Intuitionistic Fuzzy Calculus Research Articles

Series based on the new order in intuitionistic fuzzy environment

Intuitionistic fuzzy number (IFN) is an effective tool for dealing with the uncertain information, and it has been applied to various fields. According to IFNs, the intuitionistic fuzzy calculus has been developed, which can effectively integrate the continuous uncertain information. Series in intuitionistic fuzzy environment is a part of the intuitionistic fuzzy calculus theory, of which core idea is limit. However, the order used in the existing limit theory is not the one used in intuitionistic fuzzy calculus, causing the separation of the limit theory and intuitionistic fuzzy calculus. Thus, series in intuitionistic fuzzy environment is not closely related to the intuitionistic fuzzy calculus. In order to solve the above problem, we construct the related theories. There are mainly the following three aspects: (1) the limit theory including the sequence limit and the function limit is studied based on the new order. (2) we re-examine the numerical series according to the new tool of researching IFNs: the basis and the coordinates. (3) we discuss the function series and put forward the uniform convergence in intuitionistic fuzzy environment.

Limit Theory and Differential Calculus of Intuitionistic Fuzzy Functions With Several Variables

Intuitionistic fuzzy set (IFS) can depict an object simultaneously from both the superiority and inferiority aspects, and its basic element is an intuitionistic fuzzy number (IFN). Based on four basic operational laws of IFNs, the intuitionistic fuzzy calculus (IFC) has been proposed recently. However, there is no research on the limit theory and differential calculus of intuitionistic fuzzy functions with several variables based on the four basic operational laws of IFNs. Hence, the aim of this article is to systematically investigate them. In this article, we first introduce the concepts of limits and iterated limits, and then discuss their properties, based on which, we reveal their relationships among the limits, the iterated limits, and continuities. Finally, we offer the notions of the differentials by the intuitionistic fuzzy infinitesimals, and especially, we obtain the concrete values of partial derivatives. In addition, we discuss in detail the relationships between the total differentials and partial derivatives, which are thoroughly different from the corresponding relationships in the IFC of one variable.

Foundation of Interval‐Valued Intuitionistic Fuzzy Limit and Differential Theory and an Application to Continuous Data

The intuitionistic fuzzy calculus (IFC), based on the basic operational laws of intuitionistic fuzzy numbers (IFNs), has been put forward. However, the interval‐valued IFC (IVIFC), based on the basic operational laws of interval‐valued IFNs (IVIFNs), is only in the original stage. To further develop the theory of the IVIFC and make it be rigorous, the primary task is to systematically investigate the characteristics of the limits and differentials, which is a foundation of the IVIFC. Moreover, there is quite a lot of literature on IVIFNs; however, the scholars did not reveal the relationships between IFNs and the IVIFNs. To do that, we first investigate the limit of interval‐valued intuitionistic fuzzy sequences, and then, we focus on investigating the limit, the continuity, and the differential of IVIFFs in detail and reveal their relationships. After that, due to the fact that the IFC and the IVIFC are based on the basic operational laws of IFNs and IVIFNs, respectively, we reveal the relationships between the IFNs and the IVIFNs via some homomorphic mappings. Finally, a case study about continuous data of IVIFNs is provided to illustrate the advantages of continuous data.

Simplified Interval-Valued Intuitionistic Fuzzy Line Integrals and Their Application

In the past few years, the intuitionistic fuzzy calculus (IFC) has been proposed, which is based on some basic operational laws of intuitionistic fuzzy numbers (IFNs). The interval-valued intuitionistic fuzzy sets have received great attention of scholars since they can describe the characters of things comprehensively. However, so far, the interval-valued IFC is in the original stage. In this paper, we focus on investigating the simplified interval-valued intuitionistic fuzzy line integrals (SIVIFLIs). First, we construct the SIVIFLIs step by step, and then provide their concrete values. In order to further describe the characteristic of the SIVIFLIs, we discuss their basic properties, and particularly, we investigate a kind of the order in intuitionistic fuzzy circumstance in detail. Finally, we offer an application and address it by the proposed SIVIFLIs to demonstrate their effectiveness and practicability.

Intuitionistic Fuzzy Double Integrals and Their Fundamental Properties

Intuitionistic fuzzy set (IFS), as a significant generalization of the fuzzy set (FS), takes the membership function and the non-membership function into consideration simultaneously. The core components of an IFS are the ordered pairs which are called intuitionistic fuzzy numbers (IFNs). More recently, the intuitionistic fuzzy calculus (IFC) has been established, which is based on the basic operational laws of IFNs. As far as the IFC with several variables is concerned, its theory is still in the initial stage. The purpose of this paper is to systematically establish the theory of the intuitionistic fuzzy double integrals (IFDIs). To accomplish this, we first construct the IFDIs, and then assign them concrete values. To understand the IFDIs in depth, we investigate their fundamental properties in detail and offer simple proofs for them. Next, we give a counterexample to verify that in the IFC the first and second mean value theorems do not hold, and interpret the reason from the perspective of the geometric figure. In addition, we compare the results in the IFC with those in the classical calculus to demonstrate that they are thoroughly different despite their forms being sometimes similar. Finally, we offer an application to show the utility of the proposed IFDIs.

