Operations In Set Theory Research Articles

Integrating ethnomathematics and ethnomodeling in Institutionalization of school mathematics concepts: A study of fishermen community activities

While previous studies have extensively explored ethnomathematics, the relationship between ethnomathematics and ethnomodeling remains underexamined. This study investigates the connections between ethnomathematics and ethnomodeling and their implications for the institutionalization of mathematics in the context of traditional fish auction activities within the Fishermen's Community in Indramayu, Indonesia. Utilizing an ethnographic design, the study involved participants in various auction roles, including auctioneers, boat owners, captains, crew members, and traders. Data were collected through interviews and observations and analyzed using content analysis. The findings reveal that ethnomathematical practices are evident in sorting and grading fish, as well as in auction transactions, with connections to mathematical concepts such as set theory, basic statistics, social arithmetic, and arithmetic operations within an ethnomodeling framework. These results suggest that the integration of ethnomathematics and ethnomodeling into the school curriculum could support the institutionalization of mathematics education and provide a foundation for developing instructional materials based on these practices in the context of traditional fish auctions.

Solving Advanced Task-Specific Problems in Measurement Sciences with Generative AI.

The Generative Pre-Trained Transformer known as ChatGPT-4 has undergone extensive pretraining on a diverse data set, enabling it to generate coherent and contextually relevant text based on the input it receives. This capability allows it to perform tasks from answering questions and has attracted significant interest in material sciences, synthetic chemistry, and drug discovery. In this work, we posed four advanced task-specific problems to ChatGPT, which were recently published in leading journals for topics in analytical chemistry, spectroscopy, bioimage super-resolution, and electrochemistry. ChatGPT-4 successfully implemented the four ideas after assigning the "persona" to the AI and posing targeted questions. We show two cases where "unguided" ChatGPT could complete the assignments with minimal human direction. The construction of a microwave spectrum from a free induction curve and super-resolution of bioimages was accomplished using this approach. Two other specific tasks, correcting a complex baseline with morphological operations of set theory and estimating the diffusion current, required additional input, e.g., equations and specific directions from the user. In each case, the MATLAB code was eventually generated by ChatGPT-4 even when the original authors did not provide any code themselves. We show that a validation test must be implemented to detect and correct mistakes made by ChatGPT-4, followed by feedback correction. These approaches will pave the way for open and transparent science and eliminate the black boxes in measurement science when it comes to advanced data processing.

Pengaplikasian Prosedur Perhitungan untuk Menentukan Produksi Batik Tulis Madura Menggunakan Metode Fuzzy Database Model Tahani

One of the subdivisions of fuzzy logic is the study of the fuzzy database with Tahani model which is described as a model used to process data searches based on operations in fuzzy set theory to obtain appropriate information so that the data search process becomes accurate. The research aims to implement the Fuzzy Database with Tahani Model in determining the appropriate variant of batik tulis with the specified criteria. The initial step of the research is fuzzification, namely determining the domain of each complete variables with intervals of numerical variables and linguistic variables. In this research some variables were used: the price of the cloth, the number of colors and the work durations. Each variable is divided based on the interval between the lowest data and the highest data so that the variable price of the cloth is divided into 4, the number of colors is 3, and the work durations is 5. Then determine the membership function, the variable price of cloth, the number of colors, and the work duration using a linear model and triangle in determining the degree of its membership. Data tabulation from strict sets to fuzzy sets with numerical variables and linguistic variables. In preparing the criteria, 60 rule bases are based on the theory of consumer needs for batik tulis. Not all rule bases occur, because the data collects only a few criteria. Determining the value of fire strength, using the AND rule means selecting the minimum degree of membership from the available. The recommendation results determine to sort the washing machine data that has the highest fire strength value to the smallest fire strength value from each criterion.

Development of a novel uncertainty model for interval-valued Q-fuzzy soft sets: Application in design-making

In actual life, dealing with uncertain information has become a challenge for researchers who strive day after day to develop more accurate mathematical tools for better dealing with this information. The Q-Fuzzy soft model can process uncertain information in two dimensions by dealing with the subjective judgments of users effectively. Therefore, this article aims to increase the effectiveness of the Q-fuzzy soft model and address the challenges of design-making under uncertain information by proposing a new model called the interval-valued Q-fuzzy soft (IV-Q-FSS) model. Under the IV-Q-FSSs, we discuss strongly set-theory operations such as subset, union of two IV-Q-FSSs, intersection of two IV-Q-FSSs, complement of IV-Q-FSS, AND operation, and OR operation for IV-Q-FSSs, and some operations like the possibility and necessity operations of an IV-Q-FSS. In addition, we hand over numerous properties held up by numerical examples that describe how they toil. Finally, this recently developed model has been successfully trying out in dealing with one of the design-making problems based on hypothetical data for a respiratory disease. This algorithm is built based on the aggregation operator for IV-Q-FSS data to break this issue (i.e., selecting the optimal alternative).

COMPLEMENTAL BINARY OPERATIONS OF SETS AND THEIR APPLICATION TO GROUP THEORY

Set theory is considered as the foundation of all mathematics since many mathematical concepts cannot be defined precisely without using set-theoretical concepts. In this study, we define new complemental binary operations, called union complements, intersection left complement, and union right complement and investigate their properties in detail. We contribute to the literature of sets by illuminating the relationships between these complemental binary operations and inclusive\exclusive complements via researching the distribution rules. Moreover, we show that the set of all the sets together with these new complemental binary operations form some algebraic structures. Finally, with the inspiration of these novel concepts, we give an application to group theory as regards subgroups by defining new type of subgroups in order to prompt the reader to think via interesting questions. Since the concept of operations of soft set theory, one of the most popular theory for uncertainty modeling in the past twenty four years, is the crucial notion for developing the theory and since all the types of soft set operations are based on the classical set operations, generation of new complemental binary operations on sets, and thus on soft sets and derivation of their algebraic properties will provide new perspectives for solving problems related to parametric data.

Efficient Framework for Operating on Data Sketches

We study the problem of analyzing massive data streams based on concise data sketches. Recently, a number of papers have investigated how to estimate the results of set-theory operations based on sketches. In this paper we present a framework that allows to estimate the result of any sequence of set-theory operations. The starting point for our solution is the solution from 2021. Compared to this solution, the newly presented sketching algorithm is much more computationally efficient as it requires on average O (log n ) rather than O ( n ) comparisons for n stream elements. We also show that the estimator dedicated to sketches proposed in that reference solution is, in fact, a maximum likelihood estimator.

A novel approach based on rough set theory for analyzing information disorder

The paper presents and evaluates an approach based on Rough Set Theory, and some variants and extensions of this theory, to analyze phenomena related to Information Disorder. The main concepts and constructs of Rough Set Theory, such as lower and upper approximations of a target set, indiscernibility and neighborhood binary relations, are used to model and reason on groups of social media users and sets of information that circulate in the social media. Information theoretic measures, such as roughness and entropy, are used to evaluate two concepts, Complexity and Milestone, that have been borrowed by system theory and contextualized for Information Disorder. The novelty of the results presented in this paper relates to the adoption of Rough Set Theory constructs and operators in this new and unexplored field of investigation and, specifically, to model key elements of Information Disorder, such as the message and the interpreters, and reason on the evolutionary dynamics of these elements. The added value of using these measures is an increase in the ability to interpret the effects of Information Disorder, due to the circulation of news, as the ratio between the cardinality of lower and upper approximations of a Rough Set, cardinality variations of parts, increase in their fragmentation or cohesion. Such improved interpretative ability can be beneficial to social media analysts and providers. Four algorithms based on Rough Set Theory and some variants or extensions are used to evaluate the results in a case study built with real data used to contrast disinformation for COVID-19. The achieved results allow to understand the superiority of the approaches based on Fuzzy Rough Sets for the interpretation of our phenomenon.

Several significant equalities on intuitionistic fuzzy operators

There are various operators and operations in intuitionistic fuzzy set theory. The roles of these operators and operations are very significant as they show a deeper interconnection between the two ordinary modal logic operators. It can be well noted that modal operators can change intuitionistic fuzzy sets into fuzzy sets easily. Considering all of these, we establish several equalities on intuitionistic fuzzy sets.

On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

In rough set theory, upper and lower approximation operators are two crucial concepts. The study of these two approximation operators in the framework of lattice theory is an important generalization from the mathematical point of view. On the other hand, overlap and grouping functions, as two types of not necessarily associative binary aggregation functions different from the common binary aggregation functions triangular norms and triangular conorms, have not only rich theoretical results but also a wide range of practical applications. Therefore, based on overlap and grouping functions over complete lattices, this paper is devoted to proposing (O,G)-fuzzy rough sets as a further generalization of the notion of rough sets. Firstly, we define a pair of O-upper and G-lower L-fuzzy rough approximation operators and investigate basic properties of them. Then, the characterizations of (O,G)-fuzzy rough approximation operators are discussed by using different kinds of L-fuzzy relations. Meanwhile, we investigate the topological properties of (O,G)-fuzzy rough sets. Furthermore, we show a brief comparison of the (O,G)-fuzzy rough sets with other common rough set models. At the end of this paper, we further propose multigranulation (O,G)-fuzzy rough sets over complete lattices from the viewpoint of multigranulation structure.

On the algebra of data sketches

We consider the problem of designing a distributed data sketch for scenario in which data stream is observed by many independent network nodes. We require that a sketch apart from being computationally and memory efficient should also be mergeable in a way that mimics set theory operations on related data sets. For example, when monitoring network traffic, one may consider how many distinct packets passed through a given node (sum of sets for different time windows) or passed through two given nodes (intersection of sets from two locations) and what is their total size (intersection of weighted sets). In this paper we propose a sketch that allows to efficiently summarize sets constructed from a sequence of set theory operations. We also provide an analytical control over the trade-off between the accuracy and storage/computational requirements. In comparison to the previous works the proposed solution 1) allows the weights of elements, 2) allows performing set theory operations simultaneous on a large number of sketches, 3) does not require computationally expensive numerical calculations and guarantees low overheads.

Methodology for assessing agricultural risks of the digital economy

In recent years, there has been increasing interest in the introduction of digital technologies in various areas of life and activity, including the agro-industrial complex. However, few studies pay attention to the issue of risk assessment that accompanies this process. It is not known how to methodically competently and quickly analyze possible risks and assess the prospects of the organization for transformation into the digital economy. The aim of the work is to develop an effective methodology for assessing agricultural risks and on their basis to determine the prospects of organizations in the direction of digital transformation. To explore the possibilities of digitalization, we use hierarchical and digital analysis, as well as operations of mathematical logic and set theory. The results of the study indicate that the evaluation method can be divided into 8 blocks, and its algorithm consists of five stages. The proposed methodology will contribute to the formation of more effective organizational and economic regulators for the transformation of economic entities into a digital economy through a preventive assessment of agricultural risks and, taking them into account, the readiness for digitalization of state authorities, economic entities and the population.

A Novel Group Decision-Making Approach for Hesitant Fuzzy Linguistic Term Sets and Its Application to VIKOR

This paper develops a novel group decision-making (GDM) approach for solving multiple-criteria group decision-making (MCGDM) problems with uncertainty. The hesitant fuzzy linguistic term sets (HFLTSs) are applied to elicit the decision makers’ linguistic preferences due to their distinguished efficiency and flexibility in representing uncertainty. However, the existing context-free grammar for linguistic description cannot allow generating the linguistic expressions completely free to limit the richness of HFLTSs, and the related methods for dealing with HFLTSs also have limitations in aggregating HFLTSs with different lengths and types. Therefore, this paper proposes extended context-free grammar and a novel GDM approach for HFLTSs, considering the advantages of the rough set theory and OWA operators. The rough set theory can manage the uncertainty existing in the fuzzy representation and deal with HFLTSs represented by the 2-tuple fuzzy linguistic model to get rough number sets. The OWA operator can aggregate these sets with different numbers of elements into an interval simply and objectively. Then, an extended VIKOR method based on the proposed GDM approach for HFLTSs is presented to solve the MCGDM problems. Finally, two examples are given to illustrate the applicability and validity of the developed GDM approach and the hesitant VIKOR method through sensitivity and comparison analysis with other existing approaches.

Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making

Intuitionistic fuzzy sets are useful for modeling uncertain data of realistic problems. In this paper, we generalize and expand the utility of complex intuitionistic fuzzy sets using the space of quaternion numbers. The proposed representation can capture composite features and convey multi-dimensional fuzzy information via the functions of real membership, imaginary membership, real non-membership, and imaginary non-membership. We analyze the order relations and logic operations of the complex intuitionistic fuzzy set theory and introduce new operations based on quaternion numbers. We also present two quaternion distance measures in algebraic and polar forms and analyze their properties. We apply the quaternion representations and measures to decision-making models. The proposed model is experimentally validated in medical diagnosis, which is an emerging application for tackling patient’s symptoms and attributes of diseases.

Approximation of real data by fuzzy sets for the classification problem

The article deals with the method of classification of real data using the apparatus of fuzzy sets and fuzzy logic as a flexible tool for learning and recognition of natural objects on the example of oil and gas prospecting sections of the Dnieper-Donetsk basin. The real data in this approach are the values for the membership function that are obtained not through subjective expert judgment but from objective measurements. It is suggested to approximate the fuzzy set membership functions by using training data to use the approximation results obtained during the learning phase at the stage of identifying unknown objects. In the first step of learning, each traditional future of a learning data is matched by a primary traditional one-dimensional set whose membership function can only take values from a binary set — 0 if the learning object does not belong to the set, and 1 if the learning object belongs to the set. In the second step, the primary set is mapped to a fuzzy set, and the parameters of the membership function of this fuzzy set are determined by approximating this function of the traditional set membership. In the third step, the set of one-dimensional fuzzy sets that correspond to a single feature of the object is mapped to a fuzzy set that corresponds to all the features of the object in the training data set. Such a set is the intersection of fuzzy sets of individual features, to which the blurring and concentration operations of fuzzy set theory are applied in the last step. Thus, the function of belonging to a fuzzy set of a class is the operation of choosing a minimum value from the functions of fuzzy sets of individual features of objects, which are reduced to a certain degree corresponding to the operation of blurring or concentration. The task of assigning the object under study to a particular class is to compare the values of the membership functions of a multidimensional fuzzy set and to select the class in which the membership function takes the highest value. Additionally, after the training stage, it is possible to determine the degree of significance of an object future, which is an indistinctness index, to remove non-essential data (object futures) from the analysis.

Some generalized operations in soft set theory and their role in similarity and decision making

Some generalized operations in soft set theory and their role in similarity and decision making

L-fuzzifying approximation operators in fuzzy rough sets

In rough set theory, lower and upper approximation operators are two primitive notions. Various fuzzy generalizations of lower and upper approximation operators have been introduced over the years. Considering L being a completely distributive De Morgan algebra, this paper mainly proposes a general framework of L-fuzzifying approximation operators in which constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper L-fuzzifying approximation operators is defined. The connections between L-fuzzy relations and L-fuzzifying approximation operators are examined. In the axiomatic approach, various types of L-fuzzifying rough sets are proposed and L-fuzzifying approximation operators corresponding to each type of L-fuzzy relations as well as their compositions are characterized by single axioms. Moreover, the relationships between L-fuzzifying rough sets and L-fuzzifying topological spaces are investigated. It is shown that there is a one-to-one correspondence between reflexive and transitive L-fuzzifying approximation spaces and saturated L-fuzzifying topological spaces.

Set theoretic approach to the concept of net sensitivity and net specificity in screening test results

Net sensitivity and net specificity have been reviewed from set theoretic approach. With a basic knowledge of set theory one can estimate the net sensitivity and specificity in an easy way in both sequential and simultaneous screening tests. Union, intersection and complementary operations of set theory have been adopted to find out the solutions.

Fuzzy Set Theory and Arithmetic Operations On Fuzzy N Umbers

In this paper we represent a fuzzy set theory and arithmetic operation on fuzzy numbers. The fuzzy set theory to derived operation on fuzzy set and fuzzy arithmetic operation. The produce with operation on closed interval.

Textural unit operations in rough set theory

In this study, we introduce a textural counterpart of the unit operation of Wybraniec-Skardowska. A unit di-operation has two parts which are called a unit co-operation and a unit operation, respectively. In this respect, we have two types of symmetry. We show that the symmetricity of direlations is equivalent to the Galois connectivity of unit di-operations. For discrete textures, we determine the image co-operation of a unit co-operation. We prove that the symmetricity and duality are equivalent concepts for unit di-operations if one of the compounds is symmetric. We consider definability in terms of unit operations and unit co-operations. Further, we present a categorical discussion defining a category UN whose structures of objects are unit operations. We show that the category Rel whose objects are approximation spaces and morphisms are relation preserving functions can be embedded into UN.

Textures and fuzzy unit operations in rough set theory: An approach to fuzzy rough set models

In this paper, an approach for the fuzzy rough set models is presented using textures and a fuzzy version of the unit operations of Wybraniec-Skardowska. First, a fuzzy unit operation and fuzzy unit co-operation on fuzzy lattices are defined. It is proved that the well-known fuzzy rough set upper approximations as the approximation of Dubois and Prade are fuzzy unit operation. Using fuzzy direlations, an axiomatic system for fuzzy unit operations is studied and it is shown that fuzzy rough set systems obtained by different fuzzy logical connectives as Kleene-Dienes and Gödel implicators can be generated by the same textural fuzzy direlation. Finally, it is observed that the approximations of two different fuzzy rough set models together constitute two different Galois connections.