**7. Conclusions**

In this paper, we give general conclusions on the problems related to the interval ranges of the resulted fuzzy sets of compositions of finite arithmetic.

In [15], it is claimed that under certain conditions, the domain *I* of a gradual set *μ* is equal to the range <sup>R</sup>(*<sup>F</sup>μ*) of its induced fuzzy set *<sup>F</sup>μ*.

We show by examples that this is not valid. In fact, even under stronger conditions than in [15], there are still various possibilities in the relationship between *I* and <sup>R</sup>(*<sup>F</sup>μ*). Moreover, we give the relationship between *I* and <sup>R</sup>(*<sup>F</sup>μ*), and the relationship between *I* and *IFμ* .

In [15], it is claimed that *I*<sup>∩</sup> = *IA*◦*<sup>B</sup>* when ◦ is an arithmetic operation in {<sup>+</sup>, −, ×, /}. We show by examples that this is not valid. Furthermore, we give the relationship between *I*<sup>∩</sup> and *If*(*<sup>A</sup>*1,...,*An*). As a corollary, we give the relationship between *I*<sup>∩</sup> and *IA*◦*<sup>B</sup>* for ◦∈{<sup>+</sup>, −, ×, /}. We point out that *If*(*<sup>A</sup>*1,...,*An*) has three possibilities: {0}, *I*<sup>∩</sup> and **cl**{*I*∩}.

In [15], it is claimed that under certain conditions, R(*f* (*<sup>A</sup>*1, ... , *An*)) = *If*(*<sup>A</sup>*1,...,*An*)= *I*∩.

 We show by examples that this is not valid. In fact, even under stronger conditions than in [15], there are still various possibilities in the relationship between R(*f* (*<sup>A</sup>*1, ... , *An*)) and *I*∩, and there are still various possibilities in the relationship between *If*(*<sup>A</sup>*1,...,*An*)and *I*∩.

The conclusions of this paper show that *If*(*<sup>A</sup>*1,...,*An*) and R(*f* (*<sup>A</sup>*1, ... , *An*)) vary in the scopes determined by *I*∩, respectively. On the other hand, even under stronger conditions than in [15], these and *I*<sup>∩</sup> are not necessarily equal, and there exist various possibilities in their relationship with *I*∩.

The results in this paper can be used in the theoretical research and practical applications of gradual numbers, gradual sets and arithmetic using the gradual numbers and gradual sets introduced by Wu [15]. We will discuss the properties of this kind of arithmetic in the future.

**Author Contributions:** Formal analysis, Q.M.; methodology, H.H.; writing—original draft preparation, Q.M. and H.H.; writing—review and editing, H.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Natural Science Foundation of Fujian Province of China (grant number 2020J01706).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the anonymous reviewers for their valuable comments and suggestions which greatly improved the presentation of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
