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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]
Fuzzy Sets and Systems, 2001
In this paper, we deÿne generalized concepts of cardinality of a fuzzy-valued function and obtained some properties of these new concepts. Also, we investigate examples for the calculation of two generalized cardinality of fuzzy-valued functions and compared with concepts of cardinality of a fuzzy set and a fuzzy-valued function.
IEEE Transactions on Fuzzy Systems
To solve problems from the area of Computing with Words [12], , arithmetic operations often have to be carried out on fuzzy numbers. The author proposes to call fuzzy numbers fuzzy sets of numbers (FSofN) to emphasize the fact that they are sets of many numbers (in the continuous case of infinitely many numbers) and not one, single number as the name fuzzy number suggests, because it occasionally leads to incorrect interpretations of their concept. To realize arithmetic operations on FSofN the standard extension principle is used. This principle was intended for possibilistic FSofN. However, in practical tasks many FSofN are of probabilistic character. If for such FSofN the standard extension principle is used the results will be incorrect. In this paper a cardinality extension principle is proposed. It realizes arithmetic operations on probabilistic FSofN. This principle was proposed in 2 versions: as a normal and as a generalized version that enables context-dependent constraints to be taken into account. In addition a new form of the possibilistic, constraint-extension principle of Klirr [7], [8] is presented. In this paper many examples are given that illustrate computations with both extension principles to make them less abstract and more user-friendly.
In this paper we discuss on some different representations of the cardinality of a fuzzy set and their use in fuzzy quantification. We have considered the widely employed sigma-count, fuzzy numbers, and gradual numbers. Gradual numbers assign numbers to values of a relevance scale, typically [0, 1]. Contrary to sigma-count and fuzzy numbers, they provide a precise representation of the cardinality of a fuzzy set. We illustrate our claims by calculating the cardinality of the fuzzy set of pixels that match a certain fuzzy color in an image. For that purpose we consider fuzzy color spaces previously defined by the authors, consisting of a collection of fuzzy sets providing a suitable, conceptual quantization with soft boundaries of crisp color spaces. Finally, we show the suitability of our approaches to fuzzy quantification for different applications in image processing. First, the calculation of histograms. Second, the definition of the notion of dominant fuzzy color, and the calculation of the degree to which we can say that a certain color is dominant in an image.
Fuzzy Sets and Systems, 2015
The main topic of this paper is the notion of relative cardinality for intervalvalued fuzzy sets-its definition, properties and computation. First we define relative cardinality for interval-valued fuzzy sets following the concept of uncertainty modelling given by Mendel's Wavy-Slice Representation Theorem. We expand on previous approaches by considering relative cardinality based on different t-norms and scalar cardinalities and we initiate an investigation of its properties and possible applications. Drawing on the Nguyen-Kreinovich and Karnik-Mendel algorithms, we propose efficient algorithms to compute relative cardinality depending on a chosen t-norm. This seems to be the first such broad and consistent analysis to have been made of relative cardinality for intervalvalued fuzzy sets. As a promising application we consider using interval-valued relative cardinality to construct the family of parameterised subsethood measures.
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.
International Journal of Intelligent …, 2009
In this paper we define in an axiomatic way scalar and fuzzy cardinalities of finite crisp and fuzzy multisets, and we obtain explicit descriptions for them.
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...
In this article, our main intention is to revisit the existing definition of complementation of fuzzy sets and thereafter various theories associated with it are also commented on. The main contribution of this paper is to suggest a new definition of complementation of fuzzy sets on the basis of reference function. Some other results have also been introduced whenever possible by using this new definition of complementation.
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
Studies in Computational Intelligence, 2017
Counting belongs to the most basic and frequent mental activities of humans. This is hardly surprising as cardinality seems to be one of the most fundamental characteristics of a given collection of elements. It forms an important type of information about that collection, a basis for coming to a decision in a lot of situations and in many dimensions of our life. Wygralak [150] In this chapter we introduce not fully determined fuzzy sets i.e. interval-valued fuzzy sets and Atanassov's intuitionistic fuzzy sets. We defined notions and methods describing cardinalities of such sets together with multiple examples. We presented methods for their application in decision-making algorithms with uncertain information. The chapter finishes with analysis of algorithms efficacy in ovarian cancer differentiation.