Line Integrals of Intuitionistic Fuzzy Calculus and Their Properties

Atanassov's intuitionistic fuzzy set (A-IFS) is a powerful tool to handle uncertainty and vagueness in real life. The basic elements of an A-IFS are intuitionistic fuzzy values, based on which the intuitionistic fuzzy calculus (IFC) has been proposed recently. However, to date, there is no investigation for intuitionistic fuzzy line integrals (IFLIs), which are very important for further developing IFC. In this paper, we propose the IFLIs and give their concrete values. After that, we investigate a series of basic properties of the IFLIs in detail, moreover, in order to show the utility of the proposed IFLIs, we offer an example, and finally, we discuss the relationships among the additive IFLI, the multiplicative IFLI, and the intuitionistic fuzzy aggregation operators.

Multiple Definite Integrals of Intuitionistic Fuzzy Calculus and Isomorphic Mappings

Intuitionistic fuzzy set (A-IFS) introduced by Atanassov, a generalized form of fuzzy sets, is a useful tool to deal with vagueness and ambiguity. Each element of an A-IFS is called an intuitionistic fuzzy value, based on which the intuitionistic fuzzy calculus (IFC) has been proposed recently. However, the existing study only discusses the one-variable IFC, in order to enrich the theory of the IFC; in this paper, we put forward the intuitionistic fuzzy multiple definite integrals (IFMDI). We first define some concepts related to the IFMDI, based on which we reveal the relationships between the IFMDI and the intuitionistic fuzzy definite integrals. After that we investigate the basic properties of the IFMDI in detail, and finally, we apply the knowledge of abstract algebra to construct some isomorphic mappings, based on which, we interpret the reason why there are some parallel properties between the additive IFMDI and the multiplicative IFMDI.

Fundamental Properties With Respect to the Completeness of Intuitionistic Fuzzy Partially Ordered Set

Intuitionistic fuzzy set (A-IFS), originally introduced by Atanassov in 1983, is a generalization of a fuzzy set. The basic elements of an A-IFS are intuitionistic fuzzy values (IFVs), based on which the intuitionistic fuzzy calculus (IFC) has been proposed recently. To avoid relying too much upon the classical calculus and make the IFC to be an independent subject, it is necessary to develop the limit theory of the IFC. In this paper, we first define the concepts of the supremum and the infimum with respect to IFVs, and investigate their properties in detail. Then, we reveal the relationships among them and four types of limits, and finally, we give a series of fundamental theorems with respect to the completeness of intuitionistic fuzzy partially ordered set.

A Unification of Intuitionistic Fuzzy Calculus Theories Based on Subtraction Derivatives and Division Derivatives

Intuitionistic fuzzy calculus replaces real numbers in classical calculus with intuitionistic fuzzy numbers that are the basic elements of Atanassov's intuitionistic fuzzy sets. Intuitionistic fuzzy calculus consists of two parallel parts, which are respectively developed based on the subtraction derivatives and the division derivatives. This paper focuses on building the relationships between the two independent intuitionistic fuzzy calculus theories, and unifying them into a whole theory.

The chain and substitution rules of interval-valued intuitionistic fuzzy calculus

The interval-valued intuitionistic fuzzy set proposed by Atanassov is the extension of intuitionistic fuzzy set. It extends the membership degree and non-membership to interval values instead of a single value. So it contains more possible values and maybe more considerate. Among all the researches, the exploration on the calculus of interval-valued intuitionistic fuzzy set is entirely new. Recently, Zhao et al. (Int J Comput Intell Syst 9:36–56, 2016) proposed the concept of interval-valued intuitionistic fuzzy function (IVIFF) and gave a calculation method of derivative and differential of IVIFF. Based on this work, in this paper, firstly, we utilize a new and easier method to express the derivative and differential of IVIFF. Secondly, we propose the chain rules of derivative and the form invariance of differential in the interval-valued intuitionistic fuzzy environment. In addition, some properties of the substation rules for interval-valued intuitionistic fuzzy indefinite integrals and definite integrals are also developed.

