# Power set

In **set **theory, the **power set** is the set of all subsets of a given basic set .

The power set of a set is usually noted as . The nature of the power set was already examined by Ernst Zermelo . The compact term “power set” on the other hand - which is useful in connection with the arithmetic power - was not used by Gerhard Hessenberg in his 1906 textbook either; he uses the phrase “set of subsets” for this.

## definition

The power set of a set is a new set that consists of all subsets of . The power set is therefore a system of sets , that is, a set whose elements are themselves sets. In formula notation, the definition of a power set is

- .

It should be noted that the empty set and the set are also subsets of , i.e. elements of the power set . 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 not a total order , if at least two elements). The smallest element of order is , the greatest element is .

### Full association

The partial order is a complete association . This means that for *every* subset of there is an infimum and a supremum (in ). Concretely, for a set the infimum of is equal to the intersection of the elements of , and the supremum of is equal to the union of the elements of , thus

The largest and the smallest element are obtained as the infimum or supremum of the empty set, i.e.

### Boolean association

If you also use the complement map, it is a Boolean lattice , i.e. a distributive and complementary lattice.

### Commutative ring

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

## Characteristic functions

Each subset can be assigned the characteristic function , where applies

This assignment is a bijection between and (using the notation for the set of all functions from to ). This also motivates the notation , because in von Neumann's model of natural numbers is (in general:) .

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

## The size of the power set (cardinality)

denotes the power of a crowd .

- For finite sets applies: .
- Always applicable Cantor's theorem : .

The transition to the power set always provides greater power. Analogously to finite sets, one also writes for the cardinality of the power set of an infinite set . The generalized continuum hypothesis (GCH) says for infinite sets that the next greater thickness is:

## Restriction to smaller subsets

With the set of those subsets of which contain fewer than elements. For example : The set itself is missing because it has no less than elements.

## Potency class

The concept of the power set can be extended to classes , whereby it should be noted that real classes cannot be on the left side of the membership relation . The power (power class) of a class K is given by the **class of** all **sets** whose elements are all contained in K. The elements of the power class of K are therefore the **subsets** of K. The power of a real class K is again a real class, because it contains the units {x} for all elements x of K. It always contains the empty set ∅, but **not** the real class K itself.

## Others

- The existence of the power set for every set is required in the Zermelo-Fraenkel set theory as a separate axiom, namely by the power set axiom .
- A system of sets such as a topology or a σ-algebra over a basic set is a subset of the power set , i.e. an element of .

## literature

- Oliver Deiser:
*Introduction to set theory. Georg Cantor's set theory and its axiomatization by Ernst Zermelo.*2nd, improved and enlarged edition. Springer, Berlin a. a. 2004, ISBN 3-540-20401-6 .

## Web links

**Wikibooks: Math for Non-Freaks: Potency Set**- Learning and Teaching Materials

**Wikibooks: Evidence archive: set theory**- learning and teaching materials

**Wiktionary: power set**-

**explanations of**meanings, word origins, synonyms, translations