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 $\mathcal{P}(S)$.

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)