# 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

 $\displaystyle A \uparrow A$ $=$ $\displaystyle \relcomp {\Bbb U} {A \cap A}$ Definition of $\uparrow$ $\displaystyle$ $=$ $\displaystyle \relcomp {\Bbb U} A$ Intersection is Idempotent

$\blacksquare$