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Journal articles on the topic 'Vague Set'

Author: Grafiati

Published: 6 September 2023

Last updated: 25 July 2025

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1

Wang, Chao, Qi Hai Zhou, and Yan Li. "Transforming Model from Vague Set to Fuzzy Set Based on the DWCO Operator." Applied Mechanics and Materials 66-68 (July 2011): 2317–22. http://dx.doi.org/10.4028/www.scientific.net/amm.66-68.2317.

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In the research on transforming vague set to fuzzy set, it is the key question to carry on the judgment to the vague-degree's tendentiousness. Vague set's vague-degree expresses one kind of indefinite degree, therefore the question transforming vague set to fuzzy set may regard as the venture decision under the definite condition. When the policy-maker carries on the venture decision, the policy-maker's risk preferences decided by the subjective and objective condition is a very essential policy-making parameter. In this paper, introduce the risk-income balancing weight into the transforming m

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2

Soumitra, De, and Mishra Jaydev. "Neutrosophic Logic - A Better Approach to Execute Imprecise Queries Through the Neutrosophic Database Model." Indian Journal of Science and Technology 17, no. 29 (2024): 2981–91. https://doi.org/10.17485/IJST/v17i29.673.

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Abstract <strong>Objectives:</strong> The neutrosophic set has a more powerful ability to process imprecise information compared to the vague set. The objective of the authors is to process the uncertain query using vague as well as neutrosophic logic and compare the result. <strong>Methods:</strong> Authors have designed a new architecture to process uncertain queries using vague as well as neutrosophic sets. In the architecture at first, the imprecise query has been taken as input. From the inputted imprecise query uncertain data and fuzzy attributes are identified. Next, usin

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3

Yan, Ruixia, Jinliang Liu, and Bingxue Yao. "The connections of vague set and rough set." Kybernetes 41, no. 9 (2012): 1318–22. http://dx.doi.org/10.1108/03684921211275351.

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4

Wang, Chang. "Vague parameterized vague soft set theory and its decision making." Journal of Intelligent & Fuzzy Systems 33, no. 4 (2017): 2341–50. http://dx.doi.org/10.3233/jifs-17423.

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5

Zhang, Kun, Mei Yi Chen, and Zhuang Li. "Similarity Measure of Vague Set Based on Uncertainty." Applied Mechanics and Materials 602-605 (August 2014): 3850–53. http://dx.doi.org/10.4028/www.scientific.net/amm.602-605.3850.

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According to the experimental analysis, this work found out the defect of the formula for similarity measure between current Vague values, and redefined the similarity measure between Vague values. Therefore, a new formula subject to the new definition was put forward according to data mining and uncertainty of Vague value.

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6

Ramya Swetha, R., T. Anitha, and V. Amarendra Babu. "Vague Separation." International Journal of Engineering & Technology 7, no. 3.34 (2018): 654. http://dx.doi.org/10.14419/ijet.v7i3.34.19407.

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In this paper we are introducing VT1 space, vague haussdorff space (VT2) and then we derive every vague subspace of VT1 space is VT1 and also for VT2. And also we derive the Cartesian product of two vague closed sets is also vague closed set in the vague product topological space X x Y .Finally we define Vague limit point, Vague isolated point, Vague adherent point, Vague perfect, Vague derived set, vague exterior and also derive some theorems on this .

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7

Alhazaymeh, Khaleed, and Nasruddin Hassan. "Vague soft set relations and functions." Journal of Intelligent & Fuzzy Systems 28, no. 3 (2015): 1205–12. http://dx.doi.org/10.3233/ifs-141403.

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8

Al-Quran, Ashraf, and Nasruddin Hassan. "Neutrosophic vague soft expert set theory." Journal of Intelligent & Fuzzy Systems 30, no. 6 (2016): 3691–702. http://dx.doi.org/10.3233/ifs-162118.

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9

Alhazaymeh, Khaleed, and Nasruddin Hassan. "Generalized interval-valued vague soft set." Applied Mathematical Sciences 7 (2013): 6983–88. http://dx.doi.org/10.12988/ams.2013.310575.

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Alhazaymeh, Khaleed, and Nasruddin Hassan. "Possibility interval-valued vague soft set." Applied Mathematical Sciences 7 (2013): 6989–94. http://dx.doi.org/10.12988/ams.2013.310576.

