well-defined
A mathematical concept is well-defined (German wohldefiniert, French bien défini), if its contents is on the form or the alternative representative which is used for defining it.
For example, in defining the http://planetmath.org/FractionPowerpower 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 becauseevery input, , is assigned to precisely one output, . However, is not well-definedin 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 notwell-defined since yet while and .
One must question whether a function is well-defined whenever it is defined on a domain of equivalence classesin such a manner that each output is determined for a representative of each equivalence class. For example, thefunction was defined using the representative of the equivalence class of fractionsequivalent
to .