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1

DUBOIS, DIDIER, and HENRI PRADE. "ROUGH FUZZY SETS AND FUZZY ROUGH SETS*." International Journal of General Systems 17, no. 2-3 (1990): 191–209. http://dx.doi.org/10.1080/03081079008935107.

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2

Meng, Dan, Xiaohong Zhang, and Keyun Qin. "Soft rough fuzzy sets and soft fuzzy rough sets." Computers & Mathematics with Applications 62, no. 12 (2011): 4635–45. http://dx.doi.org/10.1016/j.camwa.2011.10.049.

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3

DEMİRCİ, MUSTAFA. "GENUINE SETS, VARIOUS KINDS OF FUZZY SETS AND FUZZY ROUGH SETS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11, no. 04 (2003): 467–94. http://dx.doi.org/10.1142/s0218488503002193.

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In this paper, deriving the type-m fuzzy sets, intuitionistic fuzzy sets, Φ-fuzzy sets, rough sets, fuzzy rough sets and rough fuzzy sets as particular genuine sets, and establishing their connections with genuine sets, it is demonstrated that the theory of genuine sets provides a powerful tool to model various different kinds of uncertainty in a mathematical way. Furthermore, it is also shown that the genuine set theoretic descriptions of type-m fuzzy sets, intuitionistic fuzzy sets and fuzzy rough sets point out new features of these set notations, originated from the peculiar characteristic
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4

WEI, LI-LI, and WEN-XIU ZHANG. "PROBABILISTIC ROUGH SETS CHARACTERIZED BY FUZZY SETS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 12, no. 01 (2004): 47–60. http://dx.doi.org/10.1142/s0218488504002643.

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Theories of fuzzy sets and rough sets have emerged as two major mathematical approaches for managing uncertainty that arises from inexact, noisy, or incomplete information. They are generalizations of classical set theory for modelling vagueness and uncertainty. Some integrations of them are expected to develop a model of uncertainty stronger than either. The present work may be considered as an attempt in this line, where we would like to study fuzziness in probabilistic rough set model, to portray probabilistic rough sets by fuzzy sets. First, we show how the concept of variable precision lo
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5

Nanda, S., and S. Majumdar. "Fuzzy rough sets." Fuzzy Sets and Systems 45, no. 2 (1992): 157–60. http://dx.doi.org/10.1016/0165-0114(92)90114-j.

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6

Pawlak, Zdzisław. "Rough sets and fuzzy sets." Fuzzy Sets and Systems 17, no. 1 (1985): 99–102. http://dx.doi.org/10.1016/s0165-0114(85)80029-4.

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7

Zhang, Zhiming. "The Parameter Reduction of Fuzzy Soft Sets Based on Soft Fuzzy Rough Sets." Advances in Fuzzy Systems 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/197435.

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Fuzzy set theory, rough set theory, and soft set theory are three effective mathematical tools for dealing with uncertainties and have many wide applications both in theory and practise. Meng et al. (2011) introduced the notion of soft fuzzy rough sets by combining fuzzy sets, rough sets, and soft sets all together. The aim of this paper is to study the parameter reduction of fuzzy soft sets based on soft fuzzy rough approximation operators. We propose some concepts and conditions for two fuzzy soft sets to generate the same lower soft fuzzy rough approximation operators and the same upper sof
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8

Zhao, Tao, and Zhenbo Wei. "On Characterization of Rough Type-2 Fuzzy Sets." Mathematical Problems in Engineering 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/4819353.

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Rough sets theory and fuzzy sets theory are important mathematical tools to deal with uncertainties. Rough fuzzy sets and fuzzy rough sets as generalizations of rough sets have been introduced. Type-2 fuzzy set provides additional degree of freedom, which makes it possible to directly handle high uncertainties. In this paper, the rough type-2 fuzzy set model is proposed by combining the rough set theory with the type-2 fuzzy set theory. The rough type-2 fuzzy approximation operators induced from the Pawlak approximation space are defined. The rough approximations of a type-2 fuzzy set in the g
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9

Çoker, Doǧan. "Fuzzy rough sets are intuitionistic L-fuzzy sets." Fuzzy Sets and Systems 96, no. 3 (1998): 381–83. http://dx.doi.org/10.1016/s0165-0114(97)00249-2.

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10

Aggarwal, Manish. "Probabilistic fuzzy rough sets." Journal of Intelligent & Fuzzy Systems 29, no. 5 (2015): 1901–12. http://dx.doi.org/10.3233/ifs-151668.

