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'well-defined'
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| Title of object: |
well-defined |
| Canonical Name: |
WellDefined |
| Type: |
Definition |
| Created on: |
2007-09-06 06:32:32 |
| Modified on: |
2007-09-06 08:56:50 |
| Classification: |
msc:00A05 |
| Synonyms: |
well-defined=well defined |
Revision comment (for changes between this and next version):
| corrected a typo and added a little on well-defined functions. |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
A mathematical concept is {\em well-defined} (German {\em wohldefiniert}, French {\em bien d\'efini}), if its contents in \PMlinkescapetext{independent} on the form or the alternative representative which is used for defining it.
For example, in defining the \PMlinkname{power}{FractionPower} $x^r$ 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. |
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