View source for Power set

{{TEX|done}}{{MSC|03E}}

''of a set $X$''

The set of all subsets of $X$, denoted $\mathcal{P}(X)$.  One has $A \in \mathcal{P}(X) \Leftrightarrow A \subseteq X$.  The power set of a finite set of $n$ elements has $2^n$ elements.  [[Cantor theorem|Cantor's theorem]] states that a set and its power set can never be put into [[one-to-one correspondence]], hence cannot have the same [[cardinality]].

The power set forms a [[Boolean algebra]] with the operations of [[union of sets]], [[intersection of sets]] and [[relative complement]].

If $f$ is a map from $X$ to $Y$ then there are associated maps $f_\vdash : \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ and $f^\dashv : \mathcal{P}(Y) \rightarrow \mathcal{P}(X)$ defined for $A \in \mathcal{P}(X)$, $B \in \mathcal{P}(Y)$ by
$$
f_\vdash(A) = \{ y \in Y : \exists a \in A\,,\, y = f(a) \} \ ;
$$
$$
f^\dashv(B) = \{ x \in X : \exists b \in B\,,\, b = f(x) \} \ .
$$ 
Alternative notations are $f[A]$, $f^{-1}[B]$ respectively.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  P. R. Halmos, ''Naive Set Theory'', Springer (1960) {{ISBN|0-387-90092-6}}</TD></TR>
</table>