Volume 13, 2001

University of Bialystok

Copyright (c) 2001 Association of Mizar Users

**Noboru Endou**- Gifu National College of Technology
**Takashi Mitsuishi**- Miyagi University
**Keiji Ohkubo**- Shinshu University, Nagano

- In this article, we introduce four fuzzy relations and the composition, and some useful properties are shown by them. In section 2, the definition of converse relation $R^{-1}$ of fuzzy relation $R$ and properties concerning it are described. In the next section, we define the composition of the fuzzy relation and show some properties. In the final section we describe the identity relation, the universe relation and the zero relation.

- Basic Properties of the Membership Function
- Definition of Converse Fuzzy Relation and some Properties
- Definition of the Composition and some Properties
- Definition of Identity Relation and Properties of Universe and Zero Relation

- [1]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [4]
Jaroslaw Kotowicz.
Convergent real sequences. Upper and lower bound of sets of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Takashi Mitsuishi, Noboru Endou, and Yasunari Shidama.
The concept of fuzzy set and membership function and basic properties of fuzzy set operation.
*Journal of Formalized Mathematics*, 12, 2000. - [7]
Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama.
Basic properties of fuzzy set operation and membership function.
*Journal of Formalized Mathematics*, 12, 2000. - [8]
Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama.
The concept of fuzzy relation and basic properties of its operation.
*Journal of Formalized Mathematics*, 12, 2000. - [9]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [10]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [11]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989.

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