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Über dieses Buch

In the beginning of 1983, I came across A. Kaufmann's book "Introduction to the theory of fuzzy sets" (Academic Press, New York, 1975). This was my first acquaintance with the fuzzy set theory. Then I tried to introduce a new component (which determines the degree of non-membership) in the definition of these sets and to study the properties of the new objects so defined. I defined ordinary operations as "n", "U", "+" and "." over the new sets, but I had began to look more seriously at them since April 1983, when I defined operators analogous to the modal operators of "necessity" and "possibility". The late George Gargov (7 April 1947 - 9 November 1996) is the "god­ father" of the sets I introduced - in fact, he has invented the name "intu­ itionistic fuzzy", motivated by the fact that the law of the excluded middle does not hold for them. Presently, intuitionistic fuzzy sets are an object of intensive research by scholars and scientists from over ten countries. This book is the first attempt for a more comprehensive and complete report on the intuitionistic fuzzy set theory and its more relevant applications in a variety of diverse fields. In this sense, it has also a referential character.



1. Intuitionistic Fuzzy Sets

The intuitionistic fuzzy set (IFS) theory is based on:
  • extensions of corresponding definitions of fuzzy set objects, and
  • definitions of new objects and their properties.
Krassimir T. Atanassov

2. Interval Valued Intuitionistic Fuzzy Sets

In this chapter, the basic definitions and properties of the interval valued intuitionistic fuzzy sets (IVIFSs) will be introduced. We will omit the majority of the proofs below, which are, in general, analogous to the proofs from Chapter 1.
Krassimir T. Atanassov

3. Other Extensions of Intuitionistic Fuzzy Sets

In this chapter we will present the following four extensions of the IFSs:
  • intuitionistic L-fuzzy sets,
  • IFSs over different universes,
  • temporal IFSs, and
  • IFSs of second type.
Krassimir T. Atanassov

4. Elements of Intuitionistic Fuzzy Logics

The definition of intuitionistic fuzzy sets will serve as a basis for further definitions of the elements of the intuitionistic fuzzy logics (IFLs). Here we shall present basic elements of:
  • Intuitionistic Fuzzy Propositional Calculus (IFPC),
  • Intuitionistic Fuzzy Predicate Logic (IFPL),
  • Intuitionistic Fuzzy Modal Calculus (IFMC), and
  • Temporal Intuitionistic Fuzzy Logic (TIFL).
Krassimir T. Atanassov

5. Applications of Intuitionistic Fuzzy Sets

The intuitionistic fuzzy sets can be used everywhere the ordinary fuzzy sets can be applied. On the other hand, this is not always necessary.
Krassimir T. Atanassov

Open Problems in Intuitionistic Fuzzy Sets Theory

The IFS theory is a relatively new branch of the fuzzy set theory and so there are many unsolved or unformulated problems in it. In the author’s opinion, when speaking of a new theory, it is harmful to discuss in advance whether it would be reasonable to define new concepts in it or not. If it is possible to define the new concepts correctly, it must be done. After some time, if they turn out to be useless, they will cease being objects of discussion. But after their definition, these new concepts can help the emergence of new ideas for development of the theory in general.
Krassimir T. Atanassov


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