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Cardinalities of Fuzzy Sets

  • Book
  • © 2003

Overview

Authors:
  1. Maciej Wygralak

    1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland

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  • First monograph presenting cardinality theory of fuzzy sets with triangular norms, including its scalar and fuzzy streams
  • Self-contained, systematic presentation equipped with many examples

Part of the book series: Studies in Fuzziness and Soft Computing (STUDFUZZ, volume 118)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Triangular Operations and Negations

    • Maciej Wygralak
    Pages 1-21

  3. Fuzzy Sets

    • Maciej Wygralak
    Pages 23-44

  4. Scalar Cardinalities of Fuzzy Sets

    • Maciej Wygralak
    Pages 45-66

  5. Generalized Cardinals with Triangular Norms

    • Maciej Wygralak
    Pages 67-179

  6. Back Matter

    Pages 181-195
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Keywords

About this book

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.

Authors and Affiliations

  • Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland

    Maciej Wygralak

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