Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990
Association of Mizar Users
The Collinearity Structure
-
Wojciech Skaba
-
Nicolaus Copernicus University, Torun
-
Supported by RPBP.III-24.B5.
Summary.
-
The text includes basic axioms and theorems concerning
the collinearity structure based on Wanda Szmielew [2],
pp. 18-20. Collinearity
is defined as a relation on Cartesian product
$\mizleftcart S, S, S \mizrightcart$ of set $S$.
The basic text is preceeded with a few auxiliary theorems
(e.g: ternary relation). Then come the two basic axioms of the collinearity
structure: A1.1.1 and A1.1.2 and a few theorems. Another axiom:
Aks dim, which states that there exist at least 3 non-collinear points,
excludes the trivial structures (i.e. pairs
$\llangle S, \mizleftcart S, S, S \mizrightcart\rrangle$).
Following it the notion of a line is included and several additional
theorems are appended.
MML Identifier:
COLLSP
The terminology and notation used in this paper have been
introduced in the following articles
[4]
[1]
[5]
[3]
Contents (PDF format)
Bibliography
- [1]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
- [2]
Wanda Szmielew.
\em From Affine to Euclidean Geometry, volume 27.
PWN -- D.Reidel Publ. Co., Warszawa -- Dordrecht, 1983.
- [3]
Andrzej Trybulec.
Domains and their Cartesian products.
Journal of Formalized Mathematics,
1, 1989.
- [4]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
- [5]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
Received May 9, 1990
[
Download a postscript version,
MML identifier index,
Mizar home page]