power set
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See also: powerset
English[edit]
Alternative forms[edit]
Noun[edit]
power set (plural power sets)
 (set theory, of a set S) The set whose elements comprise all the subsets of S (including the empty set and S itself).
 The power set of is .
 2009, Arindama Singh, Elements of Computation Theory, Springer, page 16:
 Moreover, for notational convenience, we write the cardinality of a denumerable set as . Cardinality of the power set of a denumerable set is written as . We may thus extend this notation further by taking cardinality of the power set of the power set of a denumerable set as , etc. but we do not have the need for it right now.
 2013, A. Carsetti, Epistemic Complexity and Knowledge Construction, Springer, page 94:
 Theorem 4.1. A complete Boolean algebra B has a set of (complete and atomic) cafree generators iff B is isomorphic to the power set of a power set.
 2015, Amir D. Aczel, Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers, Palgrave MacMillan, page 147:
 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!
Usage notes[edit]
The power set is more properly a family of sets (or possibly, in this case, a family of subsets), rather than a set.
Denoted using the notation P(S) with any one of several fonts for the letter "P" (usually uppercase). Examples include: , (with the Weierstrass p), and 𝒫(S).
An alternative notation is , derived from the consideration that a set in the power set is fully characterised by determining, for each element of , whether it is or is not in .
Derived terms[edit]
Translations[edit]
set of all subsets of a set

See also[edit]
Further reading[edit]
 Axiom of power set on Wikipedia.Wikipedia