The Properties Of Matroidal Structure Induced By Rough Sets And Its Applications

Rough set theory plays an important role in dealing with uncertain,inexact or incomplete knowledge in information systems.It copes with the problems by a pair of certain sets,that is,the upper approximation and the lower approximation,to describe the object set.It has been successfully used in artificial intelligence,knowledge and data mining,machine learning and so on.However,it is difficult to solve the practical problems by any single theory.Matroid is a generalization of linear algebra and graph theory,which is a unitized concept of abstract correlation and abstract independent.It has become a theoretical and practical mathematical tool and has been applied to the design of algorithm,information coding,combinatorial optimization and others.Therefore,researchers solve many practical problems by proposing the combinations of rough set theory with matroid theory.At present,there are many results about the combinations of rough set theory with matroid theory,but it is apparent.Especially,the research on the properties of matroidal structure induced by rough sets is simple.Therefore,the research on the properties of matroidal structure induced by rough sets and its applications will help matroid theory to solve the problem of rough sets and promote the huge function of the matroid induced by rough sets in real life.For this purpose,we will study from the following three aspects in this paper:(1)Firstly,we define a function by the relation and prove that this function satisfies the rank function axiom of matroids.So a matroidal structure induced by relation-based rough sets can been established.In particular,we find out an approach to construct the graph corresponding to a graphic matroid and draw a conclusion that a matroidal structure induced by a relation is a graphic matroid when the relation is a reflexive relation or a symmetric and transitive relation.In addition,we build the matroidal structure induced by the covering-based rough sets and study the connection between the covering approximation space and topology space.Namely,we give the topological properties of matroidal structure induced by covering-based rough sets.(2)We define the rough matroid based on the equivalence relation and present its baseaxioms.Especially,we have a conclusion that the rough matroid corresponding to a subset of the universe is not only one.So,the properties of rough matroid based on equivalence relation are mainly investigated and the correspondence relation between a subset of the universe and a rough matroid is presented.(3)We conclude the properties of the matroidal structure based on rough sets by means of the results of the combinations of rough set theory with matroid theory and build the matroidal structure based on rough sets on network security and the spread of an epidemic.It shows the practical application by applying the combinations of rough set theory with matroid theory to the real world.