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AIMS Mathematics

2022, Volume 7, Issue 7: 13104-13138. doi: 10.3934/math.2022724
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Research article

Generalization of rough sets using maximal right neighborhood systems and ideals with medical applications

  • 1.
    Department of Mathematics, Faculty of Science for Girls, King Khalid University, Abha 61471, Saudi Arabia
  • 2.
    Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt
  • Received: 12 March 2022 Revised: 17 April 2022 Accepted: 21 April 2022 Published: 10 May 2022

  • MSC : 03E99, 54A05, 54A10, 54E99

  • Rough set theory is a mathematical technique to address the issues of uncertainty and vagueness in knowledge. An ideal is considered to be a crucial extension of this theory. It is an efficacious tool to dispose of vagueness and uncertainties by helping us to approximate the rough set in a more general manner. Minimizing the boundary region is one of the pivotal and substantial themes for studying the rough sets which consequently aim to maximize the accuracy measure. An ideal is one of the effective and successful followed methods to achieve this goal perfectly. So, the objective of this work is to present new methods for rough sets by using ideals. Some important characteristics of these methods are scrutinized and demonstrated to show that they yield accuracy measures greater and higher than the former ones in the other approaches. Finally, two medical applications are introduced to show the significance of utilizing the ideals in the proposed methods.

    Citation: Mona Hosny. Generalization of rough sets using maximal right neighborhood systems and ideals with medical applications[J]. AIMS Mathematics, 2022, 7(7): 13104-13138. doi: 10.3934/math.2022724

    Related Papers:

  • Rough set theory is a mathematical technique to address the issues of uncertainty and vagueness in knowledge. An ideal is considered to be a crucial extension of this theory. It is an efficacious tool to dispose of vagueness and uncertainties by helping us to approximate the rough set in a more general manner. Minimizing the boundary region is one of the pivotal and substantial themes for studying the rough sets which consequently aim to maximize the accuracy measure. An ideal is one of the effective and successful followed methods to achieve this goal perfectly. So, the objective of this work is to present new methods for rough sets by using ideals. Some important characteristics of these methods are scrutinized and demonstrated to show that they yield accuracy measures greater and higher than the former ones in the other approaches. Finally, two medical applications are introduced to show the significance of utilizing the ideals in the proposed methods.


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    [1] E. A. Abo-Tabl, A comparison of two kinds of definitions of rough approximations based on a similarity relation, Inform. Sci., 181 (2011), 2587–2596. https://doi.org/10.1016/j.ins.2011.01.007 doi: 10.1016/j.ins.2011.01.007
    [2] E. A. Abo-Tabl, M. K. El-Bably, Rough topological structure based on reflexivity with some applications, AIMS Math., 7 (2022), 9911–9925. https://doi.org/10.3934/math.2022553 doi: 10.3934/math.2022553
    [3] H. M. Abu-Doniaa, Comparison between different kinds of approximations by using a family of binary relations, Knowl. Based Syst., 21 (2008), 911–919. https://doi.org/10.1016/j.knosys.2008.03.046 doi: 10.1016/j.knosys.2008.03.046
    [4] H. M. Abu-Doniaa, Multi knowledge based rough approximations and applications, Knowl. Based Syst., 26 (2012) 20–29. https://doi.org/10.1016/j.knosys.2011.06.010
    [5] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for basic rough set concepts, In: D. Ślȩzak, G. Wang, M. Szczuka, I. Düntsch, Y. Yao, International workshop on rough sets, fuzzy sets, data mining, and granular computing, Lecture Notes in Computer Science, Springer, 3641 (2005), 64–73. https://doi.org/10.1007/11548669_7
    [6] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for closure spaces by relations, Acta Math. Acad. Paedagog. Nyregyhziensis, 22 (2006), 285–304.
    [7] T. M. Al-shami, Maximal rough neighborhoods with a medical application, J. Ambient Intell. Human. Comput., 2022. https://doi.org/10.1007/s12652-022-03858-1
    [8] T. M. Al-shami, Topological approach to generate new rough set models, Complex Intell. Syst., 2022. https://doi.org/10.1007/s40747-022-00704-x
    [9] T. M. Al-shami, H. Işık, A. S. Nawar, R. A. Hosny, Some topological approaches for generalized rough sets via ideals, Math. Prob. Eng., 2021 (2021), 5642982. https://doi.org/10.1155/2021/5642982 doi: 10.1155/2021/5642982
    [10] 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
    [11] J. Dai, S. Gao, G. Zheng, Generalized rough set models determined by multiple neighborhoods generated from a similarity relation, Soft Comput., 13 (2018), 2081–2094. https://doi.org/10.1007/s00500-017-2672-x doi: 10.1007/s00500-017-2672-x
    [12] J. Dai, W. Wang, Q. Xu, H. Tian, Uncertainty measurement for interval-valued decision systems based on extended conditional entropy, Knowl Based Syst., 27 (2012), 443–450. https://doi.org/10.1016/j.knosys.2011.10.013 doi: 10.1016/j.knosys.2011.10.013
    [13] J. Dai, Q. Xu, Approximations and uncertainty measures in incomplete information systems, Inform. Sci., 198 (2012), 62–80. https://doi.org/10.1016/j.ins.2012.02.032 doi: 10.1016/j.ins.2012.02.032
    [14] M. K. El-Bably, T. M. Al-shami, Different kinds of generalized rough sets based on neighborhoods with a medical application, Int. J. Biomath., 14 (2021), 2150086. https://doi.org/10.1142/S1793524521500868 doi: 10.1142/S1793524521500868
    [15] M. A. El Safty, S. Al-Zahrani, Topological modeling for symptom reduction of Corona virus, Punjab Univ. J. Math., 53 (2021), 47–59.
    [16] M. Hosny, Topological approach for rough sets by using J-nearly concepts via ideals, Filomat, 34 (2020), 273–286. https://doi.org/10.2298/FIL2002273H doi: 10.2298/FIL2002273H
    [17] M. Hosny, Idealization of $j$-approximation spaces, Filomat, 34 (2020), 287–301. https://doi.org/10.2298/FIL2002287H doi: 10.2298/FIL2002287H
    [18] M. Hosny, Rough sets theory via new topological notions based on ideals and applications, AIMS Math., 7 (2021), 869–902. https://doi.org/10.3934/math.2022052 doi: 10.3934/math.2022052
    [19] 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), 4149368. https://doi.org/10.1155/2021/4149368 doi: 10.1155/2021/4149368
    [20] D. Jankovic, T. R. Hamlet, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295–310. https://doi.org/10.1080/00029890.1990.11995593 doi: 10.1080/00029890.1990.11995593
    [21] J. J$\ddot{a}$rinen, Approximations and rough sets based on tolerances, In: W. Ziarko, Y. Yao, Rough sets and current trends in computing, Lecture Notes in Computer Science, Springer, 2005 (2001), 182–189. https://doi.org/10.1007/3-540-45554-X_21