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11

Singh, Prem Kumar. "Complex vague set based concept lattice." Chaos, Solitons & Fractals 96 (March 2017): 145–53. http://dx.doi.org/10.1016/j.chaos.2017.01.019.

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12

De, Soumitra, and Jaydev Mishra. "Neutrosophic Logic - A Better Approach to Execute Imprecise Queries Through the Neutrosophic Database Model." Indian Journal Of Science And Technology 17, no. 29 (2024): 2981–91. http://dx.doi.org/10.17485/ijst/v17i29.673.

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Objectives: The neutrosophic set has a more powerful ability to process imprecise information compared to the vague set. The objective of the authors is to process the uncertain query using vague as well as neutrosophic logic and compare the result. Methods: Authors have designed a new architecture to process uncertain queries using vague as well as neutrosophic sets. In the architecture at first, the imprecise query has been taken as input. From the inputted imprecise query uncertain data and fuzzy attributes are identified. Next, using the truth membership algorithm the truth membership valu

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13

Zhang, Hai Dong, and Yan Ping He. "Aximatics for Rough Set Model Based on Vague Relations." Advanced Materials Research 282-283 (July 2011): 283–86. http://dx.doi.org/10.4028/www.scientific.net/amr.282-283.283.

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This paper presents a general framework for the study of rough set approximation operators in vague environment in which both constructive and axiomatic approaches are used. In constructive approach, by means of a vague relation defined by us, a new pair of vague rough approximation operators is ﬁrst deﬁned. Also some properties about the approximation operators are then discussed. In axiomatic approach, an operator-oriented characterization of vague rough sets is proposed, that is, vague rough approximation operators are defined by axioms.

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14

Sakr, Hanan H., Abdisalam Hassan Muse, Mohamed S. Mohamed, and Saieed F. Ateya. "Applications on Bipolar Vague Soft Sets." Journal of Mathematics 2023 (March 31, 2023): 1–25. http://dx.doi.org/10.1155/2023/5467353.

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The purpose of this research is to interpolate bipolarity into the definition of the vague soft set. This gives a new more applicable, flexible, and generalized extension of the soft set, the fuzzy soft set, or even the vague soft set, which is the bipolar vague soft set. In addition, types of bipolar vague soft sets, as well as some new related concepts and operations are established with examples. Moreover, properties of bipolar vague soft sets including absorption, commutative, associative, distributive, and De Morgan’s laws are discussed in detail. Furthermore, a bipolar vague soft set-des

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15

Yavuz, Sanem, Serkan Onar, and Bayram Ali Ersoy. "More on the Vague Complex Subhypergroups." Fundamentals of Contemporary Mathematical Sciences 5, no. 2 (2024): 94–122. http://dx.doi.org/10.54974/fcmathsci.1373528.

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The concept of a vague complex set provides a more comprehensive perspective than that of a vague set. This article aims to investigate vague complex and anti-vague complex subhypergroups (Hv−subgroups), supported by various examples with the help of vague complex sets and hyperstructures. Moreover, we explore their properties and relationships with vague and anti-vague subhypergroups.

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16

Wang, Jian Hong, and Lei Liu. "Modeling Fuzzy Decision Fusion Based on Vague Set." Applied Mechanics and Materials 197 (September 2012): 7–12. http://dx.doi.org/10.4028/www.scientific.net/amm.197.7.

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As a further generalization of fuzzy set theory, the vague set theory can overcome the shortcomings of fuzzy set by describing the membership from two sides of both TRUE and FALSE, rather than only by a single membership value. Since vague sets can provide more information than fuzzy sets, it is superior in mathematical analysis of system with uncertainty. Thus, vague set is more powerful in the describing and processing of uncertain, inaccurate, even conflicting information. In this paper, a new method vague set-based is proposed to deal with fuzzy decision fusion problem. Compared with tradi

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17

Al-shboul, Abedallah, Khaleed Alhazaymeh, Kah Lun Wang, and Kok Bin Wong. "Fermatean Vague Soft Set and Its Application in Decision Making." Mathematics 12, no. 23 (2024): 3699. http://dx.doi.org/10.3390/math12233699.

