Cardinalities of Fuzzy Sets

Front Cover
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

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