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We continue the formal development of the application of piecewise linear functions and centroids in the area of fuzzy set theory. The corresponding piecewise linear functions are symmetrical and composed by absolute function. In this paper we prove that the membership functions of isosceles triangle type and isosceles trapezoid type can be constructed by functions of this type.

Since isosceles triangular and trapezoidal membership functions [4] are easy to manage, they were applied to various fuzzy approximate reasoning [10], [13], [14]. The centroids of isosceles triangular and trapezoidal membership functions are mentioned in this article [16], [9] and formalized in [11] and [12]. Some propositions of the composition mapping (f+ · g, or f +* g using Mizar formalism, where f, gare a ne mappings), are proved following [3], [15]. Then different notations for the same isosceles triangular and trapezoidal membership function are formalized.

Academic failure is a persistent challenge in education. Despite the limited available data, in this study, we focus on identifying at-risk students in a blended learning (BL) course. Several motivational variables are analyzed to determine their effect on student performance. We use a machine-learning classifier to compare two approaches: 1) The benchmark study, which uses data from the same academic year for both training and testing; and 2) a prospective study, which focuses on extrapolating a model trained on the previous fall semester to predict outcomes for the upcoming fall semester. We categorize the motivational variables, including time management and event-occurrence frequency. The window-expansion strategy is adopted to enhance performance through periodic evaluation to facilitate timely intervention. Consequently, the prospective approach is consistent with the benchmark results, thus demonstrating its generalizability across academic years and student populations. This approach is promising for identifying at-risk students in the BL course early. The Shapley additive explanations (SHAP) method emphasizes the importance of time-management variables, particularly study in time, for identifying at-risk students. This study enhances our understanding of early detection methods for at-risk students. By analyzing time-related behaviors, we establish a robust foundation for data-driven decisions to improve future BL implementations as well as support motivation and self-regulated learning practices.

In this study, using the Mizar system [1], [2], we reuse formalization e orts in fuzzy sets described in [5] and [6]. This time the centroid method which is one of the fuzzy inference processes is formulated [10]. It is the most popular of all defuzzied methods ([11], [13], [7]) – here, defuzzified crisp value is obtained from domain of membership function as weighted average [8]. Since the integral is used in centroid method, the integrability and bounded properties of membership functions are also mentioned to fill the formalization gaps present in the Mizar Mathematical Library, as in the case of another fuzzy operators [4]. In this paper, the properties of piecewise linear functions consisting of two straight lines are mainly described.

Summary. IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [7]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [12]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [9], [10], [4]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions.

This article presents a framework for studying the existence of optimal control based on fuzzy rules. The framework consists of two propositions: (1) A set of fuzzy membership functions which are selected out of L2[a,b] is convex and compact metrizable for the weak topology. (2) The defuzzification function is continuous on the set above. Then, the existence of fuzzy optimal control is proved.