Well-defined

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In mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc.) is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy the required properties. Sometimes however, it is economical to state a definition in terms of an arbitrary choice; one then has to check that the definition is independent of that choice. On other occasions, the required properties might not all be obvious; one then has to verify them. These issues commonly arise in the definition of functions.

For instance, in group theory, the term well-defined is often used when dealing with cosets, where a function on a coset space is often defined by choosing a representative: it is then as important that we check that we get the same result regardless of which representative of the coset we choose as it is that we always get the same result when we perform arithmetical operations (e.g., whenever we add 2 and 3, we always get the answer 5).

More generally, given a set X, an equivalence relation ~ on X, and a function f from X to another set Y, one may be interested to know whether f can be viewed as a function on the quotient set X/~. That is to say, if [x] is an equivalence class in X/~, then one may attempt to define f([x]) = f(x). If the function satisfies f(x₁)=f(x₂) whenever x₁~x₂, then the definition makes sense, and f is well-defined on X/~. Although the distinction is often ignored, the function on X/~, having a different domain, should be viewed as a distinct map . In this view, one says that is well-defined if the diagram shown commutes. That is, that f factors through π, where π is the canonical projection map X → X/~, so that .

As an example, consider the equivalence relation between real numbers defined by θ₁~θ₂ if there is an integer n such that θ₁-θ₂ = 2πn, where π (not italicized) denotes Pi. The quotient set X/~ may then be identified with a circle, as an equivalence class [θ] represents an angle. (In fact this is the coset space R/2πZ of the additive subgroup 2πZ of R.) Now if f:R→R is the cosine function, then is well-defined, whereas if f(θ) = θ then is not well-defined.

Two other issues of well-definition arise when defining a function f from a set X to a set Y. First, f should actually be defined on all elements of X. For example, the function f(x) = 1/x is not well-defined as a function from the real numbers to itself, as f(0) is not defined. Secondly, f(x) should be an element of Y for all x∈X. For example, the function f(x) = x² is not well-defined as a function from the real numbers to the positive real numbers, as f(0) is not positive.

The concept of well-definedness is important for mathematics and sciences not to rely on human intuition, which is subjective and imprecise. For example, you might say an object can have the property of being "red"; however, this property is not well-defined because there is a wide variety of colours that some individuals would perceive as a shade of red, while others would insist that it is orange. Such a property would only be well-defined if strict rules were laid out that determine what frequencies of visible light the object were allowed to emit or reflect for it to be "red".

Another example would be that most people would certainly agree that 999 is almost as much as 1000. However, there is no clear boundary as to where almost as much begins or ends. (There is, however, a well-defined notion of infinite sets being almost another.)