Power set

As the power set is defined in set theory the set of all subsets of a given basic set. One lists the power set of a set usually as. The essence of the power set has already been examined by Ernst Zermelo. The compact term " power set ", however - which is ideal in the context of the power arithmetic operation - has not yet been used by Gerhard Hessenberg in his textbook of 1906; he used for the phrase " set of all subsets ".

Definition

The power set of a set is a new set consisting of all subsets of. The power set is thus a quantity system, ie, a set whose elements are themselves sets. In formula notation is the definition of a power set

It should be noted that both the empty set and the set itself subsets is. Other common notations for the power set are and.

Examples

Structures on the power set

Partial order

The inclusion relation is a partial order on ( and no total order if at least two elements has ). The smallest element of order is the largest element.

Complete Lattice

The partial order is a complete lattice. This means that there is for every subset of an infimum and a supremum ( in ). Specifically, for a lot of the infimum equal to the average of the elements of, and the supremum of is equal to the union of the elements of, ie

The largest and the smallest element is obtained as the infimum and supremum of the empty set, so

Boolean Association

If we take even the Komplementabbildung approach, is a Boolean association, ie a distributive and complementary dressing.

Commutative ring

Each Boolean association induces clearly a commutative ring structure, the so-called Boolean ring. Here the ring is given by adding the symmetric difference of quantities of the ring is the average of the multiplication. The empty set is neutral for addition and is neutral for multiplication.

Characteristic Functions

Each subset can be assigned to the characteristic function, which applies

This assignment is a bijection between, and ( where the notation for the set of functions used by after ). This motivates for the spelling, because in the von Neumann model of natural numbers is (in general ).

The correspondence is initially a pure bijection, but can be easily detected as isomorphism with respect to each of the above structures considered on of the power set.

The size of the power set ( cardinality)

Denotes the cardinality of a set.

  • For finite sets, the following applies:.
  • Always the set of Cantor applies.

The transition to the power set so always provides a greater power. Similar to finite sets you also writes for the cardinality of the power set of an infinite set. The generalized continuum hypothesis ( GCH ) states for infinite sets, that after next larger cardinality is:

Restriction on smaller subsets

With the set of subsets is referred to as containing less elements. For example: The set even missing because it has not less than elements.

Others

  • The existence of the power set at any amount called for in the Zermelo -Fraenkel set theory as a separate axiom, namely the power set axiom.
  • A lot of system such as a topology or a σ - algebra over a ground set is a subset of the power set, ie an element of.
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