Convex and Concave Soft Sets and Some Properties

In this study, after given the definition of soft sets and their basic operations we define convex soft sets which is an important concept for operation research, optimization and related problems. Then, we define concave soft sets and give some properties for the concave sets. For these, we will use definition and properties of convex-concave fuzzy sets in literature. We also give different some properties for the convex and concave soft sets

Join and Meet Operations for Type-2 Fuzzy Sets With Nonconvex Secondary Memberships

In this paper, we will present two theorems for the join and meet operations for general type-2 fuzzy sets with arbitrary secondary memberships, which can be nonconvex and/or nonnormal type-1 fuzzy sets. These results will be used to derive the join and meet operations of the more general descriptions of interval type-2 fuzzy sets presented in a paper by Bustince Sola et al. ('Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Towards a wider view on their relationship,' IEEE Trans. Fuzzy Syst., vol. 23, pp. 1876-1882, 2015), where the secondary grades can be nonconvex. Hence, this study will help to explore the potential of type-2 fuzzy logic systems which use the general forms of interval type-2 fuzzy sets which are not equivalent to interval-valued fuzzy sets. Several examples for both general type-2 and the more general forms of interval type-2 fuzzy sets are presented

A new extension of fuzzy sets using rough sets: R-fuzzy sets

This paper presents a new extension of fuzzy sets: R-fuzzy sets. The membership of an element of a R-fuzzy set is represented as a rough set. This new extension facilitates the representation of an uncertain fuzzy membership with a rough approximation. Based on our definition of R-fuzzy sets and their operations, the relationships between R-fuzzy sets and other fuzzy sets are discussed and some examples are provided

Fuzzy in 3-D: Contrasting Complex Fuzzy Sets with Type-2 Fuzzy Sets

CCIComplex fuzzy sets come in two forms, the standard form, postulated in 2002 by Ramot et al., and the 2011 innovation of pure complex fuzzy sets, proposed by Tamir et al.. In this paper we compare and contrast both forms of complex fuzzy set with type-2 fuzzy sets, as regards their rationales, applications, definitions, and structures. In addition, pure complex fuzzy sets are compared with type-2 fuzzy sets in relation to their inferencing operations. Complex fuzzy sets and type-2 fuzzy sets differ in their roles and applications. Their definitions differ also, though there is equivalence between those of a pure complex fuzzy set and a type-2 fuzzy set. Structural similarity is evident between these three-dimensional sets. Complex fuzzy sets are represented by a line, and type-2 fuzzy sets by a surface, but a surface is simply a generalisation of a line. This similarity is particularly evident between pure complex fuzzy sets and type-2 fuzzy sets, which are both mappings from the domain onto the unit square. Type-2 fuzzy sets were found not to be isomorphic to pure complex fuzzy sets

Notes on soft sets and aggregation operators

[EN]Under uncertainty, traditional sets may not be sufficient to represent real-world phenomena, and fuzzy sets can provide a more flexible and natural approach. The concept of fuzzy sets has been widely used in various fields, including artificial intelligence, control theory, decision-making, and pattern recognition. Fuzzy sets can also be combined with other mathematical tools, such as probability theory, to provide a more comprehensive approach to uncertainty management. In these notes, we explore the concept of fuzzy sets under uncertainty, and their applications in various fields. We discuss the fundamental concepts of fuzzy sets, including fuzzy membership functions, fuzzy operations, and fuzzy relations. We also examine different types of uncertainty, including epistemic and aleatory uncertainty, and how fuzzy sets can be used to model and manage uncertainty in these cases

Операции пересечения и объединения нечеткого множества нечетких множеств

В статье рассмотрены операции пересечения и объединения нечеткого множества нечетких множеств. Показано, что результатом этих операций являются нечеткие множества типа 2. Разработано конструктивное представление их функций принадлежности. Установлена взаимосвязь между операциями объединения и пересечения. Рассмотрены случаи объединения и пересечения нечеткого множества четких множеств. Построены функции принадлежности результирующих нечетких множеств типа 2, исследована их внутренняя структура.У статті розглянуті операції перетину та об’єднання нечіткої множини нечітких множин. Показано, що результатом цих операцій є нечітка множина типу 2. Розроблене конструктивне представлення їхніх функцій належності. Встановлений взаємозв’язок між операціями об’єднання і перетину. Розглянуті випадки об’єднання і перетину нечіткої множини чітких множин. Побудовані функції належності результуючих нечетких множин типу 2, досліджена їхня внутрішня структура.The operations of intersection and union of fuzzy set of fuzzy sets are considered in the article. It is shown that type-2 fuzzy sets are the results of these operations. Structural presentation of their belonging functions is developed. Intercommunication between operations of union and intersection is set. The cases of union and intersection of fuzzy set of clear sets are considered. The belonging functions of resulting type-2 fuzzy sets are built, their underlying structure is explored