# Cardinalities of Fuzzy Sets

Springer Science & Business Media, Mar 11, 2003 - Mathematics - 195 pages
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 num bers: the natural numbers. That these complementary aspects of the counting pro cess 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 distinc tion between those objects which have to be counted - and those ones which have to be neglected for the particular counting process.

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### Contents

 Triangular Operations and Negations 1 11 Triangular Norms and Conorms 2 12 Negations 4 13 Associated TrianguIar Operations 5 14 Archimedean Triangular Operations 8 15 Induced Negations and Complementary Triangular Operations 14 16 Implications Induced by Triangular Norms 19 Fuzzy Sets 23
 411 The Corresponding Equipotency Relation 70 412 Inequalities 76 413 Arithmetical Operations 84 4132 Subtraction 97 4133 Multiplication 98 4134 Division 113 42 Generalized FLCounts 124 421 Equipotencies and Inequalities 126

 22 Operations on Fuzzy Sets 27 23 Generalized Operations 29 24 Other Elements of the Language of Fuzzy Sets 31 25 Towards Cardinalities of Fuzzy Sets 34 Scalar Cardinalities of Fuzzy Sets 45 32 Cardinality Patterns 48 33 Valuation Property and Subadditivity 53 34 Cartesian Product Rule and Complementarity 56 35 On the Fulfilment of a Group of the Properties 60 Generalized Cardinals with Triangular Norms 67
 422 Addition and Other Arithmetical Operations 131 43 Generalized FECounts 143 431 The Height of a Generalized FECount 147 432 Singular Fuzzy Sets 152 433 Equipotencies Inequalities and Arithmetical Questions 164 List of Symbols 181 Bibliography 185 Index 193 Copyright