcal{P}\webleft (X\webright ) \]

defined by

\begin{align*} D_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [-,\text{Ø}\webright ]_{X}\\ & = \webleft (-\webright )^{\textsf{c}}\end{align*}

is an involutory isomorphism of categories, making $\text{Ø}$ into a dualising object for $\webleft (\mathcal{P}\webleft (X\webright ),\cap ,X,\webleft [-,-\webright ]_{X}\webright )$ in the sense of . In particular:
1. (a)
  The diagram
  commutes, i.e. we have
  
  \[ \underbrace{D_{X}\webleft (D_{X}\webleft (U\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft [U,\text{Ø}\webright ]_{X},\text{Ø}\webright ]_{X}}=U \]
  
  for each $U\in \mathcal{P}\webleft (X\webright )$.
2. (b)
  The diagram
  commutes, i.e. we have
  
  \[ \underbrace{D_{X}\webleft (U\cap D_{X}\webleft (V\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U\cap \webleft [V,\text{Ø}\webright ]_{X},\text{Ø}\webright ]_{X}}=\webleft [U,V\webright ]_{X} \]
  
  for each $U,V\in \mathcal{P}\webleft (X\webright )$.
6.
Interaction With the Empty Set III. Let $f\colon X\to Y$ be a function.
1. (a)
  Interaction With Direct Images. The diagram
  commutes, i.e. we have
  
  \[ f_{!}\webleft (D_{X}\webleft (U\webright )\webright )=D_{Y}\webleft (f_{*}\webleft (U\webright )\webright ) \]
  
  for each $U\in \mathcal{P}\webleft (X\webright )$.
2. (b)
  Interaction With Inverse Images. The diagram
  commutes, i.e. we have
  
  \[ f^{-1}\webleft (D_{Y}\webleft (U\webright )\webright )=D_{X}\webleft (f^{-1}\webleft (U\webright )\webright ) \]
  
  for each $U\in \mathcal{P}\webleft (X\webright )$.
3. (c)
  Interaction With Codirect Images. The diagram
  commutes, i.e. we have
  
  \[ f_{*}\webleft (D_{X}\webleft (U\webright )\webright )=D_{Y}\webleft (f_{!}\webleft (U\webright )\webright ) \]
  
  for each $U\in \mathcal{P}\webleft (X\webright )$.
7.
Interaction With Unions of Families of Subsets I. The diagram
does not commute in general, i.e. we may have

\[ \bigcup _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W\neq \left[\bigcup _{U\in \mathcal{U}}U,\bigcup _{V\in \mathcal{V}}V\right]_{X} \]

in general, where $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
8.
Interaction With Unions of Families of Subsets II. The diagram
commutes, i.e. we have

\[ \left[\bigcup _{U\in \mathcal{U}}U,V\right]_{X}= \bigcap _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X} \]

for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ and each $V\in \mathcal{P}\webleft (X\webright )$.
9.
Interaction With Unions of Families of Subsets III. The diagram
commutes, i.e. we have

\[ \left[U,\bigcup _{V\in \mathcal{V}}V\right]_{X}= \bigcup _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} \]

for each $U\in \mathcal{P}\webleft (X\webright )$ and each $\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
10.
Interaction With Intersections of Families of Subsets I. The diagram
does not commute in general, i.e. we may have

\[ \bigcap _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W\neq \left[\bigcap _{U\in \mathcal{U}}U,\bigcap _{V\in \mathcal{V}}V\right]_{X} \]

in general, where $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
11.
Interaction With Intersections of Families of Subsets II. The diagram
commutes, i.e. we have

\[ \left[\bigcap _{U\in \mathcal{U}}U,V\right]_{X}= \bigcup _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X} \]

for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ and each $V\in \mathcal{P}\webleft (X\webright )$.
12.
Interaction With Intersections of Families of Subsets III. The diagram
commutes, i.e. we have

\[ \left[U,\bigcap _{V\in \mathcal{V}}V\right]_{X}= \bigcap _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} \]

for each $U\in \mathcal{P}\webleft (X\webright )$ and each $\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
13.
Interaction With Binary Unions. We have equalities of sets

