Difference between revisions of "Complement"
From Maths
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Latest revision as of 13:28, 18 March 2015
\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
[Expand]
Theorem: [A\times B]^c=[A^c\times B^c]\udot[A^c\times B]\udot[A\times B^c]
