Difference between revisions of "Complement"

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(Created page with "{{Extra Maths}} ==Definition== The complement of a set is everything not in it. For example given a set {{M|A}} in a space {{M|X}} the complement of {{M|A}} (often denoted {{M...")
 
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{{Definition|Set Theory}}

Latest revision as of 13:28, 18 March 2015

[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: