Theory of Fuzzy Sets: An Overview

2015

On two important counts, the Zadehian theory of fuzzy sets urgently needs to be restructured. First, it can be established that for a normal fuzzy number N = [α, β, γ] with membership function Ψ 1 (x), if α ≤ x ≤ β, Ψ 2 (x), if β ≤ x ≤ γ, and 0, otherwise, Ψ 1 (x) is in fact the distribution function of a random variable defined in the interval [α, β], while Ψ 2 (x) is the complementary distribution function of another random variable defined in the interval [β, γ]. In other words, every normal law of fuzziness can be expressed in terms of two laws of randomness defined in the measure theoretic sense. This is how a normal fuzzy number should be constructed, and this is how partial presence of an element in a fuzzy set has to be defined. Hence the measure theoretic matters with reference to fuzziness have to be studied accordingly. Secondly, the field theoretic matters related to fuzzy sets are required to be revised all over again because in the current definition of the complement of a fuzzy set, fuzzy membership function and fuzzy membership value had been taken to be the same, which led to the conclusion that the fuzzy sets do not follow the set theoretic axioms of exclusion and contradiction. For the complement of a normal fuzzy set, fuzzy membership function and fuzzy membership value are two different things, and the complement of a normal fuzzy set has to be defined accordingly. We shall further show how fuzzy randomness should be explained with reference to two laws of randomness defined for every fuzzy observation so as to make fuzzy statistical conclusions. Finally, we shall explain how randomness can be viewed as a special case of fuzziness defined in our perspective with reference to normal fuzzy numbers of the type [α, β, β]. Indeed every probability distribution function is a Dubois-Prade left reference function, and probability can be viewed in that way too.

2011

On two important counts, the Zadehian theory of fuzzy sets urgently needs to be restructured. First, it can be established that for a normal fuzzy number N = [α, β, γ] with membership function Ψ 1 (x), if α ≤ x ≤ β, Ψ 2 (x), if β ≤ x ≤ γ, and 0, otherwise, Ψ 1 (x) is in fact the distribution function of a random variable defined in the interval [α, β], while Ψ 2 (x) is the complementary distribution function of another random variable defined in the interval [β, γ]. In other words, every normal law of fuzziness can be expressed in terms of two laws of randomness defined in the measure theoretic sense. This is how a normal fuzzy number should be constructed, and this is how partial presence of an element in a fuzzy set has to be defined. Hence the measure theoretic matters with reference to fuzziness have to be studied accordingly. Secondly, the field theoretic matters related to fuzzy sets are required to be revised all over again because in the current definition of the complement of a fuzzy set, fuzzy membership function and fuzzy membership value had been taken to be the same, which led to the conclusion that the fuzzy sets do not follow the set theoretic axioms of exclusion and contradiction. For the complement of a normal fuzzy set, fuzzy membership function and fuzzy membership value are two different things, and the complement of a normal fuzzy set has to be defined accordingly. We shall further show how fuzzy randomness should be explained with reference to two laws of randomness defined for every fuzzy observation so as to make fuzzy statistical conclusions. Finally, we shall explain how randomness can be viewed as a special case of fuzziness defined in our perspective with reference to normal fuzzy numbers of the type [α, β, β]. Indeed every probability distribution function is a Dubois-Prade left reference function, and probability can be viewed in that way too.

International Journal of Energy, Information and Communications, 2015

How exactly the membership function of a normal fuzzy number should be determined mathematically was not explained by the originator of the theory. Further, the definition of the complement of a fuzzy set led to the conclusion that fuzzy sets do not form a field. In this article, we would put forward an axiomatic definition of fuzziness such that fuzzy sets can be seen to follow classical measure theoretic and field theoretic formalisms.

1993

The theory of@zzy sets is known to be an instrument of the management of uncertainty. The objective of this paper is o introduce the concept of fuzziness degree (a measure of uncertainty) of some fuzzy set's collection (FSC) and to describe its practical applications. The collection offirzzv sets is deJned on the same universum. Such structures can be interpreted as a set of values of some fizzy linguistic scale FU) or a set of different alternatives in problem solving and decisionmaking or a descriptions of classes in fuzzy class$cation and clustering ora representation of term-sets of linguistic values and etc.

The present fuzzy arithmetic based on Zadeh's possibilistic extension principle and on the classic definition of a fuzzy set has many essential drawbacks. Therefore its application to the solution of practical tasks is limited. In the paper a new definition of the fuzzy set is presented. The definition allows for a considerable fuzziness decrease in the number of arithmetic operations in comparison with the results produced by the present fuzzy arithmetic.

Fuzzy Sets and Systems, 2003

This is an unusual book. Why and in which respect? It is amazing how Prof. Klir is able to explain a number of nontrivial facts using simple and succinct means. In 19 chapters of this small book he has succeeded in describing the essential theory of fuzzy set and fuzzy logic theory including their applications as well as explaining the main philosophical problems araising when dealing with uncertainty and vagueness. Thus, the book is very informative-one can ÿnd everything relevant there, mostly only outlined. However, the core of the problem is completed by enough references to be able to ÿnd missing information.

