Finite Set - Basic Properties

Basic Properties

Any proper subset of a finite set S 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 ZFC axioms for set theory, every Dedekind-finite set is also finite, but this requires the axiom of choice (or at least the axiom of dependent choice).

Any injective function between two finite sets of the same cardinality is also a surjection, and similarly any surjection between two finite sets of the same cardinality is also an injection.

The union of two finite sets is finite, with

In fact:

More generally, the union of any finite number of finite sets is finite. The Cartesian product of finite sets is also finite, with:

Similarly, the Cartesian product of finitely many finite sets is finite. A finite set with n elements has 2n distinct subsets. That is, the power set of a finite set is finite, with cardinality 2n.

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. (However, some authors use "countable" to mean "countably infinite", and thus 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 operation being given by set union.