الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
(1) Improve some of different approximations.
(2) Generalize these approximations by different ways.
(3) Make a comparison between all results and give the best approximation.
We introduce a new method for approximation of sets by
using a finite number of information systems. Some fundamental properties and characterizations are given and we obtain a comparison between this type of approximations and Pawlak’s approximations. Also, we use Pawlak’s definitions and Yao’s definitions for lower and upper approximations to introduce two new generalized definitions of lower and upper approximations by using a family of any (resp. reflexive, tolerance, dominance and equivalence) relations. We redefine some of Pawlak’s concepts by using our generalized second definition and we indicate the relationship between them. We use a special neighborhood to define lower and upper approximations of any set. Also, we make a comparison between all these definitions and we introduce these comparisons in tables. Finally, we determine the best generalized definition of them. As the boundary region is decreased by this definition more than any other definition and we introduce all these results in tables and diagrams.