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 xr with x a positive real and r a rational numberPlanetmathPlanetmathPlanetmath, we can freely choose the fraction form mn (m,  n+) of r and take


and be sure that the value of xr does not depend on that choice (this is justified in the entry fraction power). So, the xr is well-defined.

In many instances well-defined is a synonym for the formal definition of a function between sets. For example, the function  f(x):=x2  is a well-defined function from the real numbers to the real numbers because every input, x, is assigned to precisely one output, x2. However,  f(x):=±x  is not well-defined in that one input x can be assigned any one of two possible outputs, x or -x.

More subtle examples include expressions such as


Certainly every input has an output, for instance,  f(1/2)=3. However, the expression is not well-defined since  1/2=2/4  yet  f(1/2)=3  while  f(2/4)=6  and  36.

One must question whether a function is well-defined whenever it is defined on a domain of equivalence classesMathworldPlanetmathPlanetmath in such a manner that each output is determined for a representative of each equivalence class. For example, the function  f(a/b):=a+b  was defined using the representative a/b of the equivalence class of fractions equivalentMathworldPlanetmathPlanetmathPlanetmath to a/b.

Title well-defined
Canonical name Welldefined
Date of creation 2013-03-22 17:31:32
Last modified on 2013-03-22 17:31:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Definition
Classification msc 00A05
Synonym well defined
Related topic function
Related topic WellDefinednessOfProductOfFinitelyGeneratedIdeals