Maciej Wygralak

Cardinalities of Fuzzy Sets

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Maciej Wygralak

Cardinalities of FuzzySets

, Springer

Prof. Dr. Maciej Wygralak Adam Mickiewicz University Faculty of Mathematics and Computer Science Umultowska 87 61-614 Po zn an Poland

E-mail: wygralak@math.amu.edu.pl

ISSN 1434-9922

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ISBN 978-3-642-53514-7 ISBN 978-3-540-36382-8 (eBook) DOI 10.1007/978-3-540-36382-8

To the memory of my Parents

Foreword

Counting is one of the basic elementary mathematical activities. It comes with two complementary aspects: to determine the number of elements of a set - and to create an ordering between the objects of counting just by counting them over.

For finite sets of objects these two aspects are realized by the same type of num­bers: the natural numbers. That these complementary aspects of the counting pro­cess may need different kinds of numbers becomes apparent if one extends the process of counting to infinite sets. As general tools to determine numbers of elements the cardinals have been created in set theory, and set theorists have in parallel created the ordinals to count over any set of objects.

For both types of numbers it is not only counting they are used for, it is also the strongly related process of calculation - especially addition and, derived from it, multiplication and even exponentiation - which is based upon these numbers.

For fuzzy sets the idea of counting, in both aspects, looses its naive foundation: because it is to a large extent founded upon of the idea that there is a clear distinc­tion between those objects which have to be counted - and those ones which have to be neglected for the particular counting process.

In fuzzy sets, however, there are objects which "belong" to a fuzzy set only partial­Iy, Le. only to some degree. How to count them? Also only partially? And what may this mean, Le. does this vague idea make any true technical sense?

Even more as in classical set theory, in fuzzy set theory there seems to be a diffe­rence between the two aspects of the determination of the number of elements of a fuzzy set A, i.e. between the determination of a cardinality for A, and the crea­tion of an ordering between the objects of A. So it is quite natural for fuzzy sets to discuss these two aspects really separately.

The present book is completely devoted to the first one of these aspects: to the problem of the development of a cardinality theory for fuzzy sets. Besides all the

viii Foreword

mathematical techniques, such a theory has to start from a quite simple, but rather important idea about what kind of things such cardinals for fuzzy sets should be. And it seems that there are just two quite different kinds of approach available: The first idea is that a cardinality of a fuzzy set should be some kind of standard number known from classical mathematics. The second idea is that the fact that one has for some objects only a gradual knowledge about their membershiphood in a fuzzy set A forces that the "number of elements" of Ais determined also only gradually - which, formally, is understood as meaning that the cardinality of a fuzzy set is itself a fuzzy set (of standard numbers).

The author of this mono graph , since years one of the leading researchers in the field of fuzzy cardinality theory, explains in detail both these different ideas to cope with the notion of cardinality of fuzzy sets, and develops the corresponding mathematical theories of cardinals for fuzzy sets. This includes the development of the basic arithmetical operations of addition, multiplication, and exponentiation for these cardinals, and the study of their arithmetical properties, as weil as of the relationships between these arithmetical operations and the basic set algebraic ope­rations for fuzzy sets.

In all these fields, the author does not only offer the reader the actual develop­ments, he even extends the known results here in an essential way paving a path to a higher degree of abstraction, and hence to more flexibility for applications.

Sieg/ried Gottwald Leipzig University

Preface

The title of this monograph determines cIearly its subject: fuzzy sets and their cardinality theory. It is rather trivial to say that cardinality belongs to the most fundamental characteristics and attributes of a set. Exactly the same concems fuzzy sets, a generalization of sets founded upon many-valued logic instead of cIassical two-valued one. If one likes to define the notion of cardinality of a fuzzy set, the essential difficulty and difference in comparison with sets is, however, that to be an element of a fuzzy set is generally a matter of degree. Consequently, counting and cardinal calculus for fuzzy sets become a task which is much more advanced than in the case of sets, even if one deals with finite fuzzy sets. This, on the other hand, is why cardinality seems to be one of the most fascinating and enigmatic aspects of fuzzy sets. The growing interest in and development of the ideas of granulation, computing with words and computational theory of perceptions, pro­posed and advocated by Professor Lotfi A. Zadeh, make that aspect a yet more important one in recent years.

This book presents the state of the art in cardinality theory for fuzzy sets with tri angular norm-based operations. The two main approaches to the question of what kind of mathematical object a cardinality of a fuzzy set should be: the scalar ap­proach and the "fuzzy" one, are studied in detail. Using the scalar perception of cardinality offered by the former approach, that object is a single ordinary cardinal number, a single nonnegative integer or real number if the fuzzy set is finite. The alternative "fuzzy" perception in the latter approach lies in viewing the cardinality of a fuzzy set as a fuzzy set of ordinary cardinals (of nonnegative integers in the finite case). Dur attention will be focused on cardinalities of finite fuzzy sets. It is rather evident that just those fuzzy sets do play a leading role in applications of fuzzy sets in various areas of computer and information sciences, mathematical modeling, decision-making, control, system theory, engineering, etc. Their cardi­nality theory thus deserves special attention and treatment in the form of aseparate presentation. This monograph can, therefore, be of interest not only to mathemati­cians and scientists, but also to engineers, practitioners, lecturers and students from

x Preface

many fields who use or are interested in fuzzy set-based methods and techniques and like to become familiar with cardinal calculus for (finite) fuzzy sets. The presentation is self-contained and no prior knowledge of fuzzy sets and triangular norms is required to understand the book. However, the reader should have basic mathematical knowledge of classical set theory, logic, mathematical analysis and general algebra.

The book is divided into four chapters. The first two ones are of general intro­ductory and preparatory character. The remaining chapters do form the principal part of the monograph and are completely devoted to cardinalities of fuzzy sets. In order to obtain a simple numbering system for theorems, definitions and formu­lae, a consecutive numbering within each chapter is carried on, disregarding its division into sections, subsections and units.

Chapter 1 presents the fundamentals of the theory of tri angular operations with special reference to those notions, aspects and facts which will be useful in the next chapters. In particular, much attention is paid to Archimedean triangular ope­rations, induced negations, and associated and complementary operations.

In Chapter 2, the fundamentals of fuzzy sets and their language are described, including triangular norm-based operations and elements of many-valued sentential calculus. The last section contains an introduction to the problem of cardinalities of fuzzy sets. Motivations and a review of approaches to and concepts of those cardinalities are given therein. Among other constructions, three types of "fuzzy" cardinals for fuzzy sets with triangular norm-based operations are defined. Their theory, encompassing questions of equipotency, ordering relations and arithmetical operations, is developed in Chapter 4.

Chapter 3 is devoted to a general axiomatic theory of scalar cardinalities for fuzzy sets with triangular operations.

Maciej Wygralak Poznan, October 2002

Acknowledgments

There are several people whom I would like to thank. In the first place, as each book project causes a disturbance in family life of its author(s), it is my pleasant duty to thank my wife, Renata, and our daughters, Karolina and Agata, for their patience, understanding and support.

I am deeply indebted to Professor Aleksander Waszak, my department head, for his friendliness, support, encouragement and understanding. Let me thank my Faculty of Mathematics and Computer Science for a partial support of the final stage of this book project by Grant GN-01l2002, and Dr. Krzysztof Dyczkowski from my department for his kind technical assistance.

Also, I am very much appreciative of the constant and dependable support and friendliness of Professor lerzy Albrycht, Poznan.

My sincere thanks go to Professor Lotfi A. Zadeh of the University of Califomia at Berkeley, the founder of fuzzy set theory and fuzzy logic, for his frequent words of personal encouragement, interest and inspiring advice which I always appreciate enormously.

Further, I am very grateful to Professor Siegfried Gottwald, Leipzig University, for his friendliness and support since many years. Stimulating discussions and critical remarks he always offered me during my research stays in Leipzig are difficult to overestimate.

My special thanks for kind cooperation are due to Professor lanusz Kacprzyk, the series editor, who has encouraged me to write this book. I am also grateful to Ms Katharina Wetzel-Vandai, M. A., Dr. Thomas Ditzinger and Ms ludith Kripp from Springer-Verlag for kind and efficient cooperation.

Macie} Wygralak

Table of Contents

Foreword . . . . . vii

Preface ...... ix

Acknowledgments . xi

1. Triangular Operations and Negations (Allegro) 1

1.1. Triangular Norms and Conorms . 2

1.2. Negations. . . . . . . . . . . . . 4

1.3. Associated Triangular Operations 5

1.4. Archimedean Triangular Operations 8

1.5. Induced Negations and Complementary Triangular Operations. 14

1.6. Implications Induced by Triangular Norms . . . . . . . . . . . 19

2. Fuzzy Sets (Andante spianato) 23

2.1. The Concept of a Fuzzy Set 23

2.2. Operations on Fuzzy Sets . 27

2.3. Generalized Operations. . . 29

2.4. Other Elements of the Language of Fuzzy Sets 31

2.5. Towards Cardinalities of Fuzzy Sets . . . . . . 34

3. Scalar Cardinalities of Fuzzy Sets (Scherzo) 45

3.1. An Axiomatic Viewpoint . . . . . . . 45

3.2. Cardinality Patterns. . . . . . . . . . 48

3.3. Valuation Property and Subadditivity . 53

xiv Table 0/ Contents

3.4. Cartesian Product Rule and Complementarity

3.5. On the Fulfilment of a Group of the Properties

3.5.1. VAL and CART .

3.5.2. CART and COMP

3.5.3. VAL and COMP .

3.5.4. VAL, CART and COMP .

4. Generalized Cardinals with Triangular Norms (Rondeau dia polonaise)

56

60

60

64

65

65

67

4.1. Generalized FGCounts . . . . . . . . . 67

4.1.1. The Corresponding Equipotency Relation 70

4.1.2. Inequalities . . . . . . . 76

4.1.3. Arithmetical Operations 84

4.1.3.1. Addition . . . 84 4.1.3.2. Subtraction . . 97 4.1.3.3. Multiplication 98 4.1.3.4. Division . . . 113 4.1.3.5. Exponentiation 117

4.1.4. Some Derivative Concepts of Cardinality 122

4.2. Generalized FLCounts . . . . . . . . . . . . . 124

4.2.1. Equipotencies and Inequalities . . . . . . 126

4.2.2. Addition and Other Arithmetical Operations 131

4.3. Generalized FECounts . . . . . . . . . . . 143

4.3.1. The Height of a Generalized FECount 147

4.3.2. Singular Fuzzy Sets ........ . 152

4.3.3. Equipotencies, Inequalities and Arithmetical Questions 164

List of Symbols 181

Bibliography 185

Index . . . . 193