A rough fuzzy set is the result of approximation of a fuzzy set with respect to a crisp approximation space. It is mathematical tool for the knowledge discovery in the fuzzy information systems. In this paper, we introduce the concepts of rough standard neutrosophic sets, standard neutrosophic information system and give the knowledge discovery on standard neutrosophic information system based on rough standard neutrosophic sets

BUI CONG CUONG

Florentin Smarandache

NGUYEN XUAN THAO

English

ZENODO

J Fundam Appl Sci. 2018, 10(4S), 615-622                                                                                             615  Rough Standard Neutrosophic Sets: an Application on standard Neutrosophic Information Systems  NGUYEN XUAN THAO1*, BUI CONG CUONG2, FLORENTIN SMARANDACHE3  1 Faculty of information technology, Vietnam national university of agriculture.  E-mail:nxthao2000@gmail.com  2 Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam.  E-mail: bccuong@gmail.com  3 Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA.  E-mail: fsmarandache@gmail.com   Published online: 5 March 2018 Abstract:    A rough fuzzy set is the result of approximation of a fuzzy set with respect to a crisp approximation space. It is mathematical tool for the knowledge   discovery   in   the   fuzzy   information systems. In this paper, we introduce the concepts of rough standard neutrosophic sets, standard neutrosophic information system and give the knowledge discovery on standard neutrosophic information system based on rough standard neutrosophic sets.  Keywords:  rough  set,  standard  neutrosophic  set, rough  standard  neutrosophic  set,  standard neutrosophic information systems  1.   INTRODUCTION  Rough set theory was introduced by Pawlak in 1980s [1]. It becomes a usefully mathematical tool for data mining, especially for redundant and uncertain data.  At  first,  the  establishment  of  the rough  set theory is  based  on  equivalence  relation.  The  set  of equivalence classes of the universal set, obtained by an equivalence    relation,    is    the    basis    for    the construction of upper and lower approximation of the subset of universal set.  Fuzzy set theory was introduced by Zadeh since 1965 [2]. Immediately, it became a useful method to study in the problems of imprecision and uncertainty. Since, a lot of new theories treating imprecision and uncertainty   have   been   introduced.   For   instance,      doi: http://dx.doi.org/10.4314/jfas.v10i4s.84Intuitionistic fuzzy sets were introduced in1986, by K. Atanassov [3], which is a generalization of the notion of a fuzzy set. When fuzzy set give the degree of membership of an element in a given set, Intuitionistic fuzzy set give a degree of membership and a degree of non-membership of an element in a given set. In 1999 [14], Sarandache gave the concept of neutrosophic set which generalized fuzzy set and intuitionistic fuzzy set. This new concept is difficult to apply in the real appliction.  It  is  a  set  in which  each  proposition  is estimated to have a degree of truth (T), adegree of indeterminacy (I) and a degree of falsity (F).  Over time, the subclass of neutrosophic sets was proposed. They are also more advantageous in the practical application. Wang et al. [15] proposed interval neutrosophic sets and some operators of them. Wang et al. [16] proposed a single valued neutrosophic set as an instance of the neutrosophic set accompanied with various set theoretic operators and properties. Ye [17] defined the concept of simplified neutrosophic sets, It is  a  set  where each element  of  the  universe  has a degree of truth, indeterminacy and falsity respectively and which lies between [0, 1] and some operational laws for simplified neutrosophic sets and to propose two aggregation operators, including a simplified neutrosophic  weighted  arithmetic  average  operator and a simplified neutrosophic weighted geometric average operator. In 2013, Cuong and Kreinovich introduced the concept of picture fuzzy set [4], in which   a   given   set   were   to   be   in   with   three  Research Article Special Issue J Fundam Appl Sci. 2018, 10(4S), 615-622                                                                                             616  memberships: a degree of positive membership, a degree  of  negative  membership,  and  a  degree  of neutral membership of an element in this set. After that, Son given the application of the picture fuzzy set in the clustering problem [5]. We also regard picture fuzzy set as standard neutrosophic set.  In addition, combining rough set and fuzzy set has also many interesting results. The approximation of rough (or fuzzy) sets in fuzzy approximation space give us the fuzzy rough set [6, 7, 8]; and the approximation of fuzzy sets in crisp approximation space give us the rough fuzzy set [6, 7]. Wu et al, [8] present a general framework for the study of fuzzy rough sets in both constructive and axiomatic approaches. By the same, Wu and Xu were investigated the fuzzy topological structures on the rough fuzzy sets [9], in which both constructive and  axiomatic  approaches are  used.  In 2012, Xu and Wu were also investigated the rough intuitionistic  fuzzy  set  and  the  intuitionistic  fuzzy topologies  in  crisp  approximation  spaces  [10].  In 2015, Thao et al. introduces the rough picture fuzzy set is the result of approximation of a picture fuzzy set with respect to a crisp approximation space [12].      