Comparative characteristics of methods for synthesizing canonical forms of Zhegalkin algebra

Abstract

Abstract. The article substantiates the important role of the "Discrete Mathematics" educational component in the professional training of higher education students in the field of study E "Natural Sciences, Mathematics and Statistics," specialty E7 "Mathematics."Mastering this discipline facilitates not only the acquisition of the theoretical apparatus but also the formation of key cognitive skills: the ability to select appropriate mathematical methods for solving applied problems and the verification of the applicability conditions for mathematical statements. The purpose of the article is to conduct a comparative analysis of methods for synthesizing canonical forms of Zhegalkin algebra and to substantiate methodological approaches for developing students' critical thinking while studying this topic. Research methods include the analysis of scientific literature, the generalization of pedagogical experience in higher education institutions, surveys, and questionnaires. The approbation of the proposed teaching techniques was conducted through a pedagogical experiment at the Faculty of Information Technologies and Mathematics of Lesya Ukrainka Volyn National University. Results. The study analyzes various methods for constructing Zhegalkin polynomials (the method of undetermined coefficients, the method of identity transformations, and the method of construction based on the PCNF of a Boolean function) and provides relevant examples. It is demonstrated that the efficiency of each method is not absolute and depends on the number of variables $n$, the format of the input data, and the target implementation environment. The authors propose the introduction of problem-based tasks that allow for various solution methods and the organization of discussions regarding the results. This approach fosters students' ability to generate ideas, defend them, and, if necessary, adjust them under the reasoned influence of opponents. To activate critical thinking during practical classes, the following techniques are highlighted: "deliberate error," comparative analysis, and complexity analysis. Conclusions. The research findings emphasize that the formation of students' critical thinking occurs most effectively in decision-making situations. Instruction should not be limited to the reproduction of a single algorithm; instead, students must independently evaluate the complexity of a task and choose the most rational synthesis method.