Finite set

In set theory, a branch of mathematics, is a finite set with a lot of finitely many elements. For example, the amount of

A finite set with four elements. The empty set has no elements, by definition, ie the number of elements ( cardinality or cardinality ) is, therefore, it is also considered finite set. The cardinality (written for a lot ) of a finite set is identified by a natural number (including zero ), for example, to write then to express that consists of four elements.

A lot of that is not finite is called infinite amount.

Definition

A set is finite if there is a natural number, so is a bijection (a " one-to -one mapping " )

Between and the set of all natural numbers less than, exists.

So In particular, for the empty set is finite, since a bijection between the empty set and the empty set (all natural numbers less than, those do not exist ) exists.

For example, the amount of

Finally, there is a bijection to the amount

Exist, see for example figure opposite.

For the set of all natural numbers

However, there is no such bijection on a finite set, the set is therefore infinite.

Basic properties of finite sets

• Every subset of a finite set is also finite.
• Are finite sets, then their union, their intersection and their difference set are finite. The difference amount is even finite if only finite.
• For the cardinality of the union; and are disjoint, then one has.
• The power set of a finite set has a higher cardinality than the set itself, but is still finite, it is.
• The Cartesian product of finite sets is finite. Its thickness is higher than that of all the factors involved, if no factor is empty and at least two factors have a thickness greater than 1. For finite sets.

Dedekind - finiteness

Another distinction between finite and infinite sets derived from Dedekind. He defined:

We speak today of Dedekind finiteness or Dedekind infinity.

In order to show that any finite set of Dedekind -finite is, it suffices to show the following:

(Point 1 is clear, since the empty set has no proper subsets. Regarding point 2 must point out that one can gain from a bijection between the set and a proper subset of a bijection between and a proper subset. )

Conversely, every Dedekind - finite set and finite, because would be infinite, so you could use the axiom of choice can find a sequence of pairwise distinct elements. The figure

Then shows that the proper subset equally powerful and therefore not Dedekind - finite. Contradiction!

Hereditary finite sets

A set is hereditarily finite if the transitive closure is finite. That is, not only finite, but also all of the elements of finite quantities, and their elements are also finite quantities, and so on.

By definition, all hereditary finite sets are finite. The converse is not true, it is about a finite set, since it contains a single element, but the element itself is not finite.

In the abstract set theory, the natural numbers are introduced as hereditarily finite sets:

Thus the natural numbers themselves are finite sets, even hereditarily finite, and it is true for any natural number, where the vertical lines are not available for the absolute value function here, but for the thickness. That is the reason why above in the introduction, a DC thickness mentioned at the place of. The latter would even have been right, but the choice made better fits the definition of natural numbers. After a lot has the power, if it is to equally powerful.

Averages, associations and products hereditarily finite sets are hereditarily finally back. The set of all hereditarily finite sets is exactly the level of the von Neumann hierarchy of Zermelo -Fraenkel set theory.