# Union of Union of Cartesian Product

## Theorem

Let $A$ and $B$ be sets such that $A \ne \O$ and $B \ne \O$.

Let the ordered pair $\tuple {a, b}$ be defined using the Kuratowski formalization:

$\tuple {a, b} := \set {\set a, \set {a, b} }$

Then:

$\ds \bigcup \bigcup \paren {A \times B} = A \cup B$

where:

$\cup$ denotes union
$\times$ denotes Cartesian product.

## Proof

 $\ds \bigcup \bigcup \paren {A \times B}$ $=$ $\ds \bigcup \bigcup \set {\tuple {a, b}: a \in A, b \in B}$ Definition of Cartesian Product $\ds$ $=$ $\ds \bigcup \paren {\bigcup \set {\set {\set a, \set {a, b} }: a \in A, b \in B} }$ Definition of Kuratowski Formalization of Ordered Pair $\ds$ $=$ $\ds \bigcup \set {\set {a, b}: a \in A, b \in B}$ Definition of Set Union: $\ds \bigcup \set {\set a, \set {a, b} } = \set {a, b}$ $\ds$ $=$ $\ds \set {x: x \in A \text { or } x \in B}$ Definition of Set Union $\ds$ $=$ $\ds A \cup B$ Definition of Set Union

$\blacksquare$