In this paper, we introduce the concept of standard neutrosophic  information  system,  study  the knowledge discovery of standard neutrosophic information system based on rough standard neutrosophic sets. The remaining part of this paper is organized as following: we recall basic notions of rough   set,   standard   neutrosophic   set   and   rough standard neutrosophic set on the crisp approximation    space,  respectively,  in  section  2  and  section  3.  In                 section 4, we introduce the basic concepts of standard  neutrosophic    information    system.    Finally,    we investigate   the   knowledge   discovery   of   standard neutrosophic information system and the knowledge reduction and extension of the standard neutrosophic information  system  in   section   5  and   section  6, respectively.  II. BASIC NOTIONS OF SN SET AND ROUGH SET  In  this  paper,  we  denote      be  a  nonempty set called  the  universe  of  discourse.  The  class  of  all subsets of    will be denoted by          and the class of all fuzzy subsets of    will be denoted by  Definition 1. [4]. A standard neutrosophic set (SN - set)     on the universe    is an object of the form     J Fundam Appl Sci. 2018, 10(4S), 615-622                                                                                             617     J Fundam Appl Sci. 2018, 10(4S), 615-622                                                                                             618       J Fundam Appl Sci. 2018, 10(4S), 615-622                                                                                             619              J Fundam Appl Sci. 2018, 10(4S), 615-622                                                                                             620          J Fundam Appl Sci. 2018, 10(4S), 615-622                                                                                             621            VII. CONCLUSION  In this paper, we introduce the concept of SN- information system, study the knowledge discovery of standard  neutrosophic  information  system based  on rough SN- sets. We investigate some problems of the knowledge discovery of SNIS and the knowledge reduction and extension of the SNIS in section 6. In the future, we introduce the application of this study in the practical problems.  References  [1] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, vol. 11, no.5 , pp 341 - 356, 1982.  [2] L. A. Zadeh, Fuzzy Sets, Information and Control, Vol. 8, No. 3 (1965), p 338-353.  [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy set and systems, vol.20, pp.87-96, 1986.  [4] B.C. Cuong, V. Kreinovick, Picture fuzzy sets - a new concept for computational intelligence problems, Published in proceedings of the third world congress on information and communication technologies WICT'2013, Hanoi, Vietnam, December 15-18, pp 1- 6, 2013.  [5]  L.H.Son,  DPFCM:  A  novel  distributed  picture fuzzy clustering method on picture fuzzy sets, Expert systems with applications 42, pp 51-66, 2015.  [6] D. Dubois, H Prade, Rough fuzzy sets and fuzzy rough sets, International journal of general systems, Vol. 17, p 191-209, 1990. [7]  Y.Y.Yao,  Combination  of rough  and fuzzy sets based on   level sets, Rough sets and Data mining: analysis   for   imprecise   data,   Kluwer   Academic Publisher, Boston, p 301 - 321, 1997.  [8] W. Z. Wu, J. S. Mi, W. X. Zhang, Generalized fuzzy rough sets, Information Sciences 151, p. 263- 282, 2003.  [9] W. Z. Wu, Y. H. Xu, On fuzzy topological structures of rough fuzzy sets, Transactions on rough sets  XVI,  LNCS  7736,  Springer  - Verlag  Berlin Heidelberg, p 125-143, 2013. [10] Y.H. Xu, W.Z. Wu, Intuitionistic fuzzy topologies in  crisp  approximation  spaces,  RSKT  2012,  LNAI 7414, © Springer - Verlag Berlin Heidelberg, pp 496- 503, 2012.  [11] B. Davvaz, M. Jafarzadeh, Rough intuitionistic fuzzy information systems, Fuzzy information and Engineering, vol.4, pp 445-458, 2013. J Fundam Appl Sci. 2018, 10(4S), 615-622                                                                                             622  [12] N.X. Thao, N.V. Dinh, Rough picture fuzzy set and picture fuzzy topologies, Science computer and Cybernetics, Vol 31, No 3 (2015), pp 245-254.  [13] B. Sun, Z. Gong, Rough fuzzy set in generalized approximation space, Fifth Int. Conf. on Fuzzy Systems and Knowledge Discovery, IEEE computer society 2008, pp 416-420. [14] F. Smarandache, A unifying field in logics. Neutrosophy: Neutrosophic probability, set and logic, American Research Press, Rehoboth 1999.  [15] H. Wang, F. Smarandache, Y.Q. Zhang et al., Interval  neutrosophic  sets  and  logic:  Theory  and applications in computing, Hexis, Phoenix, AZ 2005. [16] H. Wang, F. Smarandache, Y.Q. Zhang, et al., Single   valued   neutrosophic   sets,   Multispace   and Multistructure 4 (2010), 410-413.  [17] J. Ye,  A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets, Journal of Intelligent & Fuzzy Systems 26 (2014) 2459-2466.  [18] P. Majumdar, Neutrosophic sets and its applications to decision making, Computation intelligentce for big data analysis (2015), V.19, pp 97- 115.  [19] J.  Peng,  J.  Q.  Wang, J.  Wang,  H.  Zhang,  X. Chen, Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems, International journal of systems science (2016), V.47, issue 10, pp 2342-2358. 

Rough Standard Neutrosophic Sets: an Application on standard Neutrosophic Information Systems

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Rough Standard Neutrosophic Sets: an Application on standard Neutrosophic Information Systems

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