rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Cardinalities of Fuzzy Sets
  name:
    name part: Wygralak, Maciej.
    role:
      role term: creator
    role:
      role term: author.
    role:
      role term: aut
    role:
      role term: http://id.loc.gov/vocabulary/relators/aut
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: gw
    date issued: 2003
    edition: 1st ed. 2003.
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XIV, 195 p. online resource.
  abstract: 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.
  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.
  note: by Maciej Wygralak.
  subject:
    topic: Discrete mathematics
  subject:
    topic: Computational complexity
  subject:
    topic: Computer science-Mathematics
  subject:
    topic: Group theory
  subject:
    topic: System theory
  subject:
    topic: Discrete Mathematics
  subject:
    topic: Complexity
  subject:
    topic: Math Applications in Computer Science
  subject:
    topic: Group Theory and Generalizations
  subject:
    topic: Systems Theory, Control
  classification: QA150-272
  classification: 511.1
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: Studies in Fuzziness and Soft Computing, 118
  identifier: 9783540363828
  identifier: https://doi.org/10.1007/978-3-540-36382-8
  record info:
    record creation date: 121227
    record change date: 20191022023038.0
    record identifier: 978-3-540-36382-8