| This thesis discusses some relevant problems of rough set theory and rough function model. The main content includes deepness and generalization of basic theory about rough function model, discussion of rough membership functions in rough function model and their properties, some mathematical characteristics of rough Cauchy sequences in rough function model, as well as discussion of rough continuity of discrete functions, rough calculus and its applications in rough function model, and so on.Chapter 1 is a brief summary of foundational theory on rough sets. It mainly introduces the background of proposing rough sets and the development survey, research fields and current situation of rough set theory, as well as basic concepts of rough set theory.Chapter 2 deepens and generalizes rough function model based on rough set theory. Pawlak rough function model did not proposed the discrete forms in two metrics of real line and scale. The definition of rough function in it could not reflect the obvious characteristics of rough functions which are defining and valuing in integral sets. This kind of definition is not strict from mathematical point of view; While from applied point of view, rough functions in this kind of definition are not applicable for computer and rough control, etc.. In this chapter, basic concepts of Pawlak rough function model are improved. In section 2.2, the concepts of double approximation operators, which are scale upper (lower) approximation and real line upper (lower) approximation, are defined. Their properties and antithesis characteristics are analyzed. Scale bijection theorem as well as relative propositions and conclusions are proposed furthermore. In section 2.3, based on the indiscernibility relation, the new real line-scale rough function model (RL-S rough function model) is established by generalizing the double approximation operators into two-dimensional plane.Chapter 3 discusses rough intervals and rough membership functions in rough function model and their properties. In the new theoretic scheme of real line-scale rough function model (RL-S rough function model), section 3.2 defines the basic concepts in rough function model including rough numbers, rough intervals, etc.. Operating properties of rough intervals similar to rough sets are obtained. Section 3.3 discusses rough membership functions and their properties, and by which another form of equivalent definition about rough intervals are given. In section 3.4 the relationship of rough inclusion and rough equality of rough intervals are defined by two kinds of tools which are respectively the lower (upper) approximation operator in real numbers domain and rough membership functions. Their relative properties and two theorems analyzed and proved strictly indicate that the two definition forms of rough inclusion and rough equality of rough intervals are equivalent taking into account some additional information from the boundary regions of rough intervals.Chapter 4 discusses some mathematical characteristics of rough Cauchy sequences in rough function model. Rough function model, in which the indiscernibility relation is the starting point of our considerations, divides the axis of reals into equivalent classes composed of points and open intervals, i.e., makes the axis of reals discrete. There also exists the notion of sequences on discretized axis of reals. Then what is the influence of discretion of real axis on sequences, what are the same and different properties belong to the sequences on discretized real axis? These problems have not been involved yet in the theory of rough function model. To solve the above problems, section 4.2 in this chapter defines and investigates the notion of rough convergence for sequences on discretized axis of reals. A completely opposite property after discretizing axis of reals, i.e., the non-uniqueness of rough limit of rough convergences, is drawn. The logic relationships between convergence and rough convergence, along with divergence and rough divergence, are discussed respectively. Furthermore, the direct perceptible significance of relevant conclusions is illustrated. Section 4.3 defines the concepts of rough boundedness, etc., and analyzes the relationship of boundedness and rough boundedness. The necessary condition and the sufficient condition of rough convergence are proposed respectively. In section 4.4, the relation theorem and its corollary about convergent sequences and subsequences on discretized axis of reals are derived. Moreover, corresponding examples are illustrated with the indiscernibility relation as the starting point.Chapter 5 studies roughly continuous discrete functions and their properties in rough function model. For general real functions defined and valued on real line axis, a category of important functions among them are continuous functions. While in rough function model, the rough continuity is also an essential important property of discrete functions. On the rough continuity of discrete functions. Pawlak only give its definition and proposed the intermediate value theorem of roughly continuous functions without proof in the form of a necessary and sufficient condition. While other quite few relative references did not investigate this theory either. Therefore, the completeness of related theory and applications about rough function model and the rough continuity is badly needed. In fact, the definition of the rough continuity Pawlak proposed is not comparable with the continuity definition of real functions. Moreover, the necessary and sufficient condition in the intermediate value theorem Pawlak modified is not certain to be satisfying, namely, only the necessary condition of rough continuity is valid, while the sufficient is not. So we propose the following problems. What is the relationship between Pawlak rough continuity and the continuity of real functions? What operating properties does the rough continuity satisfy? Whether the property theorems of continuous real functions on closed intervals are still valid for roughly continuous discrete functions, and how to prove them? In rough function model, does there also exist pertinent concept and theory about fix-points of discrete functions? etc.. This chapter discusses the above problems. In section 5.2, the concept of the rough continuity of discrete functions is proposed which is similar to the classical e-8 continuity definition. Moreover, theε-δdefinition of the rough continuity is proved to be consistent with two other concepts of Pawlak rough continuity. In section 5.3, a series of operating properties of roughly continuous functions are discussed, such as maximization, minimization, complementarity, etc.. In section 5.4, generalizing the properties of continuous functions on closed intervals, the maximin theorem, boundedness theorem and a new intermediate value theorem of discrete functions on closed intervals are proposed. The invalidation of the sufficiency in Pawlak intermediate value theorem is illustrated by an opposite example. Furthermore, the concept of connectivity functions closely connected with the rough continuity is introduced, by which the new intermediate value theorem is proved strictly. In section 5.5, a notion of rough fix-points of discrete functions is defined. The rough fix-point theorem of roughly continuous functions is proposed and studied.Chapter 6 carries the investigation on rough calculus and its applications in rough function model. The particular study of this chapter is as follows.On the rough derivative of discrete functions in rough function model, Pawlak proposed rough derivatives definition, put forward four operations rules about rough derivatives of two discrete functions and higher order rough derivative formula, and pointed out that Fermat theorem and Rolle theorem were not valid for general discrete functions. In the scheme of rough derivative theory Pawlak gave, section 6.2 makes corresponding improvement and development for this part of context. Analyzing the functional features of roughly derived functions and higher order roughly derived functions, section 6.2.1 points out the imperfectness in the definition of higher order rough derivative Pawlak proposed. The notion of generalized rough functions is defined to improve the former definition of higher order rough derivatives. Pawlak only put forward the conclusions of four operations rules of rough derivatives and higher order rough derivative formula but did not prove them. However, other few documents only proved the four operations formulas. In this section, the higher order rough derivative formula is verified by the notions of unit mapping and identity mapping in the difference principle of numerical analysis theory. In fact, Fermat theorem and Rolle theorem are not valid even for roughly continuous discrete functions. To perfect the theoretical foundation of rough derivative applications in rough function model, the rough extremum notion is defined, and Fermat theorem and Rolle theorem of roughly smooth functions are proposed and proved in section 6.2.2. The concepts that are rough monotone and rough convexity of discrete functions are defined in section 6.2.3. By comparing with the derivative applications of real continuous function, a series of theorems are put forward which are the relation theorem of rough derivatives and rough monotone, the two sufficient conditions of rough extrema, the two relation theorems of rough derivatives and rough convexity. Moreover, some new results are achieved such as the sufficient condition of rough smoothing of discrete functions, etc..On the rough integration of discrete functions in rough function model, Pawlak did not probe thoroughly. In the research of this subject, only a definition of the rough integral for discrete functions was proposed, by which a proposition and a recurrence formula were derived. Then was the advice of detailed discussion for the reader which was the end of the relating research. In fact, the roughly integral upper limit is variable in the definition of rough integral Pawlak proposed. So the integral is not the rough integral in common sense, but a rough integral with a variable upper limit generalized by the variable upper limit integral. Section 6.3 improves the definition of rough integral Pawlak proposed. In section 6.3.1. the new notion of the rough integral is proposed which is on a constant interval. Contrasting the rough integral definition of discrete functions with the definite integral definition of real functions, the summation method for rough integration is derived which is suitable for rough integration on a relatively small interval. By comparing with the definite integral of real functions, the properties of rough integrals are analyzed in section 6.3.2. Giving the concept of the mean value of discrete functions, the mean value method for rough integration is derived. At the same time, the intermediate value theorem of rough integration is proposed and its geometric significance is analyzed, which provides dependable theoretical tool for rough integral operation, etc.. In section 6.3.3, the new definition of the roughly integral upper limit function is given. Conclusions including the existence theorem of rough primitives and the fundamental formula of rough calculus are proposed. By the representative of the discrete function, the method of computing a primitive is given, by which basic formulas for rough integration in common use are derived, and the method of rough direct integration is obtained. There is also the method of rough integration by parts for rough integrals which is similar to that of definite integrals. Thus the recurrence formula for rough integration is deduced, in which the integrand of the rough integral is in the shape of the product of a rough power function and a rough exponential function. Finally, it is pointed out that integration by substitution is not applicable for rough integral operation. The reason is analyzed and an example is illustrated.Chapter 7 summarizes main innovation opinions and conclusions of this thesis. Moreover, the future successive research work closely connected with this thesis is prospected. The fundamental thinking and directions of future research work is cleared.
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