Definition:Power Set

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Definition

The power set of a set $S$ is the set defined and denoted as:

$\powerset S := \set {T: T \subseteq S}$

That is, the set whose elements are all of the subsets of $S$.


Note that this is a set all of whose elements are themselves sets.


It is clear from the definition that:

$T \in \powerset S \iff T \subseteq S$


Axiomatic Set Theory

The concept of the power set is axiomatised in the Axiom of Powers in Zermelo-Fraenkel set theory:


For every set, there exists a set of sets that contains amongst its elements all the subsets of the given set.

$\forall x: \exists y: \left({\forall z: \left({z \in y \iff \left({w \in z \implies w \in x}\right)}\right)}\right)$


Also known as

The rendition powerset is frequently seen.

Some sources do not use the term power set, merely referring to the term set of all subsets.


Variants of $\mathcal P$ are seen throughout the literature: $\mathfrak P, P, \mathscr P, \mathrm P, \mathbf P$, etc.

Some sources, for example J.A. Green: Sets and Groups, use $\mathscr B$


Another significant notation is:

$2^S := \set {T: T \subseteq S}$

This is used by, for example, 1971: Allan Clark: Elements of Abstract Algebra.

The relevance of this latter notation is clear from the fact that if $S$ has $n$ elements, then $2^S$ has $2^n$ elements‎.


Examples

Set of 3 Elements

Let $S = \set {a, b, c}$.

Then the power set of $S$ is:

$\powerset S = \set {\O, \set a, \set b, \set c, \set {a, b}, \set {b, c}, \set {a, c}, S}$

and so has $2^3 = 8$ elements.

Note that while $\set a \in \powerset S$, $a \notin \powerset S$.


Also see

  • Results about power sets can be found here.


Sources