The Mathematical Foundation Study Of Rough Sets And The Study Of Two General Rough Sets Models

In this paper, the mathematical foundation study of rough sets and the study of two general rough sets models are mainly researched.In chapter 1, the connection of rough sets and topology is researched. At first the approximate operators of rough sets are modified, so the new operators of union, intersection and complement and approximate power sets are presented. This has enriched rough sets theory from the operator-oriented and set-oriented views. Then by combining topology, rough topology and approximate topologies are defined and researched. Knowledge universe and knowledge topology are presented and researched by parting universe of knowledge. In the system of universe, knowledge and topology, set neighborhood and approximate neighborhood, the deep notions, are presented. The combination of rough sets and topology is researched, and the deep connection of them is achieved. At last, rough sets and topology are unified by covering space.In chapter 2, the connection and combination of rough sets and other mathematical subjects, such as fuzzy sets, measure, integral, lattice, group, are researched. In section 1, the connection of rough sets and fuzzy sets is researched, and the relationship function is defined to unify them. In section 2, based on knowledge class open intervals are constructed, then a kind of specific Lebesgue measures, called knowledge measure, is defined and researched. In section 3, inknowledge base of rough sets, knowledge is reinforced to class of measure, and knowledge integral sum and knowledge integral are defined and researched. That the entropy of knowledge is a kind of knowledge integral sums in nature is proved, and by knowledge integral sum and knowledge integral the entropy of knowledge is researched. In section 4, knowledge sets, and knowledge equivalent relation sets and knowledge class sets are defined respectively on knowledge, equivalent relation and class parts. The connection of union-intersection lattice and fine lattice of them is researched, and the combination of rough sets and lattice is achieved. In section 5, a system of notions are redefined in rough groups theory, and in the traditional and the redefined rough group theory systems, the properties of roughness of group based on subgroup are researched.In chapter 3, the relationship, combination and unification of graded rough sets and variable rough sets are researched. The more general graded approximations and variable precision approximations are defined in approximate space. The relationship between them is researched, and the important formula of conversion is achieved. The union, intersection, complement and product approximations of grade and precision are defined. Their constructions and some basic properties, such as the power actions of approximation operators, are researched. Graded rough set model and variable precision rough set model are unified by the combination models, so their old and new properties are achieved from the new model, the combination and unification model.In chapter 4, based on rough membership value a vital algorithm of getting rules is presented, and the practical example shows that this algorithm has a content result. This has showed an aspect of the practice of the rough sets theory.At last, the future of rough sets theory is prospected. The conclusion is reached and some question with thinking value is presented.