Well-defined: Encyclopedia - Well-defined

Well-defined

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.

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 ).

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.)

Category: Mathematical terminology

Adapted from the Wikipedia article "Well-defined", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki