C-fuzzy numbers and a dual of extension principle

1994

Two dierent denitions of a Fuzzy number may b e found in the literature. Both fulll Goguen's Fuzzication Principle but are dierent in nature because of their dierent starting points. The rst one was introduced b y Z adeh and has well suited arithmetic and algebraic properties. The second one, introduced by Gantner, Steinlage and Warren, i s a good and formal representation of the concept from a topological point of view. The objective of this paper is to analyze these denitions and discuss their main features.

The use of new concept of fuzzy number as an equivalence class on for solving fuzzy equation is proposed. This approach of solving fuzzy equations by new concept of fuzzy number has a considerable advantage. Through theoretical analysis, an illustrative example and computational results the paper shows that the proposed approach is more general and straightforward for solving fuzzy algebraic equations.

Journal of Intelligent & Fuzzy Systems, 2017

Starting from the study of the papers [1, 3, 4, 5, 13, 14], we have seen that we can change some concepts and give some new definitions on fuzzy algebraic structures. Our goal by introducing these definitions is to get the classical and well known ones when the fuzzy set is just the characteristic function. As a second purpose we were tempted to generalize as long as it is possible some results known in the classical set theory. We get many interesting results which concern fuzzy relations, fuzzy subgroups of a given group, fuzzy ideals of a ring. We also define ideal generated by an element in the commutative case, prime and maximal ideals. Some relationships have been made and many other still open and which will be of our interest in next studies.

Mathematics and Statistics, 2022

In this paper, a new hypothesis of fuzzy number has been proposed which is more precise and direct. This new proposed approach is considered as an equivalence class on set of real numbers 𝑅 with its algebraic structure and its properties along with theoretical study and computational results. Newly defined hypothesis provides a well-structured summary that offers both a deeper knowledge about the theory of fuzzy numbers and an extensive view on its algebra. We defined field of newly defined fuzzy numbers which opens new era in future for fuzzy mathematics. It is shown that, by using newly defined fuzzy number and its membership function, we are able to solve fuzzy equations in an uncertain environment. We have illustrated solution of fuzzy linear and quadratic equations using the defined new fuzzy number. This can be extended to higher order polynomial equations in future. The linear fuzzy equations have numerous applications in science and engineering. We may develop some iterative methods for system of fuzzy linear equations in a very simple and ordinary way by using this new methodology. This is an innovative and purposefulness study of fuzzy numbers along with replacement of this newly defined fuzzy number with ordinary fuzzy number.

Information Sciences, 1991

This paper deals with algebraic equations involving generalized fuzzy numbers with continuous membership functions. The generalized fuzzy number defined in this paper is a general name for fuzzy numbers, fuzzy intervals, crisp numbers, and interval numbers. Tbree important properties of the generalized fuzzy numbers with continuous membership functions are introduced as the basis for deriving the solvability criterion. The sufficient and necessary condition for solving A & X = C is derived. The sufficient and necessary condition for solving A X X = C and A + X = C when A and C are nonzero generalized fuzzy numbers is also derived in this paper. Instead of the inverse operation method, which fails for fuzzy sets, inverse operations on the boundary points of cy-level set intervals for continuous membership functions are used when the derived criteria of solvability are satisfied. A unique solution rather than a solution set is obtained.

Journal of Computer Science and Cybernetics

Following our investigations on some fuzzy algebraic structures started in [6--8], and [9], in the present work, we revisit fuzzy groups and fuzzy ideals and introduce some new examples and then define the notion of fuzzy relation modulo a fuzzy subgroup and modulo a fuzzy ideal. As a consequence, we introduce the right and left cosets modulo a fuzzy relation. This work and the previously cited ones can be considered as a continuation of investigations initiated in [1--5]. Toward our investigation, we have in mind that by introducing these new definitions, the results that we can get should represent generalization of classical and commonly known concepts of algebra.

The objective of this paper is to develop arithmetic operations between generalized trape-zoidal (triangular) fuzzy numbers so that the drawbacks of the existing works are overcome. In this respect, the extension principle has been used to calculate diierent arithmetic op-erations.

Kybernetika

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In this article, our main intention is to revisit the existing definition of complementation of fuzzy sets and thereafter various theories associated with it are also commented on. The main contribution of this paper is to suggest a new definition of complementation of fuzzy sets on the basis of reference function. Some other results have also been introduced whenever possible by using this new definition of complementation.

Communications in Computer and Information Science, 2010

We present new tools for fuzzy arithmetic with fuzzy numbers, based on the parametric representation of fuzzy numbers and new fuzzy operations, the generalized difference and the generalized division of fuzzy numbers. The new operations are described in terms of the parametric LR and LU representations of fuzzy numbers and the corresponding algorithms are described. An application to the solution of simple fuzzy equations is illustrated.