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

$$X$$