Yet Another Kind of Rough Sets Induced by Coverings

  • Ryszard JanickiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10313)


A new model of rough sets induced by coverings is proposed. In this new model, the elementary sets are defined as set components generated by a given covering of universe. The new model is compared with two other existing models of rough sets induced by covering and with a standard rough sets where elementary sets are defined by a given equivalence relation. The concept of optimal approximation is also introduced and analyzed for all models discussed in the paper. It is shown that, for a given covering of a universe, our model provides better approximations than the other ones.


Rough sets induced by coverings Set components Lower, upper and optimal approximation 



The authors gratefully acknowledge four anonymous referees, whose comments significantly contributed to the final version of this paper.

This research was partially supported by a Discovery NSERC grant of Canada.


  1. 1.
    Bogobowicz, A.D., Janicki, R.: On approximation of relations by generalized closures and generalized kernels. In: Flores, V., Gomide, F., Janusz, A., Meneses, C., Miao, D., Peters, G., Ślęzak, D., Wang, G., Weber, R., Yao, Y. (eds.) IJCRS 2016. LNCS (LNAI), vol. 9920, pp. 120–130. Springer, Cham (2016). doi: 10.1007/978-3-319-47160-0_11CrossRefGoogle Scholar
  2. 2.
    Bonikowski, Z., Bryniarski, E., Wybraniec, U.: Extensions and intentions in the rough set theory. Inf. Sci. 107, 149–167 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bretcher, O.: Linear Algebra with Applications. Prentice Hall, Englewood Cliffs (1995)Google Scholar
  4. 4.
    D’eer, L., Cornelis, C., Yao, Y.Y.: A semantically sound approach to Pawlak rough sets and covering-based rough sets. Int. J. Approx. Reason. 78, 62–72 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Deza, M.M., Deza, E.: Encyclopedia of Distances. Springer, Berlin (2012)zbMATHGoogle Scholar
  6. 6.
    Huth, M., Ryan, M.: Logic in Computer Science. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  7. 7.
    Jaccard, P.: Étude comparative de la distribution florale dans une portion des Alpes et des Jura. Bulletin de la Société Vaudoise des Sciences Naturalles 37, 547–549 (1901)Google Scholar
  8. 8.
    Janicki, R.: Approximations of arbitrary binary relations by partial orders. Classical and rough set models. Trans. Rough Sets 13, 17–38 (2011)zbMATHGoogle Scholar
  9. 9.
    Janicki, R.: Property-driven rough sets approximations of relations. In: [20], pp. 333–357Google Scholar
  10. 10.
    Janicki, R., Lenarčič, A.: Optimal approximations with rough sets. In: Lingras, P., Wolski, M., Cornelis, C., Mitra, S., Wasilewski, P. (eds.) RSKT 2013. LNCS, vol. 8171, pp. 87–98. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-41299-8_9CrossRefGoogle Scholar
  11. 11.
    Janicki, R., Lenarčič, A.: Optimal approximations with rough sets and similarities in measure spaces. Optimal approximations with rough sets. In: Lingras, P., Wolski, M., Cornelis, C., Mitra, S., Wasilewski, P. (eds.) Proceedings of Rough Sets and Knowledge Technology, RSKT 2013. Int. J. Approx. Reason. 71, 1–14 (2016)Google Scholar
  12. 12.
    Kuratowski, K., Mostowski, A.: Set Theory. North-Holland, Amsterdam (1967)zbMATHGoogle Scholar
  13. 13.
    Liu, G., Sai, Y.: A comparison of two types of rough sets induced by coverings. Int. J. Approx. Reason. 50, 521–528 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Marczewski, E., Steinhaus, H.: On a certain distance of sets and corresponding distance of functions. Colloq. Math. 4, 319–327 (1958)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pawlak, Z.: Rought sets. Int. J. Comput. Inform. Sci. 34, 557–590 (1982)Google Scholar
  16. 16.
    Pawlak, Z.: Rough Sets. Kluwer, Dordrecht (1991)CrossRefGoogle Scholar
  17. 17.
    Pomykała, J.A.: Approximation operations in approximation space. Bull. Acad. Pol. Sci. 35, 653–662 (1987)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Saquer, J., Deogun, J.S.: Concept approximations based on rough sets and similarity measures. Int. J. Appl. Math. Comput. Sci. 11(3), 655–674 (2001)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Skowron, A., Stepaniuk, J.: Tolarence approximation spaces. Fundam. Inform. 27, 245–253 (1996)zbMATHGoogle Scholar
  20. 20.
    Skowron, A., Suraj, Z. (eds.): Rough Sets and Intelligent Systems. Intelligent Systems Reference Library, vol. 42. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  21. 21.
    Słowiński, R., Vanderpooten, D.: A generalized definition of rough approximations based on similarity. IEEE Trans. Knowl. Data Eng. 12(2), 331–336 (2000)CrossRefGoogle Scholar
  22. 22.
    Xu, W.H., Zhang, W.X.: Measuring roughness of generalized rough sets induced by a covering. Fuzzy Sets Syst. 158, 2443–2455 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yao, Y.Y., Wang, T.: On rough relations: an alternative formulation. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) RSFDGrC 1999. LNCS, vol. 1711, pp. 82–90. Springer, Heidelberg (1999). doi: 10.1007/978-3-540-48061-7_12CrossRefGoogle Scholar
  24. 24.
    Yao, Y.Y., Yao, B.X.: Covering based rough set approximations. Inf. Sci. 200, 91–107 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefGoogle Scholar
  26. 26.
    Zhu, W.: Topological approaches to covering rough sets. Inf. Sci. 177(6), 1499–1508 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Żakowski, W.: Approximations in the space \((U, \Pi )\). Demonstr. Math. 16, 761–769 (1983)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada

Personalised recommendations