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

The Concept of Fuzzy Relation and Basic Properties of its Operation

Takashi Mitsuishi
Shinshu University, Nagano
Katsumi Wasaki
Shinshu University, Nagano
Yasunari Shidama
Shinshu University, Nagano


This article introduces the fuzzy relation. This is the expansion of usual relation, and the value is given at the fuzzy value. At first, the definition of the fuzzy relation characterized by membership function is described. Next, the definitions of the zero relation and universe relation and basic operations of these relations are shown.

MML Identifier: FUZZY_3

The terminology and notation used in this paper have been introduced in the following articles [2] [5] [6] [1] [3] [4]

Contents (PDF format)

  1. Definition of Fuzzy Relation
  2. Zero Relation and Universe Relation


[1] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[3] Jaroslaw Kotowicz. Partial functions from a domain to the set of real numbers. Journal of Formalized Mathematics, 2, 1990.
[4] 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.
[5] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[6] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.

Received September 15, 2000

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