Well-defined

From Example Problems
Jump to navigation Jump to search

In mathematics, the term well-defined is used to specify that a certain concept (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. Usually definitions are stated unambiguously, and there is no question about their well-definition. Occasionally, 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.

One of the most common places in mathematics in which the term well-defined is used is in dealing with cosets in group theory. It is 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., that it never happens that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2+3\neq5} ).

Error creating thumbnail: Unable to save thumbnail to destination

More generally, given a set X, an equivalence relation ~ on X, and a function f on X, 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(x)=f(y) whenever x~y, 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{f}} . In this view, one says that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{f}} is well-defined if the diagram shown commutes. That is, that f factors through π, where π is the canonical projection map XX/~, so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=\tilde{f}\pi} .

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.)