Definition: Power Set

Let $X$ be a set. The axiom of power set and the axiom of empty ensure * the existence of the power set, \(\mathcal P(X) :=\{A\mid A\subseteq X\}\), i.e. the set of all possible subsets of \(X\)1, and
* that the empty set \(\emptyset\) is a subset any non-empty set \(X\).

Applications: 1
Axioms: 2
Corollaries: 3
Definitions: 4 5 6
Examples: 7 8
Explanations: 9
Parts: 10
Proofs: 11 12 13 14
Propositions: 15 16

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  1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition


  1. Please note that the power set of the empty set is \(\mathcal P(\emptyset)=\{\emptyset,\{\emptyset\}\}\).