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11

Wu, W. "Generalized fuzzy rough sets." Information Sciences 151 (May 2003): 263–82. http://dx.doi.org/10.1016/s0020-0255(02)00379-1.

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12

Satirad, Akarachai, Ronnason Chinram, Pongpun Julatha, and Aiyared Iampan. "Rough Pythagorean Fuzzy Sets in UP-Algebras." European Journal of Pure and Applied Mathematics 15, no. 1 (2022): 169–98. http://dx.doi.org/10.29020/nybg.ejpam.v15i1.4254.

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This paper aims to apply the concept of rough sets to Pythagorean fuzzy sets in UP-algebras. Then we introduce fifteen types of rough Pythagorean fuzzy sets in UP-algebras and study their generalization. In addition, we will also discuss $t$-level subsets of rough Pythagorean fuzzy sets in UP-algebras to study the relationships between rough Pythagorean fuzzy sets and rough sets in UP-algebras.
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13

Liu, Guilong. "Axiomatic systems for rough sets and fuzzy rough sets." International Journal of Approximate Reasoning 48, no. 3 (2008): 857–67. http://dx.doi.org/10.1016/j.ijar.2008.02.001.

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14

Mathew, Bibin, Sunil Jacob John, and José Carlos R. Alcantud. "Multi-Granulation Picture Hesitant Fuzzy Rough Sets." Symmetry 12, no. 3 (2020): 362. http://dx.doi.org/10.3390/sym12030362.

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We lay the theoretical foundations of a novel model, termed picture hesitant fuzzy rough sets, based on picture hesitant fuzzy relations. We also combine this notion with the ideas of multi-granulation rough sets. As a consequence, a new multi-granulation rough set model on two universes, termed a multi-granulation picture hesitant fuzzy rough set, is developed. When the universes coincide or play a symmetric role, the concept assumes the standard format. In this context, we put forward two new classes of multi-granulation picture hesitant fuzzy rough sets, namely, the optimistic and pessimist
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15

Zhang, Haidong, Lan Shu, and Shilong Liao. "Intuitionistic Fuzzy Soft Rough Set and Its Application in Decision Making." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/287314.

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The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. In this paper, we present concepts of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets, and investigate some properties of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets in detail. Furthermore, classical representations of intuitionistic fuzzy soft rough approximation operators are presented. Finally, we develop an approach to intuitionistic fuzzy soft rough sets based on decision making and a numerical examp
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16

Prasertpong, Rukchart. "Roughness of soft sets and fuzzy sets in semigroups based on set-valued picture hesitant fuzzy relations." AIMS Mathematics 7, no. 2 (2022): 2891–928. http://dx.doi.org/10.3934/math.2022160.

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<abstract><p>In the philosophy of rough set theory, the methodologies of rough soft sets and rough fuzzy sets have been being examined to be efficient mathematical tools to deal with unpredictability. The basic of approximations in rough set theory is based on equivalence relations. In the aftermath, such theory is extended by arbitrary binary relations and fuzzy relations for more wide approximation spaces. In recent years, the notion of picture hesitant fuzzy relations by Mathew et al. can be considered as a novel extension of fuzzy relations. Then this paper proposes extended ap
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17

Xu, Yao-liang, Dan-dan Zou, and Ling-qiang Li. "A Further Study on L-Fuzzy Covering Rough Sets." Journal of Mathematics and Informatics 22 (2022): 49–59. http://dx.doi.org/10.22457/jmi.v22a05208.

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For L L = ∗ → ( , , ) a complete residuated lattice, a type of L -fuzzy covering rough sets was defined by Li [8] in 2017. In this paper, a further study on rough sets was given. Precisely, a single axiomatic characterization of the L -fuzzy covering rough sets was presented, and the relationships between the -fuzzy covering rough sets and L -fuzzy relation rough sets were established.
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18

Abdallah, A. A., O. R. Sayed, E. El-Sanousy, Y. H. Ragheb Sayed, M. N. Abu_Shugair, and Salahuddin Salahuddin. "Multi-Granulation Double Fuzzy Rough Sets." Symmetry 15, no. 10 (2023): 1926. http://dx.doi.org/10.3390/sym15101926.

