L-fuzzy Sets and Intuitionistic Fuzzy Sets

DOI: 10.1007/978-3-540-72687-6_16 In book: Computational Intelligence Based on Lattice Theory, Publisher: Springer, pp.325-339


Summary. In this article we firstly summarize some notions on L-fuzzy sets, where L denotes a complete lattice. We then study a special case of L-fuzzy sets, namely the “intuitionistic fuzzy sets”. The importance of these sets comes from the fact that the negation is
being defined independently from the fuzzy membership function. The latter implies both flexibility and e.ectiveness in fuzzy
inference applications. We additionally show several practical applications on intuitionistic fuzzy sets, in the context of
computational intelligence.

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Available from: Anestis Hatzimichailidis, Oct 02, 2014
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    • "[0; 1] and A : S ! [0; 1] denote the degree of membership (namely A (x)) and the degree of non-membership (namely A (x)) of each element x 2 S to the set A respectively, and 0 A (x) + A (x) 1 for all x 2 S (cf.[5]). In particular, 0 and 1 denote the intuitionistic fuzzy empty set and the intuitionistic fuzzy whole set in a set X de…ned by 0 (x) = (0; 1) and 1 (x) = (1; 0) for each x 2 X, respectively. "

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    • "Both approaches, IFS and IVFS theory, have the virtue of complementing fuzzy sets, that are able to model vagueness, with an ability to model uncertainty as well. 2 IVFSs reflect this uncertainty by the length of the interval membership degree [l 1 ; l 2 ], while in IFS theory for every membership degree ðl; mÞ, the value p ¼ 1 À l À m denotes a measure of non-determinacy (or un- decidedness). Each approach has given rise to an extensive literature covering their respective applications, but surprisingly very few people seem to be aware of their equivalence, stated first in [2] and later in [31] [63]. Indeed, take any IVFS A in a universe X , and assume that the membership degree of x in A is given as the interval [l 1 ; l 2 ]. "
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    ABSTRACT: With the demand for knowledge-handling systems capable of dealing with and distinguishing between various facets of imprecision ever increasing, a clear and formal characterization of the mathematical models implementing such services is quintessential. In this paper, this task is undertaken simultaneously for the definition of implication within two settings: first, within intuitionistic fuzzy set theory and secondly, within interval-valued fuzzy set theory. By tracing these models back to the underlying lattice that they are defined on, on one hand we keep up with an important tradition of using algebraic structures for developing logical calculi (e.g. residuated lattices and MV algebras), and on the other hand we are able to expose in a clear manner the two models’ formal equivalence. This equivalence, all too often neglected in literature, we exploit to construct operators extending the notions of classical and fuzzy implication on these structures; to initiate a meaningful classification framework for the resulting operators, based on logical and extra-logical criteria imposed on them; and finally, to re(de)fine the intuititive ideas giving rise to both approaches as models of imprecision and apply them in a practical context.
    International Journal of Approximate Reasoning 01/2004; 35(1-35):55-95. DOI:10.1016/S0888-613X(03)00072-0 · 2.45 Impact Factor
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    • "Following the idea of the L-fuzzy sets, in 1984 Stefka Stoeva and the author introduced the concept of " Intuitionistic L- Fuzzy Set. Late, in [10] Dogan Coker proved that Pawlak's fuzzy rough sets are intuitionistic L-fuzzy sets, while Guo-jun Wang and Ying-Yu He in [21], and Chris Cornelis, Etienne Kerre and Glad Deschrijver in [11] discussed relations between L-fuzzy sets and IFSs. "
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    ABSTRACT: Remarks on history, theory, and appli- cations of intuitionistic fuzzy sets are given. Some open problems are intro- duced.
    Proceedings of the 3rd Conference of the European Society for Fuzzy Logic and Technology, Zittau, Germany, September 10-12, 2003; 01/2003
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