# Cardinalities of Fuzzy Sets

• Maciej Wygralak
Book

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

1. Front Matter
Pages i-xiv
2. Maciej Wygralak
Pages 1-21
3. Maciej Wygralak
Pages 23-44
4. Maciej Wygralak
Pages 45-66
5. Maciej Wygralak
Pages 67-179
6. Back Matter
Pages 181-195

### Introduction

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.

### Keywords

Cardinality of Fuzzy Sets Computer Fuzzy Fuzzy Sets Information Many-Valued Logic Symbol Triangular Norm cardinality cardinals computer science fuzzy set modeling set theory

#### Authors and affiliations

• Maciej Wygralak
• 1
1. 1.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-540-36382-8
• Copyright Information Springer-Verlag Berlin Heidelberg 2003
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-53514-7
• Online ISBN 978-3-540-36382-8
• Series Print ISSN 1434-9922
• Series Online ISSN 1860-0808