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 $x^r$ with $x$ a positive real and $r$ a rational number, we can freely choose the fraction form $\frac{m}{n}$ ($m\in \mathbb{Z}$,  $n\in {\mathbb{Z}}_{+}$) of $r$ and take

$$x^r:=\sqrt[n]{x^m}$$

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

In many instances well-defined is a synonym for the formal definition of a function between sets. For example, the function  $f(x):=x^2$  is a well-defined function from the real numbers to the real numbers because every input, $x$, is assigned to precisely one output, $x^2$. However,  $f(x):=\pm \sqrt{x}$  is not well-defined in that one input $x$ can be assigned any one of two possible outputs, $\sqrt{x}$ or $-\sqrt{x}$.

More subtle examples include expressions such as

$$f\left(\frac{a}{b}\right):=a+b,\frac{a}{b}\in \mathbb{Q}.$$

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  $3\ne 6$.

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  $f(a/b):=a+b$  was defined using the representative $a/b$ of the equivalence class of fractions equivalent 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