Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information .In this paper we develop an hybrid structure called “ rough neutrosophic sets” and studied their properties

Broumi, Said

Dhar, Mamoni

Smarandache, Florentin

Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information. In this paper we develop an hybrid structure called rough neutrosophic sets and studied
their properties

Said Broumi

Florentin Smarandache

Mamoni Dhar

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

Rough Neutrosophic Sets

Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information. In this paper we develop an hybrid structure called “ rough neutrosophic sets” and studied their properties

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Neutrosophic Sets and Systems, Vol. 3, 2014 Said Broumi, and Florentin Smarandache, Mamoni Dhar, Rough Neutrosophic  Sets Rough Neutrosophic Sets Said Broumi1,  Florentin Smarandache2 and Mamoni Dhar3 1 Faculty of Arts and Humanities, Hay El Baraka Ben M'sik Casablanca B.P. 7951, Hassan II University Mohammedia-Casablanca , Morocco 2Department of Mathematics, University of New Mexico,705 Gurley Avenue, Gallup, NM 87301, USA 3Department of Mathematics, Science College, Kokrajhar-783370, Assam, India Abstract. Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information .In this paper we develop an hybrid structure called “ rough neutrosophic sets” and studied their properties.Keywords: Rough set, rough neutrosophic set. 1 Introduction In 1982, Pawlak [1] introduced the concept of rough set (RS), as a formal tool for modeling and processing incomplete information in information systems. There are two basic elements in rough set theory, crisp set and equivalence relation ,which constitute the mathematical basis of RSs. The basic idea of rough set is based upon the approximation of sets by a pair of sets known as the lower approximation and the upper approximation of a set .Here, the lower and upper approximation operators are based on equivalence relation. After  Pawlak, there  has been many models built upon different aspect, i.e, universe, relations, object, operators by many scholars [2, 3,4,5,6,7]. Various notions that combine rough sets and fuzzy sets, vague set  and intuitionistic fuzzy sets are introduced ,such as rough fuzzy sets, fuzzy rough sets, generalize fuzzy rough, intuitionistic fuzzy rough sets, rough intuitionistic fuzzy sets, rough vagues sets. The theory of rough sets is based upon the classification mechanism, from which the classification can be viewed as an equivalence relation and knowledge blocks induced by it be a partition on universe.  One of the interesting generalizations of the theory of fuzzy  sets and intuitionistic fuzzy sets is the theory of neutrosophic sets introduced  by F.Smarandache [8,9]. Neutrosophic sets described by three functions : a membership function  indeterminacy function and a non-membership function that are independently related. The theorie of neutrosophic set have achieved great success in various areas such as medical diagnosis [10], database [11,12], topology[13],image processing [14,15,16], and decision making problem[17]. While the neutrosophic  set is a powerful tool to deal with indeterminate and inconsistent data, the theory of rough sets is a powerful mathematical tool to deal with incompleteness. Neutrosophic sets and rough sets are two  different topics ,none conflicts the other. Recently many researchers applied the notion of  neutrosophic sets to relations, group theory, ring theory,Soft set theory [23,24,25,26,27,28,29,30,31,32] and so on.The main objective of this study was to introduce a new hybrid intelligent structure called “ rough neutrosophic sets”. The significance of introducing hybrid set structures is that the computational techniques based on any one of theses structures alone will not always yield the best results but a fusion of two or more of them can often give better results. The rest of this paper is organized as follows. Some preliminary concepts required in our work are briefly recalled in section 2. In section 3 , the concept of rough neutrosophic sets is investigated. Section 4 concludes the paper. 