Powerset
https://en.wikipedia.org/wiki/Power_set
The powerset of any set is the set of all its subsets (including the empty set and itself) denoted by .

𝒫(S) ordered by inclusion
Given the consist of all possible subsets of :
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.
Identifying the powerset of with the set of all functions from to a given set of two elements, .
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 contains all the possible subsets of a set (including the empty set), then is a powerset of , denoted as .
For example, if , then
The cardinality of a power set of a set is .
Notation
ASCII: P(S),
Unicode glyphs: 𝒫(S), ℘(S), ℙ(S)
latex keywords:
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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