Set Intersection 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:
- $A \cap B = \paren {A \uparrow B} \uparrow \paren {A \uparrow B}$
Proof
\(\ds A \cap B\) | \(=\) | \(\ds \relcomp {\Bbb U} {\relcomp {\Bbb U} {A \cap B} }\) | Relative Complement of Relative Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \relcomp {\Bbb U} {A \uparrow B}\) | Intersection Complement of Set with Itself is Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {A \uparrow B} \uparrow \paren {A \uparrow B}\) | 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 $16$