2 .Preliminaries In this section we present some preliminaries which will be useful to our work in the next section . For more details the reader may refer to [1, 8,9,] Definition 2.1[8]. Let X be an universe of discourse, with a generic element in X denoted by x,  the neutrosophic (NS) set  is an object having the form  60Neutrosophic Sets and Systems, Vol. 3, 2014 Said Broumi, and Florentin Smarandache, Mamoni Dhar, Rough Neutrosophic  Sets A = {< x: , , >,x X}, where the functions , ,  : X→ ]−0, 1+[  define respectively the degree of membership (or Truth) , the degree of indeterminacy, and the degree of non-membership (or Falsehood) of the element x X to the set A with the condition.     −0 ≤  + + ≤ 3+.   (1)     From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of ]−0, 1+[. So instead of ] −0, 1+[ we need to take the interval [0, 1] for technical applications, because ]−0, 1+[ will be difficult to apply in the real applications  such as in scientific and engineering problems. For two NS ,  ={ <x ,  , > |  }   And ={<x ,  , > | } the relations are defined as follows: i.  if and only if ,  ,ii. ,  =,  =  and  = iii. ={ <x ,min(  , , max( > |  } iv. ={ <x ,max(  , , min( > |  } v. =={ <x ,  ,1- > |  } vi.  =(0 , 1, 1) and   =(1 , 0, 0) As an illustration ,let us consider the following example. Example 2.2 Assume that the universe of discourse U={x1,x2,x3},where  characterizes the capability, characterizes the trustworthiness and  indicates the prices of the objects. It may be further assumed that the values of ,  and  are in [0,1] and they are obtained from some questionnaires of some experts. The experts may impose their opinion in three components viz. the degree of goodness,  the degree of indeterminacy and that of poorness to explain the characteristics of the objects. Suppose A is a neutrosophic set (NS) of U, such that, A = {< , (0.3, 0.5 ,0.6) >,< , (0.3, 0.2 ,0.3)>,< , (0.3, 0.5 ,0.6)>}, where the degree of goodness of capability is 0.3, degree of indeterminacy of capability is 0.5 and degree of falsity of capability is 0.6 etc. Definition 2.3 [1] Let U be any non-empty set . Suppose R is an equivalence relation over U. For any non-null subset X of U, the sets (x)={x:   X} and (x)={x:   X  } are called the lower approximation and upper approximation, respectively of X, where the pair S = (U,R) is called an approximation space. This equivalent relation R is called indiscernibility relation. The pair A (X)=  ( (x), (x))  is called the rough set of X in S. Here  denotes the equivalence class of R containing x. Definition 2.4 [1]. Let A = ( , ) and B = ( , ) be two rough sets in the approximation space S = (U, R) . Then,  A ∪ B = ( ∪ ,  ∪ ), A ∩ B = (  ∩ ,  ∩ ), A ⊂ B if A ∩B = A , A = { U –  , U – }. 3 Rough Neutrosophic  Sets In this section we introduce the notion of rough neutrosophic sets by combining both rough sets and nuetrosophic sets. and some operations viz. union, intersection , inclusion and equalities over them. Rough neutrosophic set are the generalization of rough fuzzy sets [2 ] and rough intuitionistic fuzzy sets [ 22 ]. Definition 3.1. Let U be a non-null set and R be an equivalence relation on U. Let F be neutrosophic set in U with the membership function  ,indeterminacy function  and non-membership function  . The lower and the upper approximations of F in the approximation (U ,R) denoted by  (F) and F) are respectively defined asfollows: 61Neutrosophic Sets and Systems, Vol. 3, 2014 Said Broumi, and Florentin Smarandache, Mamoni Dhar, Rough Neutrosophic  Sets  (F)= { <x, (x) , (x) , (x)> |  ,x  U}, F) ={ <x, (x) , (x) , (x)> |  , x  U}, where: (x)=, (x)=  , = , (x)=   , (x)=, = , So     0  (x) + (x) + (x) 3 And    0  (x) + (x) + (x) 3 Where “  “ and “  “ mean  “max” and “min “ operators respectively, ,  and   are the membership ,indeterminacy and non-membership of y with respect to F. It is easy to see  that F) and (F) are twoneutrosophic sets in U ,thus NS mapping  ,  :N(U)  N(U) are, respectively, referred to as the upper and lower rough NS approximation operators, and the pair ( (F), F)) is called the rough neutrosophic set in ( U, R). From the above definition , we can see that (F) and F) have constant membership on the equivalence clasesof U . if (F) = F) ;i .e =, =  and = . For any  x  U. we call F a definable neutrosophic set in the approximation ( U ,R). it is easily to be proved that  Zero  neutrosophic set and unite neutrosophic sets are definable neutrosophic sets.  Let us consider a simple example in the following  Example 3.2 . Let U = { , , , , , , , } be the universe of discourse. Let R be an equivalence relation, where its partition of U is given by U/R ={{ , }, { , , }, { }, { , }}. Let N(F)  ={( , (0.2, 0.3, 0.4), ( , (0.3, 0.5, 0.4)), ( , (0.4, 0.6, 0.2)),( , (0.1, 0.3, 0.5)) be a neutrosophic set of U. By the definition 3.1, we obtain: (F) ={( , (0.2, 0.5, 0.4)), ( , (0.2, 0.5, 0.4)), ( , (0.4, 0.6, 0.2))}; F) ={( , (0.3, 0.3, 0.4)), ( , (0.3, 0.3, 0.4)), ( ,(0.4, 0.6, 0.2)),( , (0.1, 0.3, 0.5)), ( , (0.1, 0.3, 0.5))}: For another neutrosophic set N(G) ={( , (0.2, 0.3, 0.4)), ( , (0.2, 0.3, 0.4)), ( ,(0.4, 0.6, 0.2)} The lower approximation and upper approximation of N(G) are calculated as (G) ={( , (0.2, 0.3, 0.4)), ( , (0.2, 0.3, 0.4)), ( , (0.4, 0.6, 0.2))}; G) ={( , (0.2, 0.3, 0.4)), ( , (0.2, 0.3, 0.4)), ( ,(0.4, 0.6, 0.2)) Obviously (G) = G) is a definable neutrosophic set inthe approximation space (U, R) Definition 3.3. If N(F)= ( (F), F) is a roughneutrosophic set in (U,R) ,the rough complement of N(F) is the rough neutrosophic set denoted   N(F)=(, ) ,where  ,  are the complements of neutrosophic sets  (F) and F) respectively. ={ <x,  ,1- (x) , (x) > |  xU},And 62Neutrosophic Sets and Systems, Vol. 3, 2014 Said Broumi, and Florentin Smarandache, Mamoni Dhar, Rough Neutrosophic  Sets F) ={ <x, (x) ,1- (x) , (x)> |  xU}, Definition 3.4. If N( ) and N( ) are two rough neutrosophic set  of the neutrosophic sets and respectively in U ,then we define the following: i. N( ) = N( ) iff ( )= ( ) and )= ) ii. N( )  N( ) iff    ( ) ( ) and ) ) iii. N( )  N( ) = < ( ) ( ) , ) ) iv. N( )  N( ) = < ( ) ( ) , ) ) v. N( )  N( ) = < ( ) ( ) , )) vi. N( )  N( ) = < ( ) ( ) , )) If N , M, L are rough  neutrosophic sets in (U,R) ,then the results in the following proposition are straightforward from definitions Proposition 3.5: i. N(  N)=Nii. N  M =M  , N  M =M iii. (N  M )  L= N  (M  L )  and (N  M ) L= N  (M  L ) iv. (N  M )  L= (N  M ) (N  L )   and (N M )  L= (N  M ) (N  L ) De Morgan ‘s Laws are satisfied for neutrosophic sets : Proposition 3.6 i. (N( )  N( ))=(  N( ))   (  N( )  ii. (N( )  N( ))=(  N( )  )  (  N( )  Proof : i. (N( )  N( )) =  ( { ( ) ( )}  , { ) )} ) = ( { ( ) ( )}}  ,  {{ ) )}) =(  , ) =  ( } , }) = N( ))   (  N( )  ii. Similar to the proof of (i)Proposition 3.7: If  and  are two neutrosophic sets in U such that , then N( )   N( )  i. N ( N( )  N( ) ii. N( N( )  N( ) Proof: (x) = inf {  (x)  | x }         = inf ( max { (x), (x)  | x )  max { inf { (x) | x }, inf { (x) | x }} = max {  , } = (Similarly ,  ( (Thus,   (  ( )  ( ) We can also see that  (  ( )  ( ) 63Neutrosophic Sets and Systems, Vol. 3, 2014 Said Broumi, and Florentin Smarandache, Mamoni Dhar, Rough Neutrosophic  Sets Hence, N ( N( )  N( ) (ii) proof is similar to the proof of (i) Proposition 3.8: i.  ( ) = ) ii. ) =  ( ) iii.  ( ) ) Proof , according to definition 3.1, we can obtain i) ={ <x ,  , > |  }  ={ <x ,  ,1- > |  } )=  { <x, (x) ,1- (x) , (x)> |  , x  U} ) ={ <x, (x) ,1-(1- (x)) , (x)> |  , x  U}   = { <x, (x) , (x) , (x)> |  , x  U}, where =  , (x)=and (x) = Hence   ( ) = ) ii) The proof is similar to iiii) For any  ( ), we can have (x) =(x)=And =    Hence,  ( ) ) 4 Conclusions In this paper we have defined the notion of rough neutrosophic sets. We have also studied some properties on them and proved some propositions. The concept combines two different theories which are rough sets theory and neutrosophic theory. While neutrosophic set theory is mainly concerned with ,indeterminate and inconsistent information, rough set theory is with incompleteness; but both the theories deal with imprecision. Consequently , by the way they are defined, it is clear that rough neutrosophic sets can be utilized for dealing with both of indeterminacy  and incompleteness. References [1] Z. Pawlak , Rough Sets , Int. J. Comput. Inform. Sci. 11(1982),341-356. [2]  D. Dubios and H. Prade, “Rough fuzzy sets and fuzzy roughsets,” Int. J. Gen. Syst., vol. 17, pp. 191-208, 1990 [3] Z. Gong, B. Sun, and D. Chen, “Rough set theory  for theinterval-valued fuzzy information systems,” Inf. Sci., vol. 178, pp. 1968-1985, 2008. [4]  B. Sun, Z. Gong, and D. Chen, “Fuzzy rough set theory forthe interval-valued fuzzy information systems,” Inf. Sci., vol. 178, pp. 2794-2815, 2008. [5]  W.