Definition:Power Set

Definition

The power set (or powerset) of a set $S$, denoted $\mathcal P \left({S}\right)$, is the set defined as follows:

$\mathcal P \left({S}\right) := \left\{{T: T \subseteq S}\right\}$

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:

$T \in \mathcal P \left({S}\right) \iff T \subseteq S$.

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

Axiomatic Set Theory

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

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

Alternative notations

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

Another significant notation is:

$2^S := \left\{ {T: T \subseteq S}\right\}$

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

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

Also see

• Cardinality of Power Set

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 1$
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 5$: Complements and Powers
• W.E. Deskins: Abstract Algebra (1964): $\S 1.2$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.8$: Example $25$
• Seth Warner: Modern Algebra (1965): $\S 1$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $6$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 14$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 6.7$
• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.2$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.4$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.6$