Power set

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The power set of a set [math]X[/math] is denoted by [math]\mathcal{P}(X)[/math], sometimes [math]2^X[/math] (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 [math]2^{|X|}[/math])

The characteristic property of the power set is that [math]\forall U\subset X:U\in\mathcal{P}(X)[/math]

It is the set of all subsets of [math]X[/math]