Väldefinierbarhet

Väldefinierbarhet is a property of a mathematical definition or rule meaning that the output is uniquely determined by the input, independent of any arbitrary representation or choice used to describe that input. In practice, it ensures that a function or construction is unambiguous.

A common setting is when inputs are given as equivalence classes. If a rule assigns a value to each class based on a representative, the rule is well-defined only if the result is the same for every representative of the same class. More generally, a rule f on a quotient structure A/~ to B is well-defined if f([a]) is determined by the class [a], not by a particular representative a.

Criterion for well-definedness: Suppose the input is an equivalence class [a] with a ~ a'. The derived value must satisfy f([a]) = f([a']), i.e., any two representatives yield the same output. Equivalently, if a function g: A -> B satisfies g(a) = g(a') whenever a ~ a', then the induced rule on A/~ is well-defined.

Examples: A well-defined case is f: Z/nZ -> Z/nZ given by f([k]) = [2k], since if k ~ k + tn, then 2k ~ 2k + 2tn and 2tn ≡ 0 mod n. A non-well-defined case is g([k]) = k, because replacing k by k + tn changes the output unless n divides tn.

Significance: Well-definedness is essential for quotient constructions, operations on equivalence classes, and any definition where inputs are described up to an equivalence or representation. It guards against ambiguity and ensures mathematical rigor.