power setPowerSet2001-10-06 17:06:002007-07-25 10:56:14Definitionfinite power setfinite powersetSetPowerCardinality% this is the default PlanetMath preamble. as your knowledge
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{\bf Definition}
If $X$ is a set, then the \emph{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 The above property also holds when $X$ is not finite.
For a set $X$, let $|X|$ be the cardinality of $X$.
Then $|\powset{X}|=2^{|X|}=|2^X|$,
where $2^X$ is the set of all functions from $X$ to $\{0,1\}$.
\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*{Related definition}
If $X$ is a set, then the \emph{finite power set of $X$}, denoted by $\mathcal{F}(X)$, is the
set whose elements are the {\bf finite} subsets of $X$.
\subsubsection*{Remark}
Due to the canonical correspondence between elements of $\powset{X}$ and elements of $2^X$, the power set is sometimes also denoted by $2^X$.