Infinite Intuitionistic Fuzzy Series and Product

Since the Atanassov's intuitionistic fuzzy set theory was introduced, many intuitionistic fuzzy aggregation operators have been proposed for the integration of the intuitionistic fuzzy information. It is necessary to extend them to accommodate the infinite situations so that we can deal with the large amount of intuitionistic fuzzy data, and, at the same time, we must solve an important issue that whether the sum of an infinite number of intuitionistic fuzzy numbers is convergent or not. Thus, in this paper, we first put forward the infinite intuitionistic fuzzy series and product. Then, we investigate the properties and the convergences of the infinite intuitionistic fuzzy series and product, respectively. These results greatly enrich the intuitionistic fuzzy calculus theory.

Relationships Between Two Types of Intuitionistic Fuzzy Definite Integrals

Intuitionistic fuzzy numbers (IFNs) are the basic components of Atanassov's intuitionistic fuzzy sets, which are very convenient and comprehensive in depicting the fuzzy characters of things in real life. In the existing literature, there are mainly two different types of definite integrals with respect to IFNs, which were developed from completely different aspects. The main purpose of this paper is to demonstrate their relationships in detail by deriving the definite integrals of two novel intuitionistic fuzzy functions. It is worth pointing out that one of the two types of definite integrals is shown to be a special case of another one, and finally, we also derive some other useful results for intuitionistic fuzzy calculus based on the division derivative.

Chain and Substitution Rules of Intuitionistic Fuzzy Calculus

The intuitionistic fuzzy set, originally given by Atanassov, can be considered as a set that consists of intuitionistic fuzzy numbers (IFNs), each of which is described by a membership degree, a nonmembership degree, and a hesitancy degree. Recently, the calculus of IFNs has been proposed, and the derivatives, differentials, indefinite integrals, and definite integrals with respect to IFNs have also been developed in previous research studies. In this paper, some novel results of intuitionistic fuzzy calculus are derived, which include mainly the chain rule of derivatives, the form invariance of differentials, and the substitution rules for indefinite integrals and definite integrals in intuitionistic fuzzy environment.

Limit properties and derivative operations in the metric space of intuitionistic fuzzy numbers

Intuitionistic fuzzy numbers (IFNs) are a useful tool to depict the uncertain information in real life. Based on IFNs, the intuitionistic fuzzy calculus (IFC) has been put forward recently. To further develop the IFC theory, in this paper, we investigate the limit properties of IFCs, and study the intuitionistic fuzzy infinitesimals and their orders. We also discuss the continuity, the derivatives and the differentials of intuitionistic fuzzy functions in detail, and reveal their relationships. Additionally, we define the metric space of the IFNs, based on which, a series of desirable results are obtained. These results are similar to the ones in the classical calculus.

Fundamental properties of intuitionistic fuzzy calculus

The intuitionistic fuzzy set (A-IFS) introduced by Atanassov (1986) is a generalization of fuzzy set (Zadeh, 1965). The basic elements of an A-IFS are intuitionistic fuzzy numbers (IFNs) (Xu and Yager, 2006), each of which is described by a membership degree, a non-membership degree and a hesitancy degree. The IFN is an effective tool in expressing fuzzy information of things. Based on IFNs and their basic operations, the paper first defines the indefinite integral and antiderivative of intuitionistic fuzzy functions (IFFs), and then gives the concept of definite integral of IFFs. Finally, the paper deduces the Newton–Leibniz formula (the fundamental theorem of calculus) in Atanassov’s intuitionistic fuzzy environment, and studies some basic properties of intuitionistic fuzzy calculus. Finally, this paper presents an aggregation method, which is based on the definite integral of IFFs, to deal with intuitionistic fuzzy information, and analyzes some basic properties of the method.

Approximations on intuitionistic fuzzy predicate calculus through rough computing

In 1986, Attansov introduced the intuitionistic fuzzy set which is the generalized approach of fuzzy set model introduced by Zadeh. Considering the importance of these two theories, various hybridizations with the available mathematical tools have been derived and they are in practice. Related to the concept of rough sets, the hybridized work of Nakamura, Dubois and Prade are noteworthy. In 2005, G. Ganesan et al., have introduced the concept of thresholds in Rough Fuzzy Hybridizations. Later, in 2008, G. Ganesan et al, derived a naive approach on approximating the connectives in fuzzy predicate calculus through rough sets. This concept was developed by introducing a threshold on the membership of the arguments in the fuzzy predicates. This work yielded the Two way approximations for the fuzzy connectives. In this work, it is noticed that the complement and the negation of the predicates are found in a unique way, whereas in case of intuitionistic fuzziness, the complement and the negation are to be found in two different ways namely membership based and non membership based. Using this concept, in this paper, we introduce Two way approach on Intuitionistic Fuzzy Logic. Also, in this paper, we derive Two way approximations on each intuitionistic fuzzy connectives using rough sets.