Fuzzy Sets, Near Sets, and Rough Sets for Your Computational Intelligence Toolbox

  • James F. Peters
Part of the Studies in Computational Intelligence book series (SCI, volume 202)


This chapter considers how one might utilize fuzzy sets, near sets, and rough sets, taken separately or taken together in hybridizations as part of a computational intelligence toolbox. These technologies offer set theoretic approaches to solving many types of problems where the discovery of similar perceptual granules and clusters of perceptual objects is important. Perceptual information systems (or, more concisely, perceptual systems) provide stepping stones leading to nearness relations and properties of near sets. This work has been motivated by an interest in finding a solution to the problem of discovering perceptual granules that are, in some sense, near each other. Fuzzy sets result from the introduction of a membership function that generalizes the traditional characteristic function. Near set theory provides a formal basis for observation, comparison and classification of perceptual granules. Near sets result from the introduction of a description-based approach to perceptual objects and a generalization of the traditional rough set approach to granulation that is independent of the notion of the boundary of a set approximation. Near set theory has strength by virtue of the strength it gains from rough set theory, starting with extensions of the traditional indiscernibility relation. This chapter has been written to establish a context for three forms of sets that are now part of the computational intelligence umbrella. By way of introduction to near sets, this chapter considers various nearness relations that define partitions of sets of perceptual objects that are near each other. Every perceptual granule is represented by a set of perceptual objects that have their origin in the physical world. Objects that have the same appearance are considered perceptually near each other, i.e., objects with matching descriptions. Pixels, pixel windows, and segmentations of digital images are given by way of illustration of sample near sets. This chapter also briefly considers fuzzy near sets and near fuzzy sets as well as rough sets that are near sets.The main contribution of this chapter is the introduction of a formal foundation for near sets considered in the context of fuzzy sets and rough sets.


Description fuzzy sets near sets perceptual granule perceptual system rough sets 


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  1. 1.
    Grimaldi, D., Engel, M.: Evolution of the Insects. Cambridge University Press, Cambridge (2005)Google Scholar
  2. 2.
    Gupta, S., Patnaik, K.: Enhancing performance of face recognition system by using near set approach for selecting facial features. Journal of Theoretical and Applied Information Technology 4(5), 433–441 (2008), Google Scholar
  3. 3.
    Hassanien, A., Abraham, A., Peters, J., Schaefer, G.: Rough sets and near sets in medical imaging: A review. IEEE Trans. on Information Technology in Biomedicine (submitted) (2008)Google Scholar
  4. 4.
    Henry, C., Peters, J.: Image pattern recognition using approximation spaces and near sets. In: An, A., Stefanowski, J., Ramanna, S., Butz, C.J., Pedrycz, W., Wang, G. (eds.) RSFDGrC 2007. LNCS (LNAI), vol. 4482, pp. 475–482. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Henry, C., Peters, J.: Near set image segmentation quality index. In: GEOBIA 2008 Pixels, Objects, Intelligence. GEOgraphic Object Based Image Analysis for the 21st Century, pp. 284–289. University of Calgary, Alberta (2008), Google Scholar
  6. 6.
    Henry, C., Peters, J.: Perception based image classification. IEEE Transactions on Systems, Man, and Cybernetics–Part C: Applications and Reviews (submitted) (2008)Google Scholar
  7. 7.
    Iturralde-Vinent, M., MacPhee, R.: Age and paleogeographical origin of dominican amber. Science 273, 1850–1852 (1996)CrossRefGoogle Scholar
  8. 8.
    Jähne, B.: Digital Image Processing, 6th edn. Springer, Heidelberg (2005)Google Scholar
  9. 9.
    Murray, J., Bradley, H., Craigie, W., Onions, C.: The Oxford English Dictionary. Oxford University Press, Oxford (1933)Google Scholar
  10. 10.
    Orłowska, E. (ed.): Incomplete Information: Rough Set Analysis. Studies in Fuzziness and Soft Computing, vol. 13. Physica-Verlag, Heidelberg (1998)Google Scholar
  11. 11.
    Orłowska, E., Pawlak, Z.: Representation of nondeterministic information. Theoretical Computer Science 29, 27–39 (1984)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Pavel, M.: Fundamentals of Pattern Recognition, 2nd edn. Marcel Dekker, Inc., N.Y. (1993)MATHGoogle Scholar
  13. 13.
    Pawlak, Z.: Classification of objects by means of attributes. Polish Academy of Sciences 429 (1981)Google Scholar
  14. 14.
    Pawlak, Z., Peters, J.: Jak blisko. Systemy Wspomagania Decyzji I, 57 (2007)Google Scholar
  15. 15.
    Pawlak, Z., Skowron, A.: Rough sets and boolean reasoning. Information Sciences 177, 41–73 (2007)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Pawlak, Z., Skowron, A.: Rough sets: Some extensions. Information Sciences 177, 28–40 (2007)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Pawlak, Z., Skowron, A.: Rudiments of rough sets. Information Sciences 177, 3–27 (2007)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Pedrycz, W., Gomide, F.: An Introduction to Fuzzy Sets. Analysis and Design. MIT Press, Cambridge (1998)MATHGoogle Scholar
  19. 19.
    Peters, J.: Classification of objects by means of features. In: Proc. IEEE Symposium Series on Foundations of Computational Intelligence (IEEE SCCI 2007), Honolulu, Hawaii, pp. 1–8 (2007)Google Scholar
  20. 20.
    Peters, J.: Near sets. General theory about nearness of objects. Applied Mathematical Sciences 1(53), 2029–2609 (2007)Google Scholar
  21. 21.
    Peters, J.: Near sets, special theory about nearness of objects. Fundamenta Informaticae 75(1-4), 407–433 (2007)MathSciNetMATHGoogle Scholar
  22. 22.
    Peters, J.: Near sets. toward approximation space-based object recognition. In: Yao, J., Lingras, P., Wu, W.-Z., Szczuka, M.S., Cercone, N.J., Ślȩzak, D. (eds.) RSKT 2007. LNCS (LNAI), vol. 4481, pp. 22–33. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  23. 23.
    Peters, J.: Classification of perceptual objects by means of features. Int. J. of Info. Technology & Intelligent Computing 3(2), 1–35 (2008)Google Scholar
  24. 24.
    Peters, J.: Discovery of perceputally near information granules. In: Yao, J. (ed.) Novel Developments in Granular Computing: Applications of Advanced Human Reasoning and Soft Computation. Information Science Reference, Hersey, N.Y., U.S.A. (to appear) (2008)Google Scholar
  25. 25.
    Peters, J., Ramanna, S.S.: Feature selection: A near set approach. In: ECML & PKDD Workshop on Mining Complex Data, Warsaw (2007)Google Scholar
  26. 26.
    Peters, J., Shahfar, S., Ramanna, S., Szturm, T.: Biologically-inspired adaptive learning: A near set approach. In: Frontiers in the Convergence of Bioscience and Information Technologies, Korea (2007)Google Scholar
  27. 27.
    Peters, J., Skowron, A.: Zdzisław pawlak: Life and work. Transactions on Rough Sets V, 1–24 (2006)CrossRefGoogle Scholar
  28. 28.
    Peters, J., Skowron, A., Stepaniuk, J.: Nearness in approximation spaces. In: Proc. Concurrency, Specification and Programming (CS&P 2006), Humboldt Universität, pp. 435–445 (2006)Google Scholar
  29. 29.
    Peters, J., Skowron, A., Stepaniuk, J.: Nearness of objects: Extension of approximation space model. Fundamenta Informaticae 79(3-4), 497–512 (2007)MathSciNetMATHGoogle Scholar
  30. 30.
    Peters, J., Wasilewski, P.: Foundations of near sets. Information Sciences (submitted) (2008)Google Scholar
  31. 31.
    Polkowski, L.: Rough Sets. Mathematical Foundations. Springer, Heidelberg (2002)MATHGoogle Scholar
  32. 32.
    Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)MathSciNetMATHGoogle Scholar
  33. 33.
    Zadeh, L.: Fuzzy sets. Information and Control 8, 338–353 (1965)CrossRefMathSciNetMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • James F. Peters
    • 1
  1. 1.Computational Intelligence Laboratory, Department of Electrical & Computer EngineeringUniversity of ManitobaManitobaCanada

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