Intersection Complement of Set with Itself is 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:
- $A \uparrow A = \relcomp {\Bbb U} A$
Proof
\(\ds A \uparrow A\) | \(=\) | \(\ds \relcomp {\Bbb U} {A \cap A}\) | Definition of $\uparrow$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \relcomp {\Bbb U} A\) | Set Intersection is Idempotent |
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $16$