\begin{align*} \webleft [U\cap V,W\webright ]_{X} & = \webleft [U,W\webright ]_{X}\cup \webleft [V,W\webright ]_{X},\\ \webleft [U,V\cap W\webright ]_{X} & = \webleft [U,V\webright ]_{X}\cap \webleft [U,W\webright ]_{X} \end{align*}

for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.
14.
Interaction With Binary Intersections. We have equalities of sets

\begin{align*} \webleft [U\cup V,W\webright ]_{X} & = \webleft [U,W\webright ]_{X}\cap \webleft [V,W\webright ]_{X},\\ \webleft [U,V\cup W\webright ]_{X} & = \webleft [U,V\webright ]_{X}\cup \webleft [U,W\webright ]_{X} \end{align*}

for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.
15.
Interaction With Differences. We have equalities of sets

\begin{align*} \webleft [U\setminus V,W\webright ]_{X} & = \webleft [U,W\webright ]_{X}\cup \webleft [V^{\textsf{c}},W\webright ]_{X}\\ & = \webleft [U,W\webright ]_{X}\cup \webleft [U,V\webright ]_{X},\\ \webleft [U,V\setminus W\webright ]_{X} & = \webleft [U,V\webright ]_{X}\setminus \webleft (U\cap W\webright )\end{align*}

for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.
16.
Interaction With Complements. We have equalities of sets

\begin{align*} \webleft [U^{\textsf{c}},V\webright ]_{X} & = U\cup V,\\ \webleft [U,V^{\textsf{c}}\webright ]_{X} & = U\cap V,\\ \webleft [U,V\webright ]^{\textsf{c}}_{X} & = U\setminus V \end{align*}

for each $U,V\in \mathcal{P}\webleft (X\webright )$.
17.
Interaction With Characteristic Functions. We have

\[ \chi _{\webleft [U,V\webright ]_{\mathcal{P}\webleft (X\webright )}}\webleft (x\webright )=\operatorname*{\operatorname {\mathrm{max}}}\webleft (1-\chi _{U}\ (\mathrm{mod}\ 2),\chi _{V}\webright ) \]

for each $U,V\in \mathcal{P}\webleft (X\webright )$.
18.
Interaction With Direct Images. Let $f\colon X\to Y$ be a function. The diagram
commutes, i.e. we have an equality of sets

\[ f_{!}\webleft (\webleft [U,V\webright ]_{X}\webright )=\webleft [f_{*}\webleft (U\webright ),f_{!}\webleft (V\webright )\webright ]_{Y}, \]

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
19.
Interaction With Inverse Images. Let $f\colon X\to Y$ be a function. The diagram
commutes, i.e. we have an equality of sets

\[ f^{-1}\webleft (\webleft [U,V\webright ]_{Y}\webright )=\webleft [f^{-1}\webleft (U\webright ),f^{-1}\webleft (V\webright )\webright ]_{X}, \]

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
20.
Interaction With Codirect Images. Let $f\colon X\to Y$ be a function. We have a natural transformation
with components

\[ \webleft [f_{!}\webleft (U\webright ),f_{*}\webleft (V\webright )\webright ]_{Y}\subset f_{*}\webleft (\webleft [U,V\webright ]_{X}\webright ) \]

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

Proof of Proposition 4.4.7.1.3.

Item 1: Functoriality

Since $\mathcal{P}\webleft (X\webright )$ is posetal, it suffices to prove Item 1a, Item 1b, and Item 1c.

1.
Proof of Item 1a: We have

\begin{align*} \webleft [A,V\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A^{\textsf{c}}\cup V\\ & \subset U^{\textsf{c}}\cup V\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,V\webright ]_{X}, \end{align*}

where we have used:
1. (a)
  Item 1 of Proposition 4.3.11.1.2, which states that if $U\subset A$, then $A^{\textsf{c}}\subset U^{\textsf{c}}$.
2. (b)
  Item 1a of Item 1 of Proposition 4.3.11.1.2, which states that if $A^{\textsf{c}}\subset U^{\textsf{c}}$, then $A^{\textsf{c}}\cup K\subset U^{\textsf{c}}\cup K$ for any $K\in \mathcal{P}\webleft (X\webright )$.
2.
Proof of Item 1b: We have

\begin{align*} \webleft [U,V\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}\cup V\\ & \subset U^{\textsf{c}}\cup B\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,B\webright ]_{X}, \end{align*}

where we have used Item 1b of Item 1 of Proposition 4.3.11.1.2, which states that if $V\subset B$, then $K\cup V\subset K\cup B$ for any $K\in \mathcal{P}\webleft (X\webright )$.
3.
Proof of Item 1c: We have

\begin{align*} \webleft [A,V\webright ]_{X} & \subset \webleft [U,V\webright ]_{X}\\ & \subset \webleft [U,B\webright ]_{X}, \end{align*}

where we have used Item 1a and Item 1b.

This finishes the proof.

Item 2: Adjointness

This is a repetition of Item 2 of Proposition 4.3.9.1.2 and is proved there.

Item 3: Interaction With the Empty Set I

We have

\begin{align*} \webleft [U,\text{Ø}\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}\cup \text{Ø}\\ & = U^{\textsf{c}}, \end{align*}

where we have used Item 3 of Proposition 4.3.8.1.2, and we have

\begin{align*} \webleft [\text{Ø},V\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Ø}^{\textsf{c}}\cup V\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\setminus \text{Ø}\webright )\cup V\\ & = X\cup V\\ & = X, \end{align*}

where we have used:

1.
Item 12 of Proposition 4.3.10.1.2 for the first equality.
2.
Item 5 of Proposition 4.3.8.1.2 for the last equality.

Since $\mathcal{P}\webleft (X\webright )$ is posetal, naturality is automatic (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2).

Item 4: Interaction With $X$

We have

\begin{align*} \webleft [U,X\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}\cup X\\ & = X, \end{align*}

where we have used Item 5 of Proposition 4.3.8.1.2, and we have

\begin{align*} \webleft [X,V\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X^{\textsf{c}}\cup V\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\setminus X\webright )\cup V\\ & = \text{Ø}\cup V\\ & = V, \end{align*}

where we have used Item 3 of Proposition 4.3.8.1.2 for the last equality. Since $\mathcal{P}\webleft (X\webright )$ is posetal, naturality is automatic (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2).

Item 5: Interaction With the Empty Set II

We have

\begin{align*} D_{X}\webleft (D_{X}\webleft (U\webright )\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft [U,\text{Ø}\webright ]_{X},\text{Ø}\webright ]_{X}\\ & = \webleft [U^{\textsf{c}},\text{Ø}\webright ]_{X}\\ & = \webleft (U^{\textsf{c}}\webright )^{\textsf{c}}\\ & = U, \end{align*}

where we have used:

1.
Item 3 for the second and third equalities.
2.
Item 3 of Proposition 4.3.11.1.2 for the fourth equality.

Since $\mathcal{P}\webleft (X\webright )$ is posetal, naturality is automatic (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2), and thus we have

\[ \webleft [\webleft [-,\text{Ø}\webright ]_{X},\text{Ø}\webright ]_{X}\cong \operatorname {\mathrm{id}}_{\mathcal{P}\webleft (X\webright )} \]

This finishes the proof.

Item 6: Interaction With the Empty Set III

Since $D_{X}=\webleft (-\webright )^{\textsf{c}}$, this is essentially a repetition of the corresponding results for $\webleft (-\webright )^{\textsf{c}}$, namely Item 5, Item 6, and Item 7 of Proposition 4.3.11.1.2.

Item 7: Interaction With Unions of Families of Subsets I

By Item 3 of Proposition 4.4.7.1.3, we have

\begin{align*} \webleft [\mathcal{U},\text{Ø}\webright ]_{\mathcal{P}\webleft (X\webright )} & = \mathcal{U}^{\textsf{c}},\\ \webleft [U,\text{Ø}\webright ]_{X} & = U^{\textsf{c}}. \end{align*}

With this, the counterexample given in the proof of Item 10 of Proposition 4.3.6.1.2 then applies.

