ISSN 2456-0235

International Journal of Modern Science and Technology

INDEXED IN 

​​​​​International Journal of Modern Science and Technology, Vol. 2, No. 2, 2017, Pages 41-49.

 

Historical Development on Soft Set Theory

R. Karpagam, V. S. Selvi
Department of Mathematics, Theivanai Ammal College for Women (Autonomous)Villupuram, 605 602, Tamil Nadu, India
*Corresponding author’s e-mail: karpagam12dgl@gmail.com

Abstract
Molodtsov originated the idea of soft sets, some research on soft sets has been done in the literature. The concept of soft set is fundamentally important in almost every scientific field. Soft set theory is a new mathematical tool for dealing with uncertainties and is a set associated with parameters and has been applied in several directions. In particular, the work demonstrates that soft set theory can be applied to problems that contain uncertainties especially in decision making problems. These applications explain the voluminous work in this field within a short period of time. We emphasize that soft set has enough developed basic supporting structures through which various algebraic structures in theoretical point of view could be developed. We define the notion of soft sets, and the study that are interesting and valuable in the theory of soft sets, which emphasis on a series of applications especially in decision making problems. In this paper, studying the concept of soft set theory, types of soft set development and review its existing literature.

​​Keywords: BCI-algebra; BCK-Algebra; FCM - Fuzzy cognitive map; Fuzzy Set; Fuzzy Soft Set; Soft Rough Set; Soft Semi Rings; Soft Set; Uncertainty.

References

  1. D. Molodtsov.  Soft Set Theory-First Results. Computers and Mathematics with Applications. 1999;37:19-31
  2. L. A. Zadeh. Fuzzy Set. Information and Control. 1965;8:338-353.
  3. H. J. Zimmerman. Fuzzy Set Theory and its Applications. Kluwer Academic Publishers, Boston, MA. 1996.
  4. K. Atanassov. Intuitionistic Fuzzy Sets. Fuzzy Sets and Systems.1986;20:87-96
  5. Z. Pawlak. Rough Sets. International Journal of Computer and Information Science. 1982;11:341-356.
  6. P. K. Maji, R. Biswas, A.R. Roy. Soft Set Theory. Computer and Mathematics with Application. 2003;45:555-562.
  7. D. Pei, D. Miao. From Soft Sets to Information Systems. Proceedings of the IEEE International Conference on Granular Computing. 2009;2:617-621.
  8. M. Irfan Ali, Feng Feng, Xiaoyan Liub, Won Keun Min, M. Shabir. On Some New Operations in Soft Set Theory. Computers and Mathematics with Applications. 2009;57:1547-1553.
  9. Xibei Yang, Dongjun Yu, Jingyu Yang, Chen Wu. Generalization of Soft Set Theory: From Crisp to Fuzzy Case. Proceedings in Fuzzy Information and Engineering ICFIE. ASC. 40,2007;345-354.
  10. Kharal, B. Ahmad. Mappings on Soft Classes. New Mathematics and Natural Computation. 2011;7(3):471-481.
  11. P. Majumdar, S. K. Samanta. On Similarity Measures of Fuzzy Soft Sets. International Journal Advance Soft Computing Application, 2011;3(2):1-8.
  12. P. Majumdar, S. K. Samanta. Generalised Fuzzy Soft Sets. Computers & Mathematics with Applications. 2010;59:1425-1432.
  13. P. Majumdar, S. K. Samanta. On Soft Mappings. Computers & Mathematics with Applications. 2010;60:2666-2674.
  14. M. Pal, S. Mondal. Soft Matrices. African Journal of Mathematics and Computer Science Research. 2011;4(13):379-388.
  15. A. R. Roy, P. K. Maji. A fuzzy soft set theoretic approach to decision making problem. Journal of Computational and Applied Mathematics 2007;3:412-418.
  16. Saakshi saraf. The SIJ Transaction on computer science Engineering & its Application (CSEA). 2013;1(3).
  17. Z. Pawlak. Hard Set and Soft Sets. ICS Research Report, Institute of Computer Science, Poland, 1994.
  18. T. Y. Lin. A Set Theory for Soft Computing a Unified View of Fuzzy Sets via Neighborhoods. Proceedings of the IEEE International Conference on Fuzzy Systems. 1996;2:1140-1146.
  19. P. K. Maji, R. Biswas, A. R. Roy. Fuzzy Soft Sets. Journal of Fuzzy Mathematics. 2009;9(3):589-602.
  20. Zhi Xiao, Ling chen, Bo Zhong, Shijie Ye. .Recognition for Soft Information based on the Theory of Soft Sets. Proceedings of ICSSSM '05, 2005;1104-1106.
  21. Milind M. Mushrif, S. Sengupta, A. K. Ray. Texture Classification using a Novel Soft Set Theory based Classification Algorithm. LNCS, 2006;3851:246-254.
  22. H. Aktas, N. Cagman. Soft Sets and Soft Groups. Information Sciences. 2007; 1: 2726–2735.
  23. D.V. Kovkov, V. M. Kolbanov, D. A. Molodtso. Soft Sets Theory-based Optimization. Journal of Computer and Systems Sciences. 2007;46(6):872-880.
  24. Feng Feng, Young Bae Jun, Xianzhong Zhao. Soft Semirings. Computers & Mathematics with Applications. 2008; 56:2621-2628.
  25. Young  Bae jun, Kyoung Ja Lee, Chul Hwan Park. Soft Set Theory Applied to Commutative Ideals in BCK-Algebras. Journal Application Mathematics and Informatics. 2008;26(34):707-720.
  26. Zhi Kong, Liqun Gao, Lifu Wang, Steven Li. The Normal Parameter Reduction of Soft Sets and its Algorithm. Computers and Mathematics with Applications. 2008;56:3029-3037.
  27. Qiu-Mei Sun, Zi-Long Zhang, Jing Liu. Soft Set and Soft Modules. Proceedings of Rough Sets and Knowledge Technology, Third International Conference RSKT, 2008; 403–409.
  28. Bing-xue Yao, Jin-Ling Liu, Rui-xia Yan. Fuzzy Soft Set and Soft Fuzzy Set. Proceedings of the IEEE Fourth International Conference on Natural Computation. 2008.
  29. Y. Zou, Z. Xiao. Data Analysis Approaches of Soft Sets under Incomplete Information. Knowledge based Systems. 2008;21:941-945.
  30. Tutut Herawan, Iwan Tri Riyadi Yanto, Mustafa Mat Deris. SMARViz: Soft Maximal Association Rules Visualization, IVIC, Springer-Verlag Berlin Heidelberg, 2009; 5857:664-674.
