Research On Some Cross Problems Of Rough Sets,Fuzzy Sets And Matroids

In recent years,uncertainty and incompleteness have become the research objects in many disciplines.With the complexity and diversification of problem,there are many vague,incomplete and uncertain information in the process of establishing the model.In order to solve these problems,scholars generalized the classical set theory and proposed many effective tools to deal with uncertainty,such as rough sets,fuzzy sets,concept lattices and so on.Rough sets describe knowledge approximately by means of a pair of exact sets(i.e.upper and lower approximations).Fuzzy sets describe uncertainties by membership function.This paper intends to study some hot issues of rough set and fuzzy set,the specific contents are as follows:Firstly,matroids are the generalization of some independence in linear algebra and graph theory.They have developed into an important part of combinatorial mathematics and have been widely used in optimization theory and coding theory.In this paper,matroidal method is used to explore rough sets,and matroidal structure in topology is defined.Furthermore,according to the axiom of independent sets in matroids,establishment of proposed structure is verified,and then its related properties and propositions are given.Finally,the matroidal structure is extended to the field of rough sets,and the related properties and propositions are verified.The characteristic functions and relational matrices of the matroid structure are also given.This method improves the original rough set model and is conducive to further research on the cross-fusion of rough set and matroid theory.Secondly,both rough theory and fuzzy theory generalize the classical set theory in dealing with uncertainties,but they have different starting points and emphases.Fuzzy theory focuses on categorical relationship,while rough theory considers the indistinguishable relationship between elements.Between the two can be described by a special transformation function.In this paper,rough sets are studied by means of the method of fuzzy sets.The fuzziness of rough sets on double universes is studied concretely,and the related properties are given.At last,an example of medical diagnosis is given.Finally,there are many extended models of fuzzy sets,such as intuitionistic fuzzy sets,interval fuzzy sets,type-2 fuzzy sets and so on.Compared with fuzzy sets,pythagorean fuzzy sets take into account both the membership and non-membership relations between elements and sets,which can describe the ambiguity of the objective world more accurately,and its descriptive range is wider than that of intuitionistic fuzzy sets.In this paper,cartesian product and modal operators of pythagorean fuzzy sets are defined,and their related operations and geometric interpretation are discussed.In summary,this paper mainly studies the matroidal structure,fuzziness of rough sets and the extension theory of fuzzy sets.Firstly,the matroidal structure on rough sets is defined by combining rough sets with matroids,and its related properties and propositions are studied.Secondly,the fuzziness of rough sets is studied,and rough sets are described laterally by means of fuzzy sets and related properties.At the same time,the cartesian product of the extension theory of fuzzy sets is also discussed.These results not only enrich the theory of fuzzy sets and rough sets,but also provide new methods and ideas for related research.The conclusions drawn above have certain theoretical significance and application prospects.