    [22] A. Kandil, S. A. El-Sheikh, M. Hosny, M. Raafat, Bi-ideal approximation spaces and their applications, Soft Comput., 24 (2020), 12989–3001. https://doi.org/10.1007/s00500-020-04720-2 doi: 10.1007/s00500-020-04720-2
    [23] M. Kondo, On the structure of generalized rough sets, Inform. Sci., 176 (2005), 589–600. https://doi.org/10.1016/j.ins.2005.01.001 doi: 10.1016/j.ins.2005.01.001
    [24] A. M. Kozae, On topology expansions by ideals and applications, Chaos Soliton. Fract., 13 (2002), 55–60. https://doi.org/10.1016/S0960-0779(00)00224-1 doi: 10.1016/S0960-0779(00)00224-1
    [25] M. Kryszkiewicz, Rough set approach to incomplete information systems, Inform. Sci., 112 (1998), 39–49. https://doi.org/10.1016/S0020-0255(98)10019-1 doi: 10.1016/S0020-0255(98)10019-1
    [26] K. Kuratowski, Topology, Vol. I, New York: Academic Press, 1966.
    [27] 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 (2021), 2479–2497. https://doi.org/10.3934/math.2022139 doi: 10.3934/math.2022139
    [28] J. Nieminen, Rough tolerance equality and tolerance black boxes, Fund. Inform., 11 (1988), 289–296. https://doi.org/10.3233/FI-1988-11306 doi: 10.3233/FI-1988-11306
    [29] Z. Pawlak, Rough sets, Int. J. Inform. Comput. Sci., 11 (1982), 341–356. https://doi.org/10.1007/BF01001956
    [30] Z. Pawlak, Rough concept analysis, Bull. Pol. Acad. Sci. Math., 33(1985), 9–10.
    [31] D. Pei, Z. Xu, Transformation of rough set models, Knowl.-Based Syst., 20 (2007), 745–751. https://doi.org/10.1016/j.knosys.2006.10.006 doi: 10.1016/j.knosys.2006.10.006
    [32] J. A. Pomykala, About tolerance and similarity relations in information systems, In: J. Alpigini, J. Peters, A. Skowron, N. Zhong, Rough sets and current trends in computing, Lecture Notes in Computer Science, Springer, 2475 (2002), 175–182. https://doi.org/10.1007/3-540-45813-1_22
    [33] A. Skowron, J. Stepaniuk, Tolerance approximation spaces, Fund. Inform., 27 (1996), 245–253. https://doi.org/10.3233/FI-1996-272311 doi: 10.3233/FI-1996-272311
    [34] A. Skowron, D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE T. Knowl. Data En., 12 (2000), 331–336. https://doi.org/10.1109/69.842271 doi: 10.1109/69.842271
    [35] R. Slowinski, J. Stefanowski, Rough-set reasoning about uncertain data, Fund. Inform., 27 (1996), 229–243. https://doi.org/10.3233/FI-1996-272310 doi: 10.3233/FI-1996-272310
    [36] R. Vaidynathaswamy, The localization theory in set topology, Proc. Indian Acad. Sci., 20 (1945), 51–61. https://doi.org/10.1007/BF03048958 doi: 10.1007/BF03048958
    [37] Y. Y. Yao, Two views of the theory of rough sets in finite universes, Int. J. Approx. Reason., 15 (1996), 291–317. https://doi.org/10.1016/S0888-613X(96)00071-0 doi: 10.1016/S0888-613X(96)00071-0
    [38] Y. Y. Yao, Generalized rough set models, In: L. Polkowski, A. Skowron, Rough sets in knowledge discovery, Heidelberg: Physica Verlag, 1998,286–318.
    [39] Y. Y. Yao, Constructive and algebraic methods of the theory of rough sets, Inform. Sci., 109 (1998), 21–47. https://doi.org/10.1016/S0020-0255(98)00012-7 doi: 10.1016/S0020-0255(98)00012-7
    [40] Y. Y. Yao, On generalized Pawlak approximation operators, In: L. Polkowski, A. Skowron, Rough sets and current trends in computing, Lecture Notes in Computer Science, Springer, 1424 (1998), 298–307. https://doi.org/10.1007/3-540-69115-4_41
    [41] Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 111 (1998), 239–259. https://doi.org/10.1016/S0020-0255(98)10006-3 doi: 10.1016/S0020-0255(98)10006-3
    [42] X. Zhang, J. Dai, Y. Yu, On the union and intersection operations of rough sets based on various approximation spaces, Inform. Sci., 292 (2015), 214–229. https://doi.org/10.1016/j.ins.2014.09.007 doi: 10.1016/j.ins.2014.09.007
    [43] W. Zhu, Generalized rough sets based on relations, Inform. Sci., 177 (2007), 4997–5011. https://doi.org/10.1016/j.ins.2007.05.037 doi: 10.1016/j.ins.2007.05.037
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