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 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 .
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 |