Theories of soft sets and rough sets are two different approaches to analyzing vagueness. A possible fusion of rough sets and soft sets was proposed in 2011. At this time the concept of soft rough sets was introduced, where parametrized subsets of a universal set are basic building blocks for lower and upper approximations of a subset. The main purpose of soft rough sets is to reduce the soft boundary region by increasing the lower approximation and decreasing the upper approximation. In this paper, we present two new approaches for soft rough sets that is related to the notion of ideals. The main characteristics of these recent approaches are explained and interpreted through the use of suitable propositions and examples. These recent approaches satisfy most of the conditions of well known properties of Pawlak's model. Comparisons between our methods and previous ones are introduced. In addition, we prove that our approaches produce a smaller boundary region and greater value of accuracy than the corresponding defined definitions. Furthermore, two new styles of approximation spaces related to two distinct ideals, called soft bi-ideal approximation spaces, are introduced and studied. Analysis of the fulfilled and the non-fulfilled properties is presented, and many examples to ensure and explain the advantages and the disadvantages between our styles and the previous ones are provided.
Citation: Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, K. A. Aldwoah, Ismail Ibedou. New soft rough approximations via ideals and its applications[J]. AIMS Mathematics, 2024, 9(4): 9884-9910. doi: 10.3934/math.2024484
Theories of soft sets and rough sets are two different approaches to analyzing vagueness. A possible fusion of rough sets and soft sets was proposed in 2011. At this time the concept of soft rough sets was introduced, where parametrized subsets of a universal set are basic building blocks for lower and upper approximations of a subset. The main purpose of soft rough sets is to reduce the soft boundary region by increasing the lower approximation and decreasing the upper approximation. In this paper, we present two new approaches for soft rough sets that is related to the notion of ideals. The main characteristics of these recent approaches are explained and interpreted through the use of suitable propositions and examples. These recent approaches satisfy most of the conditions of well known properties of Pawlak's model. Comparisons between our methods and previous ones are introduced. In addition, we prove that our approaches produce a smaller boundary region and greater value of accuracy than the corresponding defined definitions. Furthermore, two new styles of approximation spaces related to two distinct ideals, called soft bi-ideal approximation spaces, are introduced and studied. Analysis of the fulfilled and the non-fulfilled properties is presented, and many examples to ensure and explain the advantages and the disadvantages between our styles and the previous ones are provided.
| [1] |
L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
|
| [2] |
K. T. Atanassov, S. Stoeva, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
|
| [3] | Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci., 11 (1982), 341–356. |
| [4] | Z. Pawlak, Rough concept analysis, Bull. Pol. Acad. Sci. Math., 33 (1985), 9–10. |
| [5] | M. B. Gorzałczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Set. Syst., 21 (1987), 1–17. |
| [6] |
R. Slowinski, D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE T. Knowl. Data Eng., 12 (2000), 331–336. https://doi.org/10.1109/69.842271 doi: 10.1109/69.842271
|
| [7] | Y. Y. Yao, S. K. M. Wong, Generalization of rough sets using relationships between attribute values, In: Proceedings of the 2nd annual Joint Conference on Information sciences, 1995, 30–33. |
| [8] |
J. A. Pomykala, Approximation operations in approximation space, Bull. Pol. Acad. Sci., 35 (1987), 653–662. https://doi.org/10.1016/0165-0114(87)90148-5 doi: 10.1016/0165-0114(87)90148-5
|
| [9] | A. Skowron, J. Stepaniuk, Tolerance approximation spaces, Fund. Inform., 27 (1996), 245–253. |
| [10] | D. Molodtsov, Soft set theory—first results, Comput. Math. Appl., 37 (1999), 19–31. |
| [11] |
P. K. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (1996), 1077–1083. https://doi.org/10.1016/S0898-1221(02)00216-X doi: 10.1016/S0898-1221(02)00216-X
|
| [12] |
R. Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, I. Ibedou, Soft closure spaces via soft ideals, AIMS Math., 9 (2024), 6379–6410. https://doi.org/10.3934/math.2024311 doi: 10.3934/math.2024311
