Set Union expressed as Intersection Complement

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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

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

$\blacksquare$


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