complement
1 Definition
Let $A$ be a subset of $X$. The complement of $A$ in $X$ (denoted ${A}^{\mathrm{\complement}}$ when the larger set $X$ is clear from context) is the set difference^{} $X\setminus A$.
The Venn diagram^{} below illustrates the complement of $A$ in red.
2 Properties

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${({A}^{\mathrm{\complement}})}^{\mathrm{\complement}}=A$

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${\mathrm{\varnothing}}^{\mathrm{\complement}}=X$

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${X}^{\mathrm{\complement}}=\mathrm{\varnothing}$

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If $A$ and $B$ are subsets of $X$, then $A\setminus B=A\cap {B}^{\mathrm{\complement}}$, where the complement is taken in $X$.
3 de Morgan’s laws
Let $X$ be a set with subsets ${A}_{i}\subset X$ for $i\in I$, where $I$ is an arbitrary indexset. In other words, $I$ can be finite, countable^{}, or uncountable. Then
${\left({\displaystyle \bigcup _{i\in I}}{A}_{i}\right)}^{\mathrm{\complement}}$  $=$  $\bigcap _{i\in I}}{A}_{i}^{\mathrm{\complement}},$  
${\left({\displaystyle \bigcap _{i\in I}}{A}_{i}\right)}^{\mathrm{\complement}}$  $=$  $\bigcup _{i\in I}}{A}_{i}^{\mathrm{\complement}}.$ 
Title  complement 

Canonical name  Complement 
Date of creation  20130322 12:18:51 
Last modified on  20130322 12:18:51 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03E99 
Related topic  DeMorgansLaws 