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 contribution of this paper is the seven formalized definitions of the concept of solution for the linear equations with uncertain components in the interval settings that are interpretable in the real-world tasks. It is shown that these definitions have analogies in probability and fuzzy set terms too. These new formalized concepts of solutions are generalized for difference and differential equations under uncertainty