In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well-defined or ambiguous. A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well-defined (and thus: not a function). The term well-defined is also used to indicate whether a logical statement is unambiguous.
Let be sets, let and define as if and if . Then is well-defined if . If however then is not well-defined because is ambiguous for .
In group theory, the term well-defined is often used when dealing with cosets, where a function f on a quotient group may be defined in terms of a coset representative. Here, a necessary requirement for f to be considered a function is that the output must be independent of which coset representative is chosen. The phrase f is well-defined is used to indicate that this requirement has been verified.
For example, consider , the integers modulo 2. Since 4 and 6 are congruent modulo 2, a function f whose domain is must give the same output when the input is represented by 4 that it gives when the input is represented by 6.
A function that is not well-defined is not the same as a function that is undefined. For example, if f(x) = 1/x, then f(0) is undefined, but this has nothing to do with the question of whether f(x) = 1/x is well-defined. It is; 0 is simply not in the domain of the function.
In particular, the term well-defined is used with respect to (binary) operations on cosets. In this case one can view the operation as a function of two variables and the property of being well-defined is the same as that for a function. For example, addition on the integers modulo some n can be defined naturally in terms of integer addition.
The fact that this is well-defined follows from the fact that we can write any representative of as , where k is an integer. Therefore,
and similarly for any representative of .
For real numbers, the product is unambiguous because . In this case this notation is said to be well-defined. However, if the operation (here ) did not have this property, which is known as associativity, then there must be a convention for which two elements to multiply first. Otherwise, the product is not well-defined. The subtraction operation, , is not associative, for instance. However, the notation is well-defined under the convention that the operation is understood as addition of the opposite, thus is the same as . Division is also non-associative. However, does not have an unambiguous conventional interpretation, so this expression is ill-defined.
Other uses of the term
- Equivalence relation § Well-definedness under an equivalence relation
- Uniqueness quantification
- Weisstein, Eric W. "Well-Defined". From MathWorld--A Wolfram Web Resource. Retrieved 2 January 2013.
- Joseph J. Rotman, The Theory of Groups: an Introduction, p. 287 "... a function is "single-valued," or, as we prefer to say ... a function is well defined.", Allyn and Bacon, 1965.