# power set

## English

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### 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 ${\displaystyle \{1,2\}}$ 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!

#### 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: ${\displaystyle {\mathcal {P}}(S)}$, ${\displaystyle \wp (S)}$ (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 ${\displaystyle T}$.