power set

See also: powerset

English

Alternative forms

• power-set (attributive use)
• powerset

Noun

power set (plural power sets)

1. (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 $\{1,2\}$ is $\left\{\emptyset ,\{1\},\{2\},\{1,2\}\right\}$.

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

     Moreover, for notational convenience, we write the cardinality of a denumerable set as $\aleph _{0}$. Cardinality of the power set of a denumerable set is written as $\aleph _{1}$. We may thus extend this notation further by taking cardinality of the power set of the power set of a denumerable set as $\aleph _{2}$, 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) ca-free 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

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: ${\mathcal {P}}(S)$, $\wp (S)$ (with the Weierstrass p), $\mathbb {P} (S)$ and 𝒫(S).
An alternative notation is $2^{S}\!\!$, derived from the consideration that a set $T$ in the power set is fully characterised by determining, for each element of $S$, whether it is or is not in $T$.

Derived terms

• hyper-power set

Translations

set of all subsets of a set

• Chinese:

  Mandarin: 冪集／幂集 (mìjí)

• Czech: potenční množina f
• Dutch: machtenverzameling
• French: ensemble des parties m
• German: Potenzmenge (de) f
• Hebrew: קבוצת חוזקה \ קְבוּצַת חָזְקָה f (k'vutzát khozká)
• Hungarian: hatványhalmaz (hu)
• Icelandic: veldismengi n
• Italian: insieme delle parti m, insieme potenza m
• Japanese: 冪集合 (ja) (bekishūgō)
• Polish: zbiór potęgowy m
• Portuguese: conjunto de partes m
• Russian: булеан (ru) (bulean)
• Serbo-Croatian: partitivni skup m
• Spanish: conjunto de partes m
• Swedish: potensmängd (sv) c

See also

• axiom of power set

Further reading

• Axiom of power set on Wikipedia.Wikipedia