|
|
|
Viewing Version
9
of
'power set'
|
[ view 'power set'
|
back to history
]
| Title of object: |
power set |
| Canonical Name: |
PowerSet |
| Type: |
Definition |
| Created on: |
2001-10-06 17:06:00 |
| Modified on: |
2004-02-29 16:20:05 |
| Classification: |
msc:03E10, msc:03E99 |
| Keywords: |
Set, Power, Cardinality |
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
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
\newcommand{\powset}[1]{\mathcal{P}(#1)} |
Content:
\PMlinkescapeword{states}
\PMlinkescapeword{property}
{\bf Definition}
If $X$ is a set, then the {\bf power set of $X$}, denoted by $\powset{X}$, is the
set whose elements are the subsets of $X$.
\subsubsection*{Properties}
\begin{enumerate}
\item If $X$ is finite, then $|\powset{X}|=2^{|X|}$.
\item For an arbitrary set $X$, Cantor's theorem states:
a) there is no bijection between $X$ and $\powset{X}$, and
b) the cardinality of $\powset{X}$ is greater than the cardinality of $X$.
\end{enumerate}
\subsubsection*{Example}
Suppose $S=\{a,b\}$. Then $\powset{S}=\{\emptyset, \{a\}, \{b\}, S\}$.
In particular, $|\powset{S}|=2^{|S|}=4$.
\subsubsection*{Remark}
In view of the first property above, the power set is sometimes also
denoted by $2^X$. |
|
|
|
|
|
