Abstract
In this paper, vertical slice and horizontal slice of conditional fuzzy set (CFS) are defined and derived, which are different from those of type-2 fuzzy set (T2 FS) with type-1 (T1), blur and weight representation. First, we give the formal definition of CFS which is generalization of the definition given by Wang. And then, within the framework of T2 fuzzy theory, we also give the definitions of embedded CFS, vertical slice and horizontal slice. Different from those of T2 FS with T1, blur and weight representation that is well known by T2 fuzzy community, vertical slices of CFS are extremely complicated, which often obey unconventional fuzzy distributions that need to be calculated by extension principle (EP). Comfortingly, mathematical expressions of its horizontal slices are simple and easy-to-use with a small number of parameters. Finally, two examples are proposed to show the effectiveness and validity of these concepts above. In conclusion, they provide fundamental mathematical methods for further applications.
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