On type-2 fuzzy sets and type-2 fuzzy systems

One of the advantages of systems based on fuzzy logic (fuzzy systems) is the possibility of a soft switch from one set of values of input parameters of the system to another, when different conclusions are drawn for different sets of these values. A fuzzy set of type $2$ is a direct generalization of an ordinary fuzzy set. In this paper, we review some branches of the theory of type-$2$ fuzzy sets and the theory of type-$2$ fuzzy systems. We discuss operations on type-$2$ fuzzy sets, type-$2$ fuzzy relations, and the centroids of type-$2$ fuzzy sets and describe type-$2$ functional fuzzy systems and type-$2$ relational fuzzy systems.