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In this article, we introduce two new rough set models based on the concept of double fuzzy relations. These models are called optimistic and pessimistic multi-granulation double fuzzy rough sets. We discuss their properties and explore the relationship between these new models and double fuzzy rough sets. Our study focuses on the lower and upper approximations of these models, which generalize the conventional rough set model. In addition, we suggest that the development of the multi-granulation double fuzzy rough set model is significant for the generalization of the rough set model.
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19

Hu, Jun Hong, Guo Dong Gu, and Fu Xian Liu. "The Intuitionistic Fuzzy S-Rough Sets Model and Dynamic Transfer Degree." Applied Mechanics and Materials 513-517 (February 2014): 4352–56. http://dx.doi.org/10.4028/www.scientific.net/amm.513-517.4352.

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The Intuitionistic Fuzzy S-Rough Sets (IFS-RS) is the intuitionistic fuzzy extension of S-Rough sets theory. It has dynamic characteristic of S-Rough sets, as well as intuitionistic fuzzy characteristic of Intuitionistic Fuzzy sets. Based on S-Rough sets theory, this paper introduced the membership and non-membership concepts of Intuitionistic Fuzzy sets, builded the model of IFS-RS under general equivalence relation, put forward the rough property and transfer degree concepts of IFS-RS. By calculating the rough property and transfer degree of IFS-RS, Thus being able to describe the transforma
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20

Gafai, Mustapha Mannir, and Zulkifilu Muazu. "Fuzzy Product Rough Sets as a Special Case of Fuzzy T-Rough Sets." Dutse Journal of Pure and Applied Sciences 9, no. 2b (2023): 70–75. http://dx.doi.org/10.4314/dujopas.v9i2b.8.

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Fuzzy T-rough set consists of a set ???? and a T-similarity relation on ????, where T is a lower semicontinuous triangular norm. In this paper, axiomatic definition for fuzzy ????-rough sets and its upper approximation operator were proposed. The method employed was by relaxing the arbitrary T and adopting its special case ???????? (product triangular norm). The results obtained suggests an easier way of being specific to the product case of fuzzy rough sets and computations regarding its upper approximation operators. Some important propositions and examples were also provided.
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21

Wang, Qiumei, and Jianming Zhan. "Rough semigroups and rough fuzzy semigroups based on fuzzy ideals." Open Mathematics 14, no. 1 (2016): 1114–21. http://dx.doi.org/10.1515/math-2016-0102.

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AbstractIn this paper, we firstly introduce a special congruence relation U(μ, t) induced by a fuzzy ideal μ in a semigroup S. Then we define the lower and upper approximations based on a fuzzy ideal in semigroups. We can establish rough semigroups, rough ideals, rough prime ideals, rough fuzzy semigroups, rough fuzzy ideals and rough fuzzy prime ideals according to the definitions of rough sets and rough fuzzy sets. Furthermore, we shall consider the relationships among semigroups and rough semigroups, fuzzy semigroups and rough fuzzy semigroups, and some relative properties are also discusse
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22

Li, Wentao, Xiaoyan Zhang, and Wenxin Sun. "Further Study of MultigranulationT-Fuzzy Rough Sets." Scientific World Journal 2014 (2014): 1–18. http://dx.doi.org/10.1155/2014/927014.

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The optimistic multigranulationT-fuzzy rough set model was established based on multiple granulations underT-fuzzy approximation space by Xu et al., 2012. From the reference, a natural idea is to consider pessimistic multigranulation model inT-fuzzy approximation space. So, in this paper, the main objective is to make further studies according to Xu et al., 2012. The optimistic multigranulationT-fuzzy rough set model is improved deeply by investigating some further properties. And a complete multigranulationT-fuzzy rough set model is constituted by addressing the pessimistic multigranulationT-
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23

Wang, Jingqian, Xiaohong Zhang, and Qingqing Hu. "Three-Way Fuzzy Sets and Their Applications (II)." Axioms 11, no. 10 (2022): 532. http://dx.doi.org/10.3390/axioms11100532.

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Recently, the notion of a three-way fuzzy set is presented, inspired by the basic ideas of three-way decision and various generalized fuzzy sets, including lattice-valued fuzzy sets, partial fuzzy sets, intuitionistic fuzzy sets, etc. As the new theory of uncertainty, it has been used in attribute reduction and as a new control method for the water level. However, as an extension of a three-way decision, this new theory has not been used in multi-criteria decision making (MCDM for short). Based on the previous work, in this paper, we present rough set models based on three-way fuzzy sets, whic
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24

Bağırmaz, Nurettin. "Fuzzy Rough Subgroups on Approximation Space." Journal of Advances in Applied & Computational Mathematics 10 (October 11, 2023): 65–70. http://dx.doi.org/10.15377/2409-5761.2023.10.6.

