> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/200-set-theory/topics/powerset.md).

# Powerset

<https://en.wikipedia.org/wiki/Power_set>

The **powerset** of any set $$S$$ is the set of all its subsets (including the empty set and $$S$$ itself) denoted by $$\mathcal{P}(S)$$.

![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/317px-Hasse_diagram_of_powerset_of_3.svg.png)

*𝒫(S) ordered by inclusion*

Given $$S = {x, y, z}$$ the $$\mathcal{P}(S)$$ consist of all possible subsets of $$S$$:

$$\mathcal{P}(S) = { \ \quad {}, \qquad \small\text{ the empty set is a subset of any set } \ \quad {x}, \qquad \quad\small\text{ for each element } \ \quad {y}, \qquad \quad\small\text{ α of S: {α} is an } \ \quad {z}, \qquad \quad\small\text{ element of 𝒫(S) } \ \quad {x, y}, \qquad \ \quad {x, z}, \qquad \ \quad {y, z}, \qquad \ \quad {x, y, z} \quad \small\text{ any set is an element & subset of its 𝒫 } \ }$$

## Powerset properties

**The empty set** is a subset of any set, ∅ ∈ S, but not an explicit member.

hence besides always being a subset of any set, , it is also an element and a subset of the powerset of any set.

$$
S = {x,y,z}  \\
\mathcal{P}(S)={{},{x},{y},{z},{x,y},{x,z},{y,z},{x,y,z}} \\
\ \\
\quad \star \mathcal{P}(S) \ni S \\
\quad \star \mathcal{P}(S) \ni \varnothing \\
\ \\
\quad \star \varnothing \in \mathcal{P}(S) \\
\quad \star S \in \mathcal{P}(S) \\
\quad \star \forall s \in S . {s} \in \mathcal{P}(S)  \\
$$

Identifying the powerset of $$S$$ with the set of all functions from $$S$$ to a given set of two elements, $$2S$$.

In axiomatic set theory (e.g. ZFC), the existence of the power set of any set is postulated by **the axiom of power set**.

Any subset of P(S) is called a family of sets over S.

If a set $$\mathcal{P}$$ contains all the possible subsets of a set $$X$$ (including the empty set), then $$\mathcal{P}$$ is a **powerset** of $$X$$, denoted as $$\mathcal{P}(X)$$.

For example, if $$X={a,b}$$, then $$\mathcal{P}(X) ={{a},{b},{ab},{\varnothing}}$$

The cardinality of a power set of a set $$X$$ is $$2^{|X|}$$.

## Notation

* ASCII: P(S),
* Unicode glyphs: 𝒫(S), ℘(S), ℙ(S)
* latex keywords: $$\mathcal{P}(S),\wp(S),\mathbb{P}(S),\mathscr{P}(S)$$

## Leftovers

r ℘ ℘

```
{}           ... the empty set is a subset of any set
{x}           \
{y}            > individual elements of S
{z}           /
{x, y}
{x, z}
{y, z}
{x, y, z}    ... any set S is an element of 𝒫(S)
```