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Expression whose definition assigns it a unique interpretation

In mathematics, a well defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill 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 takes real numbers as input, and if does not equal then is not well defined (and thus not a function). The term well defined can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well defined is not the same as a function that is undefined. For example, if , then even though is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain of .

Let be sets, let and "define" as if and if
Then is well defined if For example, if and , then would be well defined and equal to
However, if , then would not be well defined because is "ambiguous" for For example, if and , then would have to be both 0 and 1, which makes it ambiguous. As a result, the latter is not well defined and thus not a function.

Questions regarding the well definedness of a function often arise when the defining equation of a function refers not only to the arguments themselves, but also to elements of the arguments, serving as representatives. This is sometimes unavoidable when the arguments are cosets and when the equation refers to coset representatives. The result of a function application must then not depend on the choice of representative.

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 is an integer. Therefore,

similar holds for any representative of , thereby making the same, irrespective of the choice of representative.

For real numbers, the product is unambiguous because ; hence the notation is said to be well defined. In the programming language APL there is only one rule: from right to left – but parentheses first.

A solution to a partial differential equation is said to be well defined if it is continuously determined by boundary conditions as those boundary conditions are changed.