Journal of Formalized Mathematics
Volume 12, 2000
University of Bialystok
Copyright (c) 2000 Association of Mizar Users

## Lower Tolerance. Preliminaries to Wroclaw Taxonomy

Mariusz Giero
University of Bialystok
Roman Matuszewski
University of Bialystok

### Summary.

The paper introduces some preliminary notions concerning the Wroclaw taxonomy according to [14]. The classifications and tolerances are defined and considered w.r.t. sets and metric spaces. We prove theorems showing various classifications based on tolerances.

This work has been partially supported by the European Community TYPES grant IST-1999-29001 and CALCULEMUS grant HPRN-CT-2000-00102.

#### MML Identifier: TAXONOM1

The terminology and notation used in this paper have been introduced in the following articles [18] [8] [20] [2] [19] [7] [21] [23] [5] [22] [6] [13] [16] [12] [11] [10] [17] [3] [4] [15] [1] [9]

#### Contents (PDF format)

1. Preliminaries
2. The Notion of Classification
3. The Tolerance on a Non Empty Set
4. The Partitions Defined by Lower Tolerance
5. The Classification on a Non Empty Set
6. The Classification on a Metric Space

#### Acknowledgments

The authors thank Prof. Andrzej Trybulec for his introduction to this topic. We thank Dr. Artur Kornilowicz for his advice on this article. We also thank Robert Milewski and Adam Naumowicz for their helpful comments.

#### Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. Reduction relations. Journal of Formalized Mathematics, 7, 1995.
[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[9] Patricia L. Carlson and Grzegorz Bancerek. Context-free grammar --- part I. Journal of Formalized Mathematics, 4, 1992.
[10] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[11] Alicia de la Cruz. Totally bounded metric spaces. Journal of Formalized Mathematics, 3, 1991.
[12] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[13] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Journal of Formalized Mathematics, 10, 1998.
[14] Roman Matuszewski and Andrzej Trybulec. \em Certain algorithm of classification in metric spaces, volume V, Number 20 of \em Mathematical Papers. Warsaw University, Bialystok Campus, 1977.
[15] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Journal of Formalized Mathematics, 6, 1994.
[16] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[17] Konrad Raczkowski and Pawel Sadowski. Equivalence relations and classes of abstraction. Journal of Formalized Mathematics, 1, 1989.
[18] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[19] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[20] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[21] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[22] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.
[23] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Journal of Formalized Mathematics, 1, 1989.