Overview

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.

Full Product Details

Author:   Maciej Wygralak
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   Softcover reprint of the original 1st ed. 2003
Volume:   118
Dimensions:   Width: 15.50cm , Height: 1.10cm , Length: 23.50cm
Weight:   0.332kg
ISBN:  

9783642535147

ISBN 10:   3642535143
Pages:   195
Publication Date:   21 July 2012
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

1. Triangular Operations and Negations (Allegro).- 1.1. Triangular Norms and Conorms.- 1.2. Negations.- 1.3. Associated Triangular Operations.- 1.4. Archimedean Triangular Operations.- 1.5. Induced Negations and Complementary Triangular Operations.- 1.6. Implications Induced by Triangular Norms.- 2. Fuzzy Sets (Andante spianato).- 2.1. The Concept of a Fuzzy Set.- 2.2. Operations on Fuzzy Sets.- 2.3. Generalized Operations.- 2.4. Other Elements of the Language of Fuzzy Sets.- 2.5. Towards Cardinalities of Fuzzy Sets.- 3. Scalar Cardinalities of Fuzzy Sets (Scherzo).- 3.1. An Axiomatic Viewpoint.- 3.2. Cardinality Patterns.- 3.3. Valuation Property and Subadditivity.- 3.4. Cartesian Product Rule and Complementarity.- 3.5. On the Fulfilment of a Group of the Properties.- 4. Generalized Cardinals with Triangular Norms (Rondeau à la polonaise).- 4.1. Generalized FGCounts.- 4.2. Generalized FLCounts.- 4.3. Generalized FECounts.- List of Symbols.

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