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'irreflexive'
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| Title of object: |
irreflexive |
| Canonical Name: |
Irreflexive |
| Type: |
Definition |
| Created on: |
2006-02-20 00:58:19 |
| Modified on: |
2006-09-07 19:45:17 |
| Classification: |
msc:03E20 |
Revision comment (for changes between this and next version):
| Changes for correction #9630 ('linking (reflexive)'). |
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 |
Content:
A binary relation $\mathcal{R}$ on a set $A$ is said to be \emph{irreflexive} if $\forall a\in A$, $\neg a\mathcal{R} a$. In other words, ``no element is $\mathcal{R}$-related to itself."
For example, the relation $<$ (``less than") is an irreflexive relation on the set of natural numbers.
Note that ``irreflexive" is not simply the negation of ``\PMlinkname{reflexive}{Reflexive}
." Although it is impossible for a relation (on a nonempty set) to be both \PMlinkname{reflexive}{Reflexive}
and irreflexive, it is easy to come up with relations that are neither. |
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