Encyclopedia of Optimization

2009 Edition
| Editors: Christodoulos A. Floudas, Panos M. Pardalos

Boolean and Fuzzy Relations

  • Ladislav J. Kohout
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-74759-0_53
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Article Outline

Keywords

Boolean Relations

  Propositional Form

  Heterogeneous and Homogeneous Relations

  The Satisfaction Set

  The Extensionality Convention

  The Digraph Representation

  Foresets and Aftersets of Relations

  Matrix Representation

Operations and Inclusions in ℛ(A ↝ B)

  Unary Operations

Binary Operations on Successive Relations

  Matrix Formulation of the Binary Operations

  Non-Associative Products of Relations

Characterization of Special Properties of Relations Between Two Sets

Relations on a Single Set: Special Properties

Partitions IN and ON a Set

Tolerances and Overlapping Classes

Hierarchies in and on a Set: Local and Global Orders and Pre-orders

Fuzzy Relations

  Definitions

Operations and Inclusion on ℛ F (X ↝ Y)

  Fuzzy Relations with Min, Max Connectives

  Fuzzy Relations Based on Łukasiewicz Connectives

  Fuzzy Relations With t-Norms and Co-Norms

Products: ℛ F (X ↝ Y) × ℛ F (Y ↝ Z) → ℛ F (X ↝ Z)

   N-ary Relations

Special Properties of Fuzzy Relations

  Alpha-cuts of Fuzzy Relations

  Fuzzy Partitions, Fuzzy Clusters and Fuzzy Hierarchies

Closures and Interiors with Special Properties

Applications of Relational Methods in Engineering, Medicine and Science

Brief Review of Theoretical Development

Basic Books and Bibliographies

See also

References

Keywords

Fuzzy relations Local relational properties Closures Interiors Pre-order Tolerances Equivalences BK-products Relational compositions Nonassociative products Generalized morphism Universal properties of relations n-ary relation Scientific applications Medicine Psychology Engineering applications Artificial intelligence Value analysis Decision theory 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Ladislav J. Kohout
    • 1
  1. 1.Department Computer Sci.Florida State UniversityTallahasseeUSA