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A mathematical concept is well-defined (German wohldefiniert, French bien défini), if its content is independent on the form or the alternative representative which is used for defining it.
For example, in defining the power  with  a positive real and  a rational number, we can freely choose the fraction form
 (
 ,
 ) of  and take
and be sure that the value of  does not depend on that choice (this is justified in the entry fraction power). So, the  is well-defined.
In many instances well-defined is a synonym for the formal definition of a function between sets. For example, the function
 is a well-defined function from the real numbers to the real numbers because every input,  , is assigned to precisely one output,  . However,
 is not well-defined in that one input  can be assigned any one of two possible outputs,  or  .
More subtle examples include expressions such as
Certainly every input has an output, for instance,
 . However, the expression is not well-defined since  yet
 while
 and  .
One must question whether a function is well-defined whenever it is defined on a domain of equivalence classes in such a manner that each output is determined for a representative of each equivalence class. For example, the function
 was defined using the representative  of the equivalence class of fractions equivalent to  .
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![$\displaystyle x^r := \sqrt[n]{x^m}$](http://images.planetmath.org:8080/cache/objects/9921/l2h/img8.png "$\displaystyle x^r := \sqrt[n]{x^m}$"){kind=small}
