A note on neutrosophic soft set with rough set

: Rough set is a very powerful invention to the whole world for dealing with uncertain, incomplete and imprecise problems. Also soft set theory and neutrosophic set theory both are advance mathematical tools to handle these uncertain, incomplete, inconsis-tent information in a better way. The purpose of this article is to expand the scope of rough set, soft set and neutrosophic set theory. We have introduced the concept of neutrosophic soft set with roughness without using full soft set. Some deﬁnition, properties and examples have been established on neutrosophic soft rough set. Moreover, dispensable and equalities are written on roughness with neutrosophic soft set.


Introduction
The rough set theory was introduced by Z. Pawlak [13] in 1982, which is enlightened to the researcher of Artificial Intelligence, Mathematics and Computer Science. Here an inexact set "S" is categorized by two exact sets that is the lower approximation and upper approximation of the set "S" through an equivalence relation.
Rough set is based on the knowledge about ones ability to discern the objects, data, phenomenon etc. In 1983, W. Zakowski [17] defines the rough set using the covering instead of equivalence relation (or partition) where the lower approximation of the set "S' is the interior of "S" and upper approximation of "S" is the closure of "S". Later T.Y. Lin ([9], 1988) defines rough set through neighborhood operators(a new covering). Many researcher found different approximation operators based on the covering and 1-neighborhood operators.
In 1999, F. Smarandache [16] introduced the concept of neutrospohic set(NS). Neutro- Hence, M X = M X. So, the set X is rough with respect to knowledge M . Proposition 2.3. [14] Suppose that (U, M ) is an approximation space and X, Y ⊆ U . Then

Soft Set
The concept of soft set was introduced by Molodtsov [12] in 1999. Here we discuss about the soft set theory with some properties.
Definition 3.1. ( [7,10,12]) Let U be an initial universe, E be the set of parameters related to U . Let P (U ) denotes the power set of U , A ⊆ E and F be a mapping given by  (2) Soft Equality: Two soft sets (F, A) and (G, B) over a common universe U are said to be soft equal, (3) Soft Union: The union of two soft sets (F, A) and (G, B) over the common universe U is the soft set (H, C), where C = A ∪ B and for all e ∈ C,  Let E = e 1 , e 2 , · · · , e n be a set of parameters. The NOT set of E denoted by E and is defined by E = e 1 , e 2 , · · · , e n , where e i = not e i , for all i, 1 ≤ i ≤ n.
(6) Soft Complement: The complement of a soft set (F, A) is denoted by (F, A) c and is defined by . But in general the complement of a soft set (F, A) that is, (F c , A) is not a soft set, since Let U = x 1 , x 2 , · · · , x 7 be the set of houses under consideration, E = a 1 , a 2 , a 3 , a 4 , a 5 be set of parameters on U that is a 1 stands for expensive, a 2 stands for beautiful, a 3 stands for wooden, a 4 stands for cheap and a 5 stands for green surrounding. Let a mapping F : E → P (U ) be given by that is an expert is giving his views as F (a 1 ) = x 5 , x 6 , F (a 2 ) = φ, F (a 3 ) = x 4 , F (a 4 ) = x 3 , x 7 , F (a 5 ) = x 1 , x 6 and G : E → P (U ) be a mapping(that is another expert giving his views) given by G( G(a 4 ) = x 6 , x 7 , G(a 5 ) = x 1 , x 6 . Let A = a 1 , a 4 ⊆ E, B = a 1 , a 4 , a 5 ⊆ E then the soft set Let U = x 1 , x 2 , · · · , x 8 be the initial universe, E = a 1 , a 2 , · · · , a 5 be set of parameters with respect to U . Let F : E → P (U ) be a mapping given by F ( The intersection of two soft set (F, A) and (G,

Neutrosophic Set
It is some how a generalization of fuzzy set and intuitionistic fuzzy set theories known as neutrosophic set theory, introduced by F. Smarandache. In this section we present the definition and some operations on neutrosophic set.
Definition 4.1. [3,6] The neutrosophic set A is a set of objects which defines on the universe of discourse U as where the function µ, ν, ω : Here, µ A (x), ν A (x), and ω A (x) are named as the degree of membership(or Truthness), the degree of indeterminacy, and the degree of non-membership(or Falsehood) of the element x ∈ U to the set A. For two neutrosophic sets A and B, the relations on neutrosophic set are given as follows: (1) Subset: The neutrosophic set A is a subset of neutrosophic set B if and only if µ (2) Equality: The neutrosophic set A is equal to neutrosophic set B if and only if µ (3) Intersection: The intersection of these two neutrosophic sets A and B is given by (4) Union: The union of these two neutrosophic sets A and B is given by The complement of neutrosophic sets A is denoted by A c and defined by 0 n = (0, α, 1) and 1 n = (1, 0, 0) for 0 ≤ α ≤ 1, are called null neutrosophic condition and unit neutrosophic condition respectively.

Example 4.2.
Let U = x 1 , x 2 , x 3 , x 4 , x 5 be a set of quality of features that is, x 1 is for sharpness, x 2 is for sound, x 3 is for color, x 4 is for internet facilities and x 5 is for video. Here A is LG TV and B is Sony TV.
Here, for the sharpness x 1 in LG TV, the degree of quality of goodness is 0.9, the degree of quanlity of indeterminacy is 0.5 and the degree of worstness is 0.4 and so on. The video quality x 5 is not considered in LG TV, because the degree of membership is 0 and degree of non-membership is 1.
Here, for the sharpness x 1 in Sony TV, the degree of quality of goodness is 0.9, the degree of quanlity of indeterminacy is 0.5 and the degree of worstness is 0.3 and so on. Hence. A ⊆ B. Let U = x 1 , x 2 , x 3 , x 4 , x 5 be different treatments that is, x 1 stands for physical therapy, x 2 stands for radiology, x 3 stands for immunotherapy, x 4 stands for phototherapy and x 5 stands for chemotherapy. Here A is Delhi city and B is Mumbai.
be neutrosophic sets. Then union, intersection and compliment of two neutrosophic sets A and B is For the city Delhi and Mumbai, we get the degree of quality of goodness in physical therapy is atleast 0.7, degree of indeterminacy is atmost 0.2 and degree of quality of worstness is atmost 0.5.

Neutrosophic soft set(NSS)
Maji [8] introduced the concept of neutrosophic soft set in 2013, also see more details K. Bhutani and S. Aggarwal [2], D.Mohanty and N. Kalia [11].  Let U be the set of dresses under consideration and E be the set of parameters. Let Now we define neutrosophic soft subset, equal, union and intersection on U .
For example see P.K. Maji [8] 6 Neutrosophic soft set with roughness In this section, N S A -lower and N S A -upper approximations are introduced and their properties are deduced and illustrated by examples. We can find the notation S A (X), for X ⊆ U and E is a set of parameters, A ⊆ E, We note here that, neutrosophic soft rough set is defined without using full soft set.
If apr N S A (X) = apr N S A (X), then X is neutrosophic soft rough set. Otherwise X is called neutrosophic soft definable set.  Now neutrosophic soft set over U is  Thus, X is rough with respect to knowledge N S A , since apr N S A (X) = apr N S A (X).
Proof. From definition of N S A -lower and N S A upper approximation, (1) and (2) are straightforward. So we prove the remaining.
Assume that X ⊆ Y . Let u ∈ apr N S A (X), by definition there exists one a ∈ A such that u ∈ F (a) ⊆ X and This implies u ∈ apr N S A (Y ). Hence, apr N S A (X) ⊆ apr N S A (Y ). This proves (3).
Remaining properties comes directly.
then S is called intersection complete neutrosophic soft set.

Proof. We have only to show apr N S
, then there exists e 1 , e 2 ∈ A such that u ∈ F (e 1 ) ⊆ X, . By definition of intersection complete soft set , there exists e 3 ∈ A such that Therefore, Example 6.6.
Let U = h 1 , h 2 , · · · , h 10 be universe of discourse and E = e 1 , e 2 , · · · , e 7 be a set of parameters. Let F : E → P (U ) be a mapping given by F (e 1 ) = h 1 , h 5 , F (e 2 ) = φ, Hence, apr N S A (X) = φ = apr N S A (φ) (2) Given, X N S A U and X ⊆ Y , then apr N S A (X) = apr N S A (U ) and apr N S A (X) ⊆ apr N S A (Y ). But we know that Remaining properties comes directly.
We note here that

Dispensable
In this section, we shall discuss about dispensable and indispensable of NSS. Let Let U = {x 1 , x 2 , · · · , x 6 } be six most affected states in India due to Corona virus infection.
Here x 1 is a group of persons from the state Maharashtra whose corona positive is detected, x 2 is a group of persons from the state Kerala whose corona positive is detected, x 3 is from Tamil Nadu, x 4 is from Delhi, x 5 is from Uttar Pradesh and x 6 is from Karnnataka.
Let E = {e 1 , e 2 , · · · , e 11 } be the set of parameters with respect to corona virus infection in the human body such that e 1 is aches, e 2 is difficult in breathing, e 3 is tiredness, e 4 is chill. e 5 is fever and cough, e 6 is sore throat, e 7 is loss of smell, e 8 is loss of taste, e 9 is headache, e 10 is diaarhea and e 11 is severe vomiting.

Conclusion
In this note it is defined the notion of neutrosophic soft rough set in new manner which is a combination of three theories that is rough set theory, soft set theory and neutrosophic set theory. We have studied some of their basic properties like union, intersection and complement. Some authors have defined neutrosophic soft rough set using full soft set which is not convenient to handle indeterminant and incomplete data. In this article neutrosophic soft rough set is established without using full soft set and also equality and dispensability on neutrosophic soft rough set are illustrated with examples.

Compliance with Ethical Standards
Funding: The authors (authors including corresponding author) declare that they have no funding for this study.