Rough Sets pp 465-500 | Cite as

From Rough to Fuzzy

  • Lech Polkowski
Part of the Advances in Soft Computing book series (AINSC, volume 15)


We have witnessed the development of logical calculi with truth values ranging continuously from 0 to 1 and in the development of the fuzzy sentential logic we have treated the set of logical axioms as a fuzzy set. The fuzzy sentential calculus in Chapter 13 has been developed in the framework of residuated lattices and an essential usage has been made of the adjoint pair (⊗, →). In the wider perspective of fuzzy calculi on sets it has turned useful to extend the notion of an adjoint pair to the notion of a pair (T,→ T ) where T is a triangular norm (or, tnorm) and → T is the induced residuated implication. t — norms and duals of them, tconorms, may be applied in the development of algebra of fuzzy sets viz. t — norms determine intersections of fuzzy sets according to the formula X A∩B (x) = T ( XA (x), XB (x)) where T is a t — norm and XA is the fuzzy characteristic (membership) function of the fuzzy set A while t — conorms may be used in determining unions of fuzzy sets via X A∪B (x) = T ( XA (x), XB (x)) where S is a t — conorm.


Residuated Lattice Fuzzy Partition Triangular Norm Adjoint Pair Fuzzy Similarity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Lech Polkowski
    • 1
    • 2
  1. 1.Polish-Japanese Institute of Information TechnologyWarsawPoland
  2. 2.Department of Mathematics and Computer ScienceUniversity of Wormia and MazuryOlsztynPoland

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