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Fuzzy rough sets are a mathematical concept that combines fuzzy sets and rough sets to deal with uncertainty and incompleteness in data and information. In this study, different from the definition of Dubois and Prade (1990), the fuzzy rough set is defined within the framework of the rough group concept defined by Biswas and Nanda (1994), and some of its algebraic properties are discussed. Then, the concepts of fuzzy rough subgroup and fuzzy rough normal subgroup are introduced in the rough group. In addition, some basic features and examples of these concepts are given. MSC (2010): Primary: 0
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25

Kumar, Virendra, and Surabhi Tiwari. "L-Fuzzy rough point-wise proximity spaces." Applied General Topology 26, no. 1 (2025): 255–70. https://doi.org/10.4995/agt.2025.21682.

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In this paper, we study the concept of nearness between L-fuzzy rough sets and L-fuzzy points. The nearness between two L-fuzzy rough sets are viewed in terms of this new concept. Also, we investigate the relationship of L-fuzzy rough point-wise proximity space with L-fuzzy rough grills, L-fuzzy rough δℛ -clan and L-fuzzy rough δℛ-clusters. Moreover, we introduce the L-fuzzy rough point-wise LO-proximity spaces and study a few of its fundamental properties. The research is supported by a sufficient number of examples.
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26

Tiwari, S. P., and Arun K. Srivastava. "Fuzzy rough sets, fuzzy preorders and fuzzy topologies." Fuzzy Sets and Systems 210 (January 2013): 63–68. http://dx.doi.org/10.1016/j.fss.2012.06.001.

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27

Abushaaban, Amal T., O. A. Embaby, and Abdelfattah A. El-Atik. "Modern classes of fuzzy $ \alpha $-covering via rough sets over two distinct finite sets." AIMS Mathematics 10, no. 2 (2025): 2131–62. https://doi.org/10.3934/math.2025100.

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<p>Following the research of Yang and Atef proposing new classes of fuzzy $ \beta $-covering via rough sets types over 2-featured universes, we present some modern classes of fuzzy $ \alpha $-covering via rough sets over two distinct finite sets using fuzzy $ \alpha $-neighborhoods for two distinct points over 2-distinct finite universes. Throughout this research, we present the ideas of the fuzzy $ \alpha $-neighborhood system and the fuzzy $ \alpha $-neighborhood for two distinct points over two distinct finite sets and investigate the relations of the fuzzy $ \alpha $-neighborhood sys
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28

Polkowski, Lech, and Maria Semeniuk-Polkowska. "Where Rough Sets and Fuzzy Sets Meet." Fundamenta Informaticae 142, no. 1-4 (2015): 269–84. http://dx.doi.org/10.3233/fi-2015-1294.

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29

Masulli, Francesco, and Alfredo Petrosino. "Advances in fuzzy sets and rough sets." International Journal of Approximate Reasoning 41, no. 2 (2006): 75–76. http://dx.doi.org/10.1016/j.ijar.2005.06.010.

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30

A. A. Allam, SH. S. Abd-Allah, and M. Y. Bakier. "Rough fuzzy sets via multifunction." ANNALS OF FUZZY MATHEMATICS AND INFORMATICS 19, no. 1 (2020): 89–94. http://dx.doi.org/10.30948/afmi.2020.19.1.89.

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31

Davvaz, Bijan, Imam Mukhlash, and Soleha Soleha. "Himpunan Fuzzy dan Rough Sets." Limits: Journal of Mathematics and Its Applications 18, no. 1 (2021): 79. http://dx.doi.org/10.12962/limits.v18i1.7705.

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32

Diker, Murat. "Textures and Fuzzy Rough Sets." Fundamenta Informaticae 108, no. 3-4 (2011): 305–36. http://dx.doi.org/10.3233/fi-2011-425.

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33

Xu, Weihua, Qiaorong Wang, and Shuqun Luo. "Multi-granulation fuzzy rough sets." Journal of Intelligent & Fuzzy Systems 26, no. 3 (2014): 1323–40. http://dx.doi.org/10.3233/ifs-130818.

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34

Kong, Qing-Zhao, and Zeng-Xin Wei. "Covering-based fuzzy rough sets." Journal of Intelligent & Fuzzy Systems 29, no. 6 (2015): 2405–11. http://dx.doi.org/10.3233/ifs-151940.

