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From Rough to Fuzzy

  • Chapter
Rough Sets

Part of the book series: Advances in Soft Computing ((AINSC,volume 15))

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Abstract

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, t — norm) and → T is the induced residuated implication. t — norms and duals of them, t — conorms, 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.

Nature can only raid Reason to kill; but Reason can invade Nature to take prisoners and even to colonize

C. S. Lewis, Miracles, 4

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Authors and Affiliations

  1. Polish-Japanese Institute of Information Technology, ul. Koszykowa 86, 02-008, Warsaw, Poland

    Prof. Dr. Lech Polkowski

  2. Department of Mathematics and Computer Science, University of Wormia and Mazury, ul. Żofnierska 14a, 10-561, Olsztyn, Poland

    Prof. Dr. Lech Polkowski

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  1. Prof. Dr. Lech Polkowski
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© 2002 Springer-Verlag Berlin Heidelberg

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Polkowski, L. (2002). From Rough to Fuzzy. In: Rough Sets. Advances in Soft Computing, vol 15. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1776-8_14

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  • DOI: https://doi.org/10.1007/978-3-7908-1776-8_14

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-7908-1510-8

  • Online ISBN: 978-3-7908-1776-8

  • eBook Packages: Springer Book Archive

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