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 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.
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References
Aktaş H, Çağman N (2007) Soft sets and soft groups. Inf Sci 177:2726–2735
Ali MI, Feng F, Liu XY, Min WK, Shabir M (2009) On some new operations in soft set theory. Comput Math Appl 57:1547–1553
Chen D, Tsang ECC, Yeung DS, Wang X (2005) The parametrization reduction of soft sets and its applications. Comput Math Appl 49:757–763
Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209
Feng F, Jun YB (2009) Inductive semimodules and the vector modules over them. Soft Comput 13:1113–1121
Feng F, Zhao XZ, Jun YB (2005) *-\(\mu\) -semirings and *-\(\lambda\) -semirings. Theoret Comput Sci 347:423–431
Feng F, Jun YB, Zhao XZ (2007) On *-\(\lambda\) -semirings. Inf Sci 177:5012–5023
Feng F, Jun YB, Zhao XZ (2008) Soft semirings. Comput Math Appl 56:2621–2628
Ganter B, Wille R (1999) Formal concept analysis—mathematical foundations. Springer, New York
Jun YB (2008) Soft BCK/BCI-algebras. Comput Math Appl 56:1408–1413
Jun YB, Park CH (2008) Applications of soft sets in ideal theory of BCK/BCI-algebras. Inf Sci 178:2466–2475
Kong Z, Gao LQ, Wang LF, Li S (2008) The normal parameter reduction of soft sets and its algorithm. Comput Math Appl 56:3029–3037
Maji PK, Biswas R, Roy AR (2001) Fuzzy soft sets. J Fuzzy Math 9(3):589–602
Maji PK, Roy AR, Biswas R (2002) An application of soft sets in a decision making problem. Comput Math Appl 44:1077–1083
Maji PK, Biswas R, Roy AR (2003) Soft set theory. Comput Math Appl 45:555–562
Molodtsov D (1999) Soft set theory—first results. Comput Math Appl 37:19–31
Molodtsov D (2004) The theory of soft sets. URSS Publishers, Moscow (in Russian)
Pawlak Z (1982) Rough sets. Int J Inf Comp Sci 11:341–356
Pawlak Z (1991) Rough sets—theoretical aspects of reasoning about data. Kluwer, Dordrecht
Pawlak Z, Skowron A (2007) Rudiments of rough sets. Inf Sci 177:3–27
Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155
Roy AR, Maji PK (2007) A fuzzy soft set theoretic approach to decision making problems. J Comput Appl Math 203:412–418
Slowinski R, Stefanowski J (1999) Medical information systems-problems with analysis and way of solution. In: Pal SK, Skowron A (eds) Rough fuzzy hybridization: a new trend in decision-making. Springer, Singapore, pp 301–315
Yao YY (1998) A comparative study of fuzzy sets and rough sets. Inf Sci 109:227–242
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Zou Y, Xiao Z (2008) Data analysis approaches of soft sets under incomplete information. Knowl Based Syst 21:941–945
Acknowledgments
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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Feng, F., Li, C., Davvaz, B. et al. Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput 14, 899–911 (2010). https://doi.org/10.1007/s00500-009-0465-6
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DOI: https://doi.org/10.1007/s00500-009-0465-6