Rough Bipolar Neutrosophic Set

Bipolar neutrosophic set theory and rough neutrosophic set theory are emerging as powerful tool for dealing with uncertainty, and indeterminate, incomlete, and inprecise information. In the present study we develop a hybrid structure called “rough bipoar neutrsophic set”. In the study, we define rough bipoar neutrsophic set and define union, complement, intersection and containment of rough bipolar neutrosophic sets. INTRODUCTION The notion of fuzzy set which is non-statistical in nature was introduced by Zadeh in 1965[1] to deal with uncertainty. Fuzzy set has been applied in many real applications to handle uncertainty. In 1986, Atanassov [2] extended the concept of fuzzy set to intuitionistic fuzzy set by introducing the degree non-membership as an independent component. In 1998, Smarandache [3] grounded the concept of degree of indeterminacy as independent component and defined neutrosophic set. In 2005, Wang et al. [4] introduced the concept of single valued neutrosophic set (SVNS) which is an instance of a neutrosophic set to deal real scientific and engineering applications. Lee [5] introduced the concept of bipolar fuzzy sets, as an extension of fuzzy sets. In bipolar fuzzy sets the degree of membership is extended from [0, 1] to [-1, 1]. In a bipolar fuzzy set, if the degree of membership of an element is zero, then we say the element is unrelated to the corresponding property, the membership degree (0, 1] of an element specifies that the element somewhat satisfies the property, and the membership degree [−1, 0) of an element implies that the element somewhat satisfies the implicit counter-property [6]. In 2014, Broumi et al. [7, 8] presented the concept rough neutrosophic set to deal indeterminacy in more flexible way. Pramanik and Mondal [9, 10] and Mondal and Pramanik [11, 12, 13] studied different applications of rough neutrosophic sets in decision making. Deli et al. [14] defined bipolar neutrosophic set and showed numerical example for multi-criteria decision making problem. In this paper we combine bipolar neutrosophic set and rough neutrosophic set and define rough bipolar neutrosophic set. We define the union, complement, intersection and containment of rough bipolar neutrosophic sets. The rest of the paper has been organized as follows. Section 2 presents mathematical preliminaries of fuzzy bipolar set, neutrosophic set, single valued neutrosophic set, bipolar neutrosophic set, and rough neutrosophic set. Section 3 is devoted to define rough bipolar neutrosophic set. SOME RELEVANT DEFINITIONS In this section we recall some basic definitions of bipolar valued fuzzy set, neutrosophic set, single valued neutrosophic sets and rough neutrosophic set. [Pramanik et al., 3(6): June, 2016] ISSN 2349-4506 Impact Factor: 2.545 Global Journal of Engineering Science and Research Management http: // www.gjesrm.com © Global Journal of Engineering Science and Research Management [72] Definition 2.1 Bipolar valued fuzzy set [5] Let U be the universe of discourse. Then a bipolar valued fuzzy set B on U is defined by positive membership function , B   i.e. B : ] 1 , 0 [ U and a negative membership function , B  i.e. : -B  ] 0 , 1 [ U   . Mathematically a bipolar valued fuzzy set is represented by B =   ) z ( ), z ( : z B B   U z  Definition 2.2 Neutrosophic set [3] Let U be the universe of discourse. Then a neutrosophic set S in U with generic elements x is characterized by a truth membership function TS(x), an indeterminacy membership function IS(x) and a falsity membership function FS(x). There is no restriction on TS(x), IS(x) and FS(x) other than they are subsets of     1 , 0 that is TS(x):  N     1 , 0 ; IS(x):  N     1 , 0 ; FS(x):  N     1 , 0 Therefore,          3 ) X ( F sup ) X ( I sup ) X ( T sup ) X ( F inf ) X ( I inf ) X ( T inf 0 S S S S S S . Definition 2.3 Single-valued neutrosophic set [4] Let U be a universal space of points (objects) with a generic element of U denoted by x. A single valued neutrosophic set S is characterized by a truth membership function ) x ( TS , a falsity membership function ) x ( FS and indeterminacy function ) x ( IS with ) x ( TS , ) x ( FS , ) x ( IS    1 , 0 for all x in U. When U is continuous, a SNVS S can be written as follows:           x S S S U x , x x F , x I , x T