الفهرس 
Fundamental notions of rough set theoryOnly 14 pages are availabe for public view
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Abstract
Preface i Preface The process of decision making uses Mathematics as a base for analyzing information to extract hidden knowledge that help in finding possible choices for decision makers. In the age of information revolution and the possibility of collecting large amount of data which in almost cases have uncertainty, new approaches for data analysis have been appeared depending on topology and set theories of uncertainty. The rough set theory was firstly proposed by pawlak (1982). It is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. Equivalence relation is one of the key notions in pawlak rough set model. The equivalence classes are the building blocks for the construction of the lower and upper approximations. However, equivalence relations are too restrictive for many applications. To address this issue, many directions have been proposed, such as tolerance relations, similarity relation, preorder relations and arbitrary binary relations. The concepts of core and reduct are two fundamental concepts of the rough sets theory in the case of attributes and knowledge [8]. The reduct is the set of attributes with minimum cardinality which represent the information system such that the accuracy is the same as all attributes [2,5]. The core is the common part of all reducts. To compute reducts and core, the discernibility matrix is used pawlak method [12]. The set of all indispensable attributes is called core. Logical rules derived from experimental data may be used to support Preface ii new reduction. We often face a question whether we can remove some data from a data table preserving its basic properties, that is, whether a table contains some superfluous data. The proposed algorithm introduces indeterminacy by removing conditional attributes in a controlled manner. The selection of attributes to be removed is made from the factors in the discerinibility function, thereby removing information needed to discern classes in the original information system. In 2000, V. Popa and T. Noiri [36] introduced the notion of minimal structure. Also, they introduced the notion of MS-open sets, MS-closed sets and characterized those sets using MS-closure and MS-interior operator respectively. T. Noiri and V. Popa [29] obtained the definitions and characterizations of separation axioms by using the concept of minimal structure. Graph theory had be introduced by Euler in 1736 which considered the general case of the Konigsberg bridge problem. For these reasons, we choose the title of thesis: (On Topological Approximations of Imprecise Concepts In Information Systems) The main objective is to use generalized Topological concepts in the process of information analysis. . The Thesis consists of five chapters: In Chapter1, we give some basic concepts to be used throughout this thesis. These concepts contains information systems, rough sets, topological spaces, minimal structure and graph theory. Preface iii Chapter 2, is based on general relation in constructing neighborhood system from the available data tables known by information system. The general relation in this work is used a generalization for all types of relations in approximations. Moreover, many forms of lower approximation and upper approximation concepts in information systems helps in improving the quality of decisions by using the resulted entropy measures. We use the neighborhood systems to discuss and obtain results in some examples of information systems. Also, we introduce a new application of these concepts in DNA recombination. Finally, we study some properties of concepts. Some of the results of Chapter 2 are published in Journal of computational and Theoretical Nano science [9]. The aim of Chapter 3, is devoted to introduce and investigate a new space which is called minimal structure approximation space (briefly, MSAS). This space based on binary relation in generalized neighborhood system on a nonempty set by using minimal structure. This space is helped to get a new classification for definability. This classification depends on the topological concepts in minimal topological space. We note that our classification is more related to real life problems, and properties of approximation in (MSAS) are investigated. Also, we define the lower and upper approximation on the space. Finally, we produce the use of separation properties in the process of approximation and definability. Some of these results of Chapter 3 are submitted for publications. The main goal of this Chapter 4 is to use the notions of minimal structure approximation space (short, MSAS) and the notion of near open sets to introduce new approximations of uncertain sets as a Preface iv mathematical tool to modify the approximations. The study and research about near open and closed sets have specific importance, it helps in the modifications of approximation space via adding new concepts and facts. The most specific modifications are the decomposition of boundary region to new subdivisions instead of one part in the original model. This decomposition introduces many alternative choices that help decision makers and takers. Some of the results in Chapter 4 are published in Journal of the Egyptian Mathematical Society[10]. The purpose of Chapter 5, is to represent information systems using graph theory and use the properties of graph to discover some hidden properties of information systems. Our representation depend on the classes of objects resulted from binary relations. List of Publications: 1. New Approximation of Uncertain Concepts Using Neighborhood Systems and Application, Journal of computational and Theoretical and Nano science. 2. Some New Types of Approximations via Minimal Structure, Journal of the Egyptian Mathematical Society. 3. Minimal Structure Approximation Space and some of its Applications, (Submitted).