Fuzzy Sets and Systems, 2002
The existing methods to assess the cardinality of a fuzzy set with finite support are intended to preserve the properties of classical cardinality. In particular, the main objective of researchers in this area has been to ensure the convexity of fuzzy cardinalities, in order to preserve some properties based on the addition of cardinalities, such as the additivity property. We have found that in order to solve many real-world problems, such as the induction of fuzzy rules in Data Mining, convex cardinalities are not always appropriate. In this paper, we propose a possibilistic and a probabilistic cardinality of a fuzzy set with finite support. These cardinalities are not convex in general, but they are most suitable for solving problems and, contrary to the generalizing opinion, they are found to be more intuitive for humans. Their suitability relies mainly on the fact that they assume dependency among objects with respect to the property “to be in a fuzzy set”. The cardinality measures are generalized to relative ones among pairs of fuzzy sets. We also introduce a definition of the entropy of a fuzzy set by using one of our probabilistic measures. Finally, a fuzzy ranking of the cardinality of fuzzy sets is proposed, and a definition of graded equipotency is introduced.
Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology, 2015
Recently, some extensions of the classical fuzzy sets are studied in depth due to the good properties that they present. Among them, in this paper finite interval-valued hesitant fuzzy sets are the central piece of the study, as they are a generalization of more usual sets, so the results obtained can be immediately adapted to them. In this work, the cardinality of finite intervalvalued hesitant fuzzy sets is studied from an axiomatic point of view, along with several properties that this definition satisfies, being able to relate it to the classical definitions of cardinality given by Wygralak or Ralescu for fuzzy sets.
Fuzzy Sets and Systems, 2006
We present meta-theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification, when adopting a scalar approach to fuzzy set cardinality. The conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection. In particular, this conjunctor should fulfil a number of Bell-type inequalities. The advantage of these meta-theorems is that repetitious calculations can be avoided. This is illustrated in the demonstration of the Łukasiewicz transitivity of fuzzified versions of the simple matching coefficient and the Jaccard coefficient, or equivalently, the triangle inequality of the corresponding dissimilarity measures.
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.
Communications in computer and information science, 2020
In this contribution the concept how to solve the problem of comparability in the interval-valued fuzzy setting and its application in medical diagnosis is presented. Especially, we consider comparability of interval-valued fuzzy sets cardinality, where order of its elements is most important. We propose an algorithm for comparing interval-valued fuzzy cardinal numbers (IVFCNs) and we evaluate it in a medical diagnosis decision support system.
2009
Studies in Fuzziness and Soft Computing, Volume 244 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 228. Adam Kasperski Discrete Optimization with Interval Data, 2008 ISBN 978-3-540-78483-8 Vol. 229. Sadaaki Miyamoto, Hidetomo Ichihashi, Katsuhiro Honda Algorithms for Fuzzy Clustering, 2008 ISBN 978-3-540-78736-5 Vol. 230.
elm.az
In this paper we first outline the shortcomings of classical binary logic and Cantor's set theory in order to handle imprecise and uncertain information. Next we briefly introduce the basic notions of Zadeh's fuzzy set theory among them: definition of a fuzzy set, operations on fuzzy sets, the concept of a linguistic variable, the concept of a fuzzy number and a fuzzy relation. The major part consists of a sketch of the evolution of the mathematics of fuzziness, mostly illustrated with examples from my research group during the past 35 years. In this evolution I see three overlapping stages. In the first stage taking place during the seventies only straightforward fuzzifications of classical domains such as general topology, theory of groups, relational calculus, . . . have been introduced and investigated w.r.t. the main deviations from their binary originals. The second stage is characterized by an explosion of the possible fuzzifications of the classical structures which has lead to a deep study of the alternatives as well as to the enrichment of the structures due to the non-equivalence of the different fuzzifications. Finally some of the current topics of research in the mathematics of fuzziness are highlighted. Nowadays fuzzy research concerns standardization, axiomatization, extensions to lattice-valued fuzzy sets, critical comparison of the different so-called soft computing models that have been launched during the past three decennia for the representation and processing of incomplete information.
1994
Two dierent denitions of a Fuzzy number may b e found in the literature. Both fulll Goguen's Fuzzication Principle but are dierent in nature because of their dierent starting points. The rst one was introduced b y Z adeh and has well suited arithmetic and algebraic properties. The second one, introduced by Gantner, Steinlage and Warren, i s a good and formal representation of the concept from a topological point of view. The objective of this paper is to analyze these denitions and discuss their main features.
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