Concepts of solutions of uncertain equations with intervals, probabilities and fuzzy sets for applied tasks

The focus of this paper is to clarify the concepts of solutions of linear equations in interval, probabilistic, and fuzzy sets setting for real world tasks. There is a fundamental difference between formal definitions of the solutions and physically meaningful concepts of solution in applied tasks, when equations have uncertain components. For instance, a formal definition of the solution in terms of Moore interval analysis can be completely irrelevant for solving a real world task. We show that formal definitions must follow a meaningful concept of the solution in the real world. The paper proposed several formalized definitions of the concept of solution for the linear equations with uncertain components in the interval, probability and fuzzy set terms that can be interpreted in the real world tasks. The proposed concepts of solutions generalized for difference and differential equations under uncertainty. Keywords—interval linear equations, fuzzy linear equations, stochastic linear equations, quantifiers, solutions of uncertain equations, difference equation, differential equations.