# Set Union expressed as Intersection Complement

## Theorem

Let $A$ and $B$ be subsets of a universal set $\Bbb U$.

Let $\uparrow$ denote the operation on $A$ and $B$ defined as:

$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$

where $\relcomp {\Bbb U} A$ denotes the complement of $A$ in $\Bbb U$.

Then:

$A \cup B = \paren {A \uparrow A} \uparrow \paren {B \uparrow B}$

## Proof

 $\ds A \cup B$ $=$ $\ds \relcomp {\Bbb U} {\relcomp {\Bbb U} A \cap \relcomp {\Bbb U} B}$ De Morgan's Laws : Complement of Union $\ds$ $=$ $\ds \relcomp {\Bbb U} {\paren {A \uparrow A} \cap \paren {B \uparrow B} }$ Intersection Complement of Set with Itself is Complement $\ds$ $=$ $\ds \paren {A \uparrow A} \uparrow \paren {B \uparrow B}$ Intersection Complement of Set with Itself is Complement

$\blacksquare$