Pythagorean Soft Sets and Hypersoft Sets: A Comprehensive Framework for Advanced Uncertainty Modeling in Decision Making

This paper introduces a comprehensive mathematical framework that integrates Pythagorean fuzzy sets (PyFS), soft sets (SS), and hypersoft set theory to establish robust methodologies for addressing complex uncertainties in multi-attribute decision-making (MADM). We present two formal approaches: Pythagorean soft sets (PySS) and Pythagorean hypersoft sets (PyHSS), which enhance the conventional soft set and hypersoft set frameworks by incorporating the expressiveness of Pythagorean fuzzy sets (PyFS). The theoretical framework demonstrates that these structures preserve all fundamental set-theoretic operations and facilitate the representation of the membership degree (MD) and non-membership degree (n-MD) with sums not exceeding 1, contingent upon satisfying the Pythagorean condition ψ² + φ² ≤ 1. We illustrate the applicability of our approach through extensive digital implementations in technology selection, cloud-based configuration, and educational recommendation systems. The mathematical functionalities exhibited, including operation closure and generalisation hierarchies, render PySS and PyHSS formidable decision support system instruments for managing imprecise and ambiguous information across several criteria.