The theory of fuzzy sets is presented in the applied mathematics section, which is devoted to methods for analyzing uncertain data describing the uncertainties of real events and processes using concepts about sets without clear boundaries.
The classical theory of sets determines the belonging of a particular element to a particular population. In this case, concepts in binary terms are accepted as belonging, i.e. there is a clear condition: the element in question either belongs or does not belong to the set.
The theory of sets with respect to fuzziness provides a graded understanding of the belonging of the element in question to a particular set, and the degree of its belonging is to be described using the corresponding function. In other words, the transition from belonging to a given set of some elements to non-belonging does not occur abruptly, but gradually using a probabilistic approach.
Sufficient experience of foreign and domestic researchers indicates the unreliability and inadequacy of the probabilistic approach used as a tool for solving problems of a loosely structured type. The use of statistical methods in solving this type of problem leads to a significant distortion of the original statement of the problem. It is the shortcomings and limitations associated with the application of classical methods for solving problems of a loosely structured form that are the result of the "principle of incompatibility", which is formulated in the theory of fuzzy sets developed by L.A. Zade.
Therefore, some foreign and domestic researchers have developed methods for assessing the risk of investment projects and efficiency using the tools of the theory of fuzzy sets. In them, the probability distribution method came to replace the probability distribution method, which is described by the membership function of a fuzzy type number.
The basics of set theory are based on tools that are relevant to decision-making methods in uncertain conditions. When using them, it is assumed that the initial parameters and performance indicators of the target orientation are formalized as a vector of a fuzzy interval (interval values). The hit in each such interval can be characterized by the degree of uncertainty.
Using arithmetic when working with such fuzzy intervals, experts can obtain a fuzzy interval for a specific target. Based on the initial information, experience and intuition, experts can give qualitative and quantitative characteristics of the boundaries (intervals) of the possible values of the region and the parameters of their possible values.
Set theory can be actively used in practice and in the theory of systems management , in finance and economics to solve problems, provided the key indicators are uncertain. For example, appliances such as cameras and some washing machines are equipped with fuzzy controllers.
In mathematics, set theory proposed by L.A. Zade, allows you to describe fuzzy knowledge and concepts, operate with them and draw fuzzy conclusions. Thanks to the methods of constructing fuzzy systems based on this theory using computer technology, the field of application of computers is expanding significantly. Recently, the management of fuzzy sets is one of the productive areas of research. The usefulness of fuzzy control is manifested in a certain complexity of technological processes from the position of analysis using quantitative methods. Also, fuzzy set management is used in the qualitative interpretation of various sources of information.