ORIGINAL PAPER Soft sets combined with fuzzy sets and rough sets: a tentative approach Feng Feng Æ Changxing Li Æ B. Davvaz Æ M. Irfan Ali Published online: 27 June 2009 Ó Springer-Verlag 2009 Abstract Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for rea- soning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft–rough fuzzy sets, which extends Dubois and Prade’s rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory. Keywords Soft set  Fuzzy set  Rough set  Rough fuzzy set  Approximation space  Approximation operator 1 Introduction In some sense almost all concepts we are meeting in everyday life are vague rather than precise. On the con- trary, it is interesting to see that classical mathematics requires that all mathematical notions must be exact, otherwise precise reasoning would be impossible (Pawlak and Skowron 2007). This gap between the real word full of vagueness and the traditional mathematics purely con- cerning precise concepts becomes smaller in recent years. In fact, philosophers and recently scientists as well as engineers are showing increasing interests in vague con- cepts, due to the fact that many practical problems emerging within fields such as economics, ecology, engi- neering, environmental science, social science, and medi- cal science require us to deal with the complexity of data containing uncertainties. The nature of the vagueness arising in these fields can be very different. Among many mathematical theories designed for modelling various types of vague concepts, fuzzy and rough sets have received much attention and been actively studied by a number of researchers worldwide. While some authors argue that one theory is more general then the other, it is accepted by majority that these two theories are closely related, but distinct in essence because they model different types of uncertainties. In general, a fuzzy set may be viewed as a class with unsharp boundaries, whereas a rough set is a coarsely described crisp set (Yao 1998). Over the years, the theories of fuzzy sets and rough sets have become much closer to each other for practical needs to use both of these two theories complementarily F. Feng (&)  C. Li Department of Applied Mathematics and Applied Physics, Xi’an Institute of Posts and Telecommunications, 710061 Xi’an, People’s Republic of China e-mail: fengnix@hotmail.com B. Davvaz Department of Mathematics, Yazd University, Yazd, Iran e-mail: davvaz@yazduni.ac.ir M. I. Ali Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan e-mail: mirfanali13@yahoo.com 123 Soft Comput (2010) 14:899–911 DOI 10.1007/s00500-009-0465-6