# 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 $2^n$ elementsâ€Ž.

## 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