Definition of the Complement of Fuzzy Number: An Exposition

2011

It has been accepted that the fuzzy sets do not form a field. In this article, we are going to put forward an extension of the definition of fuzziness. With the help of this extension, we would be able to define the complement of a fuzzy set properly. This in turn would allow us to assert that fuzzy sets do form a field. In fact, the fuzzy membership value and the fuzzy membership function for the complement of a fuzzy set are two different things. This confusion has created a stumbling block towards accepting the theory of fuzzy sets as a generalization of the classical theory of sets.

Mathematics Letters, 2021

Overtime, mathematics had been used as a tool in modeling real life phenomenon. In some cases, these problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty. The uncertainty could either be due to lack of clear boundaries in the description of the object or perhaps due to randomness. In this article, we study a mathematical tool discovered in 1965 by Zadeh suitable for modeling real life phenomenon and examined operations on such a tool. Motivated by the work of Zadeh, we studied operators on Type-1 Fuzzy Sets (T1FSs) and Type-2 Fuzzy sets (T2FSs) and provided examples, one of which is a variant of the Yager complement function for which the complement operator was graphically illustrated. The joint and the meet operators were also studied and examples provided. Non-standard operators were defined on T1FSs and T2FSs and also classified into two groups; the triangular-norm (t-norm) and triangular-conorm (t-conorm). Using tnorm and t-conorm, an example was adopted from Castillo and Aguilar to illustrate the computation of the standard operation on T2FSs. Finally, future research direction was provided based on what is yet to be achieved in fuzzy set theory.

2007

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Soft Computing, 2014

Your article is protected by copyright and all rights are held exclusively by Springer-Verlag Berlin Heidelberg. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com".

Fuzzy Sets and Systems, 2003

This is an unusual book. Why and in which respect? It is amazing how Prof. Klir is able to explain a number of nontrivial facts using simple and succinct means. In 19 chapters of this small book he has succeeded in describing the essential theory of fuzzy set and fuzzy logic theory including their applications as well as explaining the main philosophical problems araising when dealing with uncertainty and vagueness. Thus, the book is very informative-one can ÿnd everything relevant there, mostly only outlined. However, the core of the problem is completed by enough references to be able to ÿnd missing information.

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.

Iranian Journal of Fuzzy Systems, 2009

The operations in the set of fuzzy numbers are usually obtained by the Zadeh extension principle. But these definitions can have some disadvantages for the applications both by an algebraic point of view and by practical aspects. In fact the Zadeh multiplication is not distributive with respect to the addition, the shape of fuzzy numbers is not preserved by multiplication, the indeterminateness of the sum is too increasing. Then, for the applications in the Natural and Social Sciences it is important to individuate some suitable variants of the classical addition and multiplication of fuzzy numbers that have not the previous disadvantage. Here, some possible alternatives to the Zadeh operations are studied.

Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011), 2011

The aim of this investigation is to reconsider two notions of fuzzy function, namely: a fuzzy function as a special fuzzy relation and a fuzzy function as a mapping between fuzzy spaces. We propose to combine both notions in such a way that a fuzzy function as a relation determines a fuzzy function as a mapping. We investigate conditions which guarantee that dependent values of the related fuzzy functions coincide. Moreover, we investigate properties and relationship of the related fuzzy functions in the case when both of them are "fuzzified" versions of the same ordinary function.

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.

Iranian Journal of Fuzzy Systems, 2015

Measurement results contain different kinds of uncertainty. Besides systematic errors and random errors individual measurement results are also subject to another type of uncertainty, so-called fuzziness. It turns out that special fuzzy subsets of the set of real numbers R are useful to model fuzziness of measurement results. These fuzzy subsets x * are called fuzzy numbers. The membership functions of fuzzy numbers have to be determined. In the paper first a characterization of membership function is given, and after that methods to obtain special membership functions of fuzzy numbers, so-called characterizing functions describing measurement results are treated.

Results in Control and Optimization, 2023

Fuzzy set theory is a generalized form of crisp set theory where elements are binary inclusion forms. In fuzzy set, it differs with degree of membership for every element in the set. There are several strategies for arithmetic operations on fuzzy numbers. Previous studies show that there are many approaches, such as the α-cut technique, extension principle, vertex method, etc., to execute arithmetic operations on fuzzy numbers. In this study we perform details analysis and interpretation on arithmetic operations based on the α-cut method in a new way.

Fuzzy Information and Engineering, 2011

In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T , S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multifuzzy sets, a method for obtaining Atanassov's intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov's intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh's fuzzy set theory, Atanassov's intuitionsitic fuzzy set theory and L-fuzzy set theory.

Fuzzy Sets and Systems, 1992

Using the methodological approach of 'rational reconstruction' some possible explications of the notion of 'fuzzy set' are critically discussed with respect to their intuitive meaning, plausibility and consistency. Together with a historical survey we try to provide a better understanding of the development of various concepts within traditional fuzzy set theory. Based on the approaches of [39] and [18] we show that Zadeh's original intuition of a fuzzy set and its formalization as generalized 'membership' function can be reinterpreted in a plausible and consistent way, and thus are supported not only on a formal level by mathematical results, but can also be justified in principle on a semantical level. Hence, both the operational as well as the traditional valuational viewpoint have their own rights in their corresponding areas of modeling reality, and it is shown how the latter can be given a sound foundation based upon the former.

Encyclopedia of Database Systems, 2009

Fuzzy Sets and Systems, 2001

A key issue in operationalizing fuzzy set theory (particularly in decision analysis) is how to compare fuzzy numbers. In this paper, the case of L-R fuzzy numbers, i.e. the most general form of fuzzy numbers, is considered. In particular here, L-R fuzzy numbers represented by continuous, convex membership functions allowing also deÿnite integration is taken into consideration, normality is not required. Traditional comparison methods are generally limited to the use of triangular fuzzy numbers, and often the shape of the membership function is not taken into account or only a part of it is used (leading to a loss of information). Most of the approaches one can ÿnd in the literature are characterised by the use of -cuts and credibility levels, the use of areas for comparing fuzzy numbers has been proposed only recently. In particular, in the so-called NAIADE method a new semantic distance able to compare crisp numbers, fuzzy numbers and density functions has been developed. The basic idea underlying this paper is that, if only L-R fuzzy numbers are considered, other methodologies for comparing fuzzy numbers can be developed. Three indices based on the use of areas are studied, i.e. the expected value, the variance (with its decomposition into positive and negative semivariances) and the degree of coincidence of two fuzzy numbers. A justiÿcation of the use of these indices and a ÿrst tentative of axiomatisation is given. A short discussion on the issue of possible aggregation conventions of these indices is presented, and an empirical example is examined too.