Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

80

Neutrosophic Sets and Systems, Vol. 14, 2016

University of New Mexico

Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems Nguyen Xuan Thao1, Bui Cong Cuong2, Florentin Smarandache3 2

1 Faculty of Information Technology, Vietnam National University of Agriculture. E-mail: nxthao2000@gmail.com Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam. E-mail: bccuong@gmail.com 3 Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA. E-mail: smarand@unm.edu

Abstract: A rough fuzzy set is the result of the approximation of a fuzzy set with respect to a crisp approximation space. It is a mathematical tool for the knowledge discovery in the fuzzy information systems. In this paper, we introduce the concepts of rough standard

neutrosophic sets and standard neutrosophic information system, and give some results of the knowledge discovery on standard neutrosophic information system based on rough standard neutrosophic sets.

Keywords: rough set, standard neutrosophic set, rough standard neutrosophic set, standard neutrosophic information systems

1 Introduction Rough set theory was introduced by Z. Pawlak in 1980s [1]. It became a useful mathematical tool for data mining, especially for redundant and uncertain data. At first, the establishment of the rough set theory is based on the equivalence relation. The set of equivalence classes of the universal set, obtained by an equivalence relation, is the basis for the construction of upper and lower approximation of the subset of universal set.

Fuzzy set theory was introduced by L. Zadeh since 1965 [2]. Immediately, it became a useful method to study in the problems of imprecision and uncertainty. Ever since, a lot of new theories treating imprecision and uncertainty have been introduced. For instance, intuitionistic fuzzy sets were introduced in1986, by K. Atanassov [3], which is a generalization of the notion of a fuzzy set. While the fuzzy set gives the degree of membership of an element in a given set, intuitionistic fuzzy set gives a degree of membership and a degree of non-membership of an element in a given set. In 1999 [17], F. Smarandache introduced the concept of neutrosophic set which generalized fuzzy set and intuitionistic fuzzy set. It is a set in which each proposition is estimated to have a degree of truth (T), a degree of indeterminacy (I) and a degree of falsity (F). After a while, the subclass of neutrosophic sets was proposed. They are more advantageous in the practical application. Wang et al.

[18] proposed the interval neutrosophic sets, and some of their operators. Smarandache [17] and Wang et al. [19] introduced a single valued neutrosophic set as an instance of

the neutrosophic set accompanied with various set theoretic operators and properties. Ye [20] defined the concept of simplified neutrosophic set. It is a set where each element of the universe has a degree of truth, indeterminacy and falsity respectively, stretching between [0, 1]. Ye also suggested some operational laws for simplified neutrosophic sets, and two aggregation operators, including a simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator.

In 2013, B.C. Cuong and V. Kreinovich introduced the concept of picture fuzzy set [4, 5], in which a given set has three memberships: a degree of positive membership, a degree of negative membership, and a degree of neutral membership of an element in this set. After that, L. H. Son gave the application of the picture fuzzy set in the clustering problems [7, 8]. We regard picture fuzzy sets as particular cases of the standard neutrosophic sets [6].

In addition, combining rough set and fuzzy set enhanced many interesting results. The approximation of rough (or fuzzy) sets in fuzzy approximation space give us the fuzzy rough set [9,10,11]; and the approximation of fuzzy sets in crisp approximation space give us the rough fuzzy set [9,10]. W. Z. Wu et al. [11] presented a general framework for the study of the fuzzy rough sets in both constructive and axiomatic approaches. Moreover, W. Z. Wu and Y. H. Xu investigated the fuzzy topological structures on the rough fuzzy sets [12], in which both constructive and axiomatic approaches are used. In 2012, Y. H. Xu and W. Z. Wu investigated the rough intuitionistic

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

81

Neutrosophic Sets and Systems, Vol. 14, 2016

fuzzy set and the intuitionistic fuzzy topologies in crisp approximation spaces [13]. In 2013, B. Davvaz and M. Jafarzadeh studied the rough intuitionistic fuzzy information system [14]. In 2014, X. T. Nguyen introduced the rough picture fuzzy sets. It is the result of approximation of a picture fuzzy set with respect to a crisp approximation space [15].

In this paper, we introduce the concept of standard neutrosophic information system, and study some problems of the knowledge discovery of standard neutrosophic information system based on rough standard neutrosophic sets.

The remaining part of this paper is organized as follows: we recall the basic notions of rough set, standard neutrosophic set and rough standard neutrosophic set on the crisp approximation space, respectively, in Sections 2 and 3. In Section 4, we introduce the basic concepts of standard neutrosophic information system. Finally, we investigate some problems of the knowledge discovery of standard neutrosophic information system: the knowledge reduction and extension of the standard neutrosophic information system, in Section 5 and Section 6, respectively.

2 Basic notions of standard neutrosophic set and rough set

may be identified with the standard neutrosophic set in the form

A  { x,Îź A  x  ,0,  γ A  x  X  | x ďƒŽ U}

A = {(x, ÎźA (x), ÎłA (x), 0)|x ∈ U}.

The operators on PFS(U): A ďƒ? B , A ďƒ‡ B , A ďƒˆ B were introduced in [4].

Now we define some special PF sets: a constant PF set is the PF set (Îą,Ě‚ β, θ) = {(x, Îą, β, θ)|x ∈ U}; the PF universe set is Ě‚ = {(x, 1,0,0)|x ∈ U} and the PF empty U = 1U = (1,0,0) Ě‚ = {(x, 0,0,1)|x ∈ U}∅ = 0U = set is ∅ = 0U = (0,0,1) Ě‚ = {(x, 0,1,0)|x ∈ U}.

(0,1,0) For any

x ďƒŽ U , standard neutrosophic set 1x and 1U-{x}

are, respectively, defined by: for all y ďƒŽ U

ďƒŹ1 if y  x , ďƒŹ0 if y  x , Îź1x  y   ďƒ­ ď ¨1  y   ďƒ­ ďƒŽ0 if y ď‚š x ďƒŽ0 if y ď‚š x x

ďƒŹ0 if y  x ; ďƒŹ0 if y  x , Îź1U {x}  y   ďƒ­ ďƒŽ1 if y ď‚š x ďƒŽ1 if y ď‚š x

ď §1  y   ďƒ­ x

ď ¨1

U {x }

ďƒŹ0 if y  x , ďƒŹ1 if y  x ď § 1U {x}  y   ďƒ­ ďƒŽ0 if y ď‚š x ďƒŽ0 if y ď‚š x

 y  ďƒ­

Definition 2. (Lattice (D* , ≤D* )). Let D* = {(x1 , x2 , x3 ) ∈ [0,1]3 : x1 + x2 + x3 ≤ 1}.

We define a relation ≤D* on D∗ as follows: ∀(x1 , x2 , x3 ), (y1 , y2 , y3 ) ∈ D* then Definition 1. [6]. A standard neutrosophic (PF) set A on the  x , x , x  ď‚Ł  y , y , y  (x1 , x2 , x3 ) ≤ * (y1 , y2 , y3 ) D 1 2 3 1 2 3 D* universe U is an object of the form if only if In this paper, we denote by U a nonempty set called the universe of discourse. The class of all subsets of U will be denoted by P(U) and the class of all fuzzy subsets of U will be denoted by F(U).

A  { x,Îź A  x  , Ρ A  x  ,  γ A  x   | x ďƒŽ U}

(or (x1 ď&#x20AC;ź y1 , x 3 ď&#x201A;ł y3 ) (x1 < y1 , x3 â&#x2030;Ľ y3 ) or

(x1 =

'

where ÎźA (x)(â&#x2C6;&#x2C6; [0,1]) is called the â&#x20AC;&#x153;degree of positive membership of x in A â&#x20AC;?, ΡA (x)(â&#x2C6;&#x2C6; [0,1]) is called the â&#x20AC;&#x153;degree of neutral membership of x in A â&#x20AC;? and

y1 , x3 > y3 )(x = x , y > y') or (x1 = y1 , x3 = y3 , x2 â&#x2030;¤ y2 )(x = x ' , y = y ' , z â&#x2030;¤ z')) (x1 , x2 , x3 ) =D* (y1 , y2 , y3 ) â&#x;ş (x1 = y1 , x2 = and y2 , x3 = y3 ).

negative mem-bership of x in Aâ&#x20AC;?, where ÎźA, ΡA ÎźA , ÎłA and

We have D* , ď&#x201A;Ł * D

γ A  ΡA satisfy the following condition:

1D* = (1,0,0) Now, we define some operators on Dâ&#x2C6;&#x2014; .

Îź A ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;Ť Ρ A ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;Ť  γ A ď&#x20AC;¨ x ď&#x20AC;Š ď&#x201A;Ł 1,  ď&#x20AC;¨ ď&#x20AC;˘Â x ď&#x192;&#x17D; X ď&#x20AC;Š ÎźA (x) + ÎłA (x) +

Definition 3.

Îł A ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;¨ď&#x192;&#x17D; ď &#x203A;0,1ď ?ď&#x20AC;Š ÎłA (x)(â&#x2C6;&#x2C6; [0,1]) is called the â&#x20AC;&#x153;degree of

ΡA (x)) â&#x2030;¤ 1, (â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; X).

The family of all standard neutrosophic set in U is denoted

by PFS(U). The complement of a picture fuzzy set A is

~ A ď&#x20AC;˝ {ď&#x20AC;¨ x, Îł A ď&#x20AC;¨ x ď&#x20AC;Š ,  Ρ A ď&#x20AC;¨ x ď&#x20AC;Š ,  Ο A ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;Š | ď&#x20AC;˘x ď&#x192;&#x17D; U} .

