power set

plural power sets · alternative power-set · alternative powerset

1. mathematicsThe set whose elements comprise all the subsets of S (including the empty set and S itself).

   The power set of #92;#123;1,2#92;#125; is #92;left#92;#123;#92;empty,#92;#123;1#92;#125;,#92;#123;2#92;#125;,#92;#123;1,2#92;#125;#92;right#92;#125;.

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   Moreover, for notational convenience, we write the cardinality of a denumerable set as #92;aleph#95;0. Cardinality of the power set of a denumerable set is written as #92;aleph#95;1. We may thus extend this notation further by taking cardinality of the power set of the power set of a denumerable set as #92;aleph#95;2, etc. but we do not have the need for it right now.

   2009, Arindama Singh, Elements of Computation Theory, Springer, page 16

   Theorem 4.1. A complete Boolean algebra B has a set of (complete and atomic) ca-free generators iff B is isomorphic to the power set of a power set.

   2013, A. Carsetti, Epistemic Complexity and Knowledge Construction, Springer, page 94

   Exponentiation is essentially a move to the power set—the set of all subsets of a given set. This is one of the reasons why Bertrand Russell's paradox is indeed a paradox: We cannot find a universal set because no set can contain its own power set!

   2015, Amir D. Aczel, Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers, Palgrave MacMillan, page 147