The American Journal of Psychology, 1993

Properties of the Min-Max Composition 79 PREFACE TO THE FOURTH EDITION xxv around fuzzy control, a concept that was very applicable, easy to understand, and, therefore, attractive to many industrial practitioners and the broad public. Since the start of computational intelligence theoretical as well as applicationoriented developments have become much more diversified and clear lead-times between theoretical development and application can no longer be recognized. I have used the opportunity of a fourth edition of this textbook, for which I am very grateful to Kluwer Academic Publishers, to adapt the book to the new developments, without exceeding the scope of a basic textbook, as follows: All chapters have been updated . The scope of part I has only been extended with respect to t-norms, other operators and uncertainty modeling because I am convinced that chapters 2 to 7 are still sufficient as a mathematical basis to understand all new developments in this area and also for part II of the book, where the major changes and extensions of this edition can be found : In chapter lathe modeling of uncertainty in expert systems was extended because this component has gained importance in practice. In chapter II primarily a section for defuzzification has been added for the same reason. Chapter 12 has been added because the application of fuzzy technology in information processing is already important and will certainly increase in importance in the future. Chapter 13 has been extended by explaining new methodological developments in dynamic fuzzy data analysis, which will also be of growing importance in the future. Eventually applications in chapter 15 have been completely restructured by deleting some, adding others and classifying all of them differently. This was necessary because the focus of applications here changed, for reasons explained in this chapter, strongly from "engineering intelligence" to "business intelligence". Of course, the index and the references have also been updated and extended. This time I would like to thank again Kluwer Academic Publishers for giving me the chance of a fourth edition and Dr. Angstenberger for her excellent research cooperation and for letting me use one application from her book. In particular, I would like to thank Ms. Katja Palczynski for her outstanding help to get the manuscripts ready for the publisher. I hope that this new edition of my textbook will help to keep respective courses in universities and elsewhere up-to-date and challenging and motivating for students as well as professors. It may also be useful for practitioners that want to update their knowledge of fuzzy technology and look for new applications in their area. Aachen, April 2001 H.-i. Zimmermann FUZZY SETS 1.1 Crispness, Vagueness, Fuzziness, Uncertainty Most of our traditional tools for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. By crisp we mean dichotomous, that is, yes-or-no-type rather than more-or-less type. In conventional dual logic, for instance, a statement can be true or false-and nothing in between. In set theory, an element can either belong to a set or not; and in optimization, a solution is either feasible or not. Precision assumes that the parameters of a model represent exactly either our perception of the phenomenon modeled or the features of the real system that has been modeled. Generally, precision also implies that the model is unequivocal, that is, that it contains no ambiguities. Certainty eventually indicates that we assume the structures and parameters of the model to be definitely known, and that there are no doubts about their values or their occurrence. If the model under consideration is a formal model [Zimmermann 1980, p. 127], that is, if it does not pretend to model reality adequately, then the model assumptions are in a sense arbitrary, that is, the model builder can freely decide which model characteristics he chooses. If, however, the model or theory asserts factuality [Popper 1959; Zimmermann 1980], that is, if conclusions drawn from these models have a bearing on reality and are

Texts in Computer Science, 2013

Page 1. FUZZY SETS AND FUZZY LOGIC Theory and Applications ... Fuzzy Measures Evidence Theory Possibility Theory Fuzzy Sets and Possibilty Theory Possibility Theory versus Probability Theory Exercises 8. FUZZY LOGIC 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 Notes ...

In the papers [1, 3, 4] we have initiated a nonstandard approach to fuzzy sets. In this workshop I want to summarise and make some additional remarks concerning the mathematical foundations of Fuzzy Sets. 1. Mathematical Foundations. When we say "Mathematical Foun-dations of Fuzzy Sets" we mean that, fuzzy set theory, should not be build up from scratch and using some aprioristic methods, but rather, we should start with existing foundations for classical mathe-matics, and then try to construct from then a non classical theory that contain, fuzziness and vagueness as a basic element. That is fuzzi-ness should be build up from non -fuzzy classical mathematics, the same way that non -Euclidean Geometries are based on Euclidean. Presently there are the following options: (i) Base the transition on a non-classical deformation of Cantorian set theory, e.g. ZFC, to add up with a non-Cantorian set theory, which includes vagueness, fuzziness, etc. and is expressed using many -valu...

2011

It has been accepted that the fuzzy sets do not form a field. In this article, we are going to put forward an extension of the definition of fuzziness. With the help of this extension, we would be able to define the complement of a fuzzy set properly. This in turn would allow us to assert that fuzzy sets do form a field. In fact, the fuzzy membership value and the fuzzy membership function for the complement of a fuzzy set are two different things. This confusion has created a stumbling block towards accepting the theory of fuzzy sets as a generalization of the classical theory of sets.