Finite set
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In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where <math>n</math> is a natural number. (The value n=0 is allowed; that is, the empty set is finite.) All finite sets are countable <ref>Some authors use "countable" to mean "countably infinite", and thus do not consider finite sets to be countable.</ref>, but not all countable sets are finite.
Equivalently, a set is finite if its cardinality, i.e. the number of its elements, is a natural number. For instance, the set of integers between -15 and 3 (excluding the end points) is finite, since it has 17 elements. The set of all prime numbers is not finite. Sets that are not finite are called infinite.
A set is called Dedekind finite if there exists no bijection between the set and any of its proper subsets. It is a theorem (assuming the axiom of choice) that a set is finite if and only if it is Dedekind finite.
Alternative definitions of finite
There are many definitions of "finite", including the following. Notice that these are equivalent, if the axiom of choice is true.
- A set is finite iff it can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number.
- A set is finite iff it has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time.
- [Stackel] A set is finite iff it can be given a total ordering which is both well ordered forwards and backwards. That is, a set is finite iff every non-empty subset has both a least and a greatest element in the subset.
- [Dedekind] A set is finite iff every function from the set one-to-one into itself is onto.
- A set is finite iff every function from the set onto itself is one-to-one.
- [Tarski] A set is finite iff every non-empty family of subsets of the set has a minimal element wrt inclusion.
Footnotes
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See also
it:Insieme finito nl:Eindig pl:Zbiór skończony fi:Äärellinen joukko uk:Скінченна множина zh:有限集合