Item 8: Interaction With Unions of Families of Subsets II

We have

\begin{align*} \left[\bigcup _{U\in \mathcal{U}}U,V\right]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left(\bigcup _{U\in \mathcal{U}}U\right)^{\textsf{c}}\cup V\\ & = \left(\bigcap _{U\in \mathcal{U}}U^{\textsf{c}}\right)\cup V\\ & = \bigcap _{U\in \mathcal{U}}\webleft (U^{\textsf{c}}\cup V\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X},\end{align*}

where we have used:

1.
Item 11 of Proposition 4.3.6.1.2 for the second equality.
2.
Item 6 of Proposition 4.3.7.1.2 for the third equality.

This finishes the proof.

Item 9: Interaction With Unions of Families of Subsets III

We have

\begin{align*} \bigcup _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcup _{V\in \mathcal{V}}\webleft (U^{\textsf{c}}\cup V\webright )\\ & = U^{\textsf{c}}\cup \left(\bigcup _{V\in \mathcal{V}}V\right)\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left[U,\bigcup _{V\in \mathcal{V}}V\right]_{X}. \end{align*}

where we have used Item 6. This finishes the proof.

Item 10: Interaction With Intersections of Families of Subsets I

Let $X=\left\{ 0,1\right\} $, let $\mathcal{U}=\left\{ \left\{ 0,1\right\} \right\} $, and let $\mathcal{V}=\left\{ \left\{ 0\right\} ,\left\{ 0,1\right\} \right\} $. We have

\begin{align*} \bigcap _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W & = \bigcap _{W\in \mathcal{P}\webleft (X\webright )}W\\ & = \left\{ 0,1\right\} , \end{align*}

whereas

\begin{align*} \left[\bigcap _{U\in \mathcal{U}}U,\bigcap _{V\in \mathcal{V}}V\right]_{X} & = \webleft [\left\{ 0,1\right\} ,\left\{ 0\right\} \webright ]\\ & = \left\{ 0\right\} , \end{align*}

Thus we have

\[ \bigcap _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W=\left\{ 0,1\right\} \neq \left\{ 0\right\} =\left[\bigcap _{U\in \mathcal{U}}U,\bigcap _{V\in \mathcal{V}}V\right]_{X}. \]

This finishes the proof.

Item 11: Interaction With Intersections of Families of Subsets II

We have

\begin{align*} \left[\bigcap _{U\in \mathcal{U}}U,V\right]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left(\bigcap _{U\in \mathcal{U}}U\right)^{\textsf{c}}\cup V\\ & = \left(\bigcup _{U\in \mathcal{U}}U^{\textsf{c}}\right)\cup V\\ & = \bigcup _{U\in \mathcal{U}}\webleft (U^{\textsf{c}}\cup V\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcup _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X},\end{align*}

where we have used:

1.
Item 12 of Proposition 4.3.6.1.2 for the second equality.
2.
Item 6 of Proposition 4.3.7.1.2 for the third equality.

This finishes the proof.

Item 12: Interaction With Intersections of Families of Subsets III

We have

\begin{align*} \bigcap _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{V\in \mathcal{V}}\webleft (U^{\textsf{c}}\cup V\webright )\\ & = U^{\textsf{c}}\cup \left(\bigcap _{V\in \mathcal{V}}V\right)\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left[U,\bigcap _{V\in \mathcal{V}}V\right]_{X}. \end{align*}

where we have used Item 6. This finishes the proof.

Item 13: Interaction With Binary Unions

We have

\begin{align*} \webleft [U\cap V,W\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (U\cap V\webright )^{\textsf{c}}\cup W\\ & = \webleft (U^{\textsf{c}}\cup V^{\textsf{c}}\webright )\cup W\\ & = \webleft (U^{\textsf{c}}\cup V^{\textsf{c}}\webright )\cup \webleft (W\cup W\webright )\\ & = \webleft (U^{\textsf{c}}\cup W\webright )\cup \webleft (V^{\textsf{c}}\cup W\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,W\webright ]_{X}\cup \webleft [V,W\webright ]_{X}, \end{align*}

where we have used:

1.
Item 2 of Proposition 4.3.11.1.2 for the second equality.
2.
Item 8 of Proposition 4.3.8.1.2 for the third equality.
3.
Several applications of Item 2 and Item 4 of Proposition 4.3.8.1.2 and for the fourth equality.

For the second equality in the statement, we have

\begin{align*} \webleft [U,V\cap W\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}\cup \webleft (V\cap W\webright )\\ & = \webleft (U^{\textsf{c}}\cup V\webright )\cap \webleft (U^{\textsf{c}}\cap W\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,V\webright ]_{X}\cap \webleft [U,W\webright ]_{X}, \end{align*}

where we have used Item 6 of Proposition 4.3.8.1.2 for the second equality.

Item 14: Interaction With Binary Intersections

We have

\begin{align*} \webleft [U\cup V,W\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (U\cup V\webright )^{\textsf{c}}\cup W\\ & = \webleft (U^{\textsf{c}}\cap V^{\textsf{c}}\webright )\cup W\\ & = \webleft (U^{\textsf{c}}\cup W\webright )\cap \webleft (V^{\textsf{c}}\cup W\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,W\webright ]_{X}\cap \webleft [V,W\webright ]_{X}, \end{align*}

where we have used:

1.
Item 2 of Proposition 4.3.11.1.2 for the second equality.
2.
Item 6 of Proposition 4.3.8.1.2 for the third equality.

Now, for the second equality in the statement, we have

\begin{align*} \webleft [U,V\cup W\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}\cup \webleft (V\cup W\webright )\\ & = \webleft (U^{\textsf{c}}\cup U^{\textsf{c}}\webright )\cup \webleft (V\cup W\webright )\\ & = \webleft (U^{\textsf{c}}\cup V\webright )\cup \webleft (U^{\textsf{c}}\cup W\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,V\webright ]_{X}\cup \webleft [U,W\webright ]_{X}, \end{align*}

where we have used:

1.
Item 8 of Proposition 4.3.8.1.2 for the second equality.
2.
Several applications of Item 2 and Item 4 of Proposition 4.3.8.1.2 and for the third equality.

This finishes the proof.

Item 15: Interaction With Differences

We have

\begin{align*} \webleft [U\setminus V,W\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (U\setminus V\webright )^{\textsf{c}}\cup W\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\setminus \webleft (U\setminus V\webright )\webright )\cup W\\ & = \webleft (\webleft (X\cap V\webright )\cup \webleft (X\setminus U\webright )\webright )\cup W\\ & = \webleft (V\cup \webleft (X\setminus U\webright )\webright )\cup W\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (V\cup U^{\textsf{c}}\webright )\cup W\\ & = \webleft (V\cup \webleft (U^{\textsf{c}}\cup U^{\textsf{c}}\webright )\webright )\cup W\\ & = \webleft (U^{\textsf{c}}\cup W\webright )\cup \webleft (U^{\textsf{c}}\cup V\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,W\webright ]_{X}\cup \webleft [U,V\webright ]_{X}, \end{align*}

where we have used:

1.
Item 10 of Proposition 4.3.10.1.2 for the third equality.
2.
Item 4 of Proposition 4.3.9.1.2 for the fourth equality.
3.
Item 8 of Proposition 4.3.8.1.2 for the sixth equality.
4.
Several applications of Item 2 and Item 4 of Proposition 4.3.8.1.2 and for the seventh equality.

