An N-Soft Set Approach to Rough Sets

The philosophy of soft sets is founded on the fundamental idea of parameterization, while Pawlak’s rough sets put more emphasis on the importance of granulation. As a multi-valued extension of soft sets, the newly emerging concept called N-soft sets can provide a finer granular structure with higher distinguishable power. This study offers a fresh insight into rough set theory from the perspective of N-soft sets. We reveal a close connection between N-soft sets and rough structures of various types. First, we show how the corresponding structures of Pawlak’s rough sets, tolerance rough sets and multigranulation rough sets can be derived from a given N-soft set. Conversely, we investigate the representation of these distinct rough structures using the corresponding notions derived from suitable N-soft sets. The applicability of these theoretical results is highlighted with a case study using real data regarding hotel rating. The established two-way correspondences between N-soft sets and diverse rough structures are constructive, which can bridge the gap between seemingly disconnected disciplines, and hopefully nourish the development of both rough sets and soft sets

New Challenges in Neutrosophic Theory and Applications

Neutrosophic theory has representatives on all continents and, therefore, it can be said to be a universal theory. On the other hand, according to the three volumes of “The Encyclopedia of Neutrosophic Researchers” (2016, 2018, 2019), plus numerous others not yet included in Encyclopedia book series, about 1200 researchers from 73 countries have applied both the neutrosophic theory and method. Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics lies in the introduction of the degree of indeterminacy/neutrality (I) as an independent component in the neutrosophic set. Thus, neutrosophic theory involves the degree of membership-truth (T), the degree of indeterminacy (I), and the degree of non-membership-falsehood (F). In recent years, the field of neutrosophic set, logic, measure, probability and statistics, precalculus and calculus, etc., and their applications in multiple fields have been extended and applied in various fields, such as communication, management, and information technology. We believe that this book serves as useful guidance for learning about the current progress in neutrosophic theories. In total, 22 studies have been presented and reflect the call of the thematic vision. The contents of each study included in the volume are briefly described as follows. The first contribution, authored by Wadei Al-Omeri and Saeid Jafari, addresses the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets in neutrosophic topological spaces. In the article “Design of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distribution”, the authors Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, and Abdur Razzaque Mughal discuss the use of probability distribution function of Birnbaum–Saunders distribution as a proportion of defective items and the acceptance probability in a fuzzy environment. Further, the authors Derya Bakbak, Vakkas Uluc¸ay, and Memet S¸ahin present the “Neutrosophic Soft Expert Multiset and Their Application to Multiple Criteria Decision Making” together with several operations defined for them and their important algebraic properties. In “Neutrosophic Multigroups and Applications”, Vakkas Uluc¸ay and Memet S¸ahin propose an algebraic structure on neutrosophic multisets called neutrosophic multigroups, deriving their basic properties and giving some applications to group theory. Changxing Fan, Jun Ye, Sheng Feng, En Fan, and Keli Hu introduce the “Multi-Criteria Decision-Making Method Using Heronian Mean Operators under a Bipolar Neutrosophic Environment” and test the effectiveness of their new methods. Another decision-making study upon an everyday life issue which empowered us to organize the key objective of the industry developing is given in “Neutrosophic Cubic Einstein Hybrid Geometric Aggregation Operators with Application in Prioritization Using Multiple Attribute Decision-Making Method” written by Khaleed Alhazaymeh, Muhammad Gulistan, Majid Khan, and Seifedine Kadry

A Comprehensive study on (α,β)-multi-granulation bipolar fuzzy rough sets under bipolar fuzzy preference relation

