Complement
[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]
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:
[math]A^c=\{x\in X|x\notin A\}=X-A[/math]
It may also be written using set subtraction
Examples
Take [ilmath]X=\mathbb{R}[/ilmath] and [math]A=[0,1)=\{x\in\mathbb{R}|0\le x< 1\}[/math] then [math]A^c=(-\infty,0)\cup[1,\infty)[/math]
Cartesian products
Theorem: [math][A\times B]^c=[A^c\times B^c]\udot[A^c\times B]\udot[A\times B^c][/math]
TODO:
