# Bijection between Power Set of Disjoint Union and Cartesian Product of Power Sets

## Theorem

Let $S$ and $T$ be disjoint sets.

Let $\powerset S$ denote the power set of $S$.

Then there exists a bijection between $\powerset {S \cup T}$ and $\paren {\powerset S} \times \paren {\powerset T}$.

Hence:

$\powerset {S \cup T} \sim \paren {\powerset S} \times \paren {\powerset T}$

where $\sim$ denotes set equivalence.

## Proof

Let $\phi: \paren {\powerset S} \times \paren {\powerset T} \to \powerset {S \cup T}$ be defined as:

$\forall \tuple {A, B} \in \paren {\powerset S} \times \paren {\powerset T}: \map \phi {A, B} = A \cup B$

In order to show that $\phi$ is a bijection, it needs to be shown that $\phi$ has the following properties:

$\phi$ is a mapping, that is:
$\phi$ is left-total
$\phi$ is many-to-one
$\phi$ is a surjection
$\phi$ is an injection.

Let $A \subseteq S$ and $B \subseteq T$.

Then:

 $\ds A$ $\in$ $\ds \powerset S$ Definition of Power Set $\, \ds \land \,$ $\ds B$ $\in$ $\ds \powerset T$ $\ds \leadsto \ \$ $\ds \tuple {A, B}$ $\in$ $\ds \paren {\powerset S} \times \paren {\powerset T}$ Definition of Cartesian Product

Thus:

$\forall A \subseteq S, B \subseteq T: \tuple {A, B} \in \Dom \phi$

and so $\phi$ is left-total.

$\Box$

Let $A_1, A_2 \subseteq S$ and $B_1, B_2 \subseteq T$.

Then:

 $\ds \map \phi {A_1, B_1}$ $\ne$ $\ds \map \phi {A_2, B_2}$ $\ds \leadsto \ \$ $\ds A_1 \cup B_1$ $\ne$ $\ds A_2 \cup B_2$ Definition of $\phi$ $\ds \leadsto \ \$ $\ds A_1 \cup B_1$ $\nsubseteq$ $\ds A_2 \cup B_2$ Definition of Set Equality $\, \ds \lor \,$ $\ds A_2 \cup B_2$ $\nsubseteq$ $\ds A_1 \cup B_1$

Without loss of generality, suppose $A_1 \cup B_1 \nsubseteq A_2 \cup B_2$.

Then:

 $\ds \exists x \in A_1 \cup B_1: \ \$ $\ds x$ $\notin$ $\ds A_2 \cup B_2$ $\ds \leadsto \ \$ $\ds x \in A_1 \lor x \in B_1: \ \$ $\ds x \notin A_2$ $\land$ $\ds x \notin B_2$ $\ds \leadsto \ \$ $\ds A_1 \ne A_2$ $\lor$ $\ds A_1 \ne B_2$ $\, \ds \land \,$ $\ds B_1 \ne A_2$ $\lor$ $\ds B_1 \ne B_2$ $\ds \leadsto \ \$ $\ds \tuple {A_1, B_1}$ $\ne$ $\ds \tuple {A_2, B_2}$

That is:

$\map \phi {A_1, B_1} \ne \map \phi {A_2, B_2} \implies \tuple {A_1, B_1} \ne \tuple {A_2, B_2}$

demonstrating that $\phi$ is many-to-one.

Thus $\phi$ has been shown to be a mapping.

$\Box$

Let $A \cup B \in \powerset {S \cup T}$.

Then:

$\tuple {A, B} \in \paren {\powerset S} \times \paren {\powerset T}$

by definition of power set and Cartesian product.

That is, $\phi$ is a surjection by definition.

$\Box$

It remains to be shown that $\phi$ is an injection.

Let $\tuple {A_1, B_1} \ne \tuple {A_2, B_2}$.

Then either $A_1 \ne A_2$ or $B_1 \ne B_2$.

That is, at least one of the following holds:

 $\ds A_1$ $\nsubseteq$ $\ds A_2$ $\ds A_2$ $\nsubseteq$ $\ds A_1$ $\ds B_1$ $\nsubseteq$ $\ds B_2$ $\ds B_2$ $\nsubseteq$ $\ds B_1$

Without loss of generality, suppose $A_1 \nsubseteq A_2$.

Then:

$\exists x \in A_1: x \notin A_2$

and so by Set is Subset of Union:

$x \in A_1 \cup B_1$

But we have that $S$ and $T$ are disjoint.

That is:

$x \in A_1 \implies x \notin B_2$

Thus:

$\exists x \in A_1 \cup B_1: x \notin A_2 \cup B_2$

and so:

$A_1 \cup B_1 \ne A_2 \cup B_2$

That is:

$\map \phi {A_1, B_1} \ne \map \phi {A_2, B_2}$

demonstrating that $\phi$ is an injection.

Hence the result.

$\blacksquare$