Fuzzy Set Theory and Rough Set Theory

2007 ◽  
pp. 1-32
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Set Theory

scholarly journals Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 432 ◽  
Author(s):  
Vilém Novák
Keyword(s):  
Fuzzy Logic ◽  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Sets ◽  
Type Theory ◽  
Rough Set Theory ◽  
Basic Concepts ◽  
Alternative Set

In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic notions of rough set theory have already been included in AST. Using fuzzy type theory, we generalize basic concepts of rough set theory and the topological concepts of AST to become the concepts of the fuzzy set theory. We will give mostly syntactic proofs of the main properties and relations among all the considered concepts, thus showing that they are universally valid.

On Connections of Soft Set Theory with Existing Mathematics of Uncertainties: A Short Discussion for Non-Mathematicians with Respect to Soft Set Theory

2018 ◽  
Vol 14 (01) ◽  
pp. 1-9 ◽  
Author(s):  
Santanu Acharjee
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Set Theory ◽  
Mathematical Structure ◽  
Soft Set ◽  
Hybrid Structures ◽  
Hesitant Fuzzy Set ◽  
Short Discussion

This paper focuses on two very important questions: “what is the future of a hybrid mathematical structure of soft set in science and social science?” and “why should we take care to use hybrid structures of soft set?”. At present, these are the most fundamental questions; which encircle a few prominent areas of mathematics of uncertainties viz. fuzzy set theory, rough set theory, vague set theory, hesitant fuzzy set theory, IVFS theory, IT2FS theory, etc. In this paper, we review connections of soft set theory and hybrid structures in a non-technical manner; so that it may be helpful for a non-mathematician to think carefully to apply hybrid structures in his research areas. Moreover, we must express that we do not have any intention to nullify contributions of fuzzy set theory or rough set theory, etc. to mankind; but our main intention is to show that we must be careful to develop any new hybrid structure with soft set. Here, we have a short discussion on needs of artificial psychology and artificial philosophy to enrich artificial intelligence.

Granular Computing Based Data Mining in the Views of Rough Set and Fuzzy Set

2010 ◽  
pp. 148-178 ◽  
Author(s):  
Guoyin Wang ◽  
Jun Hu ◽  
Qinghua Zhang ◽  
Xianquan Liu ◽  
Jiaqing Zhou
Keyword(s):  
Data Mining ◽  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Rule Sets ◽  
Incomplete Information Systems

Granular computing (GrC) is a label of theories, methodologies, techniques, and tools that make use of granules in the process of problem solving. The philosophy of granular computing has appeared in many fields, and it is likely playing a more and more important role in data mining. Rough set theory and fuzzy set theory, as two very important paradigms of granular computing, are often used to process vague information in data mining. In this chapter, based on the opinion of data is also a format for knowledge representation, a new understanding for data mining, domain-oriented data-driven data mining (3DM), is introduced at first. Its key idea is that data mining is a process of knowledge transformation. Then, the relationship of 3DM and GrC, especially from the view of rough set and fuzzy set, is discussed. Finally, some examples are used to illustrate how to solve real problems in data mining using granular computing. Combining rough set theory and fuzzy set theory, a flexible way for processing incomplete information systems is introduced firstly. Then, the uncertainty measure of covering based rough set is studied by converting a covering into a partition using an equivalence domain relation. Thirdly, a high efficient attribute reduction algorithm is developed by translating set operation of granules into logical operation of bit strings with bitmap technology. Finally, two rule generation algorithms are introduced, and experiment results show that the rule sets generated by these two algorithms are simpler than other similar algorithms.

A HISTORICAL OVERVIEW OF FUZZY MATHEMATICS

2005 ◽  
Vol 01 (01) ◽  
pp. 1-26 ◽  
Author(s):  
ETIENNE E. KERRE ◽  
JOHN N. MORDESON
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Set Theory ◽  
Fuzzy Mathematics ◽  
Mathematical Representation ◽  
Uncertain Information ◽  
Historical Overview ◽  
The Many

In this paper, we present a historical overview of the development of fuzzy mathematics. We mainly concentrate on the evolution of the mathematical representation of fuzziness by means of fuzzy set theory. From the many remaining recently introduced models to represent imprecise and uncertain information, we briefly treat rough set theory.

Research on Extension of the Fuzzy Rough Set Theory

2012 ◽  
Vol 532-533 ◽  
pp. 1472-1476
Author(s):  
Yu Zhang
Keyword(s):  
Set Theory ◽  
Membership Function ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Incomplete Data ◽  
Rough Set Theory ◽  
Fuzzy Rough Set ◽  
Set Membership ◽  
Lower Maximum

With the probability rules and the degree of coverage of elements in the partition set, and the combination of the Fuzzy Set Theory and Rough Set Theory, a new extension of Fuzzy Rough Set theory was proposed. It is defined the maximum of Rough Set membership function, the minimum of Rough Set membership function, the average of Rough Set membership function, the upper minimum of Rough Set membership function and the lower maximum of Rough Set membership function. The properties of the extension for the Rough Set membership functions are also given. This made Fuzzy Set Theory and Rough Set Theory complemented each other and provided a new way to deal with incomplete data.

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