الفهرس 
Chapter 1 PreliminariesOnly 14 pages are availabe for public view
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Abstract
The rough sets, published in 1982, is a famous study by Pawlak that is often cited as the beginning of the rough set theory. Many mathemati- cians, logicians, and computer scientists have developed an interest in rough set theory and have conducted extensive research on it as well as applications (see [9], [18], [19]). This is discussed in a variety of domains, including machine learning [7], data mining [8], decision sup- port and analysis [20, 26, 28], process control [27], and expert systems [37]. A topology of study rough sets is an additional intriguing orien- tation. Rough-set conceptions and topological concepts exhibit com- parable characteristics, as reported by Skowron [30] and Wiweger [33], suggesting that topological concepts may eventually replace rough-set concepts. This inspired other academics to develop some topological structures and utilize them to investigate loosely defined concepts and features.
Rough set theory is a mathematical technique for deriving knowl- edge from uncertain and imperfect data-based information (see [11], [22], [24]). The idea presupposes that we initially possess the knowl- edge or information necessary to classify all the universe’s items into distinct groups. When two items share exactly the same information, we refer to them as being indistinguishable (similar), meaning that we are unable to tell them apart based on our current knowledge. The theory of rough sets can be used to identify dependencies between data, assess the significance of attributes, identify patterns in the data,
SUMMARY
learn common decision-making principles, eliminate all redundant ob- jects and attributes, and look for the smallest subset of attributes in order to achieve satisfying classification. For further reading in rough sets check [16], [23], [24], [31],[32], [41].
This Thesis divided as follows:
In Chapter 1, the reader will find results in relation, rough sets, rough membership function, and multisets.
In Chapter 2, the concept of a rough membership function has been put forth in the context of multisets in this chapter. So, using the rough membership multiset functions, we introduced the lower and upper multiset approximations. The application for selecting the best chemical suppliers has finally been presented.
The results of this chapter has been accepted for publication in
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In Chapter 3, Two methods of rough multisets on two universes are introduced. The first structure is built on a binary relation, multi- valued mapping, and two universes. The second one is based on two universes, inverse serial relation, and a cover for one of the two uni- verses. We construct a topology for the second structure and provide various examples to help explain each strategy.
The results of this chapter has been submitted for publication.