Edustusfunktio

Edustusfunktio, in mathematics and computer science, is a term used in Finnish to describe a representation function. Broadly, it refers to a function that assigns to each element a distinguished representative from its equivalence class under a given relation. Such a function enables the construction of quotient objects and the use of canonical forms.

Formally, let X be a set with an equivalence relation ∼. The quotient set X/∼ consists of the equivalence classes [x]. A representative function (edustusfunktio) is a function r: X/∼ → X such that r([x]) ∈ [x] for every x ∈ X. Equivalently, a related but sometimes more convenient notion is a section s: X/∼ → X that selects a single representative from each class. In some contexts, an edustusfunktio may instead be described as a rule r: X → X that, for each x, assigns a representative from its class [x].

Examples illustrate the idea. In modular arithmetic modulo n, the function rep_n(x) = x mod n assigns to each integer a canonical residue in {0, 1, ..., n−1}, serving as a representative for the class of numbers congruent modulo n. More generally, any rule that picks a canonical element from each equivalence class—such as the least element with respect to a fixed order—is a representative function.

Existence and uniqueness: a representative function exists under the axiom of choice for arbitrary families of nonempty classes, but canonical choices require additional structure (e.g., a well-order or a natural rule). The non-uniqueness of representatives means that different edustusfunktio can yield different but valid canonical forms.

See also: quotient set, selection function, section of a projection, canonical form, least residue.