-Z. Wu, J.-S. Mi, and W.-X. Zhang, “Generalized  FuzzyRough Sets,” Inf. Sci., vol. 151, pp. 263-282, 2003. [6] Z. Zhang, “On interval type-2 rough fuzzy sets,” Knowledge-Based Syst., vol. 35, pp. 1-13, 2012. [7] J.-S. Mi, Y. Leung, H.-Y. Zhao, and T. Feng, “GeneralizedFuzzy Rough Sets determined by a triangular norm,” Inf. Sci., vol. 178, pp. 3203-3213, 2008. [8] F.Smarandache, “A Unifying Field in Logics. Neutrosophy: Neutrosophic  Probability, Set and Logic”. Rehoboth: American Research Press.1999. [9] F. Smarandache,  “ Linguistic Paradoxists and Tautologies”, Libertas Mathematica, University of Texas at Arlington, Vol. XIX (1999),pp.143-154. [10] Ansari, Biswas, Aggarwal, ” Proposal  for Applicability of Neutrosophic Set Theory in Medical AI”, 70 64Neutrosophic Sets and Systems, Vol. 3, 2014 Said Broumi, and Florentin Smarandache, Mamoni Dhar, Rough Neutrosophic  Sets International Journal of Computer Applications (0975 – 8887),Vo 27– No.5, (2011), pp.5-11. [11] M. Arora, R. Biswas,  U.S.Pandy, “Neutrosophic Relational Database Decomposition”, International Journal of Advanced Computer Science and Applications, Vol. 2, No. 8, (2011), pp.121-125. [12] M. Arora and R. Biswas, ” Deployment of Neutrosophic technology to retrieve answers for queries posed in natural language”, in 3rdInternational Conference on Computer Science and Information Technology ICCSIT, IEEE catalog Number CFP1057E-art, Vol. No. 3, (2010), pp.435-439. [13] F.G. Lupiáñez "On neutrosophic topology", Kybernetes, Vol. 37 Iss: 6, (2008), pp.797 - 800,Doi:10.1108/03684920810876990. [14] H. D. Cheng,& Y Guo. “A new  neutrosophic approach to image thresholding”. New Mathematics and Natural Computation, 4(3),  (2008), pp. 291–308. [15] Y.Guo & H. D. Cheng “New neutrosophic approach to image segmentation”. Pattern Recognition, 42, (2009), pp.587–595. [16]  M.Zhang,  L.Zhang, and H.D.Cheng. “A neutrosophic approach to image segmentation based on watershed method”. Signal Processing 5, 90, (2010), pp.1510-1517. [17] A. Kharal, “A Neutrosophic Multicriteria Decision Making Method”, New Mathematics & Natural Computation, to appear in Nov 2013. [20] A. Nakamura, Fuzzy rough sets, Note on Multiple-Valued Logic in Japan 9,1988,p. 1–8. [21] S. Nanda, S. Majumdar, Fuzzy rough sets, Fuzzy Sets and Systems 45,1992, p. 157–160. [ 22] K. V. Thomas, L. S. Nair, Rough intutionistic fuzzy sets in a lattice, Int. Math. Forum 6(27) ,(2011) 1327–1335 [23] S.Broumi and F. Smarandache, “Intuitionistic Neutrosophic Soft Set”, Journal of Information and Computing Science, England, UK ,ISSN 1746-7659,Vol. 8, No. 2, (2013) 130-140. [24] S.Broumi, “Generalized Neutrosophic Soft Set”, International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), ISSN: 2231-3605, E-ISSN : 2231-3117, Vol.3, No.2, (2013) 17-30 [25] S.Broumi and F.Smarandache,” More on Intuitionistic Neutrosophic Soft Sets”, Computer Science and Information Technology 1(4): 257-268, 2013  ;DOI: 10.13189/csit.2013.010404. [26] S. Broumi,I. Deli and F.Smarandache,” Relations on Interval Valued Neutrosophic Soft Sets”,Journal of New Results in Science,ISSN:1304-798,;2014,accepted. [27] S.Broumi, F.Smarandache, P. K. Maj, Intuitionistic Neutrosphic Soft Set over Rings, Mathematics and Statistics 2(3): 120-126, 2014, DOI: 10.13189/ms.2014.020303 [28] Said Broumi, and Florentin Smarandache, Several Similarity Measures of Neutrosophic Sets” ,Neutrosophic Sets and Systems,VOL1 ,2013,54-62. [29] S. Broumi, F. Smarandache , “Correlation Coefficient of Interval Neutrosophic set”, Periodical of Applied Mechanics and Materials, Vol. 436, 2013, with the title Engineering Decisions and Scientific Research in Aerospace, Robotics, Biomechanics, Mechanical Engineering and Manufacturing; Proceedings of the International Conference ICMERA, Bucharest, October 2013. [30] I. Deli, S.Broumi,” Neutrosophic soft sets and neutrosophic soft matrices based on decision making”,2014,(submitted) [31] S. Broumi,F.Smarandache,” More on Intuitionistic Neutrosophic Soft Sets”, Computer Science and Information Technology 1(4): 257-268, 2013  ;DOI: 10.13189/csit.2013.010404. [32] Deli, I.Interval-valued neutrosophic soft sets and its decision making http://arxiv.org/abs/1402.3130. Received: June 9th, 2014. Accepted: June 13th, 2014.65

Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information. In this paper we develop an hybrid structure called rough neutrosophic sets and studied their properties

ROUGH NEUTROSOPHIC SETS

PhilPapers

Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information. In this paper we develop an hybrid structure called “rough neutrosophic sets” and studied their properties

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italian journal of pure and applied mathematics { n. 32 2014 (493 502) 493ROUGH NEUTROSOPHIC SETSSaid BroumiFaculty of Arts and HumanitiesHay El Baraka Ben M'sik Casablanca B.P. 7951Hassan II University Mohammedia-CasablancaMoroccoFlorentin SmarandacheDepartment of MathematicsUniversity of New Mexico705 Gurley Avenue, Gallup, NM 87301USAMamoni DharDepartment of MathematicsScience CollegeKokrajhar-783370, AssamIndiaAbstract. Both neutrosophic sets theory and rough sets theory are emerging as power-ful tool for managing uncertainty, indeterminate, incomplete and imprecise information.In this paper we develop an hybrid structure called rough neutrosophic sets and studiedtheir properties.Keywords: Rough set, rough neutrosophic set.1. IntroductionIn 1982, Pawlak [1] introduced the concept of rough set (RS), as a formal tool formodeling and processing incomplete information in information systems. Thereare two basic elements in rough set theory, crisp set and equivalence relation,which constitute the mathematical basis of RSs. The basic idea of rough setis based upon the approximation of sets by a pair of sets known as the lowerapproximation and the upper approximation of a set. Here, the lower and upperapproximation operators are based on equivalence relation. After Pawlak, therehas been many models built upon dierent aspect, i.e., universe, relations, object,operators by many scholars [2], [3], [4], [5], [6], [7]. Various notions that combinerough sets and fuzzy sets, vague set and intuitionistic fuzzy sets are introduced,such as rough fuzzy sets, fuzzy rough sets, generalize fuzzy rough, intuitionisticfuzzy rough sets, rough intuitionistic fuzzy sets, rough vagues sets. The theory of494 s. broumi, f. smarandache, m. dharrough sets is based upon the classication mechanism, from which the classicationcan be viewed as an equivalence relation and knowledge blocks induced by it bea partition on universe.One of the interesting generalizations of the theory of fuzzy sets and intuitio-nistic fuzzy sets is the theory of neutrosophic sets introduced by F. Smarandache[8], [9]. Neutrosophic sets described by three functions: a membership functionindeterminacy function and a non-membership function that are independentlyrelated. The theory of neutrosophic set have achieved great success in variousareas such as medical diagnosis [10], database [11], [12], topology [13], image pro-cessing [14], [15], [16], and decision making problem [17]. While the neutrosophicset is a powerful tool to deal with indeterminate and inconsistent data, the theoryof rough sets is a powerful mathematical tool to deal with incompleteness.Neutrosophic sets and rough sets are two dierent topics, none conicts theother. Recently many researchers applied the notion of neutrosophic sets to re-lations, group theory, ring theory, soft set theory [23], [24], [25], [26], [27], [28],[29], [30], [31], [32] and so on. The main objective of this study was to introducea new hybrid intelligent structure called rough neutrosophic sets. The signicanceof introducing hybrid set structures is that the computational techniques basedon any one of theses structures alone will not always yield the best results but afusion of two or more of them can often give better results.The rest of this paper is organized as follows. Some preliminary conceptsrequired in our work are briey recalled in Section 2. In Section 3, the concept ofrough neutrosophic sets is investigated. Section 4 concludes the paper.2. PreliminariesIn this section we present some preliminaries which will be useful to our work inthe next section. For more details the reader may refer to [1], [8], [9].Denition 2.1. [8] Let X be an universe of discourse, with a generic element inX denoted by x, the neutrosophic (NS) set is an object having the formA = fhx : A(x); A(x); !A(x)i ; x 2 Xg;where the functions ; ; ! : X !] 0; 1+[ dene respectively the degree of member-ship (or Truth), the degree of indeterminacy, and the degree of non-membership(or Falsehood) of the element x 2 X to the set A with the condition(1)  0  A(x) + A(x) + !A(x)  3+:From a philosophical point of view, the neutrosophic set takes the value from realstandard or non-standard subsets of ] 0; 1+[. So, instead of ] 0; 1+[ we need totake the interval [0; 1] for technical applications, because ] 0; 1+[ will be dicultto apply in the real applications such as in scientic and engineering problems.rough neutrosophic sets 495For two NS,A = fhx; A(x); A(x); !A(x)i j x 2 Xg andB = fhx; B(x); B(x); !B(x)i j x 2 Xg;the relations are dened as follows:(i) A  B if and only if A(x)  B(x); A(x)  B(x), !A(x)  !B(x),(ii) A = B if and only if A(x) = B(x); A(x) = B(x), !A(x) = !B(x),(iii) A \B = fhx;min(A(x); B(x));max(A(x); B(x));max(!A(x); !B(x))i jx 2 Xg,(iv) A [B = fhx;max(A(x); B(x));min(A(x); B(x));min(!A(x); !B(x))i jx 2 Xg,(v) AC = fhx; !A(x); 1  A(x); A(x)i j x 2 Xg(vi) 0n = (0; 1; 1) and 1n = (1; 0; 0):As an illustration, let us consider the following example.Example 2.2. Assume that the universe of discourse U = fx1; x2; x3g; where x1characterizes the capability, x2 characterizes the trustworthiness and x3 indicatesthe prices of the objects. It may be further assumed that the values of x1; x2 andx3 are in [0; 1] and they are obtained from some questionnaires of some experts.The experts may impose their opinion in three components viz. the degree ofgoodness, the degree of indeterminacy and that of poorness to explain the cha-racteristics of the objects. Suppose A is a neutrosophic set (NS) of U , such that,A = fhx1; (0:3; 0:5; 0:6)i ; hx2; (0:3; 0:2; 0:3)i ; hx3; (0:3; 0:5; 0:6)ig;where the degree of goodness of capability is 0:3, degree of indeterminacy ofcapability is 0:5 and degree of falsity of capability is 0:6 etc.Denition 2.3. [1] Let U be any non-empty set. Suppose R is an equivalencerelation over U: For any non-null subset X of U , the setsA1(x) = fx : [x]R  Xg and A2(x) = fx : [x]R \X 6= ;gare called the lower approximation and upper approximation, respectively of X,where the pair S = (U;R) is called an approximation space. This equivalentrelation R is called indiscernibility relation.The pair A(X) = (A1(x); A2(x)) is called the rough set of X in S. Here [x]Rdenotes the equivalence class of R containing x.496 s. broumi, f. smarandache, m. dharDenition 2.4. [1] Let A = (A1; A2) and B = (B1; B2) be two rough sets in theapproximation space S = (U;R): Then,A [B = (A1 [B1; A2 [B2);A \B = (A1 \B1; A2 \B2);A  B if A \B = A; A = fU   A2; U   A1g:3. Rough neutrosophic setsIn this section we introduce the notion of rough neutrosophic sets by combi-ning both rough sets and nuetrosophic sets. and some operations viz. union,intersection, inclusion and equalities over them. Rough neutrosophic set are thegeneralization of rough fuzzy sets [2] and rough intuitionistic fuzzy sets [22].Denition 3.1. Let U be a non-null set and R be an equivalence relation on U .Let F be neutrosophic set in U with the membership function F , indetermi-nacy function F and non-membership function !F . The lower and the upperapproximations of F in the approximation (U;R) denoted by N(F ) and N(F ) arerespectively dened as follows:N(F ) = f< x; N(F )(x); N(F )(x); !N(F )(x) >j y 2 [x]R; x 2 Ug;N(F )) = f< x; N(F )(x); N(F )(x); !N(F )(x) >j y 2 [x]R; x 2 Ug;where:N(F )(x) =^y2[x]RF (y); N(F )(x) =_y2[x]RF (y); !N(F )(x) =_y2[x]R!F (y);N(F )(x) =_y2[x]RF (y); N(F )(x) =^y2[x]RF (y); !N(F )(x) =^y2[x]R!F (y):So0  N(F )(x) + N(F )(x) + !N(F )(x)  3andN(F )(x) + N(F )(x) + !N(F )(x)  3;where "_" and "^" mean "max" and "min" operators respectively, F (x), F (y)and !F (y) are the membership, indeterminacy and non-membership of y withrespect to F . It is easy to see that N(F ) and N(F ) are two neutrosophic sets inU , thus the NS mappings N;N : N(U ! N(U) are, respectively, referred to as theupper and lower rough NS approximation operators, and the pair (N(F ); N(F ))is called the rough neutrosophic set in (U;R):rough neutrosophic sets 497From the above denition, we can see that N(F ) and N(F ) have constantmembership on the equivalence classes of U , if N(F ) = N(F ); i.e.,N(F ) = N(F );N(F ) = N(F );!N(F ) = !N(F ):For any x 2 U; we call F a denable neutrosophic set in the approximation (U;R).It is easily to be proved that Zero ON neutrosophic set and unit neutrosophicsets 1N are denable neutrosophic