{"title":"On Ralescu's cardinality of fuzzy sets","authors":"Eduard Bartl, Radim Belohlavek","doi":"10.1016/j.fss.2024.109118","DOIUrl":null,"url":null,"abstract":"<div><p>We provide a direct formula for Ralescu's scalar cardinality. Unlike the original, iterative definition, the formula reveals intuitive shortcomings of this concept of cardinality. These are apparent from examples and reflected formally in that, as we show, the concept violates one of the axioms of cardinality of fuzzy sets. In addition, we provide a relationship of this concept to Ralescu's concept of fuzzy cardinality which unveils a tight link between the two concepts and points out another counterintuitive property of the concept of scalar cardinality. We argue that the discussed concept of fuzzy cardinality represents an interesting proposition, suggest its geometric interpretation, and provide preliminary observations as a basis for future considerations.</p></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109118"},"PeriodicalIF":3.2000,"publicationDate":"2024-09-02","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/S0165011424002641","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
We provide a direct formula for Ralescu's scalar cardinality. Unlike the original, iterative definition, the formula reveals intuitive shortcomings of this concept of cardinality. These are apparent from examples and reflected formally in that, as we show, the concept violates one of the axioms of cardinality of fuzzy sets. In addition, we provide a relationship of this concept to Ralescu's concept of fuzzy cardinality which unveils a tight link between the two concepts and points out another counterintuitive property of the concept of scalar cardinality. We argue that the discussed concept of fuzzy cardinality represents an interesting proposition, suggest its geometric interpretation, and provide preliminary observations as a basis for future considerations.
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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