We also have

\begin{align*} \webleft [U\setminus V,W\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (U\setminus V\webright )^{\textsf{c}}\cup W\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\setminus \webleft (U\setminus V\webright )\webright )\cup W\\ & = \webleft (\webleft (X\cap V\webright )\cup \webleft (X\setminus U\webright )\webright )\cup W\\ & = \webleft (V\cup \webleft (X\setminus U\webright )\webright )\cup W\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (V\cup U^{\textsf{c}}\webright )\cup W\\ & = \webleft (V\cup U^{\textsf{c}}\webright )\cup \webleft (W\cup W\webright )\\ & = \webleft (U^{\textsf{c}}\cup W\webright )\cup \webleft (V\cup W\webright )\\ & = \webleft (U^{\textsf{c}}\cup W\webright )\cup \webleft (\webleft (V^{\textsf{c}}\webright )^{\textsf{c}}\cup W\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,W\webright ]_{X}\cup \webleft [V^{\textsf{c}},W\webright ]_{X}, \end{align*}

where we have used:

1.
Item 10 of Proposition 4.3.10.1.2 for the third equality.
2.
Item 4 of Proposition 4.3.9.1.2 for the fourth equality.
3.
Item 8 of Proposition 4.3.8.1.2 for the sixth equality.
4.
Several applications of Item 2 and Item 4 of Proposition 4.3.8.1.2 and for the seventh equality.
5.
Item 3 of Proposition 4.3.11.1.2 for the eighth equality.

Now, for the second equality in the statement, we have

\begin{align*} \webleft [U,V\setminus W\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}\cup \webleft (V\setminus W\webright )\\ & = \webleft (V\setminus W\webright )\cup U^{\textsf{c}}\\ & = \webleft (V\cup U^{\textsf{c}}\webright )\setminus \webleft (W\setminus U^{\textsf{c}}\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (V\cup U^{\textsf{c}}\webright )\setminus \webleft (W\setminus \webleft (X\setminus U\webright )\webright )\\ & = \webleft (V\cup U^{\textsf{c}}\webright )\setminus \webleft (\webleft (W\cap U\webright )\cup \webleft (W\setminus X\webright )\webright )\\ & = \webleft (V\cup U^{\textsf{c}}\webright )\setminus \webleft (\webleft (W\cap U\webright )\cup \text{Ø}\webright )\\ & = \webleft (V\cup U^{\textsf{c}}\webright )\setminus \webleft (W\cap U\webright )\\ & = \webleft (V\cup U^{\textsf{c}}\webright )\setminus \webleft (U\cap W\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,V\webright ]_{X}\setminus \webleft (U\cap W\webright ) \end{align*}

where we have used:

1.
Item 4 of Proposition 4.3.8.1.2 for the second equality.

Item 4 of Proposition 4.3.10.1.2 for the third equality.

Item 10 of Proposition 4.3.10.1.2 for the fifth equality.

Item 13 of Proposition 4.3.10.1.2 for the sixth equality.

Item 3 of Proposition 4.3.8.1.2 for the seventh equality.

Item 5 of Proposition 4.3.9.1.2 for the eighth equality.

This finishes the proof.

Item 16: Interaction With Complements

We have

\begin{align*} \webleft [U^{\textsf{c}},V\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (U^{\textsf{c}}\webright )^{\textsf{c}}\cup V,\\ & = U\cup V, \end{align*}

where we have used Item 3 of Proposition 4.3.11.1.2. We also have

\begin{align*} \webleft [U,V^{\textsf{c}}\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}\cup V^{\textsf{c}}\\ & = U\cap V\\ \end{align*}

where we have used Item 2 of Proposition 4.3.11.1.2. Finally, we have

\begin{align*} \webleft [U,V\webright ]^{\textsf{c}}_{X} & = \webleft (\webleft (U\setminus V\webright )^{\textsf{c}}\webright )^{\textsf{c}}\\ & = U\setminus V, \end{align*}

where we have used Item 2 of Proposition 4.3.11.1.2.

Item 17: Interaction With Characteristic Functions

We have

\begin{align*} \chi _{\webleft [U,V\webright ]_{\mathcal{P}\webleft (X\webright )}}\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{U^{\textsf{c}}\cup V}\webleft (x\webright )\\ & = \operatorname*{\operatorname {\mathrm{max}}}\webleft (\chi _{U^{\textsf{c}}},\chi _{V}\webright )\\ & = \operatorname*{\operatorname {\mathrm{max}}}\webleft (1-\chi _{U}\ (\mathrm{mod}\ 2),\chi _{V}\webright ), \end{align*}

where we have used: