Complement

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$

Definition

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

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

It may also be written using set subtraction

Examples

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

Cartesian products

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

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TODO:

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