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In this paper, we introduce the concept of Fermatean vague soft set and its operations, which are union, intersection, complement, subset, AND, and OR. An application of Fermatean vague soft set in decision-making according to the degree of preference is demonstrated. Fermatean vague soft set is profound, as it bridges gaps in traditional vague and soft set theories. By offering enhanced precision and flexibility, the Fermatean vague soft set provides decision-makers with a powerful tool to tackle challenges in fields like artificial intelligence, operational research, and engineering design,

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18

S., Cicily Flora* &. I. Arockiarani*. "SOME REMARKS ON BIPOLAR VAGUE S-SPACES." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 7, no. 4 (2018): 250–57. https://doi.org/10.5281/zenodo.1218529.

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In this work, we introduce the notion of S-space and almost S-space on bipolar vague topological space with some of its properties. The interrelations between newly defined spaces and other spaces are also discussed.

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19

Chinnadurai, V., G. Thirumurugan, and A. Arulselvam. "CUBIC VAGUE SOFT SET AND ITS APPLICATION." Advances in Mathematics: Scientific Journal 9, no. 4 (2020): 1569–75. http://dx.doi.org/10.37418/amsj.9.4.11.

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20

Vervaat, Wim. "Narrow and vague convergence of set functions." Statistics & Probability Letters 6, no. 5 (1988): 295–98. http://dx.doi.org/10.1016/0167-7152(88)90002-8.

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21

Bourahla, Mustapha. "Using Rough Set Theory for Reasoning on Vague Ontologies." International Journal of Intelligent Systems and Applications 14, no. 4 (2022): 21–36. http://dx.doi.org/10.5815/ijisa.2022.04.03.

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Web ontologies can contain vague concepts, which means the knowledge about them is imprecise and then query answering will not possible due to the open world assumption. A concept description can be very exact (crisp concept) or exact (fuzzy concept) if its knowledge is complete, otherwise it is inexact (vague concept) if its knowledge is incomplete. In this paper, we propose a method based on the rough set theory for reasoning on vague ontologies. With this method, the detection of vague concepts will insert into the original ontology new rough vague concepts where their description is define

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22

Zhang, Qinghua, Jin Wang, Guoyin Wang, and Hong Yu. "The approximation set of a vague set in rough approximation space." Information Sciences 300 (April 2015): 1–19. http://dx.doi.org/10.1016/j.ins.2014.12.023.

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23

Qin, Xiaoyan, Yi Liu, and Yang Xu. "Vague Congruences and Quotient Lattice Implication Algebras." Scientific World Journal 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/197403.

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The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given.

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24

Wang, Hai Feng, Hong E. Ren, Kun Zhang, and Hong Xu Wang. "Mining Methods Based on Vague Optimization Evaluation." Advanced Materials Research 659 (January 2013): 128–33. http://dx.doi.org/10.4028/www.scientific.net/amr.659.128.

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Method based on vague optimization evaluation is vague pattern recognition. There are six detailed steps of application. The first, Set up Techno-economic indicator system. Secondly set up preparative optimization scheme sets. Thirdly set up optimal scheme in theory. It is made up of each Techno-economic indicator optimal data. Fourthly transform techno-economic input data into vague data. The fifth, Calculating similarly measures. Similarity measures will be evaluated between preparative optimization scheme vague sets and optimal scheme in theory. The last is vague optimization evaluation. Th

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25

Alhazaymeh, Khaleed, Yousef Al-Qudah, Nasruddin Hassan, and Abdul Muhaimin Nasruddin. "Cubic Vague Set and its Application in Decision Making." Entropy 22, no. 9 (2020): 963. http://dx.doi.org/10.3390/e22090963.

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From the hybrid nature of cubic sets, we develop a new generalized hybrid structure of cubic sets known as cubic vague sets (CVSs). We also define the concept of internal cubic vague sets (ICVSs) and external cubic vague sets (ECVSs) with examples and discuss their interesting properties, including ICVSs and ECVSs under both P and R-Order. Moreover, we prove that the R and R-intersection of ICVSs (or ECVSs) need not be an ICVS (or ECVS). We also derive the different conditions for P-union (P-intersection, R and R-intersection) operations of both ICVSs (ECVSs) to become an ICVS (ECVS). Finally,

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26

Shijian, Gao, and Yu Jiankun. "Mining Spatial Co-location Patterns from Vague Set." Journal of Physics: Conference Series 1437 (January 2020): 012009. http://dx.doi.org/10.1088/1742-6596/1437/1/012009.

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27

Yatabe, Shunsuke, and Hiroyuki Inaoka. "On Evans's Vague Object from Set Theoretic Viewpoint." Journal of Philosophical Logic 35, no. 4 (2006): 423–34. http://dx.doi.org/10.1007/s10992-005-9022-7.