The rough set (RS) and multi-granulation RS (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, the bipolar fuzzy sets (BFSs) are effective tools for handling bipolarity and fuzziness of the data. In this study, with the description of the background of risk decision-making problems in reality, we present $(\alpha, \beta)$-optimistic multi-granulation bipolar fuzzified preference rough sets ($(\alpha, \beta)^o$-MG-BFPRSs) and $(\alpha, \beta)$-pessimistic multi-granulation bipolar fuzzified preference rough sets ($(\alpha, \beta)^p$-MG-BFPRSs) using bipolar fuzzy preference relation (BFPR). Subsequently, the relevant properties and results of both $(\alpha, \beta)^o$-MG-BFPRSs and $(\alpha, \beta)^p$-MG-BFPRSs are investigated in detail. At the same time, a relationship among the $(\alpha, \beta)$-BFPRSs, $(\alpha, \beta)^o$-MG-BFPRSs and $(\alpha, \beta)^p$-MG-BFPRSs is given

Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity {i.e. element, concept, idea, theory, logical proposition, etc.}, is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor .Based on neutrosophy, the neutrosophic triplets were founded, which have a similar form (x, neut(x), anti(x)), that satisfy several axioms, for each element x in a given set.This collective book presents original research papers by many neutrosophic researchers from around the world, that report on the state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets and their algebraic structures – that have been defined recently in 2016 but have gained interest from world researchers. Connections between classical algebraic structures and neutrosophic triplet / duplet / multiset structures are also studied. And numerous neutrosophic applications in various fields, such as: multi-criteria decision making, image segmentation, medical diagnosis, fault diagnosis, clustering data, neutrosophic probability, human resource management, strategic planning, forecasting model, multi-granulation, supplier selection problems, typhoon disaster evaluation, skin lesson detection, mining algorithm for big data analysis, etc

Quadripartitioned single valued neutrosophic sets with covering based rough sets and their matrix representation

The notion of a quadripartitioned single-valued neutrosophic set (in short QSVNS) is considered to be the more general mathematical framework of the neutrosophic set to model indeterminacy. In QSVNS, the indeterminacy component is divided into two parts, namely, “unknown” and “contradiction”. In a real-life scenario, while handling indeterminacy, we may have some hesitation about whether the indeterminacy occurs due to the belongingness or the non-belongingness of an object. This leads to the introduction of QSVNS. On the other hand, the theory of rough set (RS) is introduced to depict the incomplete data hidden in nature with an aid of equivalence relation. So, by combining the QSVNS and the RS, a new mathematical structure known as a quadripartitioned single-valued neutrosophic rough set is formed. The main purpose of this article is to present two types of quadripartitioned single-valued neutrosophic covering rough set models. Also, we have introduced QSVN β-covering approximation space and studying some of its properties. Based on QSVN β -covering approximation spaces, two types of QSVN covering rough set models are investigated. Furthermore, a matrix representation of the QSVN covering-based rough set model is developed. Finally, an algorithm-based model under QSVN covering-based rough set is developed and employed in a case study to diagnose a patient/s that is more likely suffering from a disease having certain symptoms

A Historical Account of Types of Fuzzy Sets and Their Relationships

In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used

CODAS method for 2-tuple linguistic Pythagorean fuzzy multiple attribute group decision making and its application to financial management performance assessment

Financial management performance evaluation (FMPE) has a significant effect on the identifying an investment chance. We can usually consider FMPE as a multiple attribute group decision making (MAGDM) issue, and the MAGDM method is needed to address it. Uncertainty may be one of the significant factors which could influence the process of MAGDM. In order to handle the uncertainty of group decision-making issues, MAGDM approaches along with 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs) have been designed. In this essay, CODAS method is extended to 2TLPFSs to tackle MAGDM issues. Linguistic variables and 2TLPFSs are also used to extend the CODAS method. An application of the presented 2-tuple linguistic Pythagorean fuzzy CODAS (2TLPF-CODAS) method to a case study of FMPE problem with 2-tuple linguistic Pythagorean fuzzy numbers (2TLPFNs) is given. To confirm the results, a comparative analysis between the fuzzy CODAS and 2TLPF-TODIM is performed. The results of the comparison illustrate that the presented 2TLPF-CODAS method offers effective and steady results

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Some New Operations of ( alpha, , ) Interval Cut Set of Interval Valued Neutrosophic Sets

In this paper, we define the disjunctive sum, difference and Cartesian product of two interval valued neutrosophic sets and study their basic properties