Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

INTERVAL-VALUED FUZZY SET MODELING UNTUK RELIABILITAS SISTEM

Abstrak Pemodelan reliabilitas sistim dalam kaitan dengan istilah teori himpunan fuzzy adalah pada dasarnya memanfaatkan himpunan-himpunan fuzzy Type I, di mana keanggotaan fuzzy ini diasumsikan sebagai fungsi positif  menurut titik yang berkisar di [0,1]. Seperti itu adalah suatu praktek yang tidak  praktis karena sebuah keanggotaan interval-valued  boleh mencerminkan ketidakjelasan dari sistim lebih baik menurut pola pemikiran manusia. Dalam paper ini, kita akan mengeksplor dasar-dasar teori Interval-valued himpunan fuzzy dan menggambarkan aplikasinya dalam kaitan dengan  istilah sebuah contoh industri. Kata Kunci : Interval-Valued, Fuzzy Sets, reliabilitas sistem   Abstract System reliability modeling in relation to the term fuzzy set theory is basically utilizing fuzzy sets Type I, in which the fuzzy membership is assumed to be a positive function by point range in [0,1]. As it is a practice that is not practical because of an interval-valued membership should reflect the uncertainty of the system better by human thought patterns. In this paper, we will explore the basics of the theory of interval-valued fuzzy set and describe its application in relation to the terms of an industrial example.   Keywords: Interval-Valued, Fuzzy Sets, system reliabilit

Bi-ideals of ordered semigroups based on the interval-valued fuzzy point

Interval-valued fuzzy set theory (advanced generalization of Zadeh’s fuzzy sets) is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, we introduce the notion of generalized quasi-coincident with (q(Formula Presented)) relation of an interval-valued fuzzy point with an interval-valued fuzzy set. In fact, this new concept is a more generalized form of quasi-coincident with relation of an interval-valued fuzzy point with an interval-valued fuzzy set. Applying this newly defined idea, the notion of an interval-valued (∈,∈vq(Formula Presented)) -fuzzy bi-ideal is introduced. Moreover, some characterizations of interval-valued (∈,∈vq(Formula Presented)) -fuzzy bi-ideals are described. It is shown that an interval-valued (∈,∈vq(Formula Presented)) -fuzzy bi-ideal is an interval-valued fuzzy bi-ideal by imposing a condition on interval-valued fuzzy subset. Finally, the concept of implication-based interval-valued fuzzy bi-ideals, characterizations of an interval-valued fuzzy bi-ideal and an interval-valued (∈,∈vq(Formula Presented)) - fuzzy bi-ideal are considered

Triangular norms which are join-morphisms in 3-dimensional fuzzy set theory

The n-dimensional fuzzy sets have been introduced as a generalization of interval-valued fuzzy sets, Atanassov's intuitionistic and interval-valued intuitionistic fuzzy sets. In this paper we investigate t-norms on 3-dimensional sets which are join-morphisms. Under some additional conditions we show that they can be represented using a representation which generalizes a similar representation for t-norms in interval-valued fuzzy set theory

Interval valued (\in,\ivq)-fuzzy filters of pseudo BLBL-algebras

We introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, we introduce the notions of interval valued (\in,\ivq)-fuzzy filters of pseudo $BL$-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval valued fuzzy filters are derived. The relationship among these generalized interval valued fuzzy filters of pseudo $BL$-algebras is considered. Finally, we consider the concept of implication-based interval valued fuzzy implicative filters of pseudo $BL$-algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed

Comments on: Interval Type-2 Fuzzy Sets are generalization of Interval-Valued Fuzzy Sets: Towards a Wider view on their relationship

This Letter makes some observations about [2] that further support the distinction between an interval type-2 fuzzy set (IT2 FS) and an interval-valued fuzzy set (IV FS), points out that all operations, methods and systems that have been developed and published about IT2 FSs are, so far, only valid in the special case when IT2 FS = IVFS, and suggests some research opportunities