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28

BHARGAVI, YELLA, AKBAR REZAE, TAMMA ESWARLAL та SISTLA RAGAMAYI. "Vague Weak Interior Ideals of Γ-Semirings". Kragujevac Journal of Mathematics 49, № 5 (2024): 711–26. http://dx.doi.org/10.46793/kgjmat2505.711b.

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The notion of a ((complete-) normal) vague weak interior ideal on a (regular) Γ-semiring is deﬁned. It is proved that the set of all vague weak interior ideals forms a complete lattice. Also, a characterization theorem for a regular Γ-semiring in terms of vague weak interior ideals is derived. Another interesting consequence of the main result is that the cardinal of a non-constant maximal element in the set of all (complete-) normal vague weak interior ideals is 2.

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29

Kamesh, Kumar, and K. Sharma M. "Generalized Fuzzy Technique and its Consistent Assessment in Multicriteria Decision Making of Medical Decisions." Indian Journal of Science and Technology 17, no. 42 (2024): 4438–48. https://doi.org/10.17485/IJST/v17i42.3115.

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Abstract <strong>Objectives:</strong> The purpose of this study is to develop a decision-making expert system to assist the diagnostic decisions effectively. Generalized fuzzy sets (Vague sets) are used to model the uncertainty that exists in the process. <strong>Method:</strong> Initially, concepts of centroid and signed distance are generalized for Trapezoidal Vague Numbers (TVNs). This model introduces some desirable properties for the proposed distance measure. A multicriteria decision-making (MCDM) is introduced using trapezoidal vague numbers (TVNs) and other intuitions in

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30

Ghorbani, Shokoofeh. "Vague Filters of Residuated Lattices." Journal of Discrete Mathematics 2014 (September 10, 2014): 1–9. http://dx.doi.org/10.1155/2014/120342.

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Notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are investigated. The characterizations of (subpositive implicative, Boolean) vague filters is obtained. We prove that the set of all vague filters of a residuated lattice forms a complete lattice and we find its distributive sublattices. The relation among subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vague filters are equivalent to Boolean vague filters.

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31

Liu, Peide. "MULTI‐ATTRIBUTE DECISION‐MAKING METHOD RESEARCH BASED ON INTERVAL VAGUE SET AND TOPSIS METHOD." Technological and Economic Development of Economy 15, no. 3 (2009): 453–63. http://dx.doi.org/10.3846/1392-8619.2009.15.453-463.

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This paper proposed a method to resolve the multi‐attribute decision‐making problem using TOPSIS method based on attribute weights and attribute values are all interval vague value. Firstly, based on the operation rules of the interval Vague value, the interval Vague attribute value is made by weighted operation, and the ideal and negative ideal solutions are calculated based on the score function. Then the distance of interval Vague value is defined, as well as the distance between each project and the ideal, and negative ideal solutions. The relative adjacent degree is calculated by TOPSIS m

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32

Sivakumar, C., P. Maragatha Meenakshi, Aiyared Iampan, N. Rajesh, and Suganthi Mariyappan. "Logarithmic Pythagorean neutrosophic vague aggregating operators and their real-life applications." International Journal of Neutrosophic Science 23, no. 3 (2024): 111–30. http://dx.doi.org/10.54216/ijns.230310.

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This article examines Pythagorean neurosophic vague set (PyNVS) problems relevant to multiple attribute decision-making (MADM). Pythagorean vague set (PyVS) and neutrosophic set (NS) can be generalized into Pythagorean neutrosophic vague set (PyNVS). We discuss log Pythagorean neutrosophic vague weighted averaging (log PyNVWA), logarithmic Pythagorean neutrosophic vague weighted geometric (log PyNVWG), log generalized Pythagorean neurosophic vague weighted averaging (log GPyNVWA) and log generalized Pythagorean neutrosophic vague weighted geometric (log GPyNVWG). In this article, we define the

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33

Mishra, Jaydev, and Sharmistha Ghosh. "Uncertain Query Processing using Vague Set or Fuzzy Set: Which One Is Better?" International Journal of Computers Communications & Control 9, no. 6 (2014): 730. http://dx.doi.org/10.15837/ijccc.2014.6.500.

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34

Zhang, Qinghua, Yu Xiao, and Guoyin Wang. "A new method for measuring fuzziness of vague set (or intuitionistic fuzzy set)." Journal of Intelligent & Fuzzy Systems 25, no. 2 (2013): 505–15. http://dx.doi.org/10.3233/ifs-2012-0571.

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35

Reddy, B. Surender, S. Vijayabalaji, N. Thillaigovindan, and P. Balaji. "A Vague Rough Set Approach for Failure Mode and Effect Analysis in Medical Accident." International Journal of Analysis and Applications 21 (December 8, 2023): 134. http://dx.doi.org/10.28924/2291-8639-21-2023-134.

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Identifying the risk factor is a key to success for the Failure model. This paper presents a new FMEA (Failure Mode and Effect Analysis) in medical accident for the hospital management by using vague rough matrix information captured through vague rough set. Using β-product and β-complement product into fuzzy relation matrix and vague set to find the new lower and upper approximation of vague rough set. This method will provide an easy methodology to find the weight of risk factor. Using an example we validate our proposed approach.

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Wang, Lihui, Desheng Sun, Qingya Liu, and Le Yu. "Matching area selection of an underwater terrain navigation database with fuzzy multi-attribute decision making method." Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment 233, no. 4 (2018): 1133–40. http://dx.doi.org/10.1177/1475090218812517.

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Selecting suitable underwater terrain navigation matching areas is a prerequisite for building an underwater terrain navigation database, which is important for vehicles operating underwater. By using information features to evaluate underwater terrain matching areas, vague sets are proposed to evaluate matching performance. Mathematical models of matching area features are built and topographic factor eigenvalues are obtained. With the topographic factor eigenvalues, fuzzy relationships between factor sets and judge sets are calculated. Vague set uses membership functions and non-membership f

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37

Wang, Chang, and Yaya Li. "Topological Structure of Vague Soft Sets." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/504021.

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We introduce vague soft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions of vague soft open sets, vague soft closed sets, vague soft interior, vague soft closure, and vague soft boundary are introduced and their basic properties and relations are investigated. Furthermore, with the help of examples they established that some properties of topological spaces and soft topological spaces do not hold in vague soft topological spaces. Vague soft connectedness and vague soft compactness are also studied.

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38

BHUTANI, KIRAN R., and JOHN N. MORDESON. "SIMILARITY RELATIONS, VAGUE GROUPS, AND FUZZY SUBGROUPS." New Mathematics and Natural Computation 02, no. 03 (2006): 195–208. http://dx.doi.org/10.1142/s1793005706000488.

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We define vague groups in terms of similarity relations rather than fuzzy equalities. This yields a bijection between the set of all right-invariant similarity relations on a group and the set of all fuzzy subgroups of the group. Under this bijection, right-invariant and left-invariant similarity relations correspond to normal fuzzy subgroups. We show how this bijection allows for the transfer of results between vague groups and fuzzy subgroups. In particular, certain numerical invariants that characterize fuzzy subgroups of an Abelian group can be used to characterize vague groups.

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S. Anitha and A. Francina Shalini. "Similarity Measure of Plithogenic Cubic Vague Sets: Examples and Possibilities." Neutrosophic Systems with Applications 11 (October 21, 2023): 39–47. http://dx.doi.org/10.61356/j.nswa.2023.81.

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The crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets are the extension of the plithogenic set, in which elements are characterized by the number of attributes and each attribute can assume many values. To achieve more accuracy and precise exclusion, a contradiction or dissimilarity degree is specified between each attribute and the values of the dominating attribute. A plithogenic cubic vague set is a combination of a plithogenic cubic set and a vague set. The key tool for resolving problems with pattern recognition and clustering analysis is the similarity measure. In this research,

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Afzal, Farkhanda, Arif Mehmood, Samer Al Ghour, Mudasar Zafar, Hamzah Sakidin, and Saeed Gul. "Characterization of Bipolar Vague Soft S -Open Sets." Journal of Function Spaces 2022 (May 5, 2022): 1–13. http://dx.doi.org/10.1155/2022/5964872.

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This paper concerns the study of the concept of bipolar vague soft s -open set, bipolar vague soft s-interior, bipolar vague soft s-closer, and bipolar vague soft s-exterior in bipolar vague soft topological spaces. By using such concepts, some results are addressed in bipolar vague soft topological spaces. The engagements among these results are also addressed by using bipolar vague soft s -open sets.

