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

## Basic Properties of Fuzzy Set Operation and Membership Function

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

### Summary.

This article introduces the fuzzy theory. The definition of the difference set, algebraic product and algebraic sum of fuzzy set is shown. In addition, basic properties of those operations are described. Basic properties of fuzzy set are a~little different from those of crisp set.

#### MML Identifier: FUZZY_2

The terminology and notation used in this paper have been introduced in the following articles [5] [1] [8] [6] [7] [2] [9] [3] [4]

#### Contents (PDF format)

1. Basic Properties of Membership Function and Difference Set
2. Algebraic Product and Algebraic Sum

#### Bibliography

[1] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[2] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Jaroslaw Kotowicz. Real sequences and basic operations on them. Journal of Formalized Mathematics, 1, 1989.
[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] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[6] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[7] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.
[8] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[9] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.