Abstract and Applied Analysis

New Rough Set Approximation Spaces

H. M. Abu-Donia

Abstract

Rough set theory was introduced by Pawlak in 1982 to handle imprecision, vagueness, and uncertainty in data analysis. Our aim is to generalize rough set theory by introducing concepts of ${\bigwedge }_{\beta }$-lower and ${\bigwedge }_{\beta }$-upper approximations which depends on the concept of ${\bigwedge }_{\beta }$-sets. Also, we study some of their basic properties.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 189208, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511776

Digital Object Identifier
doi:10.1155/2013/189208

Mathematical Reviews number (MathSciNet)
MR3035307

Zentralblatt MATH identifier
1272.68386

Citation

Abu-Donia, H. M. New Rough Set Approximation Spaces. Abstr. Appl. Anal. 2013 (2013), Article ID 189208, 7 pages. doi:10.1155/2013/189208. https://projecteuclid.org/euclid.aaa/1393511776

References

• Z. Pawlak, “Rough sets,” International Journal of Computer and Information Sciences, vol. 11, no. 5, pp. 341–356, 1982.
• Z. Pawlak, Rough Sets, Theoretical Aspects of Reasoning about Data, Kluwer Academic, Boston, Mass, USA, 1991.
• L. Polkowski and A. Skowron, Rough Sets in Knowledge Discovery. 2. Applications, vol. 19 of Studies in Fuzziness and Soft Computing, Physical, Heidelberg, Germany, 1998.
• R. Slowinski and D. Vanderpooten, “A generalized definition of rough approximations based on similarity,” IEEE Transactions on Knowledge and Data Engineering, vol. 12, no. 2, pp. 331–336, 2000.
• Z. Pawlak and A. Skowron, “Rough sets: some extensions,” Information Sciences, vol. 177, no. 1, pp. 28–40, 2007.
• Z. Pawlak and A. Skowron, “Rudiments of rough sets,” Information Sciences, vol. 177, no. 1, pp. 3–27, 2007.
• M. Novotný and Z. Pawlak, “On rough equalities,” Bulletin of the Polish Academy of Sciences. Mathematics, vol. 33, no. 1-2, pp. 99–104, 1985.
• P. Pattaraintakorn and N. Cercone, “A foundation of rough sets theoretical and computational hybrid intelligent system for survival analysis,” Computers & Mathematics with Applications, vol. 56, no. 7, pp. 1699–1708, 2008.
• Y. Y. Yao, “Constructive and algebraic methods of the theory of rough sets,” Information Sciences, vol. 109, no. 1–4, pp. 21–47, 1998.
• J. L. Kelley, General Topology, D. Van Nostrand Company, London, UK, 1955.
• A. Wiweger, “On topological rough sets,” Bulletin of the Polish Academy of Sciences. Mathematics, vol. 37, no. 1-6, pp. 89–93, 1989.
• Y. Y. Yao, “Relational interpretations of neighborhood operators and rough set approximation operators,” Information Sciences, vol. 111, no. 1–4, pp. 239–259, 1998.
• D. Boixader, J. Jacas, and J. Recasens, “Upper and lower approximations of fuzzy sets,” International Journal of General Systems, vol. 29, no. 4, pp. 555–568, 2000.
• W.-Z. Wu and W.-X. Zhang, “Neighborhood operator systems and approximations,” Information Sciences, vol. 144, no. 1–4, pp. 201–217, 2002.
• Y. Yang and R. I. John, “Generalisation of roughness bounds in rough set operations,” International Journal of Approximate Reasoning, vol. 48, no. 3, pp. 868–878, 2008.
• Y. Y. Yao, “Two views of the theory of rough sets in finite universes,” International Journal of Approximate Reasoning, vol. 15, no. 4, pp. 291–317, 1996.
• H. M. Abu-Donia, “Comparison between different kinds of approximations by using a family of binary relations,” Knowledge-Based Systems, vol. 21, no. 8, pp. 911–919, 2008.
• H. M. Abu-Donia, A. A. Nasef, and E. A. Marai, “Finite information systems,” Applied Mathematics & Information Sciences, vol. 1, no. 1, pp. 13–21, 2007.
• V. L. Fotea, “The lower and upper approximations in a hypergroup,” Information Sciences, vol. 178, no. 18, pp. 3605–3615, 2008.
