About Rough Neutrosophic Soft Sets Theory and Study Their Properties

: Earlier fuzzy set, vague set, intuitionistic fuzzy set, L-fuzzy set etc are used as a mathematical tools for solving problems based on uncertainties or ambiguous in nature. But due to more complexity involves in problems exist in nature, traditional tools are unable to handle those in a systematic manner. So we need a tool which is more flexible to handle those problems. Which leads to the invention of soft set which was introduced by Molodtsov in 1999. Soft set (SS) theory is a mathematical tool deals with parametric data which are imprecise in nature. Ithis a generalization of fuzzy set theory. On the other hand Rough set (RS) theory and Neutrosophic set (NS) theory both rising as a powerful tool to handle these uncertain, incomplete, inconsistent and imprecise information in an effective manner. Actually Neutrosophic set is a generalization of intuitionistic fuzzy set. Sometimes it is not possible to handle all sorts of uncertain problems with a single mathematical tool. Fusion of two or more mathematical tools give rise to a new mathematical concept which gives an idea how to solve such type of problems in a more sophisticated ways. Which leads to the introduction of fuzzy soft set, rough soft set, intuitionistic fuzzy soft set, soft rough set etc. Neutrosophic soft set (NSS) was established by combining the concept of Soft set and Neutrosophic set. In this paper, using the concept of Rough set and Neutrosophic soft set a new concept known as Rough neutrosophic soft set (RNSS) is developed. Some properties and operations on them are introduced.


Introduction
In 1965 [1] L. A. Zadeh introduced the concept of fuzzy set which is termed as an extension of classical set or crisp set in which every element has a degree of membership. It is the most successful theoretical approach to vagueness. Unlike classical set theory, fuzzy set theory is described with an aid of membership function where the membership value of every element belongs to the unit interval [0, 1] so that it can be used in wide range of domains. Many mathematicians and researchers worked tirelessly on fuzzy set theory in different areas and able to extend this concept by developing some other theories such as vague set [2], L-fuzzy set [3], Rough set [4], i ntuitionistic fuzzy set [5], interval-valued fuzzy set [6], interval-valued intuitionistic fuzzy set [7] etc.
But all these theories have their own limitations and it is due to the lack of parametrization in a data. That's why soft set theory was introduced by Molodtsov [8] in 1999 to handle parametric data so that we can express the uncertain problems inmore generalized form. It has several applications in many fields like economics, engineering, medical sciences etc. In that period soft set was progressing more rapidly which leadsto the introduction of fuzzy soft set [9], intuitionistic fuzzy soft set [10], interval-valued fuzzy soft set [11], interval valued intuitionistic fuzzy soft set [12], rough soft set [13] etc.
In 1982, another mathematical tool which is known as rough set was introduced by computer scientist Z. Pawlak [4]. There are two basic elements in rough set theory, crisp set and equivalence relation, which constitute the mathematical basis of rough set. Upper and lower approximation operators are based on equivalence relation. It is a powerful tool to deal with incompleteness. It gives information of hidden data. F. Smarandache [14] introduced the concept of neutrosophic set which is a generalization of intuitionistic fuzzy set. It is described by three functions: a membership function, indetermining function and a non membership function and they are independently related to each other. It is a mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. Combining neutrosophic set with soft sets, neutrosophic soft set [15] is introduced by P. K. Maji. Neutrosophic soft set and rough sets are two different terms, none contradict the other. The main objective of this study is to introduce a new hybridstructure called rough neutrosophic soft sets. The significance of introducing hybrid set structure is that thecomputation technique based on any one of these structure alone will not always yield the best results but a fusion of two or more of them can often give the better results.

Preliminaries
In this section we recall some basic definations and examples which are relevant to this work.  [14]: Let X be an universe of discourse, with a generic elementin X denoted by x, the neutrosophic set (NS) is an object having the form Where the functions , , :  define respectively the degree of membership (or Truth), the degree of indeterminacy, and the degree of non-membership (orFalsehood) of the element x X ∈ to the set A with the From a philosophical point of view, the neutrosophic set takes the value from real standard or non-standard sub sets of x 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 indeter minacy and that of poorness to explain the characteristics of the objects. Suppose A is a neutrosophic set (NS) of U , such that Where the degree of goodness of capability is 0.3, degree of indeterminacy of capability is 0.5and degree of falsity of capability is 0.6 etc. Definition 2.3 [13]: Let E andU be the set of parameters and the universe set respectively. Let R be an equivalence relation onU . If E F is a soft set then we define two soft sets ( ) ( ) : and : as follows : x denotes the equivalence class of R containing x . .
LetU be the set of houses under consideration and E be the set of parameters. Each parameter is a neutrosophic word or sentence involvingneutrosophic words.
Consider E ={beautiful, wooden, costly, very costly, moderate, green surroundings, in good repair, in bad repair, cheap, expensive}. In this case, to define a neutrosophic soft set means to point out beautiful houses, wooden houses, houses in the green surroundings and so on. Suppose that, there are five houses in the universe U given by, U = { } Where each approximation has two parts: (i) a predicate p, and (ii) an approximate value-set v (or simply to be called value-set v).
The tabular representation of the neutrosophic soft set ( ) , F A is as follows: It is useful for computer storage. Definition 2.8 [15]: G B be two neutrosophic soft sets over the common universe .

Rough Neutrosophic Soft Sets
Here we introduce the concept of rough neutrosophic soft sets by combining both rough sets and neutrosophic soft sets and perform some operations viz. union, intersection, inclusion and equality over them. Definition 3.1: Let U be a non-empty universe set, E be a set of parameters and R be an equivalence relation on U .
where the symbols ∧ and ∨ used to denote minimum and maximum operators respectivel-y and the pair Thus the lower and the upper rough neutrosophic soft set corresponding to X is given by, N F N F ∩ (ii) Similar to the proof of (i) Proposition 3.6 If 1 F and 2 F are two neutrosophic soft sets inU such that 1 We can also see that The proof of (ii) is similar to the proof of (i)

Conclusion
In this work we have introduced the notion of rough neutrosophic soft sets by using equivalence relation. We have also studied some basic operations on them and proved some properties. The new concept of rough neutrosophic soft sets is developedby using rough sets, soft sets and neutrosophic sets. Soft set theory mainly concerned with parametric data, while neutrosophic set theory deals with indeterminate and consistent information and rough set theory is with incompleteness. So rough neutrosophic soft sets can be utilized for dealing with parametrization, indeterminacy and incompleteness. So in future there is a scope of using the rough neutrosophic Soft set in various problems of uncertainties and get more vulnerable results.