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35

Morsi, Nehad N., and M. M. Yakout. "Axiomatics for fuzzy rough sets." Fuzzy Sets and Systems 100, no. 1-3 (1998): 327–42. http://dx.doi.org/10.1016/s0165-0114(97)00104-8.

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36

Huang, Bing, Chun-xiang Guo, Yu-liang Zhuang, Hua-xiong Li, and Xian-zhong Zhou. "Intuitionistic fuzzy multigranulation rough sets." Information Sciences 277 (September 2014): 299–320. http://dx.doi.org/10.1016/j.ins.2014.02.064.

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37

Pei, D., and T. Fan. "On generalized fuzzy rough sets." International Journal of General Systems 38, no. 3 (2009): 255–71. http://dx.doi.org/10.1080/03081070701553678.

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38

Du, Wen-Sheng, Bao-Qing Hu, and Yan Zhao. "On (⊥, ⊤)-Generalized Fuzzy Rough Sets." Fuzzy Information and Engineering 4, no. 3 (2012): 249–59. http://dx.doi.org/10.1007/s12543-012-0114-0.

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39

Hu, Qinghua, Daren Yu, and Maozu Guo. "Fuzzy preference based rough sets." Information Sciences 180, no. 10 (2010): 2003–22. http://dx.doi.org/10.1016/j.ins.2010.01.015.

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40

Tiwari, S. P., Shambhu Sharan, and Vijay K. Yadav. "Fuzzy Closure Spaces vs. Fuzzy Rough Sets." Fuzzy Information and Engineering 6, no. 1 (2014): 93–100. http://dx.doi.org/10.1016/j.fiae.2014.06.007.

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41

Theerens, Adnan, and Chris Cornelis. "Fuzzy rough sets based on fuzzy quantification." Fuzzy Sets and Systems 473 (December 2023): 108704. http://dx.doi.org/10.1016/j.fss.2023.108704.

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42

Mareay, R., Radwan Abu-Gdairi, and M. Badr. "Soft rough fuzzy sets based on covering." AIMS Mathematics 9, no. 5 (2024): 11180–93. http://dx.doi.org/10.3934/math.2024548.

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<abstract><p>Soft rough fuzzy sets ($ \mathcal SRFSs $) represent a powerful paradigm that integrates soft computing, rough set theory, and fuzzy logic. This research aimed to comprehensively investigate the various dimensions of $ \mathcal SRFSs $ within the domain of approximation structures. The study encompassed a wide spectrum of concepts, ranging from covering approximation structures and soft rough coverings to soft neighborhoods, fuzzy covering approximation operators, and soft fuzzy covering approximation operators. We introduced three models of $ \mathcal SRFSs $ based on
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43

Maji, Pradipta, and Sankar K. Pal. "RFCM: A Hybrid Clustering Algorithm Using Rough and Fuzzy Sets." Fundamenta Informaticae 80, no. 4 (2007): 475–96. https://doi.org/10.3233/fun-2007-80408.

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A hybrid unsupervised learning algorithm, termed as rough-fuzzy c-means, is proposed in this paper. It comprises a judicious integration of the principles of rough sets and fuzzy sets. While the concept of lower and upper approximations of rough sets deals with uncertainty, vagueness, and incompleteness in class definition, the membership function of fuzzy sets enables efficient handling of overlapping partitions. The concept of crisp lower bound and fuzzy boundary of a class, introduced in rough-fuzzy c-means, enables efficient selection of cluster prototypes. Several quantitative indices are
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44

Yaqoob, Naveed, Muhammad Aslam, and Kostaq Hila. "Rough Fuzzy Hyperideals in Ternary Semihypergroups." Advances in Fuzzy Systems 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/595687.

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The relations between rough sets and algebraic systems have been already considered by many mathematicians, and rough sets have been studied in various kinds of algebraic systems. This paper concerns a relationship between rough sets and ternary semihypergroups. We introduce the notion of rough hyperideals and rough bi-hyperideals in ternary semihypergroups. We also study fuzzy, rough, and rough fuzzy ternary subsemihypergroups (left hyperideals, right hyperideals, lateral hyperideals, hyperideals, and bi-hyperideals) of ternary semihypergroups.
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45

Tang, Weidong, Jinzhao Wu, and Dingwei Zheng. "On Fuzzy Rough Sets and Their Topological Structures." Mathematical Problems in Engineering 2014 (2014): 1–17. http://dx.doi.org/10.1155/2014/546372.

