# Complement

(Redirected from Complementation)


## Definition

The complement of a set is everything not in it. For example given a set [ilmath]A[/ilmath] in a space [ilmath]X[/ilmath] the complement of [ilmath]A[/ilmath] (often denoted [ilmath]A^c[/ilmath], [ilmath]A'[/ilmath] or [ilmath]C(A)[/ilmath]) is given by:

$A^c=\{x\in X|x\notin A\}=X-A$

It may also be written using set subtraction

## Examples

Take [ilmath]X=\mathbb{R}[/ilmath] and $A=[0,1)=\{x\in\mathbb{R}|0\le x< 1\}$ then $A^c=(-\infty,0)\cup[1,\infty)$

## Cartesian products

Theorem: $[A\times B]^c=[A^c\times B^c]\udot[A^c\times B]\udot[A\times B^c]$

TODO: