Mathematical Logic Research Articles

The process of mathematics students’ reflective abstraction in solving continuous random variable problems in the probability course

Reflective abstraction is a mental mechanism in mental constructs in which all the structures of mathematical logic are developed in the mind of the individual. The mental mechanisms in question are interiorization, coordination, reversal, encapsulation, and generalization. This research is classified as exploratory research with a qualitative descriptive approach. This research data was taken from three fifth-semester mathematics students. The instruments used in this study were researchers as key instruments, essay tests, and interview guidelines. Data analysis included: 1) preparing the data for analysis, 2) reading the entire data, 3) encoding the data, 4) describing the data and presenting it in the narrative, and 5) interpreting the data and testing the validity of the data using data triangulation. The research found that (1) The first participant went through four components of reflective abstraction: interiorization – coordination – encapsulation – generalization. (2) The second participant went through five components of reflective abstraction thinking namely interiorization – coordination – reversal – encapsulation – generalization. (3) The third participant failed to solve the problem because the thought process in his mental mechanism had not yet manifested coordination, encapsulation, and generalization. Researchers found that the mental construction that the first participant had gone through could illustrate that the minimum mental construction that must be passed in solving the problem of continuous random variables required components- interiorization, coordination, encapsulation, and generalization.

A Proof That the Set of NP-problems is Bigger Than the Set of P-problems by Using a Logical Consideration

The field of informatics is the domain that emerged by applying the mathematical logic on electronic devices called computers in order to simplify many tasks for humans. The application of informatics in all economic and scientific areas is the most important factor that made our civilization reach our current phase of development. Nowadays, the experts and even the beginners of informatics are eager to use quantum computers. However, there is still an unsolved problem of classical theoretical informatics in ordinary electronic computers. It is the famous philosophical problem of the “Millenium Prize” of the Clay Mathematics Institute concerning the complexity of problems that has been treated by many other researchers but without acceptable sufficient answers. A solution to this problem can make all the fields based on informatics make huge progress. And thus, thanks to my short studies about informatics, I present to you this mathematical proof that deals with the sets of P problems, NP-Complete problems and NP-Hard problems in the field of classical electronic computers in order to prove new formulas about the cardinals of each group of complexity problems and about the intersections of each one of these sets. The aim is to contribute to an acceptable solution for this Millenium Problem and the methodology is purely logical and mathematical. The readers won’t need any complicated notions from the background of previous informatics or mathematics research in order to understand the demonstrations of this article since the proof is based only on notions of sets by starting with easy logical considerations. Furthermore, you will find in this work a proof of an interesting theorem about the complexity of problems that allows us to identify NP-problems even if their algorithms have infinite time of execution. This paper ends by proving that the set of NP-Problems is definitely bigger than the set of PProblems. Hence, all the readers are invited to understand and develop this work by inspecting the applied logical considerations in order to succeed in finding a sufficient solution to the interesting Millenium Problem of complexity.

ASESMEN KOMPETENSI MINIMUM UPAYA DALAM MENINGKATKAN LITERASA BAHASA DAN NUMERASI DI SEKOLAH DASAR

Language and numeracy literacy are the basic skills required for students to analyze reading or numbers in different contexts. Literacy and numeracy skills must be mastered and possessed in order to increase competitiveness and improve the quality of human resources. However, what happens is the low reading ability of Indonesian school children known from the results of the PISA test. In 2018, Indonesia ranked 70th out of 78 countries. Students' difficulties in critical thinking are known due to the lack of students who can solve Higher Thinking Skills (HOTS) questions and teachers who have not been able to create HOTS-based questions. Therefore, the Ministry of Education and Culture decided to implement competency assessment for all schools. The purpose of this study is to provide insight into the concept of Minimum Competency Assessment (MCA) to improve reading, writing and arithmetic skills in primary schools. The research method used is literature research. Based on the review of the literature approach, it was found that AKM focuses more on the ability to reason using language or literacy and using mathematical logic or numeracy.

Analysis of the impact of public procurement indicators on the achievement of public policy goals

The study is devoted to the study of the dynamics of the achievement of indicators of the state macroeconomic policy of Russia during 2013—2022 and the assessment of the impact of public procurement spheres on this process. The article presents statistics of absolute and relative indicators of public policy, as well as gross indicators and effective indicators of public procurement. The article contains the results of a correlation analysis of the impact of public procurement indicators implemented under Federal Law No. 44-FZ of April 5, 2013 "On the Contract System in the Field of procurement of Goods, Works, Services for State and Municipal Needs" on the achievement of public policy targets, including the dynamics of GDP and changes in the level of budget savings in the implementation of public procurement activities.The scientific novelty of the study consists in the application of the mathematical and statistical method of correlation with respect to macroeconomic indicators of the functioning of the national economy of the Russian Federation to identify the direction and strength of the relationship between the studied areas of economic activity. An element of scientific novelty is also the justification of the time shift of the interdependent studied indicators of public policy and public procurement based on the logic of the process impact of public procurement activities on the results of public policy and information restrictions imposed by the current mechanism for collecting, summarizing and publishing official state statistics. The author's contribution to the development of scientific research devoted to the study and attempts to optimize public procurement activities as a tool for achieving the goals of public policy consists in the use of mathematical logic tools to substantiate the performance indicators of public procurement, providing the best economic results of public policy in the time period under study. Based on the results of the study, it is concluded that it is possible to establish and quantify the relationship between the gross indicators of the functioning of public procurement and the achievement of public policy indicators, including ensuring economic growth in terms of GDP dynamics and managing the level of budget savings generated as a result of public procurement activities.

Combining Building Block Process With Computational Thinking Improves Learning Outcomes of Python Programming With Peer Assessment

The capability of computer programming language logic is one of the basics of technical education. How to improve students “interest in program logic design and help overcome students” fears of coding has become vital for educators. Cultivating practical talents with information technology application and basic programming development will become one of the important topics in the department of information related science. The objective of this research is to improve the ability of learning basic programming courses by using Zuvio interactive software. Zuvio employs the mathematical logic of computational thinking to analyze problems and enhance learners’ interest in learning programming skills through a graphical interface tool with building blocks. It uses innovative interactive teaching to use peer and self-assessment to study the content of the course. Zuvio improves the design ability of different groups of class learning Python programming. In line with the innovative teaching policy of the schools and the current stage of the learner’s learning model, learning effectiveness can be achieved. The research results were analyzed by midterm and final experimental group scores, and the progress of the experimental group’s scores was examined through descriptive statistics. The average and standard deviation of the assessment were used to analyze the progress of the experimental group students in the programming course. In the classroom, assessment criteria were set up as the basis for peer assessment scoring. After the midterm and final exams, the teacher assessment and peer assessment scores were analyzed for cognitive differences, and possible learning differences were analyzed. The students’ professional ability was examined to see if it met the professional standards required by the course, and whether innovative teaching methods could improve the learning outcomes of learners with different professional backgrounds in Python programming.

Комплексная оценка трудноформализуемых вентиляционно-технологических процессов на угольных шахтах

Introduction. Automated process control systems, as a set of characteristics related to the ability of an object to satisfy certain needs, are one of the most important indicators of the quality of complex technical systems consisting of a large number of interacting mechanisms, apparatus and instruments. Their failure can lead to accidents and emergencies, resulting in large economic losses, destruction of industrial facilities, and sometimes human casualties. Methodology and research methods. The solution to the assigned problems is based on modern methods of artificial intelligence, automatic control theory, mathematical modeling of technological processes, and mathematical statistics; graph theory; decision theories; apparatus of mathematical logic; set theory and systems analysis. Discussion of research results. The article presents the results of studies of the patterns of ventilation and technological processes in coal mines. It has been established that studies of patterns in coal mines are becoming more relevant in connection with the formulation of the task of modernizing production and the Russian economy as a whole. It has been revealed that the methodological foundations for the synthesis of control decisions that ensure the achievement of a stable state make it possible to formulate acceptable organizational and managerial decisions and evaluate their effectiveness using metered characteristics of the proximity of vaguely described states. Conclusion. Based on the research carried out, new theoretical and applied problems were posed and solved, system connections and patterns of the ventilation technological process (VTP) were identified, models, methods and algorithms for automated ventilation technological process control systems (ACVTP) were developed, aimed at increasing the efficiency of the technical air supply system coal mines. The theoretical foundations for modeling and controlling ventilation and technological processes have been created, including concepts, models, methods and control algorithms. The research results can be useful in controlling ventilation and technological processes. Conclusions on the article. The article presents the results of studies of ETP conducted in coal mines. It has been established that, using the developed methodology, it is possible to: design high-tech equipment with specified quality indicators for intelligent control systems; choose from a variety of structural diagram options the best in terms of operational efficiency; carry out structural-parametric adjustment and change the types of mathematical models, both for an individual element and for the ETP as a whole; identify the weakest points in the HTP automated control system and adapt the methodology for controlling ventilation and technological processes, evaluate the effectiveness of using structural redundancy. Suggestions for practical applications and directions for future research. The research results can be useful when carrying out mining operations at all enterprises where air supply is necessary.

Учет переменного характера расчетных нормативов плана формирования поездов с использованием технологий вычислительного интеллекта

Purpose: The train formation plan is the most important logistics tool for cargo transportation management. But the use of discrete values of design standards in its development — the quantity of car-hours for the accumulation of trains and savings from passing of cars through technical station without processing — do not always guarantee the optimal solution due to the objectively existing unevenness of operational work. It is precisely this factor that generates uncertainty, which is not inherently stochastic, and necessitates adjustments to the train formation plan throughout its lifecycle. In addition to fluctuations in car traffic, it causes a change in the values of the calculated standards. To manage such uncertainty, there are special methods, one of which is fuzzy logic. The article describes a way to account for changes in all calculation standards of the train formation plan. Methods: The methods of one of the technologies of computational artificial intelligence are used — fuzzy logic, fuzzy sets and fuzzy mathematics. Results: The dependence has been established on how changes in the calculated parameters of the train formation plan are affected by fluctuations in car flows on specific destinations. The formulas obtained on the basis of known analytical expressions allow us to determine the values of the standards of the formation plan without using auxiliary tables and graphs. Practical significance: The use of the obtained dependencies in the development of the formation plan will improve the accuracy of its calculation by taking into account fluctuations not only of car traffic, but also the values of the calculated standards depending on them.

Education-oriented Proof Assistant Based on Calculational Logic: Proof Theory Algorithms and Assessment Experience

This work presents an interactive proof assistant, based on Dijkstra-Scholten logic, aimed at teaching logic and discrete mathematics in higher education. The assistant interface is web and easy to use, since inferences can be made just with the mouse. The educational experience is presented showing a correlation between the grades of the assessments in class and those made with the application web. Additionally, an algorithm proof theory for the Disjktra-Scholten system are made and the following algorithms are shown: 1) a versatile printing algorithm that allows the administrator to configure the symbols of a theory, by assigning the desired presentation with LaTeX; 2) An algorithm, based on Broda and Damas combinators, for generate monotonic or anti monotonic inferences in the Dijkstra-Scholten logic; 3) An algorithm to generate the proofs of dual theorems in Boolean Algebra theory.

Bertrand Russell’s Philosophical Logic and its Logical Forms

From 1901 to 1919, Russell persistently maintained that there were two kinds of logic and distinguished between one and the other as mathematical logic and philosophical logic. In this paper, we discuss the concept of philosophical logic, as used by Russell. This was only a tentative program that Russell did not clarify in detail; therefore, our task will be to make it explicit. We shall show that there are three (-and-a-half) kinds of Russellian philosophical logic: (i) “pure logic”; (ii) philosophical logic investigating the logical forms of propositions; (iii) philosophical logic exploring the logical forms of facts: in epistemology and in the external world. In particular, Russell’s program or philosophical logic of the facts of the external world remained less than sketchily outlined. Keywords: Russell, mathematical logic, philosophical logic, Wittgenstein

Russell Paradox Analysis: Descriptions and Logical Cases

The Russell Paradox is a classic problem in mathematical logic that challenges the underlying basics of set theory and the concept of infinity. This study provides a comprehensive analysis of the paradox, including its history, development, and recent research. The paradox arises from the set whose members are sets not containing themselves, which results in being contradictory. Bertrand Russell and Ernst Zermelo proposed early solutions, while the axiomatic set theory provided a rigorous foundation for mathematics. Recent studies suggest alternative solutions, including distinguishing between object-language and meta-language and introducing a new concept called a "Russell-class." This paper summarizes the history of the paradox, summarizes recent studies, and presents a framework for understanding and evaluating solutions. According to the analysis, it aims to deepen the understanding of the paradoxical situation and why it is of enormous importance for the underlying basics of mathematics. This paradox remains a subject of intense study and continues to inspire the development of various theories and frameworks to resolve it. Overall, these results shed light on guiding further exploration of the paradoxical situation and its enormous importance for the underlying basics of mathematics. They also present a framework for understanding and evaluating solutions, with the goal of deepening our understanding of the paradox and inspiring the development of various theories and frameworks to resolve the issue.

The umbilical cord of finite model theory

Abstract Model theory was born and developed as a part of mathematical logic. It has various application domains but is not beholden to any of them. A priori, the research area known as finite model theory would be just a part of model theory but it didn’t turn out that way. There is one application domain, relational database management, that finite model theory had been beholden to during a substantial early period when databases provided the motivation and were the main application target for finite model theory. Arguably, finite model theory was motivated even more by complexity theory. But the subject of this paper is how relational database theory influenced finite model theory. This is NOT a scholarly history of the subject with proper credits to all participants. My original intent was to cover just the developments that I witnessed or participated in. The need to make the story coherent forced me to cover some additional developments.

What Are Statistics: It's Time to set the Record Straight

The authors present their views on the modern concept of statistics as a science, academic discipline, and field of activity. The urgency of the matter lies in an adequate understanding of the essence of statistics, which determines the direction for its further evolution as a tool for increasing the effectiveness of the national socio-economic processes.According to the authors, by the beginning of the XXI century, de jure statistics are associated with activities, measured by data, that reflect a set of phenomena of diverse nature while ignoring de facto activities, associated with an assessment of the quantitative side of mass phenomena as part of information support of cognitive processes and management decision-making in the socio-economic sphere. In the former case, theoretical, methodological, and practical statistical activities are based on the consideration of the statistical population as such, for which is being developed an appropriate toolkit based primarily on the unity of methods of mathematical logic, mathematical statistics, and big data analysis. In the latter case, theoretical, methodological, and practical activities are interpreted within the framework of the concept implying that the quantitative side of mass social phenomena is the object of cognition and management as an objective reality. It is based on the following categories: a statistical indicator, a system of indicators, and an information model.Analysis of the main features of the two types of activity show that their convergence is impossible. The paper concludes that while maintaining the independent status of the first type of activity, called Statistics, it would be appropriate to acknowledge the independent status of the second type of activity, called Socio-economic statistics, as a type of occupation, a specialty and area of training in higher education, a scientific specialty – a branch of science, incorporating it in the professional standard «Statistician», the educational standard of higher education «Socio-economic statistics», and the standard of the scientific specialty «Socio-economic statistics».

Digital flipbook module in circulatory system to facilitate student different intelligent and learning style

Students acquire and improve various intellectual abilities through learning process facilitated by learning methods. The study provides information on the development process of digital flipbook modules to facilitate student intelligence and learning style for the circulatory system, determine its feasibility, and students’ responses. The method used was an R&D with 4D development models. It consisted of four stages: define, design, develop, and disseminate. However, this research was limited to the development stage. Development of digital flipbook modules based on nine types of intelligence: linguistic, mathematical logic, kinesthetic, visual-spatial, musical, interpersonal, intrapersonal, naturalist, and existential. The instruments used expert feasibility questionnaires (materials and media) and student response questionnaires. The result of the assessment test of experts in the material aspect was 96.4% in the very eligible category, and the media aspect was 92.8% in the very suitable category. The score of students’ responses was 90,14%, in the very eligible category. Based on the analysis results, the digital flipbook module was suitable biology teaching material for the circulatory system

Mathematical Applications of Negating of Conditional Sentences in the Arabic Language

This paper studies the negation of conditionalsentences in Arabic language by using mathematicalapplications with three steps. The first step is to convert theconditional sentence to be negated from the Arabic to themathematical statement. The second step is to negate themathematical conditional statement obtained in the first stepusing the negatives in mathematical logic. The third step isto convert the mathematical conditional statement obtainedin the second step to an Arabic sentence.

MATHEMATICAL MODEL OF TECHNOLOGICAL ASSEMBLY PROCESSES
 IN MACHINE-BUILDING AND REPAIR PRODUCTION

In a logical representation, the technological process of assembling any mechanism or assembly, including the crank mechanism of the engine, can always be considered as one of the forms of displaying functional and constructive signs of searching for some objective function. (Research purpose) The research purpose is substantiating the effectiveness of original approaches to mathematical modeling of assembly processes in machine-building and repair industries. (Materials and methods) Conducted research using complete information about the modeling object as part of a unified system of mathematical modeling of assembly processes formed at the levels of abstraction of structure, logical and quantitative predicates. (Results and discussion) It has been shown that modern principles of mathematical modeling are based on the basic provisions of the theory of sets, graphs and mathematical logic. It was noted that of all the considered spaces of production systems describing assembly technological processes, the space of predicates of the mechanism and technological system is most closely related to the problem of ensuring a given accuracy of the closing link in a particular real dimensional chain. As a result of the assembly process, in general, parts or assemblies are transformed into a connected system of Cartesian bodies with a limited number of degrees of freedom. Mathematical modeling of assembly processes at the level of quantitative predicates is characterized by the least degree of response, but it gives a description in a more rigorous mathematical form that allows for a well-founded optimization of accuracy parameters and methods of performing technological techniques that provide unambiguous optimization. The applied value of modeling at the level of quantitative predicates is significantly higher than the description at the levels of structural and logical properties and relationships. (Conclusions) Mathematical modeling of assembly technological processes with a cybernetic approach at the levels of multiple, logical and quantitative predicates provides the best adequacy and applied significance in the case of building analytical models.

Morpho-logic from a topos perspective – application to symbolic AI

Modal logics have proved useful for many reasoning tasks in symbolic artificial intelligence (AI), such as belief revision, spatial reasoning, among others. On the other hand, mathematical morphology (MM) is a theory for non-linear analysis of structures, that was widely developed and applied in image analysis. Its mathematical bases rely on algebra, complete lattices, topology. Strong links have been established between MM and mathematical logics, mostly modal logics. In this paper, we propose to further develop and generalize this link between mathematical morphology and modal logic from a topos perspective, i.e. categorial structures generalizing space, and connecting logics, sets and topology. Furthermore, we rely on the internal language and logic of a topos. We define structuring elements, dilations and erosions as morphisms. Then we introduce the notion of structuring neighborhoods, and show that the dilations and erosions based on them lead to a constructive modal logic, for which a sound and complete proof system is proposed. We then show that the modal logic thus defined (called morpho-logic here), is well adapted to define concrete and efficient operators for revision, merging, and abduction of new knowledge, or even spatial reasoning.

Fuzzy Logic with Deep Learning for Detection of Skin Cancer

Melanoma is the deadliest type of cancerous cell, which is developed when melanocytes, melanin producing cell, starts its uncontrolled growth. If not detected and cured in its situ, it might decrease the chances of survival of patients. The diagnosis of a melanoma lesion is still a challenging task due to its visual similarities with benign lesions. In this paper, a fuzzy logic-based image segmentation along with a modified deep learning model is proposed for skin cancer detection. The highlight of the paper is its dermoscopic image enhancement using pre-processing techniques, infusion of mathematical logics, standard deviation methods, and the L-R fuzzy defuzzification method to enhance the results of segmentation. These pre-processing steps are developed to improve the visibility of lesion by removing artefacts such as hair follicles, dermoscopic scales, etc. Thereafter, the image is enhanced by histogram equalization method, and it is segmented by proposed method prior to performing the detection phase. The modified model employs a deep neural network algorithm, You Look Only Once (YOLO), which is established on the application of Deep convolutional neural network (DCNN) for detection of melanoma lesion from digital and dermoscopic lesion images. The YOLO model is composed of a series of DCNN layers we have added more depth by adding convolutional layer and residual connections. Moreover, we have introduced feature concatenation at different layers which combines multi-scale features. Our experimental results confirm that YOLO provides a better accuracy score and is faster than most of the pre-existing classifiers. The classifier is trained with 2000 and 8695 dermoscopic images from ISIC 2017 and ISIC 2018 datasets, whereas PH2 datasets along with both the previously mentioned datasets are used for testing the proposed algorithm.

