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The Scientific World Journal
Volume 2014 (2014), Article ID 161607, 10 pages
http://dx.doi.org/10.1155/2014/161607
Research Article

Similarity Measure and Entropy of Fuzzy Soft Sets

1College of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
2Center for Radio Administration and Technology Development, Xihua University, Chengdu, Sichuan 610039, China

Received 22 March 2014; Accepted 29 May 2014; Published 15 June 2014

Academic Editor: Feng Feng

Copyright © 2014 Zhicai Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. Molodtsov, “Soft set theory—first results,” Computers and Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999. View at Google Scholar · View at Scopus
  2. P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers and Mathematics with Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002. View at Publisher · View at Google Scholar · View at Scopus
  3. M. M. Mushrif, S. Sengupta, and A. K. Roy, “Texture classification using a novel soft set theory based classification algorithm,” in Computer Vision—ACCV 2006, vol. 3851 of Lecture Notes in Computer Science, pp. 246–254, 2006. View at Google Scholar
  4. Z. Xiao, K. Gong, and Y. Zou, “A combined forecasting approach based on fuzzy soft sets,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 326–333, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. Zou and Z. Xiao, “Data analysis approaches of soft sets under incomplete information,” Knowledge-Based Systems, vol. 21, no. 8, pp. 941–945, 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. P. K. Maji, R. Biswas, and A. R. Roy, “Soft set theory,” Computers and Mathematics with Applications, vol. 45, no. 4-5, pp. 555–562, 2003. View at Publisher · View at Google Scholar · View at Scopus
  7. M. I. Ali, F. Feng, X. Liu, W. K. Min, and M. Shabir, “On some new operations in soft set theory,” Computers and Mathematics with Applications, vol. 57, no. 9, pp. 1547–1553, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. K. Qin and Z. Hong, “On soft equality,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1347–1355, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. P. K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft sets,” The Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589–602, 2001. View at Google Scholar
  10. P. Majumdar and S. K. Samanta, “Generalised fuzzy soft sets,” Computers and Mathematics with Applications, vol. 59, no. 4, pp. 1425–1432, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. P. K. Maji, R. Biswas, and A. R. Roy, “Intuitionistic fuzzy soft sets,” The Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 677–692, 2001. View at Google Scholar
  12. P. K. Maji, A. R. Roy, and R. Biswas, “On intuitionistic fuzzy soft sets,” The Journal of Fuzzy Mathematics, vol. 12, no. 3, pp. 669–683, 2004. View at Google Scholar
  13. X. Yang, T. Y. Lin, J. Yang, Y. Li, and D. Yu, “Combination of interval-valued fuzzy set and soft set,” Computers and Mathematics with Applications, vol. 58, no. 3, pp. 521–527, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. W. Xu, J. Ma, S. Wang, and G. Hao, “Vague soft sets and their properties,” Computers and Mathematics with Applications, vol. 59, no. 2, pp. 787–794, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. Y. Jiang, Y. Tang, Q. Chen, H. Liu, and J. Tang, “Interval-valued intuitionistic fuzzy soft sets and their properties,” Computers and Mathematics with Applications, vol. 60, no. 3, pp. 906–918, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. K. Qin, D. Meng, Z. Pei, and Y. Xu, “Combination of interval set and soft set,” International Journal of Computational Intelligence Systems, vol. 6, no. 2, pp. 370–380, 2013. View at Publisher · View at Google Scholar · View at Scopus
  17. Z. Pawlak, “Rough sets,” International Journal of Computer & Information Sciences, vol. 11, no. 5, pp. 341–356, 1982. View at Publisher · View at Google Scholar · View at Scopus
  18. Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic, Boston, Mass, USA, 1991.
  19. F. Feng, C. Li, B. Davvaz, and M. I. Ali, “Soft sets combined with fuzzy sets and rough sets: a tentative approach,” Soft Computing, vol. 14, no. 9, pp. 899–911, 2010. View at Publisher · View at Google Scholar · View at Scopus
  20. F. Feng, X. Liu, V. Leoreanu-Fotea, and Y. B. Jun, “Soft sets and soft rough sets,” Information Sciences, vol. 181, no. 6, pp. 1125–1137, 2011. View at Publisher · View at Google Scholar · View at Scopus
  21. D. Meng, X. Zhang, and K. Qin, “Soft rough fuzzy sets and soft fuzzy rough sets,” Computers and Mathematics with Applications, vol. 62, no. 12, pp. 4635–4645, 2011. View at Publisher · View at Google Scholar · View at Scopus
  22. X. Zhang, B. Zhou, and P. Li, “A general frame for intuitionistic fuzzy rough sets,” Information Sciences, vol. 216, pp. 34–49, 2012. View at Publisher · View at Google Scholar · View at Scopus
  23. H. Aktaş and N. Çaǧman, “Soft sets and soft groups,” Information Sciences, vol. 177, no. 13, pp. 2726–2735, 2007. View at Publisher · View at Google Scholar · View at Scopus
  24. F. Feng, Y. B. Jun, and X. Zhao, “Soft semirings,” Computers and Mathematics with Applications, vol. 56, no. 10, pp. 2621–2628, 2008. View at Publisher · View at Google Scholar · View at Scopus
  25. U. Acar, F. Koyuncu, and B. Tanay, “Soft sets and soft rings,” Computers and Mathematics with Applications, vol. 59, no. 11, pp. 3458–3463, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. Y. B. Jun, “Soft BCK/BCI-algebras,” Computers and Mathematics with Applications, vol. 56, no. 5, pp. 1408–1413, 2008. View at Publisher · View at Google Scholar · View at Scopus
  27. Y. B. Jun and C. H. Park, “Applications of soft sets in ideal theory of BCK/BCI-algebras,” Information Sciences, vol. 178, no. 11, pp. 2466–2475, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. J. Zhan and Y. B. Jun, “Soft B L-algebras based on fuzzy sets,” Computers and Mathematics with Applications, vol. 59, no. 6, pp. 2037–2046, 2010. View at Publisher · View at Google Scholar · View at Scopus
  29. W. Bandler and L. Kohout, “Fuzzy power sets and fuzzy implication operators,” Fuzzy Sets and Systems, vol. 4, no. 1, pp. 13–30, 1980. View at Google Scholar · View at Scopus
  30. J.-L. Fan and Y.-L. Ma, “Some new fuzzy entropy formulas,” Fuzzy Sets and Systems, vol. 128, no. 2, pp. 277–284, 2002. View at Publisher · View at Google Scholar · View at Scopus
  31. B. Kosko, “Fuzzy entropy and conditioning,” Information Sciences, vol. 40, no. 2, pp. 165–174, 1986. View at Google Scholar · View at Scopus
  32. Y. Li, K. Qin, and X. He, “Some new approaches to constructing similarity measures,” Fuzzy Sets and Systems, vol. 234, pp. 46–60, 2014. View at Publisher · View at Google Scholar · View at Scopus
  33. L. Xuecheng, “Entropy, distance measure and similarity measure of fuzzy sets and their relations,” Fuzzy Sets and Systems, vol. 52, no. 3, pp. 305–318, 1992. View at Google Scholar · View at Scopus
  34. A. de Luca and S. Termini, “A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory,” Information and Control, vol. 20, no. 4, pp. 301–312, 1972. View at Google Scholar · View at Scopus
  35. J. Recasens, Indistinguishability Operators-Modelling Fuzzy Equalities and Fuzzy Equivalence Relations, Springer, Heidelberg, Germany, 2010.
  36. D. Sinha and E. R. Dougherty, “Fuzzification of set inclusion: theory and applications,” Fuzzy Sets and Systems, vol. 55, no. 1, pp. 15–42, 1993. View at Google Scholar · View at Scopus
  37. X. Z. Wang, D. Ruan, and E. E. Kerre, Mathematics of Fuzziness-Basic Issues, Springer, Heidelberg, Germany, 2009.
  38. C.-P. Wei, P. Wang, and Y.-Z. Zhang, “Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications,” Information Sciences, vol. 181, no. 19, pp. 4273–4286, 2011. View at Publisher · View at Google Scholar · View at Scopus
  39. L. A. Zadeh, “Fuzzy sets and systems,” in Proceedings of the Symposium on Systems Theory, pp. 29–37, Polytechnic Institute of Brooklyn, Brooklyn, NY, USA, 1965.
  40. W. Zeng and H. Li, “Inclusion measures, similarity measures, and the fuzziness of fuzzy sets and their relations,” International Journal of Intelligent Systems, vol. 21, no. 6, pp. 639–653, 2006. View at Publisher · View at Google Scholar · View at Scopus
  41. P. Majumdar and S. K. Samanta, “Similarity measure of soft sets,” New Mathematics and Natural Computation, vol. 4, no. 1, pp. 1–12, 2008. View at Google Scholar
  42. P. Majumdar and S. K. Samanta, “On similarity measures of fuzzy soft sets,” International Journal of Advances in Soft Computing and Its Applications, vol. 3, no. 2, pp. 1–8, 2011. View at Google Scholar · View at Scopus
  43. A. Kharal, “Distance and similarity measures for soft sets,” New Mathematics and Natural Computation, vol. 6, no. 3, pp. 321–334, 2010. View at Publisher · View at Google Scholar
  44. Y. Jiang, Y. Tang, H. Liu, and Z. Chen, “Entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets,” Information Sciences, vol. 240, pp. 95–114, 2013. View at Publisher · View at Google Scholar · View at Scopus
  45. C. Wang and A. Qu, “Entropy, similarity measure and distance measure of vague soft sets and their relations,” Information Sciences, vol. 244, pp. 92–106, 2013. View at Publisher · View at Google Scholar · View at Scopus
  46. W. K. Min, “Similarity in soft set theory,” Applied Mathematics Letters, vol. 25, no. 3, pp. 310–314, 2012. View at Publisher · View at Google Scholar · View at Scopus
  47. L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965. View at Google Scholar · View at Scopus
  48. P. Z. Wang, Fuzzy Sets and Its Applications, Shanghai Science and Technology Press, Shanghai, China, 1983 (Chinese).
  49. J. C. Fodor and M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic, Dordrecht, Netherlands, 1994.
  50. A. Kolesarova, “Limit properties of quasi-arithmetic means,” Fuzzy Sets and Systems, vol. 124, no. 1, pp. 65–71, 2001. View at Publisher · View at Google Scholar · View at Scopus
  51. E. P. Klement, R. Mesiar, and E. Pap, Triangular Norms, Kluwer Academic, Dordrecht, Netherlands, 2000.
  52. H. Bustince, P. Burillo, and F. Soria, “Automorphisms, negations and implication operators,” Fuzzy Sets and Systems, vol. 134, no. 2, pp. 209–229, 2003. View at Publisher · View at Google Scholar · View at Scopus