Four-Valued Extension of Rough Sets

  • Aida Vitória
  • Andrzej Szałas
  • Jan Małuszyński
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5009)


Rough set approximations of Pawlak [15] are sometimes generalized by using similarities between objects rather than elementary sets. In practical applications, both knowledge about properties of objects and knowledge of similarity between objects can be incomplete and inconsistent. The aim of this paper is to define set approximations when all sets, and their approximations, as well as similarity relations are four-valued. A set is four-valued in the sense that its membership function can have one of the four logical values: unknown (u), false (f), inconsistent (i), or true (t). To this end, a new implication operator and set-theoretical operations on four-valued sets, such as set containment, are introduced. Several properties of lower and upper approximations of four-valued sets are also presented.


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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Aida Vitória
    • 1
  • Andrzej Szałas
    • 2
    • 3
  • Jan Małuszyński
    • 3
  1. 1.Department of Science and TechnologyLinköping UniversityNorrköpingSweden
  2. 2.Institute of InformaticsWarsaw UniversityWarsawPoland
  3. 3.Department of Computer and Information ScienceLinköping UniversityLinköpingSweden

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