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The core concepts of rough set theory are information systems and approximation operators of approximation spaces. Approximation operators draw close links between rough set theory and topology. This paper is devoted to the discussion of fuzzy rough sets and their topological structures. Fuzzy rough approximations are further investigated. Fuzzy relations are researched by means of topology or lower and upper sets. Topological structures of fuzzy approximation spaces are given by means of pseudoconstant fuzzy relations. Fuzzy topology satisfying (CC) axiom is investigated. The fact that there
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46

INTAN, ROLLY, and MASAO MUKAIDONO. "GENERALIZED FUZZY ROUGH SETS BY CONDITIONAL PROBABILITY RELATIONS." International Journal of Pattern Recognition and Artificial Intelligence 16, no. 07 (2002): 865–81. http://dx.doi.org/10.1142/s0218001402002039.

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In 1982, Pawlak proposed the concept of rough sets with a practical purpose of representing indiscernibility of elements or objects in the presence of information systems. Even if it is easy to analyze, the rough set theory built on a partition induced by equivalence relation may not provide a realistic view of relationships between elements in real-world applications. Here, coverings of, or nonequivalence relations on, the universe can be considered to represent a more realistic model instead of a partition in which a generalized model of rough sets was proposed. In this paper, first a weak f
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47

Qin, Hua Ni, and Da Rong Luo. "Rule Extraction Based on Interval-Valued Rough Fuzzy Sets." Applied Mechanics and Materials 665 (October 2014): 668–73. http://dx.doi.org/10.4028/www.scientific.net/amm.665.668.

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A model of interval-valued rough fuzzy set combining interval-valued fuzzy set and rough set is investigated in this paper. Firstly, considering the deficiency of general sorting method between any interval-valued fuzzy numbers, an improved sorting method and a pair of new approximation operators about minimum and maximum are presented. Based on the improved operators, a model of interval-valued rough fuzzy set is established. At last, by using the modified model of interval-valued rough fuzzy set, a method of knowledge discovery in interval-valued fuzzy information systems is investigated.
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48

Yang, Jie, Taihua Xu, and Fan Zhao. "Modified Uncertainty Measure of Rough Fuzzy Sets from the Perspective of Fuzzy Distance." Mathematical Problems in Engineering 2018 (August 6, 2018): 1–11. http://dx.doi.org/10.1155/2018/4160905.

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As an extension of Pawlak’s rough sets, rough fuzzy sets are proposed to deal with fuzzy target concept. As we know, the uncertainty of Pawlak’s rough sets is rooted in the objects contained in the boundary region, while the uncertainty of rough fuzzy sets comes from three regions (positive region, boundary region, and negative region). In addition, in the view of traditional uncertainty measures, the two rough approximation spaces with the same uncertainty are not necessarily equivalent, and they cannot be distinguished. In this paper, firstly, a fuzziness-based uncertainty measure is propose
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49

Li, Qiao Yan, Yan Yan Chen, and Shao Yang Li. "Research on the N-Reduction of Fuzzy Covering Rough Sets." Applied Mechanics and Materials 556-562 (May 2014): 3682–85. http://dx.doi.org/10.4028/www.scientific.net/amm.556-562.3682.

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The granular reduction plays an important role in covering rough sets, and the aim of this paper is to explore the granular reduction of covering rough sets and fuzzy covering rough set Firstly, the covering rough sets based on neighborhood element and the N-reduction are introduced, and their properties are discussed; secondly, The definitions and properties of upper and lower approximation of fuzzy covering rough based on the neighborhood are given; Lastly, the N- reduction of fuzzy covering rough based on the neighborhood element is proposed, and we can get that covering rough sets remain t
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50

Močkoř, Jiří, Petr Hurtik, and David Hýnar. "Rough Semiring-Valued Fuzzy Sets with Application." Mathematics 10, no. 13 (2022): 2274. http://dx.doi.org/10.3390/math10132274.

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Abstract:
Many of the new fuzzy structures with complete MV-algebras as value sets, such as hesitant, intuitionistic, neutrosophic, or fuzzy soft sets, can be transformed into one type of fuzzy set with values in special complete algebras, called AMV-algebras. The category of complete AMV-algebras is isomorphic to the category of special pairs (R,R*) of complete commutative semirings and the corresponding fuzzy sets are called (R,R*)-fuzzy sets. We use this theory to define (R,R*)-fuzzy relations, lower and upper approximations of (R,R*)-fuzzy sets by (R,R*)-relations, and rough (R,R*)-fuzzy sets, and w
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