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WANG, Peilong, Dandan LI, and Wei XU. "A Vague Set Based OWA Method for Talent Evaluation." 系统科学与信息学报(英文) 12, no. 4 (2024): 543–53. https://doi.org/10.21078/jssi-2023-0158.

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42

Feng, Lin, Tianrui Li, Da Ruan, and Shirong Gou. "A vague-rough set approach for uncertain knowledge acquisition." Knowledge-Based Systems 24, no. 6 (2011): 837–43. http://dx.doi.org/10.1016/j.knosys.2011.03.005.

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43

Zhang, Qingchuan, Guangping Zeng, Chaoen Xiao, and Yang Yue. "A rule conflict resolution method based on Vague set." Soft Computing 18, no. 3 (2013): 549–55. http://dx.doi.org/10.1007/s00500-013-1075-x.

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Selvachandran, Ganeshsree, and Abdul Razak Salleh. "RINGS AND IDEALS IN A VAGUE SOFT SET SETTING." Far East Journal of Mathematical Sciences (FJMS) 99, no. 2 (2015): 279–300. http://dx.doi.org/10.17654/ms099020279.

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Xu, Ke Hu, Jin Yu Chen, De Peng Kong, and Pu Fan. "Research on Vague Set of Target Threat Information Fusion Algorithm." Applied Mechanics and Materials 599-601 (August 2014): 1671–78. http://dx.doi.org/10.4028/www.scientific.net/amm.599-601.1671.

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Information fusion technology is one of the most active research areas currently, which is affected by the fact that the computer can only handle quantitative information and the result cannot reflect the actual feature of the target sometimes. This paper makes use of advantages of the vague set in dealing with uncertain information process to establish the target threat sequencing model based on vague set and take both quantitative and qualitative information of target into account. Using the improved scoring function, this paper comes up with the target threat sequence steps based on extreme

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Mehmood, Arif, Samer Al Ghour, Saleem Abdullah, Choonkil Park, and Jung Rye Lee. "A new approach to vague soft toplogical structures concerning soft points." Journal of Intelligent & Fuzzy Systems 42, no. 3 (2022): 1483–99. http://dx.doi.org/10.3233/jifs-210828.

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This paper concerns the study of the notion of vague soft β-open set and vague soft separation axioms in vague soft topological spaces. By using such notions and that of the vague soft pints, we study the separation axioms βi (with i = 0, 1, 2, 3, 4) in vague soft topological spaces. We give some peculiar examples about them and we prove some relationships between them. The relationship of βi (with i = , 1, 2, 3, 4) spaces with the closer of vague soft β-open set by means of soft points, vague soft countable spaces and their relationship with βi (with i = , 1, 2) spaces by means of soft points

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P., John Robinson, and Henry Amirtharaj E.C. "A Short Primer on the Correlation Coefficient of Vague Sets." International Journal of Fuzzy System Applications 1, no. 2 (2011): 55–69. http://dx.doi.org/10.4018/ijfsa.2011040105.

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Intuitionistic fuzzy sets and vague sets are generalizations of the concept of fuzzy sets. Various researchers have studied the vagueness of data through vague sets, and it was later demonstrated that vague sets are indeed intuitionistic fuzzy sets. Since its entry in the literature, vague set theory has received increased attention. Many real life problems involve information in the form of vague values, due to the increasing complexity of the socio-economic environment and the vagueness of the inherent subjective nature of human thinking. Instead of using point-based membership as in fuzzy s

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Yousef, Al-Qudah, Khaleed Alhazaymeh, Nasruddin Hassan, Hamza Qoqazeh, Mohammad Almousa, and Mohammad Alaroud. "Transitive Closure of Vague Soft Set Relations and its Operators." INTERNATIONAL JOURNAL of FUZZY LOGIC and INTELLIGENT SYSTEMS 22, no. 1 (2022): 59–68. http://dx.doi.org/10.5391/ijfis.2022.22.1.59.

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Hong, Dug Hun, and Chang-Hwan Choi. "Multicriteria fuzzy decision-making problems based on vague set theory." Fuzzy Sets and Systems 114, no. 1 (2000): 103–13. http://dx.doi.org/10.1016/s0165-0114(98)00271-1.

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Beaubouef, Theresa, Frederick E. Petry, and Roy Ladner. "Spatial data methods and vague regions: A rough set approach." Applied Soft Computing 7, no. 1 (2007): 425–40. http://dx.doi.org/10.1016/j.asoc.2004.11.003.

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