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

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $16$