Soft Computing

, Volume 14, Issue 9, pp 899–911 | Cite as

Soft sets combined with fuzzy sets and rough sets: a tentative approach

  • Feng Feng
  • Changxing Li
  • B. Davvaz
  • M. Irfan Ali
Original Paper


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 reasoning 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.


Soft set Fuzzy set Rough set Rough fuzzy set Approximation space Approximation operator 



We are highly grateful to the anonymous referees for their helpful comments and suggestions for improving the paper. We are indebted to Dr. Brunella Gerla and Dr. Vincenzo Marra for their kindly help. This work is supported by a grant (No. 08JK432) from the Education Department of Shaanxi Province of China, and by the Shaanxi Provincial Research and Development Plan of Science and Technology under Grant No. 2008K0133.


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Feng Feng
    • 1
  • Changxing Li
    • 1
  • B. Davvaz
    • 2
  • M. Irfan Ali
    • 3
  1. 1.Department of Applied Mathematics and Applied PhysicsXi’an Institute of Posts and TelecommunicationsXi’anPeople’s Republic of China
  2. 2.Department of MathematicsYazd UniversityYazdIran
  3. 3.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan

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