Measures of fuzziness and operations with fuzzy sets

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.

Journal of Mathematical Analysis and Applications, 1972

Some new algebraic properties of the class _Lp(I) of the "fuzzy sets" are stressed; in particular it is pointed out that the class of the generalized characteristic functions furnished with the lattice operations proposed by Zadeh is a Brouwerian lattice. The possibility of inducing other different lattice operations to the whole class s(I) or to a suitable subclass of it is considered. The problem of the relationship between "fuzzy sets" and classical set theory is finally remarked. A qualitative comparison with similar situations appearing in the axiomatic formulation of quantum mechanics and in the classical theory of probability is made. * This paper is a slightly revised version of the report LC50 of the Laboratorio di Cibemetica de1 C.N.R.

International Journal of Information Engineering and Electronic Business, 2013

In this art icle, we would like to revisit and comment on the definit ion of co mplementation of fu zzy sets and also on some of the theories and formulas associated with this. Furthermore, the existing probability-possibility consistency principles are also revisited and related results are v iewed fro m the standpoint of the Randomness-Fuzziness consistency principles. It is found that the existing definition of complementation as well as the probability-possibility consistency principles is not well defined. Consequently the results obtained from these would be inappropriate fro m our standpoints. Hence we would like to suggest some new defin itions for so me of the terms often used in the theory of fuzzy sets whenever possible.

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.

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...

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.

Bulletin of the American Mathematical Society

2003

It seems that a suitably constructed fuzzy sets of natural numbers form the most complete and adequate description of cardinality of finite fuzzy sets.(see ) Nevertheless, in many applications one needs a simple scalar evaluation of that cardinality by nonnegative real number, e.g scalar cardinality. There are many approaches to this evaluation-sigma count of fuzzy set,psigma count of fuzzy sets, cardinality of its core or support, cardinality of its αcut set,etc.(see [2]), [3], [4], [5], [7], [9], [10] for more details). Wygralak in [8]

Results in Control and Optimization, 2023

Fuzzy set theory is a generalized form of crisp set theory where elements are binary inclusion forms. In fuzzy set, it differs with degree of membership for every element in the set. There are several strategies for arithmetic operations on fuzzy numbers. Previous studies show that there are many approaches, such as the α-cut technique, extension principle, vertex method, etc., to execute arithmetic operations on fuzzy numbers. In this study we perform details analysis and interpretation on arithmetic operations based on the α-cut method in a new way.

Information and Control, 1973

The problem of making decisions to classify the objects of a certain universe into two or more suitable classes has been considered in the setting of fuzzy sets theory. A measure of the total amount of uncertainty that arises in making decisions has been proposed in the general case. This quantity reduces to the "entropy" of a fuzzy set in the case of two classes. Other quantities which play a relevant role in this theory are the "energy" and the "effective power" of a fuzzy set, defined as N N Z-,J, and * ES,, t=1 l~l respectively, where w is a nonnegative weight function and ¢ a nonnegative constant. If go = constant and ~ q= 0, the energy is proportional to the effective power and, therefore, to the "power" of the fuzzy set. The maximum of the uncertainty has been calculated in some cases of interest, keeping constant the total energy and effective power. In particular the Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distributions are derived. Some applications to decision theory are considered in the case of both deterministic and probabilistic decisions. Finally, the analogies that exist between the previous concepts and the thermodynamic ones are discussed.

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

Information Sciences, 1993

DINABANDHU BHANDARI NIKHIL K. PAL Communicated by Abe Kandel ABSTKACT After reviewing some existing measures for fuzzy sets, we introduce a new informativc mcaxurc for discrimination between two fuzzy sets. This discriminating measure reduces to the nonprobabilistic entropy of Deluca and Termini [7] under a special condition. The divcrgcnce mcasure between two sets has been defined along with a large set of properties. It has also been used to define an ambiguity (fuzziness) measure.

International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2023

This article proposes some important theoretic aspects of L-fuzzy power set of a finite set. Here, we take a non-empty finite set E and an ordered subset M of the closed interval   0,1 , then the set of mappings from E to M denoted by F is defined as L-fuzzy power set. Considering the disjunctive union '    ' operations between fuzzy subsets of F , the structure   , F  forms a groupoid, which is defined as special fuzzy groupoid. Later, we try to introduce the product of special fuzzy groupoids and their properties.

Iranian Journal of Fuzzy Systems, 2009

The operations in the set of fuzzy numbers are usually obtained by the Zadeh extension principle. But these definitions can have some disadvantages for the applications both by an algebraic point of view and by practical aspects. In fact the Zadeh multiplication is not distributive with respect to the addition, the shape of fuzzy numbers is not preserved by multiplication, the indeterminateness of the sum is too increasing. Then, for the applications in the Natural and Social Sciences it is important to individuate some suitable variants of the classical addition and multiplication of fuzzy numbers that have not the previous disadvantage. Here, some possible alternatives to the Zadeh operations are studied.

Encyclopedia of Database Systems, 2009