Noun
power set (plural power sets)
(set theory, of a set S) The set whose elements comprise all the subsets of S (including the empty set and S itself).
The power set of
{
1
,
2
}
{\displaystyle \{1,2\}}
is
{
∅
,
{
1
}
,
{
2
}
,
{
1
,
2
}
}
{\displaystyle \left\{\emptyset,\{1\},\{2\},\{1,2\}\right\}}
.
A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem : for every set S the power set of S; that is, the set of all subsets of S (here written as P(S)), has a larger cardinality than S itself. Source: Internet
From this perspective the idea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structure or algebra. Source: Internet
For example, the power set of some set, partially ordered by set inclusion, is a filter. Source: Internet
For any set A, the power set is a Boolean algebra under the operations of union and intersection. Source: Internet
For infinite Boolean algebras this is no longer true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra (see Stone's representation theorem ). Source: Internet
Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be larger than the original set). Source: Internet