АДАПТАЦІЯ ОСНОВНИХ ПОНЯТЬ ТЕОРІЇ МНОЖИН ДО ПОШУКУ ОПТИМАЛЬНИХ РІШЕНЬ ПРИ ПРОЄКТУВАННІ ОДЯГУ

The set theory, as a core component of discrete mathematics, offers a powerful foundation for formalizing complex systems, including those found in clothing design. This paper explores the application of set-theoretical models in digital garment development, emphasizing the structuring of parameters into constructive, material, functional, anthropometric, and aesthetic sets. By representing garment elements as sets and utilizing operations such as union, intersection, and Cartesian product, we build a comprehensive framework for systematizing the design process. The study highlights the inadequacy of current digital methods in effectively incorporating discrete mathematical approaches into clothing design workflows. It identifies a gap between the capabilities of modern computational systems and the lack of formalized, structured data suitable for intelligent systems. A generalized design model is proposed, in which parameter sets interact through a multivariate function that ensures compatibility and optimization of design elements. This model is further enhanced through a compatibility hypothesis, which asserts the necessity of non-empty intersection among parameter sets to achieve design harmony. To account for variability and uncertainty inherent in anthropometric and aesthetic data, fuzzy set theory is integrated into the modelling process. Membership functions are introduced to quantify parameter suitability, allowing for adaptive optimization based on consumer-specific characteristics and preferences. Practical applications of the model are discussed, including integration with 3D prototyping and AI-based systems for style evaluation and customization. The research concludes that set theory, especially when extended through fuzzy logic and weighting systems, can significantly enhance digital clothing design by improving adaptability, reducing reliance on physical prototyping, and enabling intelligent decision-making. The presented approach forms a basis for the development of advanced intelligent systems in fashion design, bridging the gap between technical constraints and aesthetic flexibility.