Editing Finite set (section)

== Basic properties ==

Any [[proper subset]] of a finite set <math>S</math> is finite and has fewer elements than ''S'' itself.  As a consequence, there cannot exist a [[bijection]] between a finite set ''S'' and a proper subset of ''S''.  Any set with this property is called [[Dedekind-finite]].  Using the standard [[Zermelo–Fraenkel set theory|ZFC]] axioms for [[set theory]], every Dedekind-finite set is also finite, but this implication cannot be [[mathematical proof|proved]] in ZF (Zermelo–Fraenkel axioms without the [[axiom of choice]]) alone.
The [[axiom of countable choice]], a weak version of the axiom of choice, is sufficient to prove this equivalence.

Any injective function between two finite sets of the same cardinality is also a [[surjective function]] (a surjection). Similarly, any surjection between two finite sets of the same cardinality is also an injection.

The [[union (set theory)|union]] of two finite sets is finite, with
{{bi|left=1.6|<math>\displaystyle |S \cup T| \le |S| + |T|.</math>}}

In fact, by the [[inclusion–exclusion principle]]:
{{bi|left=1.6|<math>\displaystyle |S \cup T| = |S| + |T| - |S\cap T|.</math>}}
More generally, the union of any finite number of finite sets is finite.  The [[Cartesian product]] of finite sets is also finite, with:
{{bi|left=1.6|<math>\displaystyle |S \times T| = |S|\times|T|.</math>}}
Similarly, the Cartesian product of finitely many finite sets is finite. A finite set with <math>n</math> elements has <math>2^n</math> distinct subsets.  That is, the [[power set]] <math>\wp(S)</math> of a finite set ''S'' is finite, with cardinality <math>2^{|S|}</math>.

Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite.

All finite sets are [[countable]], but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do not consider finite sets to be countable.)

The [[free semilattice]] over a finite set is the set of its non-empty subsets, with the [[Join and meet|join operation]] being given by set union.