Strategic variants of the razor and blade business model: an inspiring benchmark

PurposeGillette’s historically successful “razor and blade” business model (RBM) has been a promising benchmark for multiple businesses across diverse industries worldwide in the past several decades. The extant literature deals with very few nuances of this business model notwithstanding the fact that there are several variants of this business model being put to practical use by firms in diverse industries in grossly metaphorically equivalent situations.Design/methodology/approach This study adopts the 2 × 2 truth table framework from the domains of mathematical logic and combinatorics in fleshing out all possible (four logical possibilities) variants of the razor and blade business model for further analysis. This application presents four mutually exclusive yet collectively exhaustive possibilities on any chosen dimension. Two major dimensions (viz., provision of subsidy and intra- or extra-firm involvement in the making of razors or blades or both) form part of the discussion in this paper. In addition, this study synthesizes and streamlines entrepreneurial wisdom from multiple intra-industry and inter-industry benchmarks in terms of real-time firms explicitly or implicitly adopting several variants of the RBM that suit their unique context and idiosyncratic trajectory of evolution in situations that are grossly reflective of the metaphorically equivalent scenario of razor and recurrent blades. Inductive method of research is carried out with real-time cases from diverse industries with a pivotally common pattern of razor and blade model in some form or the other.Findings Several new variants of the razor and blade model (much beyond what the extant literature explicitly projects) have been discovered from the multiple metaphorically equivalent cases of RBM across industries. All of these expand the portfolio of options that relevant entrepreneurial firms can explore and exploit the best possible option chosen from them, given their unique context and idiosyncratic trajectory of growth.Research limitations/implications This study has enriched the literature by presenting and analyzing a more inclusive or perhaps comprehensive palette of explicit choices in the form of several variants of the RBM for the relevant entrepreneurial firms to choose from. Future research can undertake the task of comparing these variants of RBM with those of upcoming servitization business models such as guaranteed availability, subscription and performance-based contracting and exploring the prospects of diverse combinations.Practical implications Smart entrepreneurial firms identify and adopt inspiring benchmarks (like razor and blade model whenever appropriate) duly tweaked and blended into a gestalt benchmark for optimal profits and attractive market shares. They target diverse market segments for tied-goods with different variants or combinations of the relevant benchmarks in the form of variegated customer value propositions (CVPs) that have unique and enticing appeal to the respective market segments.Social implications Value-sensitive customers on the rise globally choose the option that best suits them from among multiple alternatives offered by competing firms in the market. As long as the ratio of utility to price of such an offer is among the highest, even a no-frills CVP may be most appealing to one market segment while a plush CVP may be tempting to yet another market segment simultaneously. While professional business firms embrace resource leverage practices consciously, amateur customers do so subconsciously. Each party subliminally desires to have the maximum bang-to-buck ratio as the optimal return on investment, given their priorities ceteris paribus.Originality/value Prior studies on the RBM have explicitly captured only a few variants of the razor and blade model. This study is perhaps the first of its kind that ferrets out many other variants (more than ten) of the razor and blade model with due simplification and exemplification, justification and demystification.

On identity, indiscernibility and (non-)individuality in the quantum domain.

The topics of identity, individuality and indiscernibility had already entered the domain of the philosophy of physics, mainly in what respects the philosophy of quantum theories. In this paper, I present the main ideas of an underlying metaphysics that considers the basic entities of quantum physics as non-individuals, that is, as entities to which the standard theory of identity of classical logic and standard mathematics (set theory) does not apply. Two things must be emphasized: the first is that we pursue Erwin Schrödinger's ideas that the notion of identity (or sameness) does not apply to quantum things; the second is to follow Heinz Post's claim that the non-individuality of quantum entities 'must be introduced right at the start' and not by taking individuals and making them pretend to be non-individuals 'by hand', say by confining them to non-rigid structures. Thus, this paper should be regarded as an attempt to justify the metaphysics of non-individuals and the logical systems that cope with them, although these systems are here only referred to. We also answer some criticisms that have been directed to such a view. The paper is directed to an audience of both physicists interested in foundations and philosophers of physics. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.

Learning Mathematical Logic and Sets Using E-Learning

Video is a learning medium that is simple to integrate into a variety of activities, both online and offline. As a result, video development, including studying mathematical logic and set material, is essential. This research aims to develop project-based educational videos on mathematical logic and mathematics material. This research is development research using the Richey Klein (DPE) development model. There are three stages of development: design, production, and evaluation. This educational video was implemented on 25 students in the third semester of the Mathematics Education Study Program (FKIP UAD). The results of the material expert assessment showed a score of 4.23 (good category) and the media expert assessment scored 4.55 (very good category). Besides that, the student’s response during implementation got a score of 3.9 (good category).