  31. Young Bae Jun, Kyoung Ja Lee, Jianming Zhan, Soft P-Ideals of Soft BCI-Algebra. Computers and Mathematics with Applications. 2009;58(10):2060-2068.
  32. Y.B. Jun & C.H. Park. Application of Soft Sets in Hilbert Algebras. Iranian Journal of Fuzzy Systems. 2009;6(2):75-86.
  33. Xibei Yang, Tsau Young Lin, Jingyu Yang, Yan Li, Dongjun Yua. Combination of Interval-Valued Fuzzy Set and Soft Set. Computers and Mathematics with Applications. 2009;58(3):521-527.
  34. Ummahan Acar, Fatih Koyuncu, Bekir Tanay. Soft Sets and Soft Rings. Computers and Mathematics with Applications. 2009; 59:3458-3463.
  35. K. V. Babitha, J. J. Sunil. Soft Set Relations and Functions.  Computers and Mathematics with Applications. 2010;60(7):1840-1849.
  36. N. Cagman, S. Enginoglu. Soft Matrix Theory and its Decision Making. Computers and Mathematics with Applications. 2010;59:3308-3314.
  37. N. Cagman. S. Enginoglu, Soft Set Theory and Uni-Int Decision Making, European Journal of Operational Research. 2010; 207:848-855.
  38. Feng Feng, Changxing Li, B. Davvaz, M. Irfan Ali.  Soft Sets Combined with Fuzzy Sets and Rough Sets: A Tentative Approach. Soft Computing. 2010;14(9): 899-911.
  39. Feng Feng, Yongming Li, Violeta Leoreanu-Fotea. Application of Level Soft Sets in Decision Making based on Interval-Valued Fuzzy Soft Sets. Computers and Mathematics with Applications. 2010;60:1756-1767.
  40. Feng Feng, Young Bae Jun, Xiaoyan Liu, Lifeng Li. An Adjustable Approach to Fuzzy Soft Set based Decision Making. Journal of Computational and Applied Mathematics, 2010;234(1):10-20.
  41. Fu Li. Soft Lattices. Global Journal of Science Frontier Research. 2010;10:57-59.
  42. Ke Gong, Zhi xiao, Xia Zhang. The Bijective Soft Set with its Operations.  Computers and Mathematics with Applications. 2010;60:2270-2278.
  43. K. Qin & Z. Hong. On Soft Equality. Journal of Computational and Applied Mathematics. 2010;234:1347-1355.
  44. Zhi Xiao, Ke Gong, Sisi Xia, Yan Zou. Exclusive Disjunctive Soft Sets. Computers & Mathematics with Applications. 2010;59: 2128-2137.
  45. Wei Xu, Jian Ma, Shouyang Wang and Gang Hao.Vague Soft Sets and their Properties. Computers & Mathematics with Applications. 2010;59(2):787-794.
  46. Muhammad Irfan Ali, Muhammad Shabir, Munazza Naz.  Algebraic Structures of Soft Sets Associated with New Operation. Computers and Mathematics with Applications. 2011;61:2647-2654.
  47. Shawkat Alkhazaleh, Abdul Razak Salleh, and Nasruddin Hassan. Soft Multisets Theory. Applied Mathematical Sciences. 2011;72:3561–3573.
  48. A. O. Atagun, A. Sezgin. Soft Substructures of Rings, Fields and Modules. Computers and Mathematics with Applications. 2011;61:592-601.
  49. K. V. Babitha, J. J. Sunil. Transitive Closure and Ordering on Soft Sets. Computers and Mathematics with Applications. 2011;60:2235-2239.
  50. Yıldıray Celik, Canan Ekiz, Sultan Yamak. A New View on Soft Rings. Hacettepe Journal of Mathematics and Statistics. 2011;40:273-286.
  51. Feng Feng. Soft Rough Sets Applied to Multicriteria Group Decision Making. Annals of Fuzzy Mathematics and Informatics. 2011;2(1):69-80.
  52. Feng Feng, Xiaoyan Liu, Violeta Leoreanu-Fotea, Young Bea Jun. Soft Sets and Soft Rough Sets. Information Sciences. 2011;181(6):1125-1137.
  53. Fu Li. Notes on the Soft Operations. ARPN Journal of System and Software. 2011;1:205-208.
  54. X. Ge, S. Yang. Investigations on Some Operations of Soft Set. World Academy of Science, Engineering and Technology. 2011;51:1112-1115.
  55. Jayanta Ghosh, Bivas Dinda, T. K. Samanta. Fuzzy Soft Rings and Fuzzy Soft Ideals. International Journal Pure Application Science Technology. 2011;2(2):66-74.
  56. T. Herawan & M.M. Deris. A Soft Set Approach for Association Rule Mining. Knowledge based Systems. 2011;24:186-195.
  57. Yungcheng Jiang, Yong Tang, Qimai Chen, Hai Liu and Jianchao Tang. Extending Fuzzy Soft Sets with Fuzzy Description Logics. Knowledge-based Systems. 2011;24: 1096-1107.
  58. Young Bae Jun, Hee Sik Kim and Chul Hwan Park. Positive Implicative Ideals of BCK-Algebras based on a Soft Set Theory. Bulletin of the Malaysian Mathematical Science. 2011;34(2):345–354.
  59. Kyoung Ja Lee, Young Bae Jun and Chul Hwan Park. Implicative Soft Ideals and Implicative Idealistic Soft BCK-Algebras. Communications of the Korean Mathematical Society. 2011;26(2):183-196.
  60. Mehmet Ali Ozturk & Ebubekir Inna. Soft-Rings and Idealistic Soft Rings. Annuals of Fuzzy Mathematics and Information. 2011;1(1):71-80.
  61. A. Sezgin, A. O. Atagun. On Operations of Soft Sets. Computers and Mathematics with Applications. 2011;60:1840-1849.
  62. A. Sezgin, A. O. Atagun. Soft Groups and Normalistic Soft Group. Computers & Mathematics with Applications. 2011; 62:685-698.
  63. H. Yang, Z. Guo. Kenels and Closures of Soft Set Relations and Soft Set Relation Mappings. Computers and Mathematics with Applications. 2011;61:651-662.
  64. Jiehua Zhou, Yuewu Li and Yunqiang Yin .Intuitionistic Fuzzy Soft Set Semigroups. Mathematica Aeterna. 2011;1(3):173-183.
  65. Faruk Karaaslan, Naim Cagman and Serdar Enginoglu. Soft Lattices.  Journal of New Results in Science. 2012;1:5-17.
  66. W. K. Min. Similarity in Soft Set Theory.  Applied Mathematics Letters. 2012;25:310-314.
  67. D. Singh, I. A. Onyeozili. Some Conceptual Misunderstandings of the Fundamentals of Soft Set Theory. ARPN Journal of Systems and Software. 2012;2(9):251-254.
  68. D. Singh, I. A. Onyeozili. Some Results on Distributive and Absorption Properties on Soft Operations. IOSR Journal of Mathematics. 2012;4(2):18-30.
  69. D. Singh, I. A. Onyeozili. On Some New Properties of Soft Set Operations. International Journal of Computer Applications. 2012;59(4):39-44.
  70. Zhi Xiao, Sisi Xia, Ke Gong and Dan Li. The Trapezoidal Fuzzy Soft Set and its Application in MCDM. Applied Mathematical Modelling. 2012;36:5844-5855.
  71. Zhi Xiao, Weijie Chen, Lingling Li. An Integrated FCM and Fuzzy Soft Set for Supplier Selection Problem based on Risk Evaluation. Applied Mathematical Modelling. 2012;36:1444-1454.
  72. Khaleed Alhazaymeh, Shafida Abdul Halim, Abdul Razak Salleh and Nasruddin Hassan. Soft Intuitionistic Fuzzy Sets.  Applied Mathematical Sciences. 2012;6(54):2669-2680.
  73. M.I. Ali. Another View on Reduction of Parameters in Soft Sets. Applied Soft Computing. 2012;12:1814-1821.
  74. Jin Han Park, Oe Hyeon Kim and Young Chel Kwun. Some Properties of Equivalence Soft Set Relations. Computers and Mathematics with Applications. 2012;63:1079-1088.
  75. Sk Nazmul and Symal Kumar Samanta. Soft topological Soft group. Nazmul and Samanta Mathamatical Science, 2012.
  76. B.A.Ersoy, S. Onar, K. Hila, and B. Davvaz. Some Properties of Intuitionistic Fuzzy Soft Rings.  Hindawi Publishing Corporation Journal of Mathematics. 2013;Article ID 650480.
  77. Dariusz Wardowski. On a soft mapping and its fixed points. Wardowski Fixed Point Theory and Applications, 2013.
  78. HacJ AktaG and Ferif Ozlu . Cyclic Soft Groups and Their Applications on Groups”, Scientific World Journal. 2014;Article ID 437324.
  79. Xiaolong Xin and Wenting Li. Soft Congruence Relations over Rings. Scientific World Journal. 2014;ArticleID541630.
  80. J. Subhashinin and Dr. C. Sekar. Soft pre T1 Space in the Soft Topological Spaces. International Journal of Fuzzy Mathematics And Systems. 2014;4(2):203-207.
  81. Zhao Haiyan & Jia Jingjing. Fuzzy soft relation and its application in decision making. 7th International Conference on Modelling, Identification and Control, 2015.
  82. Anjan Mukherjee & AjoyKanti. Interval Valued Intuitionistic Fuzzy Soft Multi Set Theoretic Approach to Decision Making Problems. IEEE International Conference on Computer, Communication and Control, 2015.
  83. Chen Bin & Guan YanYong. Hesitant fuzzy soft sets and their applications in decision-making. 12th International Conference on Fuzzy Systems and Knowledge Discovery, 2015.
  84. B.K Tripathy, R.K. Mohanty and T.R. Sooraj. On Intuitionistic Fuzzy Soft Set and its Application in Group Decision Making. International Conference on Emerging Trends in Engineering, technology and Science, IEEE Conference Publications 2016;1-5.
  85. Jia Wenhua, Kong Zhi and Wang Lifu. Sequence interval-valued fuzzy soft set and its application in decision making, Chines Control and Decision Conference, IEEE Conference Publications, 2016;3090-3095.