The Position of Rough Set in Soft Set: A Topological Approach

In this paper, the author presents the concept of topological space that must be used to show a relation between rough set and soft set. There are two main results presented; firstly, a construction of a quasi-discrete topology using indiscernibility (equivalence) relation in rough set theory is described. Secondly, the paper describes that a “general” topology is a special case of soft set. Hence, it is concluded that every rough set can be considered as a soft set. DOI: 10.4018/jamc.2012070103 34 International Journal of Applied Metaheuristic Computing, 3(3), 33-48, July-September 2012 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. employing available knowledge (Pawlak & Skowron, 2007). Thus any rough set, in contrast to a crisp set, has a non-empty boundary region. Motivation for rough set theory has come from the need to represent a subset of a universe in terms of equivalence classes of a partition of the universe. Rough set theory has attracted attention of more than 7000 researchers and practitioners all over the world, who contributed essentially to its development and applications including the work of Herawan and Deris (2009b, 2009c, 2009d, 2009e), Herawan et al. (2009a), Herawan, Yanto, and Deris (2010), and Yanto et al. (2010, 2011, 2012). Another general method for dealing with uncertain data was soft set theory which proposed by Molodtsov in 1999. As for standard soft set, it may be redefined as the classification of objects in two distinct classes, thus confirming that soft set can deal with a Boolean-valued information system. Molodtsov (1999) pointed out that one of the main advantages of soft set theory is that it is free from the inadequacy of the parameterization tools, unlike in the theories of fuzzy set, probability and interval mathematics. Sub-sequentially, Molodtsov successfully applied the theory of soft set in several directions, such as smoothness of function, game theory, operation research, Riemann integration, Perron integration, probability, theory of measurement and showed that fuzzy set and topological space can be seen as a special soft set. In recent years, research on soft set theory has been active, and great progress has been achieved (Awang et al., 2011; Herawan & Deris, 2009f, 2010, 2011; Herawan, Rose, & Deris, 2009; Herawan et al., 2009c, 2010b; Mamat et al., 2011; Xiuqin, Sulaiman, Hongwu, & Herawan, 2011; Xiuqin, Sulaiman, Hongwu, Zain, & Herawan, 2011). It was including the works of the using of fundamental soft set theory, soft set theory in abstract algebra and soft set theory for data analysis, particularly in decision making. For the theory of soft set, few research focus on the relation between rough and soft sets. In our previous paper, it is shown that Pawlak’s and Iwinski’s rough sets can be considered as soft set (Herawan & Deris, 2009a). A direct proof relation via descriptive method is proposed. In this paper, an enhancement of our previous work is proposed in presenting a relation between rough set and soft set. The idea of rough set-based topological space is proposed to show that a rough set can be considered as a soft set. It is shown that “standard” rough set is properly included in soft set. This confirms that soft set theory provides wider study both its theoretical and applications. The rest of this paper is organized as follows. Section 2 describes the constructive approach of rough set theory. Section 3 describes the descriptive definition and example of soft set theory. Section 4 describes the concept of general topological spaces. Finally, the main results and conclusion of this work are described in Sections 5 and 6, respectively.