An Introduction to Dynamic Soft Sets: A Framework for Modeling Temporal Uncertainty

Abstract

 In engineering, healthcare, and decision-making systems, effectively handling uncertainty remains a fundamental research challenge. Existing frameworks, including fuzzy sets, rough sets, and soft sets, typically operate on static parameter spaces with fixed approximation mappings, which limits their ability to model real-world environments where relevant parameters and their interpretations evolve over time. To address this limitation, this paper introduces a dynamic extension of soft set theory and makes the following principal contributions. (1) A formal definition of Dynamic Soft Sets is proposed, in which both the active parameter subsets At ⊆ E and the approximation mappings Ft : At → P(U) are explicitly indexed by time or context t ∈ T, enabling the representation of parameters that activate, deactivate, or change relevance over time as a natural extension of Molodtsov’s soft sets. (2) This framework is further generalized to Dynamic Hypersoft Sets, allowing time-varying multi-attribute Cartesian product parameter spaces At = At1 ×···×Atn, thereby extending classical hypersoft sets to dynamic environments. (3) A complete set-theoretic operator framework for Dynamic Soft Sets is developed, including union, intersection, complement, restricted and extended operators, as well as logical AND/OR operations, all defined in a time-dependent setting with fundamental algebraic properties established. (4) A systematic comparative analysis is presented to distinguish Dynamic Soft Sets from classical soft sets and hypersoft sets, highlighting differences in parameter dynamics and structural flexibility through struc tured tables. (5) The practical applicability of the proposed framework is demonstrated through a healthcare monitoring case study, showing how time-varying sensor availability and evolving clinical conditions can be naturally modeled using Dynamic Soft Sets. Unlike classical soft sets with static parameters and hypersoft sets with fixed Cartesian product domains, Dynamic Soft Sets introduce explicit time- or context-indexing of both parameter subsets and approximation mappings, achieving dual structural and temporal dynamism. The proposed framework provides a unified andmathematically rigorous foundation for time- and context-adaptive uncer tainty modeling and decision-making in dynamic real-world applications.