Powersetとは 意味・読み方・使い方

別の表記

• power-set (attributive use)
• powerset

名詞

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

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

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

使用する際の注意点

Denoted using the notation P(S) with any one of several fonts for the letter "P" (usually uppercase). Examples include: ${\displaystyle {\mathcal {P}}(S)}$, ${\displaystyle \wp (S)}$ (with the Weierstrass p), ${\displaystyle \mathbb {P} (S)}$ and 𝒫(S).
An alternative notation is ${\displaystyle 2^{S}\!\!}$, derived from the consideration that a set ${\displaystyle T}$ in the power set is fully characterised by determining, for each element of ${\displaystyle S}$, whether it is or is not in ${\displaystyle T}$.

派生語

• hyper-power set

参考

• axiom of power set

Further reading

• Axiom of power set on Wikipedia.Wikipedia