sets. Let us consider a simple example in thefollowing.Example 3.2. Let U = fp1; p2; p3; p4; p5; p6; p7; p8g be the universe of discourse.Let R be an equivalence relation its partition of U is given byU=R = ffp1; p4g; fp2; p3; p6g; fp5g; fp7; p8gg:LetN(F )=f(p1; (0:2; 0:3; 0:4); (p4; (0:3; 0:5; 0:4)); (p5; (0:4; 0:6; 0:2));(p7; (0:1; 0:3; 0:5))gbe a neutrosophic set of U . By Denition 3.1, we obtain:N(F ) = f(p1; (0:2; 0:5; 0:4)); (p4; (0:2; 0:5; 0:4)); (p5; (0:4; 0:6; 0:2))g;N(F ) = f(p1; (0:2; 0:3; 0:4)); (p4; (0:2; 0:3; 0:4)); (p5; (0:4; 0:6; 0:2));(p7; (0:1; 0:3; 0:5)); (p8; (0:1; 0:3; 0:5))g:For another neutrosophic setsN(G) = f(p1; (0:2; 0:3; 0:4)); (p4; (0:2; 0:3; 0:4)); (p5; (0:4; 0:6; 0:2))g:The lower approximation and upper approximation of N(G) are calculated asN(G) = f(p1; (0:2; 0:3; 0:4)); (p4; (0:2; 0:3; 0:4)); (p5; (0:4; 0:6; 0:2))g;N(G) = f(p1; (0:2; 0:3; 0:4)); (p4; (0:2; 0:3; 0:4)); (p5; (0:4; 0:6; 0:2))g:Obviously N(G) = N(G) is a denable neutrosophic set in the approximationspace (U;R).Denition 3.3. If N(F ) = (N(F ); N(F )) is a rough neutrosophic set in (U;R),the rough complement of N(F ) is the rough neutrosophic set denoted  N(F ) =(N(F )c; N(F )c), where N(F )c; N(F )c are the complements of neutrosophic setsN(F ) and N(F ), respectively,N(F )c = f< x; !N(F ); 1  N(F )(x); N(F )(x) >j x 2 Ug;andN(F )c = f< x; !N(F ); 1  N(F )(x); N(F )(x) >j x 2 Ug:Denition 3.4. If N(F1) and N(F2) are two rough neutrosophic set of theneutrosophic sets F1 and F2 respectively in U , then we dene the following:498 s. broumi, f. smarandache, m. dhar(i) N(F1) = N(F2) i N(F1) = N(F2) and N(F1) = N(F2).(ii) N(F1)  N(F2) i N(F1)  N(F2) and N(F1)  N(F2).(iii) N(F1) [N(F2) =N(F1) [N(F2); N(F1) [N(F2):(iv) N(F1) \N(F2) =N(F1) \N(F2); N(F1) \N(F2):(v) N(F1) +N(F2) =N(F1) +N(F2); N(F1) +N(F2):(vi) N(F1) N(F2) =N(F1) N(F2); N(F1) N(F2):If N;M;L are rough neutrosophic set in (U;R); then the results in the followingproposition are straightforward from denitions.Proposition 3.5.(i)  N( N) = N(ii) N [M =M [N; N \M =M \N(iii) (N [M) [ L = N [ (M [ L) and (N \M) \ L = N \ (M \ L)(iv) (N [M)\L = (N [M)\ (N [L) and (N \M)[L = (N \M)[ (N \L).De Morgan 's Laws are satised for neutrosophic sets:Proposition 3.6.(i)  (N(F1) [N(F2)) = ( N(F1)) \ ( N(F2))(ii)  (N(F1) \N(F2)) = ( N(F1)) [ ( N(F2)):Proof. (i) (N(F1) [ N(F2)) = (fN(F1) [ N(F2)g; fN(F1) [ N(F2)g) =(fN(F1)[N(F2)g;fN(F1)[N(F2)g)=(fN(F1)[N(F2)gc; fN(F1)[N(F2)gc)= (fN(F1) \N(F2)g;fN(F1) \N(F2)g) = (N(F1)) \ ( N(F2)).(ii) Similar to the proof of (i).Proposition 3.7. If F1 and F2 are two neutrosophic sets in U such that F1  F2,then N(F1)  N(F2)(i) N(F1 [ F2)  N(F1) [N(F2);(ii) N(F1 \ F2)  N(F1) \N(F2).rough neutrosophic sets 499Proof.N(F1[F2)(x) = inffF1[F2)(x) j x 2 Xig= inf(maxfF1(x); F2(x) j x 2 Xig) maxfinffF1(x) j x 2 Xig; inffF2(x) j x 2 Xigg= maxfN(F1)(xi); N(F2)(xi)g= N(F1) [ N(F2)(xi):Similarly,N(F1[F2)(xi)  (N(F1) [ N(F2))(xi)!N(F1[F2)(xi)  (!N(F1) [ !N(F2))(xi)Thus,N(F1 [ F2)  N(F1) [N(F2):We can also see thatN(F1 [ F2) = N(F1) [N(F2):Hence,N(F1 [ F2)  N(F1) [N(F2):(ii) The proof of (ii) is similar to the proof of (i).Proposition 3.8.(i) N(F ) =  N( F )(ii) N(F ) =  N( F )(iii) N(F )  N(F ).Proof. According to Denition 3.1, we can obtain(i) F = fhx; F (x); F (x); !F (x)i j x 2 Xg F = fhx; !F (x); 1  F (x); F (x)i j jx 2 XgN( F ) =nDx; !N(F )(x); 1  N(F )(x); N(F )(x)Ej y 2 [x]R; x 2 Uo N( F ) =nDx; N(F )(x); 1 (1 N(F )(x)); !N(F )(x)Ejy 2 [x]R; x 2 Uo=nDx; N(F )(x); N(F )(x); !N(F )(x)Ej y 2 [x]R; x 2 UowhereN(F )(x) =^y2[x]RF (y); N(F )(x) =_y2[x]RF (y), !N(F )(x) =_y2[x]R!F (y):Hence N(F ) = N( F ).(ii) The proof is similar to the proof of (i).500 s. broumi, f. smarandache, m. dhar(iii) For any y 2 N(F ), we can haveN(F )(x) =^y2[x]RF (y) _y2[x]RF (y), N(F )(x) =_y2[x]RF (y) ^y2[x]RF (y)and !N(F )(x) =_y2[x]R!F (y) ^y2[x]R!F (y):Hence N(F )  N(F ):4. ConclusionIn this paper we have dened the notion of rough neutrosophic sets. We havealso studied some properties on them and proved some propositions. The conceptcombines two dierent theories which are rough sets theory and neutrosophictheory. While neutrosophic set theory is mainly concerned with, indeterminateand inconsistent information, rough set theory is with incompleteness; but boththe theories deal with imprecision. Consequently, by the way they are dened,it is clear that rough neutrosophic sets can be utilized for dealing with both ofindeterminacy and incompleteness.References[1] Pawlak, Z., Rough Sets, Int. J. Comput. Inform. Sci., 11 (1982), 341-356.