Abstract
The arithmetic operations of fuzzy sets are completely different from the arithmetic operations of vectors of fuzzy sets. In this paper, the arithmetic operations of vectors of fuzzy intervals are studied by using the extension principle and a form of decomposition theorem. These two different methodologies lead to the different types of membership functions. We establish their equivalences under some mild conditions. On the other hand, the α-level sets of addition, difference and scalar products of vectors of fuzzy intervals are also studied, which will be useful for the different usage in applications.
Highlights
Let à and Bbe two fuzzy sets in R with the membership functions ξ à and ξ B, respectively.The arithmetic operations à ⊕ B, à B, à ⊗ Band à Bare based on the extension principle.More precisely, the membership functions are given by ξ Ã⊕ B (z) = ξ à B ( z ) = sup {( x,y):z= x −y} ξ Ã⊗ B (z) = {( x,y):z= x +y} {( x,y):z= x ∗y}
The generalized extension principle for non-normal fuzzy sets has been extensively studied in Wu [15]
The differentiation and integrals of fuzzy-number-valued functions using the form of decomposition theorem have been studied in Wu [17]
Summary
Let à and Bbe two fuzzy sets in R with the membership functions ξ à and ξ B , respectively. Alternative, we use the form of decomposition theorem to define the arithmetic operations of vectors of fuzzy intervals These two methodologies can lead to the different types of membership functions. Their equivalences are established under some mild conditions.
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