Definition:Power Set/Class Theory
Definition
The power set of a set $x$ is the class of all the subsets of $x$:
- $\powerset x := \set {y: y \subseteq x}$
It is clear from the definition that:
- $y \in \powerset x \iff y \subseteq x$
Axiom of Powers
The concept of the power set is axiomatised in the Axiom of Powers in class theory:
Let $x$ be a set.
Then its power set $\powerset x$ is also a set.
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 $\PP$ 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, 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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 6$ The power axiom