The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets

Abstract

In this paper, the cut sets, decomposition theorems and representation theorems of intuitionistic fuzzy sets and interval valued fuzzy sets are researched indail. First, new definitions of four kinds of cut sets on intuitionistic fuzzy sets are introduced, which are generalizations of cut sets on Zadeh fuzzy sets and have the same properties as that of Zadeh fuzzy sets. Second, based on these new cut sets, the decomposition theorems and representation theorems on intuitionistic fuzzy sets are established. Each kind of cut sets corresponds to two kinds of decomposition theorems and representation theorems. Thus eight kinds of decomposition theorems and representation theorems on intuitionistic fuzzy sets are obtained, respectively. At last, new definitions of cut sets on interval valued fuzzy sets are given based on the theory of cut sets on intuitionistic fuzzy sets, and eight kinds of decomposition theorems and representation theorems on interval valued fuzzy sets are also obtained. These results provide a fundamental theory for the research of intuitionistic fuzzy sets and interval valued fuzzy sets. CitationYuan X H, Li H X, Sun K B. The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets.With the development of fuzzy set theory, many L-fuzzy sets are put forward as the extension of Zadeh fuzzy sets, among which two well-known ones are intuitionistic fuzzy sets [18] and interval valued fuzzy sets [19]. Although it is proved that intuitionistic fuzzy sets are equivalent to interval valued fuzzy sets [20,21], the researches on these two kinds of fuzzy systems have been done from different points of view according to practical requirement. The cut sets of intuitionistic fuzzy sets are studied in [22], and the cut sets, the decomposition theorems and representation theorems on interval valued fuzzy sets are studied in [23][24][25]. These results play an active role in the corresponding fuzzy systems.Note that the cut sets in [22,23] are defined by comparing the interval number [λ 1 , λ 2 ] in [0, 1] (or two numbers (λ 1 , λ 2 ) with λ 1 + λ 2 1 in [0, 1]) with the membership function of interval valued (or intuitionistic) fuzzy set (it is also an interval number or two numbers in [0, 1]), which is equivalent to a comparison between two planar vectors. Since the order in interval numbers is not linearly ordered, this kind of cut sets does not satisfy Properties 5 and 6 of cut sets summarized in [16]. Moreover, the (2, 2)-cut set given in [23] does not preserve the operation "Union(∪)", and the representation theorems given in [24] do not preserve operation "Complement(c)". In order to improve the situations above and promote the research of intuitionistic fuzzy sets and interval valued fuzzy sets, it is needed to present a more reasonable definition of cut sets.In this paper, we introduce new definitions of cut sets of intuitionistic (or interval valued) fuzzy sets by the triple valued fuzzy sets obtained by using the number λ in [0, 1] to cut int...

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