[2] Dubios, D., Prade, H., Rough fuzzy sets and fuzzy rough sets, Int. J.Gen. Syst., vol. 17 (1990), 191-208.[3] Gong, Z., Sun, B., Chen, D., Rough set theory for the interval-valuedfuzzy information systems, Inf. Sci., vol. 178 (2008), 1968-1985.[4] Sun, B., Gong, Z., Chen, D., Fuzzy rough set theory for the interval-valued fuzzy information systems, Inf. Sci., vol. 178, pp. 2794-2815, 2008.[5] Wu, W.-Z., Mi, J.-S., Zhang, W.-X., Generalized Fuzzy Rough Sets,Inf. 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XIX (1999), 143-154.rough neutrosophic sets 501[10] Ansari, Biswas, Aggarwal, Proposal for Applicability of NeutrosophicSet Theory in Medical AI, International Journal of Computer Applications(0975-8887), vol. 27, no. 5 (2011), 5-11.[11] Arora, M., Biswas, R., Pandy, U.S., Neutrosophic Relational DatabaseDecomposition, International Journal of Advanced Computer Science andApplications, vol. 2, no. 8 (2011), 121-125.[12] Arora, M., Biswas, R., Deployment of Neutrosophic technology to re-trieve answers for queries posed in natural language, in 3rd InternationalConference on Computer Science and Information Technology ICCSIT, IEEEcatalog Number CFP1057E-art, 3 (2010), 435-439.[13] Lupiez, F.G., On neutrosophic topology, Kybernetes, 37 (6) (2008), 797-800, Doi:10.1108/03684920810876990.[14] Cheng, H.D., Guo, Y., A new neutrosophic approach to image threshol-ding, New Mathematics and Natural Computation, 4 (3) (2008), 291-308.[15] Guo, Y., Cheng, H.D., New neutrosophic approach to image segmenta-tion, Pattern Recognition, 42, (2009), 587-595.[16] Zhang, M., Zhang, L., Cheng, H.D., A neutrosophic approach to imagesegmentation based on watershed method, Signal Processing, 5 90, (2010),1510-1517.[17] Kharal, A., A Neutrosophic Multicriteria Decision Making Method, NewMathematics & Natural Computation, to appear in Nov 2013.[20] Nakamura, A., Fuzzy rough sets, Note on Multiple-Valued Logic in Japan,9 (1988), 1-8.[21] Nanda, S., Majumdar, S., Fuzzy rough sets, Fuzzy Sets and Systems, 45(1992), 157-160.[ 22] Thomas, K.V., Nair, L.S., Rough intutionistic fuzzy sets in a lattice, Int.Math. Forum, 6 (27) (2011), 1327-1335.[23] Broumi, S., Smarandache, F., Intuitionistic Neutrosophic Soft Set, Jour-nal of Information and Computing Science, England, UK, ISSN 1746-7659,vol. 8, no. 2 (2013), 130-140.[24] Broumi, S., Generalized Neutrosophic Soft Set, International Journal ofComputer Science, Engineering and Information Technology (IJCSEIT),ISSN: 2231-3605, E-ISSN: 2231-3117, vol. 3, no. 2 (2013), 17-30.[25] Broumi, S., Smarandache, F., More on Intuitionistic Neutrosophic SoftSets, Computer Science and Information Technology, 1 (4) (2013), 257-268;DOI: 10.13189/csit.2013.010404.502 s. broumi, f. smarandache, m. dhar[26] Broumi, S., Deli, I., Smarandache, F., Relations on Interval ValuedNeutrosophic Soft Sets, Journal of New Results in Science, ISSN: 1304-798;2014, accepted.[27] Broumi, S., Smarandache, F., Maj, P.K., Intuitionistic NeutrosphicSoft Set over Rings, Mathematics and Statistics, 2 (3) (2014), 120-126. DOI:10.13189/ms.2014.020303.[28] Broumi, S., Smarandache, F., Several Similarity Measures of Neutro-sophic Sets, Neutrosophic Sets and Systems, vol. 1 (2013), 54-62.[29] Broumi, S., Smarandache, F., Correlation Coecient of Interval Neu-trosophic set, Periodical of Applied Mechanics and Materials, vol. 436(2013), with the title Engineering Decisions and Scientic Research in Aero-space, Robotics, Biomechanics, Mechanical Engineering and Manufacturing;Proceedings of the International Conference ICMERA, Bucharest, October2013.[30] Deli, I., Broumi, S., Neutrosophic soft sets and neutrosophic soft matricesbased on decision making, 2014 (submitted).[31] Broumi, S., Smarandache, F., More on Intuitionistic Neutrosophic SoftSets, Computer Science and Information Technology, 1 (4) (2013), 257-268;DOI: 10.13189/csit.2013.010404.[32] Deli, I., Interval-valued neutrosophic soft sets and its decision making,http://arxiv.org/abs/1402.3130Accepted: 22.04.2014

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Broumi, Sadi

Dhar, Manomi

University of New Mexico

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Rough Neutrosophic Sets

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