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'complement'
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| Title of object: |
complement |
| Canonical Name: |
Complement |
| Type: |
Definition |
| Created on: |
2002-02-13 03:02:28 |
| Modified on: |
2005-02-14 21:09:45 |
| Classification: |
msc:03E99 |
Revision comment (for changes between this and next version):
| Changes for correction #7749 ('typo'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
\section{Definition}
Let $A$ be a subset of $X$. The {\em complement} of $A$ in $X$ (denoted $A^\complement$ when the larger set $X$ is clear from context) is the set difference $X \setminus A$.
\section{Properties}
\begin{itemize}
\item $(A^{\complement})^\complement=A$
\item $\emptyset^\complement = X$
\item $X^\complement = \emptyset$
\item If $A$ and $B$ are subsets of $X$, then
$A\setminus B = A\cap B^\complement$, where the complement is taken in $X$.
\end{itemize}
\section{de Morgans laws}
Let $X$ be a set with subsets $A_i \subset X$ for $i\in I$, where
$I$ is an arbitrary index-set. In other words, $I$ can be finite,
countable, or uncountable. Then
\begin{eqnarray*}
\left( \bigcup_{i\in I} A_i \right)^\complement &=& \bigcap_{i\in I} A_i^\complement, \\
\left( \bigcap_{i\in I} A_i \right)^\complement &=& \bigcup_{i\in I} A_i^\complement.
\end{eqnarray*} |
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