Ebook: Cardinalities of Fuzzy Sets
- Tags: Complexity, Math Applications in Computer Science, Group Theory and Generalizations, Systems Theory Control
- Series: Studies in Fuzziness and Soft Computing 118
- Year: 2003
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
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.
This is the first book presenting cardinality theory of fuzzy sets with triangular norms, including its scalar and "fuzzy" streams. This theory constitutes not only a powerful basis but also a useful tool for modelling and processing vague and imprecise quantitative information. The multiple application areas of the theory encompass computer science, soft computing, computing with words, and decision-making. Starting with a presentation of the fundamentals of triangular norms and fuzzy set theory, the book offers a self-contained, concise and systematic exposition of cardinalities of fuzzy sets that includes many examples.
This is the first book presenting cardinality theory of fuzzy sets with triangular norms, including its scalar and "fuzzy" streams. This theory constitutes not only a powerful basis but also a useful tool for modelling and processing vague and imprecise quantitative information. The multiple application areas of the theory encompass computer science, soft computing, computing with words, and decision-making. Starting with a presentation of the fundamentals of triangular norms and fuzzy set theory, the book offers a self-contained, concise and systematic exposition of cardinalities of fuzzy sets that includes many examples.
Content:
Front Matter....Pages i-xiv
Triangular Operations and Negations....Pages 1-21
Fuzzy Sets....Pages 23-44
Scalar Cardinalities of Fuzzy Sets....Pages 45-66
Generalized Cardinals with Triangular Norms....Pages 67-179
Back Matter....Pages 181-195
This is the first book presenting cardinality theory of fuzzy sets with triangular norms, including its scalar and "fuzzy" streams. This theory constitutes not only a powerful basis but also a useful tool for modelling and processing vague and imprecise quantitative information. The multiple application areas of the theory encompass computer science, soft computing, computing with words, and decision-making. Starting with a presentation of the fundamentals of triangular norms and fuzzy set theory, the book offers a self-contained, concise and systematic exposition of cardinalities of fuzzy sets that includes many examples.
Content:
Front Matter....Pages i-xiv
Triangular Operations and Negations....Pages 1-21
Fuzzy Sets....Pages 23-44
Scalar Cardinalities of Fuzzy Sets....Pages 45-66
Generalized Cardinals with Triangular Norms....Pages 67-179
Back Matter....Pages 181-195
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