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Rough neutrosophic sets.

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 nonmembership 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 A = {< x: [[mu].sub.[DELTA]](x), [v.sub.A](x), [[omega].sub.a](x)>, x [epsilon]X}. where the functions [mu], v, [omega] : X [right arrow] ][sup.-] 0. l [sup.+][ define respectively the degree of membership (or Truth), the degree of indeterminacy, and the degree of non-membership (or Falsehood) of the element x [epsilon]X to the set A with the condition.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of ][sup.-] 0. l [sup.+][. So instead of ] -0, [1.sup.+][ we need to take the interval [0, 1] for technical applications, because ][sup.-] 0. l [sup.+][ will be difficult to apply in the real applications such as in scientific and engineering problems. For two NS ,

A = { <x, [[mu].sub.A](x), [v.sub.A](x), [[omega].sub.A](x)> |x [member of] X}

And B={<x, [[mu].sub.B](x), [v.sub.B](x), [[omega].sub.B]{x) > | x [member of] X} the relations are defined as follows:

i. A [subset or equal to] B if and only if [[mu].sub.A](x) [less than or equal to] [[mu].sub.B](x), [v.sub.A](x) [greater than or equal to] [v.sub.B](x) ,[[omega].sub.A](x) [greater than or equal to] [[omega].sub.B](x)

ii. A = B if and only if. [[mu].sub.A](x) = [[mu].sub.B](x), [v.sub.A](x) = [v.sub.B](x) and [[omega].sub.A](x) = [[omega].sub.B](x)

iii. A [intersection] B = { < x, min ([[mu].sub.A](x), [[omega].sub.B](x)). max ([v.sub.A](x), [v.sub.B](x)), max([[omega].sub.A](x), [[omega].sub.B](x))> | x [member of] X }

iv. A U B = { <x, max([[mu].sub.A](x), [[mu].sub.B](x)). min ([v.sub.A](x), [v.sub.B](x)), min([[omega].sub.A](x), [[omega].sub.B](x)> | x [member of] X}

v. [A.sup.C] = {<x, [[omega].sub.A] {x), l- [v.sub.A](x), [[mu].sub.A](x)>| x [member of] X}

vi. [O.sub.n] =(0, 1, 1) and [l.sub.n] =(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 [x.sub.1] characterizes the capability, [x.sub.2] characterizes the trustworthiness and [x.sub.3] indicates the prices of the objects. It may be further assumed that the values of [x.sub.1], [x.sub.2] and [x.sub.3] 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 = {< [x.sub.1], (0.3, 0.5 ,0.6) >,< [X.sub.2], (0.3, 0.2 ,0.3)>,< [x.sub.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 [A.sub.1](x) = {x: [[x].sub.R] [subset or equal to] X} and [A.sub.2](x) = {x: [[x].sub.R] [intersection] X [not equal to] 0}

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 indiscemibility relation. The pair A (X)= ([A.sub.1](x), [A.sub.2](x)) is called the rough set of X in S. Here [[x].sub.R] denotes the equivalence class of R containing x.

Definition 2.4 [1]. Let A = ([A.sub.1], [A.sub.2]) and B = ([B.sub.1], [B.sub.2]) be

two rough sets in the approximation space S = (U, R) . Then,

A [union] B = ([A.sub.1] u[B.sub.1], [A.sub.2] [union] [B.sub.2]).

A [intersection] B = ([A.sub.1] [intersection][B.sub.1], [A.sub.2] [intersection] [B.sub.2]),

A [subset] B if A [intersection]B = A,

-A = {U - [A.sub.2], U - [A.sub.1]}.

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 [[mu].sub.f], indeterminacy function [V.sub.f] and non-membership function [[omega].sub.F]. The lower and the upper approximations of F in the approximation (U ,R) denoted by [N.bar] (F) and [bar.N](F) are respectively defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So 0 [less than or equal to] [[mu].sub.[bar.N](F)] (x) + [v.sub.[bar.N](F)] (x) + [[omega].sub.[bar.N](F)](x) [less than or equal to] 3 And 0 [less than or equal to] [[mu].sub.N(F)] (x) + [v.sub.N(F)] (x) + [[omega].sub.N(F)](x) [less than or equal to] 3 Where "[disjunction]" and "[conjunction]" mean "max" and "min " operators respectively, [[mu].sub.F](y), [v.sub.F](y) and [[omega].sub.F](y) are the membership ,indeterminacy and non-membership of y with respect to F. It is easy to see that [bar.N](F) and [N.bar](F) are two neutrosophic sets in U ,thus NS mapping

[bar.N], [N.bar] : N(U) [right arrow] N(U) are, respectively, referred to as the upper and lower rough NS approximation operators, and the pair ([N.bar](F), [bar.N](F)) is called the rough neutrosophic set in (U, R).

From the above definition , we can see that [N.bar](F) and [[bar.N].sub.(F)] have constant membership on the equivalence clases of U. if [N.bar](F) = [bar.N](F); i ,e [[mu].sub.[N.bar](f)] = [[mu].sub.[bar.N](f)]

[v.sub.[bar.N](f)] and = [v.sub.[N.bar](f)]. For any x [member of] U. we call F a definable neutrosophic set in the approximation ( U ,R). it is easily to be proved that Zero [O.sub.N] neutrosophic set and unite neutrosophic sets [1.sub.N] are definable neutrosophic sets. Let us consider a simple example in the following

Example 3.2. Let U = {[p.sub.1], [p.sub.2], [p.sub.3], [p.sub.4], [p.sub.5], [p.sub.6], [p.sub.7], [p.sub.8]} be the universe of discourse. Let R be an equivalence relation, where its partition of U is given by

U/R ={{[p.sub.1], [p.sub.4]}, {[P.sub.2], [P.sub.3], [P.sub.6]}, {[P.sub.5]}, {[P.sub.7], [P.sub.8]}}. Let N(F) ={([p.sub.1], (0.2, 0.3, 0.4), ([P.sub.4], (0.3, 0.5, 0.4)), ([P.sub.5], (0.4, 0.6, 0.2)),([P.sub.7], (0.1, 0.3, 0.5)) be a neutrosophic set of U. By the definition 3.1, we obtain:

[N.bar](F) ={([P.sub.1], (0.2, 0.5, 0.4)), ([p.sub.4], (0.2, 0.5, 0.4)), ([p.sub.5], (0.4, 0.6, 0.2))};

[bar.N](F) ={([P.sub.1], (0.3, 0.3, 0.4)), ([p.sub.4], (0.3, 0.3, 0.4)), ([p.sub.5]. (0.4, 0.6, 0.2)),([p.sub.7], (0.1, 0.3, 0.5)), ([p.sub.8] , (0.1, 0.3, 0.5))}:

For another neutrosophic set

N(G) ={([P.sub.1], (0.2, 0.3, 0.4)), [P.sub.4], (0.2, 0.3, 0.4)), [p.sub.5], (0.4, 0.6, 0.2)}

The lower approximation and upper approximation of N(G) are calculated as

[N.bar](G) ={([P.sub.1], (0.2, 0.3, 0.4)), ([p.sub.4], (0.2, 0.3, 0.4)), ([P.sub.5], (0.4, 0.6, 0.2))};

[bar.N](G) ={([P.sub.1], (0.2, 0.3, 0.4)), ([p.sub.4], (0.2, 0.3, 0.4)), ([p.sub.5], (0.4, 0.6, 0.2))

Obviously [N.bar](G) = [bar.N](G) is a definable neutrosopliic set in the approximation space (U, R)

Definition 3.3. If N(F)= ([N.bar](F), [bar.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.bar][(F).sup.C], [bar.N] [(F).sup.c]), where [N.bar][(F).sup.c], [bar.N][(F).sup.c] are the complements of neutrosopliic sets [bar.N](F) and [bar.N](F) respectively.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], And

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Definition 3.4. If N([F.sub.1]) and N([F.sub.2]) are two rough neutrosophic set of the neutrosophic sets [F.sub.1] and [F.sub.2] respectively in U, then we define the following:

i. N([F.sub.1]) = N([F.sub.2]) iff [N.bar]([F.sub.1]) = [N.bar]([F.sub.2]) and [bar.N]([F.sub.1]) = [bar.N]([F.sub.2])

ii. N([F.sub.1]) [subset or equal to] N([F.sub.2]) iff [N.bar]([F.sub.1]) [subset or equal to] [N.bar]([F.sub.2]) and [bar.N]([F.sub.1]) [subset or equal to] [bar.N]([F.sub.2])

iii. N([F.sub.1]) [union] N([F.sub.2]) = <[N.bar]([F.sub.1]) [union] [N.bar]([F.sub.2]). [bar.N]([F.sub.1]) [union] [bar.N]([F.sub.2])

iv. N([F.sub.1]) [intersection] N([F.sub.2]) = <[N.bar]([F.sub.1]) [intersection] [N.bar]([F.sub.2]). [bar.N]([F.sub.1]) [intersection] [bar.N]([F.sub.2])

V. N([F.sub.1]) + N([F.sub.2]) = < [N.bar]([F.sub.1]) + [N.bar]([F.sub.2]). [bar.N]([F.sub.1]) + [bar.N]([F.sub.2])

vi. N([F.sub.1]). N([F.sub.2]) = < [N.bar]([F.sub.1]). ([F.sub.2]), [bar.N]([F.sub.1]). [bar.N]([F.sub.2])

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)=N

ii. N [union] M =M [union] N, N [intersection] M = M [intersection] N

iii. (N [union] M ) [union] L = N [union] (M [union] L ) and (N [intersection] M) [intersection]

L= N [intersection] (M [intersection] L)

iv. (N [union] M) [intersection] L= (N [union] M) [intersection] (N [union] L) and (N [intersection] M) [union] L= (N [intersection] M) [union] (N [intersection] L)

De Morgan 's Laws are satisfied for neutrosophic sets : Proposition 3.6

i. ~ (N([F.sub.1]) [union] N([F.sub.2])) = (~N([F.sub.1])) [intersection] (~ N([F.sub.2])

ii. ~ (N([F.sub.1]) [intersection] N([F.sub.2])) = ( ~N([F.sub.1]) ) [union] (~N([F.sub.2])

Proof :

i. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii. Similar to the proof of (i)

Proposition 3.7:

If [F.sub.1] and [F.sub.2] are two neutrosophic sets in U such that [F.sub.1] [subset or equal to] [F.sub.2]. then N([F.sub.1]) [subset or equal to] N([F.sub.2])

i. N([F.sub.1] [union] [F.sub.2]) [contains or equal to] N([F.sub.1]) [union] N([F.sub.2])

ii. N([F.sub.1] [intersection] [F.sub.2]) [subset or equal to] N([F.sub.1]) [intersection] N([F.sub.2])

Proof:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly ,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, [N.bar] ([F.sub.1] [union] [F.sub.2]) [contains or equal to] [N.bar] ([F.sub.1]) [union] [N.bar] ([F.sub.2])

We can also see that

[bar.N]([F.sub.1] [intersection] [F.sub.2]) = [bar.N]([F.sub.1]) [intersection] [bar.N]([F.sub.2])

Hence,

N([F.sub.1] [union] [F.sub.2]) [contains or equal to] N([F.sub.1]) [union] N([F.sub.2])

(ii) proof is similar to the proof of (i)

Proposition 3.8:

i. [N.bar] (F) = ~ [bar.N](~F)

ii. [bar.N](F) = ~ [N.bar] (~F)

iii. N(F) [subset or equal to] [bar.N](F)

Proof , according to definition 3.1, we can obtain

i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [N.bar] (F) = ~ [bar.N](~F)

ii) The proof is similar to i

iii) For any y [member of] [N.bar] (F), we can have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, [N.bar] (F) [subset or equal to] [bar.N](F)

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.

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[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.

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[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.

Said Broumi (1), Florentin Smarandache (2) and Mamoni Dhar (3)

(1) Faculty of Arts and Humanities, Hay El Baraka Ben M'sik Casablanca B.P. 7951, Hassan II University Mohammedia- Casablanca, Morocco

(2) department of Mathematics, University of New Mexico,705 Gurley Avenue, Gallup, NM 87301, USA

(3) department of Mathematics, Science College, Kokrajhar-783370, Assam, India

Said Broumi, and Florentin Smarandache, Mamoni Dhar, Rough Neutrosophic Sets
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