Family Of Fuzzy Sets Research Articles

Novel Approaches Characterizing Separation in Fuzzy Topology with Quasi-Coincidence

Fuzzy sets were introduced by Zadeh in 1965, and three years later, Chang defined fuzzy topological spaces. Fuzzy topological spaces are families of fuzzy sets that satisfy the three classical axioms of topology. In this paper, we introduce and study some new notions of To separation axioms in fuzzy topological spaces using the quasi-coincident relation for fuzzy sets. Every ordinary (crisp) topological space vacuously satisfies the condition of being quasi-To. We define the quasi-separation axioms for fuzzy topological spaces, as quasi-To, quasi-T1, Quasi-T2. We have introduced and studied some new notions of To separation axioms in fuzzy topological spaces using the quasi-coincident relation for fuzzy sets. These quasi-separation axioms are weaker than the corresponding classical To separation axioms, but they are still useful for characterizing and studying fuzzy topological spaces.

Subsethood measure for picture fuzzy sets and its applications on multicriteria decision making

Picture fuzzy set is a direct generalization of intuitionistic fuzzy set and is therefore more capable of dealing with uncertainty while working on real life problems. The concept of inclusion is a subject that is frequently studied in family of fuzzy sets and has many applications in real life problems. Therefore, in this work, the measuring degree of inclusion between picture fuzzy sets is introduced. For this purpose, firstly axioms for subsethood measure are given and then a subsethood measure based on a distance measure for picture fuzzy sets is proposed. Then, a numerical example is provided to illustrate the applicability and usefulness of the presented measure. Finally, results are compared with the existing methods and aggregation operator to show validity of subsethood measure for PFS.

Exploring fuzzy set consensus analysis in IoT resource ranking

The complexity of the procedures for discovering, classifying, and selecting suitable resources to meet customer demands is related to the growing resource offers connected to the Internet. This proposal takes into account the uncertainties in the specification and processing of customer preferences, via consensual analysis. In this work we study the relationship between restricted equivalence functions, more generally, consensus measures, and the possibility of building the latter using the former. Thus, consensus measures of fuzzy values and consensus measures on fuzzy sets are both defined by aggregations, such as the arithmetic mean and the exponential mean. Based on the interval-valued fuzzy logic we consider inaccuracies related to the measurements beyond the uncertainties, modeling the imprecision of expertise in classifying a set of resources in the IoT based on the IT2FL-EXEHDA-RR Model. Several results arise from these methods. Firstly, the methodology L[0,1]-FCM, which is performed to measure how similar are the corresponding fuzzy values of a fuzzy set on [0,1]. In particular, it is applied to the superior and the inferior limits of each interval-valued membership function, modeling linguistic variables related to the attributes of the IT2FL-EXEHDA-RR Model. And the second one, based on LFχ-FSCM, providing the consensus analysis among a family of fuzzy sets. In the reported case study, this is applied to obtain the consensus measure between corresponding upper and lower bounds of each interval-valued function. This research also investigates the conditions under which we can build convex sum based on LFχ-FSCM, including the analysis of their main properties.

Fuzzy fixed point results and applications to ordinary fuzzy differential equations in complex valued metric spaces

The main purpose of this article is to discuss the existence of the common solution of second-order nonlinear boundary value problems is a continuous function and S(S) is a family of fuzzy sets. In this regard we obtain common fixed point results for two pairs of fuzzy mappings satisfying rational contractive condition in the setting of complex valued metric spaces. Our results improve those reported in the existing literature.

L-Fuzzy Sets and Isomorphic Lattices: Are All the “New” Results Really New? †

We review several generalizations of the concept of fuzzy sets with two- or three-dimensional lattices of truth values and study their relationship. It turns out that, in the two-dimensional case, several of the lattices of truth values considered here are pairwise isomorphic, and so are the corresponding families of fuzzy sets. Therefore, each result for one of these types of fuzzy sets can be directly rewritten for each (isomorphic) type of fuzzy set. Finally we also discuss some questionable notations, in particular, those of “intuitionistic” and “Pythagorean” fuzzy sets.

Some Universal Constructions For I- Fuzzy Topological Spaces

Hamouda E. H. [J. of Advances in Math. Vol 6 (2014), 973-941] has introduced some universal constructions in the category FTOP , whose objects are the I-fuzzy topological spaces where X is an ordinary set, is a fuzzy set in X and F is a family of fuzzy sets in X satisfying some axioms. In this paper we introduce the dual universal constructions, namely, fuzzy co-products, fuzzy co-equalizers and fuzzy pushouts for I-fuzzy topological spaces. Also we discuss some results concerning all such universal objects.

A hierarchical model of a linguistic variable

In this work a theoretical hierarchical model of dichotomous linguistic variables is presented. The model incorporates certain features of the approximate reasoning approach and others of the Fuzzy Control approach to Fuzzy Linguistic Variables. It allows sharing of the same hierarchical structure between the syntactic definition of a linguistic variable and its semantic definition given by fuzzy sets. This fact makes it easier to build symbolic operations between linguistic terms with a better grounded semantic interpretation. Moreover, the family of fuzzy sets which gives the semantics of each linguistic term constitutes a multiresolution system, and thanks to that any fuzzy set can be represented in terms of the set of linguistic terms. The model can also be considered a general framework for building more interpretable fuzzy linguistic variables with a high capacity of accuracy, which could be used to build more interpretable Fuzzy Rule Based Systems (FRBS).

Towards many-valued/fuzzy interpretation of quantum mechanics

In this paper, we propose a new interpretation of quantum mechanics according to which Born probabilities are interpreted as many-valued truth values of statements concerning results of not-yet-performed experiments on quantum objects, or equivalently [0,1]-valued degrees to which quantum objects possess all their properties prior to measurements. The interpretation is based on mathematical results already published by the author, according to which any Birkhoff–von Neumann quantum logic can be isomorphically represented as a family of fuzzy sets or, equivalently, many-valued propositional functions.

On a pair of fuzzy [formula omitted]-contractive mappings

We establish common fixed point theorems for fuzzy mappings under a φ -contraction condition on a metric space with the d ∞ -metric (induced by the Hausdorff metric) on the family of fuzzy sets. The study of fixed points of fuzzy set-valued mappings related to the d ∞ -metric is useful in geometric problems arising in high energy physics. Our results generalize some recent results.

Compactness of Family of Fuzzy Sets in L2 Space with Application to Optimal Control

The optimization of nonlinear feedback fuzzy system using the product-sum-gravity method is described in this paper. The fuzzy control discussed here is the nonlinear feedback control in which the feedback laws are determined by IF-THEN type fuzzy production rules through product-sum-gravity method. To prove existence of optimal control, we applied compactness of a set of membership functions in L2 space and continuity of the approximate reasoning, and prepared some propositions concerning product-sum-gravity method. By considering fuzzy optimal control problems as problems of finding the minimum (maximum) value of the integral cost (benefit) function on an appropriate set of membership functions, the existence of fuzzy optimal control is shown.

A fuzzy hybrid method for image decomposition problem

We use a hybrid approach based on a genetic algorithm and on the gradient descent method for image decomposition problem. We adopt an iterative gradient descent method, already used in a previous paper and here improved, in order to reconstruct an image by using an optimisation task based on the minimisation of a cost function. By normalising the values of its pixels with respect to the grey scale used, an image R is interpreted as a fuzzy relation. In order to obtain better results in terms of quality of the reconstructed image, we use a preprocessing genetic algorithm for determining two initial families of fuzzy sets that compose R in accordance to the concept of Schein rank of R. The experiments are executed on some images extracted from the SIDBA standard image database.

Data-driven fuzzy sets for classification

Using the mass assignment mechanism, a fuzzy classifier can be derived directly from the class relative frequency distribution. Moreover, in this framework, a family of fuzzy sets can represent a class, thus adapting the classifier to the need of classification. Graduality and the corresponding concept of error can be used to guide the process of deriving class representing fuzzy sets. The classification algorithm is attractive due to its low complexity. Successful applications include imbalanced data classification problems where the class having fewer examples is the class of interest.

Monopolistic price setting under fuzzy information

We consider a profit maximizing monopolistic firm that sets prices in the presence of improper information about the demand for its products. This information deficiency is viewed as being vagueness rather than stochastic uncertainty and modelled by a family of fuzzy sets. Applying a defuzzification strategy that explicitly takes the firm’s attitude towards deviations of actual demand from its most possible values into account, we derive explicit solutions to the maximization problem of a single-product and a general multi-product firm. Furthermore, some comparative static results are established.

On Many-Valued Partitions and Many-Valued Equivalence Relations

For a fixed integral commutative cl-monoid M=(L, ≤, *), introducing the notions of M-partitions and M-equivalence relations as the extensions of T-partitions and T-equivalence relations to the integral commutative cl-monoid M=(L, ≤, *), respectively, it is shown that the previous approaches on T-partitions and T-equivalence relations can be unified, and most of the results in these works can be easily stated based on the integral commutative cl-monoid M=(L, ≤, *). Modifying the notion of T-redundancy of finite family of fuzzy sets of a nonempty ordinary set X, it is extended to arbitrary family of L-fuzzy sets on the basis of the integral commutative cl-monoid M, and is shown that the proposed definition has more desirable properties than the previous one. Furthermore, handling the works of Höhle, Klawonn, Gebhardt and Kruse on fuzzy partitions and fuzzy equivalence relations, some of their results are improved, and several new results in this direction are pointed out.

Neighborhood systems in fuzzy topological group

In this paper, we have established a necessary and sufficient condition for a family of fuzzy sets in a group to be the family of neighborhoods of the identity element of a fuzzy topological group. As an application, a mixed fuzzy topological group is constructed from two given fuzzy topological groups with the help of neighborhood systems.

T ∞-Fuzzy observables

Observables are defined as homomorphisms from the Borelσ-algebra into a family of fuzzy sets considered with respect to the Giles connectives. Algebraic operations with observables are introduced and their relation to the corresponding operations with fuzzy random variables is explained.