Convex and Concave Sets Based on Soft Sets and Fuzzy Soft Sets

Abstract

In this study, after given some basic definitions of soft sets and fuzzy soft sets we firstly define convex-concave soft sets. Then, we investigate their properties and give some relations between convex and concave soft sets. Furthermore, we define fuzzy convex-concave soft sets and give some properties for the sets.

Keywords

• Fuzzy set,soft sets,convex sets,concave sets,strictly convex,strongly convex

References

1. D. A. Molodtsov, Soft Set Theory–First Results, Computers and Mathematics with Applications 37 (1999) 19-31.
2. L. A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) 338-353.
3. Z. Pawlak, Rough Sets, International Journal of Information and Computer Sciences 11 (1982) 341-356.
4. U. Acar, F. Koyuncu, B. Tanay, Soft Sets and Soft Rings, Computers and Mathematics with Applications 59 (2010) 3458-3463.
5. H. Aktaş, N. Çağman, Soft Sets and Soft Groups, Information Sciences 177 (2007) 2726-2735.
6. M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On Some New Operations in Soft Set Theory, Computers and Mathematics with Applications 57 (2009) 1547-1553.
7. N. Çağman, S. Enginoğlu, Soft Set Theory and uni–int Decision Making, European Journal of Operational Research 207(2) (2010) 848-855.
8. N. Çağman, F. Çıtak, H. Aktaş, Soft int-group and Its Applications to Group Theory, Neural Computing and Applications 21 (2012) 151-158.

