{"title":"Fuzzification of crisp functions using gradual elements and gradual sets","authors":"Hsien-Chung Wu","doi":"10.1016/j.fss.2025.109358","DOIUrl":null,"url":null,"abstract":"<div><div>There are two ways to fuzzify the crisp functions into fuzzy functions. One is using the extension principle. Another one is using the expression in decomposition theorem. In this paper, we present a new methodology to fuzzify the crisp functions for non-normal fuzzy sets using the concepts of gradual elements and gradual sets. The main purpose of this paper is to establish the equivalence between the fuzzy functions that are obtained using the extension principle and the gradual elements when the membership functions are assumed to be upper semi-continuous. This kind of technique is based on the idea in which the fuzzy set is formulated as consisting of gradual elements like the usual set consisting of usual elements. In this case, the operations among sets can be borrowed to set up the operations among fuzzy sets.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"509 ","pages":"Article 109358"},"PeriodicalIF":3.2000,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fuzzy Sets and Systems","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165011425000971","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
There are two ways to fuzzify the crisp functions into fuzzy functions. One is using the extension principle. Another one is using the expression in decomposition theorem. In this paper, we present a new methodology to fuzzify the crisp functions for non-normal fuzzy sets using the concepts of gradual elements and gradual sets. The main purpose of this paper is to establish the equivalence between the fuzzy functions that are obtained using the extension principle and the gradual elements when the membership functions are assumed to be upper semi-continuous. This kind of technique is based on the idea in which the fuzzy set is formulated as consisting of gradual elements like the usual set consisting of usual elements. In this case, the operations among sets can be borrowed to set up the operations among fuzzy sets.
期刊介绍：
Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies.
In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.
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