Abstract
Wu introduced the interval range of fuzzy sets. Based on this, he defined a kind of arithmetic of fuzzy sets using a gradual number and gradual sets. From the point of view of soft computing, this definition provides a new way of handling the arithmetic operations of fuzzy sets. The interval range is an important characterization of a fuzzy set. The interval range is also useful for analyses and applications of arithmetic. In this paper, we present general conclusions on crucial problems related to interval ranges of fuzzy sets induced by this arithmetic. These conclusions indicate that the corresponding conclusions in previous works should be modified: firstly, we give properties of the arithmetic and the composites of finite arithmetic. Then, we discuss the relationship between the domain of a gradual set and the range of its induced fuzzy set, and the relationship between the domain of a gradual set and the interval range of its induced fuzzy set. Based on the above results, we present the relationship between the intersection of the interval ranges of a group of fuzzy sets and the interval ranges of their resulting fuzzy sets obtained by compositions of finite arithmetic. Furthermore, we construct examples to show that even under conditions stronger than in previous work, there are still various possibilities in the relationship between the intersection of interval ranges of a group of fuzzy sets and the ranges of their resulted fuzzy sets, and there are still various possibilities in the relationship between the intersection of the interval ranges of a group of fuzzy sets and the interval ranges of their resulting fuzzy sets.
Highlights
IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
Based on the use of an interval range, he introduced the arithmetic operations of fuzzy sets in R by using gradual sets and gradual numbers
According to Wu’s definition, the resulted fuzzy sets of arithmetic operations of two fuzzy sets are defined as the induced fuzzy sets of the gradual sets on the intersection of the interval ranges of these two fuzzy sets
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Based on the use of an interval range, he introduced the arithmetic operations of fuzzy sets in R by using gradual sets and gradual numbers. According to Wu’s definition, the resulted fuzzy sets of arithmetic operations of two fuzzy sets are defined as the induced fuzzy sets of the gradual sets on the intersection of the interval ranges of these two fuzzy sets. Wu [15] introduced the arithmetic operations of two gradual numbers which are elements of two fuzzy sets in F (R), respectively. The resulted fuzzy set of the arithmetic operation is induced from this gradual set or this assignment function. We still use +, −×, / to denote the arithmetic operations of fuzzy sets in. The symbol fe denotes the corresponding composite of finite arithmetic operations of fuzzy sets in R.
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