# Power set

The power set of a set $X$ is denoted by $\mathcal{P}(X)$, sometimes $2^X$ (a number to the power of a set is not defined (as it cannot be usefully defined) leaving it free to be used as notation. This comes from the cardinality of the power set being $2^{|X|}$)
The characteristic property of the power set is that $\forall U\subset X:U\in\mathcal{P}(X)$
It is the set of all subsets of $X$