New Approaches of Rough Sets via Ideals

New Approaches of Rough Sets via Ideals

Ali Kandil (Helwan University, Egypt), M. M. Yakout (Ain Shams University, Egypt) and A. Zakaria (Ain Shams University, Egypt)
DOI: 10.4018/978-1-4666-9798-0.ch012


An ideal I on a nonempty set X is a subfamily of P(X) which is closed under finite unions and subsets. In this chapter, a new definition of approximation operators and rough membership functions via ideal has been introduced. The concepts of lower and upper approximations via ideals have been mentioned. These new definitions are comparing with Pawlak's, Yao's and Allam's definitions. It's therefore shown that the current definitions are more generally. Also, it's shown that the present method decreases the boundary region. In addition to these points, the topology generated via present method finer than Allam's one which is a generalization of that obtained by Yao's method. Finally, T1 topological spaces are obtained by relations and ideals which are not discrete.
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2. Preliminaries

  • Definition: (Pawlak, 1982) Let R be an equivalence relation on a nonempty set X, [x]R be the equivalence class containing x. For AX, a pair of lower and upper approximations, R(A) and 978-1-4666-9798-0.ch012.m01, are defined respectively as:R(A) = {xX: [x]RA}, (1)

    . (2)

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