|
| [13] |
O. Dalkılıç, N. Demirtaş, Soft somewhat open sets: soft separation axioms and medical application to nutrition, Comput. Appl. Math., 41 (2022), 22. https://doi.org/10.1007/s40314-022-01919-x doi: 10.1007/s40314-022-01919-x
|
| [14] |
M. K. El-Bably, M. I. Ali, E. S. A. Abo-Tabl, New topological approaches to generalized soft rough approximations with medical applications, J. Math., 2021 (2021), 1–16. https://doi.org/10.1155/2021/2559495 doi: 10.1155/2021/2559495
|
| [15] |
J. Sanabria, K. Rojo, F. Abad, A new approach of soft rough sets and a medical application for the diagnosis of coronavirus disease, AIMS Math., 8 (2023), 2686–2707. https://doi.org/10.3934/math.2023141 doi: 10.3934/math.2023141
|
| [16] |
S. Al Ghour, On soft generalized $\omega$-closed sets and soft T 1/2 spaces in soft topological spaces, MDPI Axioms, 11 (2022), 194. https://doi.org/10.3390/axioms11050194 doi: 10.3390/axioms11050194
|
| [17] |
T. M. Al-Shami, Compactness on soft topological ordered spaces and its application on the information system, J. Math., 2021 (2021), 1–12. https://doi.org/10.1155/2021/6699092 doi: 10.1155/2021/6699092
|
| [18] |
T. M. Al-Shami, A. Mhemdi, R. Abu-Gdairi, A novel framework for generalizations of soft open sets and its applications via soft topologies, Mathematics, 11 (2023), 16. https://doi.org/10.3390/math11040840 doi: 10.3390/math11040840
|
| [19] |
S. Al Ghour, Between the classes of soft open sets and soft omega open sets, Mathematics, 10 (2022), 10. https://doi.org/10.3390/math10050719 doi: 10.3390/math10050719
|
| [20] | R. Vaidyanathaswamy, The localisation theory in set-topology, In: Proceedings of the Indian Academy of Sciences-Section A, 20 (1944), 51–61. https://doi.org/10.1007/BF03048958 |
| [21] |
R. A. Hosny, B. A. Asaad, A. A. Azzam, T. M. Al-Shami, Various topologies generated from E_j-Neighbourhoods via ideals, Complexity, 2021 (2021), 1–11. https://doi.org/10.1155/2021/4149368 doi: 10.1155/2021/4149368
|
| [22] |
A. Kandil, S. A. El-Sheikh, M. Hosny, M. Raafat, Bi-ideal approximation spaces and their applications, Soft Comput., 24 (2020), 12989–13001. https://doi.org/10.1007/s00500-020-04720-2 doi: 10.1007/s00500-020-04720-2
|
| [23] |
A. S. Nawar, M. A. El-Gayar, M. K. El-Bably, R. A. Hosny, $\theta$$\beta$-ideal approximation spaces and their applications, AIMS Math., 7 (2022), 2479–2497. https://doi.org/10.3934/math.2022139 doi: 10.3934/math.2022139
|
| [24] |
H. I. Mustafa, Bipolar soft ideal rough set with applications in COVID-19, Turk. J. Math., 47 (2023), 1–36. https://doi.org/10.55730/1300-0098.3343 doi: 10.55730/1300-0098.3343
|
| [25] |
M. I. Ali, A note on soft sets, rough soft sets and fuzzy soft sets, Appl. Soft Comput., 11 (2011), 3329–3332. https://doi.org/10.1016/j.asoc.2011.01.003 doi: 10.1016/j.asoc.2011.01.003
|
| [26] | F. Feng, Soft rough sets applied to multicriteria group decision making, Ann. Fuzzy Math. Inform., 2 (2011), 69–80. |
| [27] |
P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6 doi: 10.1016/S0898-1221(03)00016-6
|
| [28] |
F. Feng, X. Y. Liu, V. Leoreanu-Fotea, Y. B. Jun, Soft sets and soft rough sets, Inform. Sci., 181 (2011), 1125–1137. https://doi.org/10.1016/j.ins.2010.11.004 doi: 10.1016/j.ins.2010.11.004
|
| [29] |
S. Alkhazaleh, E. A. Marei, New soft rough set approximations, Int. J. Fuzzy Log. Inte., 21 (2021), 123–134. https://doi.org/10.5391/IJFIS.2021.21.2.123 doi: 10.5391/IJFIS.2021.21.2.123
|
| [30] |
D. Janković, T. R. Hamlett, New topologies from old via ideals, Am. Math. Mon., 97 (1990), 295–310. https://doi.org/10.1080/00029890.1990.11995593 doi: 10.1080/00029890.1990.11995593
|
| [31] | T. Herawan, M. M. Deris, Soft decision making for patients suspected influenza, In: ICCSA 2010: Computational Science and Its Applications-ICCSA 2010, Springer, Berlin, Heidelberg, 6018 (2010), 405–418. https://doi.org/10.1007/978-3-642-12179-1_34 |
| [32] | K. Abbas, A. R. Mikler, R. Gatti, Temporal analysis of infectious diseases: Influenza, In: Proceedings of the 2005 ACM symposium on Applied computing, 2005,267–271. https://doi.org/10.1145/1066677.1066740 |
| [33] |
A. A. Azzam, A. M. Khalil, S. G. Li, Medical applications via minimal topological structure, J. Intell. Fuzzy Syst., 39 (2020), 4723–4730. https://doi.org/10.3233/JIFS-200651 doi: 10.3233/JIFS-200651
|
Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, K. A. Aldwoah, Ismail Ibedou. New soft rough approximations via ideals and its applications[J]. AIMS Mathematics, 2024, 9(4): 9884-9910. doi: 10.3934/math.2024484
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