Footprint of uncertainty for type-2 fuzzy sets

In the paper, a new and comprehensible definition is proposed for type-2 fuzzy sets (T2 FSs), and the primary and secondary memberships function are defined respectively by using multi valued mapping. A new definition and formula for the footprint of uncertainty (FOU) is presented, and based on the new definitions, the relation between FOU and the original definition of T2 FS is discussed. Finally, the partition method of FOU is provided to represent the primary membership grade, the FOU and the interval type-2 fuzzy sets (IT2 FS) when the upper and lower membership functions (UMF, LMF) of FOU are given.