Set-theoretical approach to finding the range of possible values of the size of a group of objects in a situation of multiple indistinguishability

Abstract

In situations where it is not possible to timely use new methods for assessing hazardous objects, traditional methods of risk assessment can be supplemented with new methods based on possible assessments of objective primary information. Evaluation of the destructive capacity and consequences of the use of high-risk systems becomes increasingly important as their structural and functional complexity grows.Despite the existence of the considered works on certain aspects, the known approaches to the consideration of the problem of indistinguishability of influencing factors are characterized by limited functionality. These capabilities do not fully provide the required level of reliability when making management decisions. In view of this, the problem of predicting equipment failure caused by an unfavourable coincidence of circumstances due to the indistinguishability of the initial data is one of the urgent tasks of risk management. Its solution is of significant theoretical and practical interest for many complex heterogeneous dynamical systems. The paper deals with a problem from the field of probabilistic modeling. The relevance of the performed research lies in the fact that the existing methods for predicting equipment failure in systems of potentially hazardous objects do not take into account the causes of accidents that occurred due to unfavourable circumstances.The aim of the work is to develop a method for the set-theoretic modeling of the occurrence of problems with equipment due to an unfavourable coincidence of circumstances, which provides a range of possible values for the size of a group with a given composite very dangerous sign. To achieve this goal, the following tasks are solved:– finding the range of possible values of the number of groups of objects modeled by the intersection of indistinguishable sets and finding the range of possible values of the number of groups of objects modeled by the union and the difference of indistinguishable sets;– finding possible values of the number of groups of objects modeled by an arbitrary composite property and generalization of the problem, features of the set-theoretic apparatus of its solution.In the course of the study, it was found that a set of objects with a compound very dangerous feature of an arbitrary type can be associated with a set representing the union of intersections (intersection of unions) of sets of objects with simple features and their complements. Moreover, the operations of intersection, union, and complement constitute a complete set of operations for the most common version of Boolean algebra of sets.