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

Tutut Herawan (Universiti Malaysia Pahang, Malaysia)
DOI: 10.4018/jamc.2012070103
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Abstract

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.
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1. Introduction

The problem of imprecise knowledge has been tackled for a long time by mathematicians. Recently it became also a crucial issue for computer scientists, particularly in the area of artificial intelligence. There are many approaches to the problem of how to understand and manipulate imprecise knowledge. The most successful one is, no doubt, the fuzzy set theory proposed by Zadeh (1965). The basic tools of the theory are possibility measures. There is extensive literature on fuzzy logic with also discusses some of the problem with this theory. The basic problem of fuzzy set theory is the determination of the grade of membership of the value of possibility (Busse, 1998).

In the 1980’s, Pawlak introduced rough set theory to deal the problem of vagueness and uncertainty. Similarly to rough set theory it is not an alternative to classical set theory but it is embedded in it. Fuzzy and rough sets theories are not competitively, but complementary each other (Pawlak & Skowron, 2007). The original goal of the rough set theory is induction of approximations of concepts. The idea consists of approximation of a subset by a pair of two precise concepts called the lower approximation and upper approximation. Intuitively, the lower approximation of a set consists of all elements that surely belong to the set, whereas the upper approximation of the set constitutes of all elements that possibly belong to the set. The difference of the upper approximation and the lower approximation is a boundary region. It consists of all elements that cannot be classified uniquely to the set or its complement, by 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.

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