Powerset

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

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

𝒫(S) ordered by inclusion

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

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,zP(S)={{},{x},{y},{z},{x,y},{x,z},{y,z},{x,y,z}} P(S)SP(S) P(S)SP(S)sS.{s}P(S)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 SS with the set of all functions from SS to a given set of two elements, 2S2S.

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

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

The cardinality of a power set of a set XX is 2X2^{|X|}.

Notation

  • ASCII: P(S),

  • Unicode glyphs: 𝒫(S), ℘(S), ℙ(S)

  • latex keywords: P(S),(S),P(S),P(S)\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)

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