• S. Greco, B. Matarazzo, and R. Slowinski, “Rough approximation by dominance relations,” International Journal of Intelligent Systems, vol. 17, no. 2, pp. 153–171, 2002.
• F. Min, Q. Liu, and C. Fang, “Rough sets approach to symbolic value partition,” International Journal of Approximate Reasoning, vol. 49, no. 3, pp. 689–700, 2008.
• A. Mousavi and P. Jabedar-Maralani, “Relative sets and rough sets,” International Journal of Applied Mathematics and Computer Science, vol. 11, no. 3, pp. 637–653, 2001.
• Y. Y. Yao, “Generalized rough set models,” in Rough sets in Knowledge Discovery, vol. 18, pp. 286–318, Physica, Heidelberg, Germany, 1998.
• Y.Y. Yao, “On generalized rough set theory,” in Proceedings of the 9th International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing (RSFDGrC '03), vol. 2639 of Lecture Notes in Artificial Intelligence, pp. 44–51, 2003.
• Y. Yao and Y. Chen, “Subsystem based generalizations of rough set approximations,” ISMIS, vol. 3488, pp. 210–218, 2005.
• M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud, “$\beta$-open sets and $\beta$-continuous mapping,” Bulletin of the Faculty of Science. Assiut University, vol. 12, no. 1, pp. 77–90, 1983.
• H. Maki, “Generalized $\Lambda$-sets and the associated clouser operator,” The special issue in commeration of prof. Kazusada Ikeda's retirement, pp. 139–146, 1986.
• T. Noiri and E. Hatir, “${\Lambda }_{sp}$-sets and some weak separation axioms,” Acta Mathematica Hungarica, vol. 103, no. 3, pp. 225–232, 2004.
• H. M. Abu-Donia and A. S. Salama, “$\beta$-approximation spaces,” Journal of Hybrid computing Research, vol. 1, no. 2, 2008.
• J. A. Pomykała, “Approximation operations in approximation space,” Bulletin of the Polish Academy of Sciences. Mathematics, vol. 35, no. 9-10, pp. 653–662, 1987.
• Y. Y. Yao and T. Y. Lin, “Generalization of rough sets using modal logic,” Intelligent Automation and Soft Computing, vol. 2, pp. 103–120, 1996.
• Y. Y. Yao, S. K. M. Wong, and T. Y. Lin, “A review of rough set models,” in Rough Sets and Data Mining: Analysis for Imprecise Data, T. Y. Lin and N. Cercone, Eds., pp. 47–75, Kluwer Academic Publishers, Boston, Mass, USA, 1997.
• U. Wybraniec-Skardowska, “On a generalization of approximation space,” Bulletin of the Polish Academy of Sciences. Mathematics, vol. 37, no. 1–6, pp. 51–62, 1989.
• Y. Y. Yao, “Information granulation and rough set approximation,” International Journal of Intelligent Systems, vol. 16, pp. 87–104, 2001.
• W. Żakowski, “Approximations in the space $(U,\Pi )$,” Demonstratio Mathematica, vol. 16, no. 3, pp. 761–769, 1983.
• Y. Y. Yao, “On generalizing Pawlak approximation operators,” in Proceedings of the 1st International Conference on Rough Sets and Current Trends in Computing (RSCTC' 98), vol. 1424 of Lecture Notes in Computer Science, pp. 298–307, 1998.
• Y. Y. Yao and T. Wang, “On rough relations: an alternative formulation,” in Proceedings of the 7th International Workshop on Rough Sets, Fuzzy Sets, Data Mining, and Granular-Soft Computing, vol. 1711 of Lecture Notes in Artificial Intelligence, pp. 82–90, 1999.
• O. Njastad, “On some classes of nearly open sets,” Pacific Journal of Mathematics, vol. 15, pp. 961–970, 1965.
• A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb, “On pre-continuous and week pre-continuous mappings,” Proceedings of the Mathematical and Physical Society of Egypt, vol. 53, pp. 47–53, 1982.
• M. Novotný and Z. Pawlak, “Characterization of rough top equalities and rough bottom equalities,” Bulletin of the Polish Academy of Sciences. Mathematics, vol. 33, no. 1-2, pp. 91–97, 1985.