9. G. Şenel, A New Approach to Hausdorff Space Theory via the Soft Sets, Mathematical Problems in Engineering (2016) Article ID 2196743, 6 pages.
10. G. Şenel, A Comparative Research on the Definition of Soft Point, International Journal of Computer Applications 163(2) (2017) 1-4.
11. G. Şenel, The Parameterization Reduction of Soft Point and Its Applications with Soft Matrix, International Journal of Computer Applications 164(1) (2017) 1-6.
12. G. Şenel, New Soft Metric Applications, The Journal of Fuzzy Mathematics 25(1) (2017) 1-12.
13. İ. Deli, S. Broumi, Neutrosophic Soft Matrices and NSM-Decision Making, Journal of Intelligent and Fuzzy Systems 28 (5) (2015) 2233-2241.
14. İ. Deli, N. Çağman, Intuitionistic Fuzzy Parameterized Soft Set Theory and Its Decision Making, Applied Soft Computing 28 (2015) 109-113.
15. İ. Deli, N. Çağman, Fuzzy Soft Games, Filomat 29(9) (2015) 1901-1917.
16. İ. Deli, S. Karataş, Interval Valued Intuitionistic Fuzzy Parameterized Soft Set Theory and Its Decision Making, Journal of Intelligent and Fuzzy Systems 30(3) (2016) 2073-2082.
17. İ. Deli, N. Çğman, Probabilistic Equilibrium Solution of Soft Games, Journal of Intelligent and Fuzzy Systems 30(3) (2016) 2237-2244.
18. İ. Deli, Interval-Valued Neutrosophic Soft Sets and Its Decision Making, International Journal of Machine Learning and Cybernetics 8(2) (2017) 665-676.
19. İ. Deli, S. Broumi, Neutrosophic Soft Relations and Some Properties, Annals of Fuzzy Mathematics and Informatics 9(1) (2015) 169-182.
20. İ. Deli, S. Broumi, M. Ali, Neutrosophic Soft Multi-Set Theory and Its Decision Making, Neutrosophic Sets and Systems 5 (2014) 65-76.
21. İ. Deli, N. Çağman, Relations on FP-Soft Sets Applied to Decision Making Problems, Journal of New Theory 3 (2015) 98-107.
22. İ. Deli, S. Broumi, F. Smarandache, On Neutrosophic Refined Sets and Their Applications in Medical Diagnosis, Journal of New Theory 6 (2015) 88-98.
23. İ. Deli, N. Çağman, Application of Soft Sets in Decision Making based on Game Theory, Annals of Fuzzy Mathematics and Informatics 11(3) (2016) 425-438.
24. İ. Deli, npn-Soft Sets Theory and Applications, Annals of Fuzzy Mathematics and Informatics 10(6) (2015) 847-862.
25. İ. Deli, N. Çağman, Similarity Measure of IFS-Sets and Its Application in Medical Diagnosis, Annals of Fuzzy Mathematics and Informatics 11(5) (2016) 841-854.
26. N. Çağman, S. Enginoğlu, F. Çıtak, Fuzzy Soft Set Theory and Its Applications, Iranian Journal of Fuzzy Systems 8(3) (2011) 137-147.
27. N. Çağman, İ. Deli, Means of FP-Soft Sets and Its Applications, Hacettepe Journal of Mathematics and Statistics 41(5) (2012) 615-625.
28. N. Çağman, İ. Deli, Product of FP-Soft Sets and Its Applications, Hacettepe Journal of Mathematics and Statistics, 41 (3) (2012), 365-374.
29. Y. Jiang, Y. Tang, H. Liu, Z. Chen, Entropy on Intuitionistic Fuzzy Soft Sets and on Interval- Valued Fuzzy Soft Sets, Information Sciences 240 (2013) 95-114.
30. P. K. Maji, R. Biswas, A. R. Roy, Fuzzy Soft Sets, Journal of Fuzzy Mathematics 9(3) (2001) 589-602.
31. M. Shabir, M. I. Ali, T. Shaheen, Another Approach to Soft Rough Sets, Knowledge-Based Systems 40 (2013) 72-80. C. Wang, A. Qu, Entropy, Similarity Measure and Distance Measure of Vague Soft Sets and Their Relations, Information Sciences 244 (2013) 92-106.
32. T. Deng, X. Wang, An Object-Parameter Approach to Predicting Unknown Data in Incomplete Fuzzy Soft Sets, Applied Mathematical Modelling 37 (2013) 4139-4146.
33. K. Gong, P. Wanga, Z. Xiao, Bijective Soft Set Decision System Based Parameters Reduction under Fuzzy Environments, Applied Mathematical Modelling 37 (2013) 4474-4485.
34. Y. Yang, X. Tan, C. Meng, The Multi-Fuzzy Soft Set and Its Application in Decision Making, Applied Mathematical Modelling 37 (2013) 4915-4923.
35. J. Mao, D. Yao, C. Wang, Group Decision Making Methods Based on Intuitionistic Fuzzy Soft Matrices, Applied Mathematical Modelling 37 (2013) 6425-6436.
36. F. Feng, Y. Li, Soft Subsets and Soft Product Operations, Information Sciences 232 (2013) 44-57.
37. Y. Wang, Z. Liu, Z. Du, Y. Huang, Mobile Cloud Computing Network Attack and Defense Learning System Based on Fuzzy Soft Sets, Procedia Computer Science 17 (2013) 214-221.
38. M. Agarwala, K. K. Biswas, M. Hanmandlu, Generalized Intuitionistic Fuzzy Soft Sets with Applications in Decision-Making, Applied Soft Computing 13 (2013) 3552-3566.
39. B. B. Chaudhuri, Concave Fuzzy Set: A Concept Complementary to the Convex Fuzzy Set, Pattern Recognition Letters 13(2) (1992) 103-106.
40. D. Sarkar, Concavoconvex Fuzzy Set, Fuzzy Sets and Systems 79(2) (1996) 267-269.
41. E. E. Ammar, Some Properties of Convex Fuzzy Sets and Convex Fuzzy Cones, Fuzzy Sets and Systems 106(3) (1999) 381-386.
42. P. Diamond, P. Kloeden, A. Vladimirov, Spikes, Broken Planes and the Approximation of Convex Fuzzy Sets, Fuzzy Sets and Systems 99 (1998) 225-232.
43. D. Dubois, Fuzzy Weighted Averages and Fuzzy Convex Sums, Fuzzy Sets and Systems 213 (2013) 106-108.
44. O. Pavlacka, Note on the Lack of Equality between Fuzzy Weighted Average and Fuzzy Convex Sum, Fuzzy Sets and Systems 213 (2013) 102-105.
45. X. Wang, D. Ruan, E. E. Kerre, Mathematics of Fuzziness Basic Issues, Springer, (2009).
46. X. Yang, Some Properties of Convex Fuzzy Sets, Fuzzy Sets and Systems 72(1) (1995) 129-132.
47. İ. Deli, Convex and Concave Soft Sets and Some Properties, arXiv:1307.4960v1 [math.GM] 17 July 2013.