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.

中文翻译：

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渐进数算术运算推导出的模糊集区间范围

吴介绍了模糊集的区间范围。在此基础上，他定义了一种使用渐进数和渐进集的模糊集算法。从软计算的角度来看，该定义提供了一种处理模糊集算术运算的新方法。区间范围是模糊集的重要表征。区间范围对于算术的分析和应用也很有用。在本文中，我们给出了与由该算法引起的模糊集区间范围相关的关键问题的一般结论。这些结论表明，应修改先前工作中的相应结论：首先，我们给出了算术的性质和有限算术的组合。然后，我们讨论渐进集的域与其诱导模糊集的范围之间的关系，以及渐进集的定义域与其诱导模糊集区间范围的关系。基于上述结果，我们给出了一组模糊集的区间区间的交集与其通过有限算术组合得到的模糊集区间区间的关系。此外，我们构造了例子来表明即使在比以前工作更强的条件下，一组模糊集的区间范围的交集与其结果模糊集的范围之间的关系仍然存在各种可能性，并且仍然存在一组模糊集的区间范围的交集与其结果模糊集的区间范围之间关系的各种可能性。