S and when U is discrete, a SVNS S can be written as follows:       U x , x x F , x I , x T S S S S     There is no restriction on TS(x), IS(x) and FS(x) other than they are subsets of [0, 1] that is TS(x):  N [0, 1]; IS(x):  N [0, 1]; FS(x):  N [0, 1] Therefore, 3 ) X ( F sup ) X ( I sup ) X ( T sup ) X ( F inf ) X ( I inf ) X ( T inf 0 S S S S S S        Definition 2.4 bipolar neutrosophic set [14] A bipolar neutrosophic set A in Z is defined as an object of the form A={< T(z), I(z), F(z), T(z), I(z), F(z) >: z ∈ Z}, where ,T, I, F : X [0, 1] and , T, I, F: X  [-1, 0]. The positive membership degree T(z), I(z), and F(z) denote the truth membership, indeterminate membership and false membership respectively of an element z ∈ Z corresponding to a bipolar neutrosophic set A. The negative membership degree T(z), I(z), and F(z) denote the truth membership, indeterminate membership and false membership respectively of an element z ∈ Z to some implicit counter-property corresponding to a bipolar neutrosophic set A. Definition 2.5 [14] Let, B1 = {x, ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T 1 B 1 B 1 B 1 B 1 B 1 B        x U} and B2 = {x, ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T 2 B 2 B 2 B 2 B 2 B 2 B        x U} be two BNSs. Then B1  B2 if and only if ) x ( T 1 B   ) x ( T 2 B  , ) x ( I 1 B   ) x ( I 2 B  , ) x ( F 1 B   ) x ( F 2 B  ; ) x ( T 1 B   ) x ( T 2 B  , ) x ( I 1 B   ) x ( I 2 B  , ) x ( F 1 B   ) x ( F 2 B  for all x U. Definition 2.6 [14] Assume that B1 = {x, ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T 1 B 1 B 1 B 1 B 1 B 1 B        x U} and B2 = {x, ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T 2 B 2 B 2 B 2 B 2 B 2 B        x U} be two BNSs. Then B1 = B2 if and only if ) x ( T 1 B  = ) x ( T 2 B  , ) x ( I 1 B  = ) x ( I 2 B  , ) x ( F 1 B  = ) x ( F 2 B  ; ) x ( T 1 B  = ) x ( T 2 B  , ) x ( I 1 B  = ) x ( I 2 B  , ) x ( F 1 B  = ) x ( F 2 B  for all x U. [Pramanik et al., 3(6): June, 2016] ISSN 2349-4506 Impact Factor: 2.545 Global Journal of Engineering Science and Research Management http: // www.gjesrm.com © Global Journal of Engineering Science and Research Management [73] Definition 2.7 [14] Assume that B = {x, ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T B B B B B B        x U} be a BNS. The complement of B is denoted by B and is defined by ) x ( T c B  = {1} ) x ( TB  , ) x ( I c B  = {1} ) x ( IB  , ) x ( F c B  = {1} ) x ( FB  ; ) x ( T c B  = {1} ) x ( TB  , ) x ( I c B  = {1} ) x ( IB  , ) x ( F c B  = {1} ) x ( FB  for all x U. Definition 2.8 [14] Assume that B1 = {x, ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T 1 B 1 B 1 B 1 B 1 B 1 B        x U} and B2 = {x, ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T 2 B 2 B 2 B 2 B 2 B 2 B        x U} be two BNSs. Then their union B1B2 is defined as follows: B1B2 = {Max ( ) x ( T 1 B  , ) x ( T 2 B  ), 2 ) x ( I ) x ( I 2 B 1 B    , Min ( ) x ( F 1 B  , ) x ( F 2 B  ), Min ( ) x ( T 1 B  , ) x ( T 2 B  ), 2 ) x ( I ) x ( I 2 B 1 B    , Max ( ) x ( F 1 B  , ) x ( F 2 B  )} for all x U. Definition 2.9 [14] Assume that B1 = {x, ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T 1 B 1 B 1 B 1 B 1 B 1 B        x U} and B2 = {x, ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T 2 B 2 B 2 B 2 B 2 B 2 B        x U} be two BNSs. Then their intersection B1B2 is defined as follows: B1B2 = {Min ( ) x ( T 1 B  , ) x ( T 2 B  ), 2 ) x ( I ) x ( I 2 B 1 B    , Max ( ) x ( F 1 B  , ) x ( F 2 B  ), Max ( ) x ( T 1 B  , ) x ( T 2 B  ), 2 ) x ( I ) x ( I 2 B 1 B    , Min ( ) x ( F 1 B  , ) x ( F 2 B  )}for all x U. Definition 2.10: Definitions of rough neutrosophic set [7, 8] Let Z be a non-null set and R be an equivalence relation on Z. Let P be neutrosophic set in Z with the membership function , TP indeterminacy function P I and non-membership function P F . The lower and the upper approximations of P in the approximation (Z, R) denoted by   P N and   P N can be respectively defined as follows:     , Z x , x z / ) x ( F ), x ( I ), x ( T , x P N R ) P ( N ) P ( N ) P ( N          Z ∈ x , x ∈ z / ) x ( F ), x ( I ), x ( T , x P N R ) P ( N ) P ( N ) P ( N    Where,     z T x ) x ( T P R z ) P ( N   ,     z I x ) x ( I P R z ) P ( N   ,     z F x ) x ( F P R z ) P ( N   ,     z T x ) x ( T P R z ) P ( N   ,     z T x ) x ( I P R z ) P ( N   ,     z I x ) x ( F P R z ) P ( N   So, 3 ) x ( F ) x ( I ) x ( T 0 ) P ( N ) P ( N ) P ( N     and 3 ) x ( F ) x ( I ) x ( T 0 ) P ( N ) P ( N ) P ( N     Where  and  denote “max” and “min’’ operators respectively,   z TP ,   z IP and   z FP are the membership, indeterminacy and non-membership of z with respect to P. It is easy to see that   P N and   P N are two neutrosophic sets in Z. Thus NS mapping , N N : N(Z) N(Z) are, respectively, referred to as the lower and upper rough NS approximation operators, and the pair )) P ( N ), P ( N ( is called the rough neutrosophic set in ( Z, R). From the above definition, it is seen that ) P ( N and ) P ( N have constant membership on the equivalence classes of R if ); P ( N ) P ( N  i.e. ), x ( T ) x ( T ) P ( N ) P ( N  ), x ( I ) x ( I ) P ( N ) P ( N  ) x ( F ) P ( N  x ( F ) P ( N ) for all x belongs to Z. [Pramanik et al., 3(6): June, 2016] ISSN 2349-4506 Impact Factor: 2.545 Global Journal of Engineering Science and Research Management http: // www.gjesrm.com © Global Journal of Engineering Science and Research Management [74] ROUGH BIPOLAR NEUTROSOPHIC SETS (RBNS) In this section we introduce the notion of rough bipolar neutrosophic sets by combining rough neutrosophic set and bipolar neutrosophic set. We define union, complement, intersection and containment of rough bipolar neutrosophic sets. Definition 3.1: Rough bipolar neutrosophic sets Let Z be a non-null set and R be an equivalence relation on Z. Let B be a bipolar neutrosophic set in Z. The positive membership degrees T(z), I(z), and F(z) respectively denote the truth membership, indeterminate membership and falsity membership respectively of an element z ∈ Z corresponding to a bipolar neutrosophic set B. The negative membership degrees ), Z ( I ), Z ( T and ) Z ( F denote the truth membership, indeterminate membership and false membership of an element z ∈ Z to some implicit counter-property corresponding to a bipolar neutrosophic set B. The lower and the upper approximations of B in the approximation (Z, R) denoted by   B N and   B N are respectively defined as follows:   B N   , Z x , x z / ) x ( F ), x ( I ), x ( T ), x ( F ), x ( I ), x ( T , x R ) B ( N ) B ( N ) B ( N ) B ( N ) B ( N ) B ( N           B N   Z ∈ x , x ∈ z / ) x ( T ), x ( T ), x ( T ), x ( T ), x ( T ), x ( T , x R ) B ( N ) B ( N ) B ( N ) B ( N ) B ( N ) B ( N      Here,     z T x ) x ( T R z ) B ( N     ,     z I x ) x ( I R z ) B ( N     ,     z F x ) x ( F R z ) B ( N     ,     z T x ) x ( T R z ) B ( N   ,     z I x ) x ( I R z ) B ( N   ,     z F x ) x ( F R z ) B ( N   ;     z T x ) x ( T R z ) B ( N     ,     z I x ) x ( I R z ) B ( N     ,     z F x ) x ( F R z ) B ( N     ,     z T x ) x ( T R z ) B ( N   ,     z I x ) x ( I R z ) B ( N       z F x ) x ( F R z ) B ( N   . Here  and  denote “max” and “min’’ operators respectively, the positive membership degrees T(z), I(z), and F(z) denote respectively the degree of truth membership, indeterminate membership and falsity membership of an element z ∈ Z corresponding to a bipolar neutrosophic set B. The negative membership degrees T(z), I(z), and F(z) denote respectively the degrees of truth membership, indeterminate membership and falsity membership of an element z ∈ Z to some implicit counter-property corresponding to a bipolar neutrosophic set B. It is easy to see that   B N and   B N are two rough bipolar neutrosophic sets in Z. Thus NS mappings , N N : N(Z) N(Z) are, respectively, referred to as the lower and upper rough bipolar NS approximation operators, and the pair )) B ( N ), B ( N ( is called the rough bipolar neutrosophic set in ( Z, R). From the above definition, it is seen that ) B ( N and ) B ( N have constant membership on the equivalence classes of R if ); B ( N ) B ( N  i.e. ), x ( T ) x ( T ) B ( N ) B ( N  ), x ( I ) x ( I ) B ( N ) B ( N  ) x ( F ) x ( F ) B ( N ) B ( N  . X x  Example 3.1 Let X={x1, x2, x3}                                           20 . 0 , 45 . 0 , 35 . 0 , 10 . 0 , 40 . 0 , 40 . 0 , 20 . 0 , 15 . 0 , 25 . 0 , 30 . 0 , 10 . 0 , 50 . 0 x 10 . 0 , 30 . 0 , 30 . 0 , 20 . 0 , 10 . 0 , 60 . 0 , 18 . 0 , 40 . 0 , 40 . 0 , 30 . 0 , 50 . 0 , 30 . 0 x 12 . 0 , 30 . 0 , 50 . 0 , 20 . 0 , 40 . 0 , 60 . 0 , 08 . 0 , 40 . 0 , 60 . 0 , 10 . 0 , 30 . 0 , 50 . 0 x A