On the relevance of some families of fuzzy sets

Lecture Notes in Computer Science, 2000

In this paper we show that Atanassov's Intuitionistic Fuzzy sets can be viewed as a classification model, that can be generalized in order to take into account more classes than the three classes considered by Atanassov's (membership, non-membership and non-determinacy). This approach will imply, on one hand, to change the meaning of these classes, so each one will have a positive definition. On the other hand, this approach implies the possibility of a direct generalization for alternative logics and additional valuation states, being consistent with Atanassov's focuss. From this approach we shall stress the absence of any structure within those three valuation states in Atanassov's model. In particular, we consider this is the main cause of the dispute about Atanassov's model: acknowledging that the name intuitionistic is not appropriate, once we consider that a crisp direct graph is defined in the valuation space, formal differences with other three-state models will appear.

2013

When Atanssov created Intuitionistic Fuzzy Sets, he imposed the condition that the sum of membership and nonmembership values for each point in the universe of discours e should be less than or equal to one. We challenge this constra int and define some new types of Intuitionistic Fuzzy Sets such th at, for any point in the universe of discourse, the sum of members hip and non-membership values can be greater than one, or their difference can be negative or positive, while one value is gr eater than the other, or the sum of their squares is less than or equa l to one. Keywords-Intuitionistic fuzzy set, Interval-valued fuzzy sets. I. I NTRODUCTION Fuzzy Sets concept was introduced by Zadeh [1] in 1965. Given an non-empty universe of discourse X , one can define a fuzzy set A based on its membership function μA : X → [0, 1], that is A is a set with ‘vague boundary’ when compared with crisp sets, where μA : X → {0, 1}. Of course, the bigger the value of μA(x) is, the greater the degree...

Fuzzy Sets and Systems, 2005

This note points out a terminological clash between Atanassov's "intuitionistic fuzzy sets" and what is currently understood as intuitionistic logic. They differ both by their motivations and their underlying mathematical structure. Furthermore, Atanassov's construct is isomorphic to interval-valued fuzzy sets and other similar notions, even if their interpretive settings and motivation are quite different, the latter capturing the idea of ill-known membership grade, while the former starts from the idea of evaluating degrees of membership and non-membership independently. This paper is a plea for a clarification of terminology, based on mathematical resemblances and the comparison of motivations between "intuitionistic fuzzy sets" and other theories .

Handbook of Granular Computing, 2008

International Journal of Approximate Reasoning, 2004

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 modelsÕ 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.

Information Sciences, 2014

We address the problem of how to measure the amount of knowledge conveyed by the Atanassov's intuitionistic fuzzy set (A-IFS for short). The problem is relevant from the point of view of many application areas, notably decision making. An amount of knowledge considered is strongly linked to its related amount of information. Our analysis is concerned with an intrinsic relationship between the positive and negative information and a lack of information expressed by the hesitation margin. Illustrative examples are shown.

Studies in Computational Intelligence, 2007

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 effectiveness in fuzzy inference applications. We additionally show several practical applications on intuitionistic fuzzy sets, in the context of computational intelligence.

Bulletin of the American Mathematical Society

Emerging Research on Applied Fuzzy Sets and Intuitionistic Fuzzy Matrices

In this chapter, we define four separations of generalized interval-valued intuitionistic fuzzy sets (GIVIFSs). In fact, all interval-valued Intuitionistic fuzzy sets (IVIFSs) are GIVIFSs but all GIVIFSs are not IVIFSs. Also, we studied some properties of the four separated subsets of GIVIFSs.

European Society for Fuzzy Logic and Technology, 2003

This paper concerns the theory of intui- tionistic fuzzy sets according to Atanassov. If triangular norms, especially nonstrict Archimedean ones, are used, we propose a revision and a flexibilizing generalization of some fundamental notions and constructions of that theory. Its application to group de- cision making is outlined.