Obviously, any intuitionistic fuzzy set: A = {(x, ÎźA (x), ÎłA (x))}

ď&#x20AC;¨

ď&#x20AC;Š is a lattice. Denote

0D* = (0,0,1) ,

đ?&#x2018;Ľ = (đ?&#x2018;Ľ1 , đ?&#x2018;Ľ2 , đ?&#x2018;Ľ3 ) â&#x2C6;&#x2C6; đ??ˇ â&#x2C6;&#x2014; is đ?&#x2018;Ľ =

(i)

Negative of (đ?&#x2018;Ľ3 , đ?&#x2018;Ľ2 , đ?&#x2018;Ľ1 )

(ii)

For all x = (x1 , x2 , x3 ) â&#x2C6;&#x2C6; D* we have

x ď&#x192;&#x2122; y ď&#x20AC;˝ ď&#x20AC;¨ x1 ď&#x192;&#x2122; y1 , x2 ď&#x192;&#x2122; y2 , x3 ď&#x192;&#x161; y3 ď&#x20AC;Š

x ď&#x192;&#x161; y ď&#x20AC;˝ ď&#x20AC;¨ x1 ď&#x192;&#x161; y1 , x2 ď&#x192;&#x2122; y2 , x3 ď&#x192;&#x2122; y3 ď&#x20AC;Š .

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

82

Neutrosophic Sets and Systems, Vol. 14, 2016

(α+ , θ+ )- level cut set of the standard neutro• sophic set A as

We have some properties of those operators.

Lemma 1.

Aθα  {x  U|μ A  x    , γ A  x   θ} 

(a) For all x = (x1 , x2 , x3 ) ∈ D* we have •

(b1) x  y  x  y x ∧ y = x ∨ y (b2) x  y  x  y x ∨ y = x ∧ y

α- level cut set of the degree of positive membership of x in A as

Aα  {x  U|μ A  x   α}

(b) For all x, y, u, v ∈ D* and x ≤D* u, y ≤D* v

the strong α- level cut set of the degree of positive membership of x in A as

we have (c1) x ∧ y ≤D* u ∧ v (c2) x ∨ y ≤D* u ∨ v

Aα  {x  U|μ A  x   α} 

Proof.

(a) We have x ∧ y = (x3 ∨ y3 , x2 ∧ y2 , x1 ∧ y1 ) = (x3 , x2 , x1 ) ∨ (y3 , y2 , y1 ) = x ∨ y Similary x ∨ y = (x3 ∧ y3 , x2 ∧ y2 , x1 ∨ y1 ) = (x3 , x2 , x1 ) ∨ (y3 , y2 , y1 ) = x ∨ y (b) For a, b, c, d ∈ [0,1] , if a ≤ b, c ≤ d then a ∧ c ≤ b ∧ d and. From definitions 2 and 3, we have the result to prove. □

•

θ- level low cut set of the degree of negative membership of x in A as

Aθ  {x  U|γ A  x   θ} the strong θ- level low cut set of the degree of negative membership of x in A as

Aθ  {x  U|γ A  x   θ}

Example 1. Given the universe U = {u1 , u2 , u3 }. Then Now, we mention the level sets of the standard neutrosophic sets, where •

 α, β, θ   D* ; we define:

is a standard neutrosophic set on

(α, β, θ)- level cut set of the standard neutrosophic set

A  { x,μ A  x  ,η A  x  , γA  x   | x  U}

A

 {x  U|  μ A  x  , η A  x  , γ A  x     α, β, θ }

= {x ∈

U|(μA (x), ηA (x), γA (x)) ≥ (α, β, θ)} •

strong (α, β, θ)- level cut set of the standard neutrosophic set A as follows:

Aθα ,β  {x  U|  μ A  x  , η A  x  , γ A  x     α, β, θ } 

•



(α+ , β, θ)-- level cut set of the standard neutrosophic set A as

Aθα •



,β

+)

(α, β, θ set A as

 {x  U|μ A  x    , γ A  x   θ} − level cut set of the standard neutrosophic

Aθα,β  {x  U|μ A  x   α, γ A  x   θ}

By β  0 we denoted

{u1 , u2 } but A0.7,0.1 = {u1 } 0.1

= U . Then A0.7,0.2 0.1

and

A0.7,0.2 = {u1 } , 0.1+

0.7 0.5 A0.7 = {u1 , u2 , u3 } , A0.5 = 0.1  u1  , A 0.1+ = ∅, A +

A = {(x, μA (x), γA (x), ηA (x))|x ∈ U}as follows: α,β θ

A    u1 , 0.8, 0.05, 0.1 ,  u2 , 0.7, 0.1, 0.2  ,  u3 , 0.5, 0.01, 0.4  

{u1 , u2 }, A0.2+ = {u1 }, A0.2 = {u1 , u2 }.

Definition 3. Let U be a nonempty universe of discourse which may be infinite. A subset R ∈ P(U×U) is referred to as a (crisp) binary relation on U. The relation R is referred to as: •

Reflexive: if for all

•

Symmetric: if for all

x  U,  x, x   R .

x,y  U,  x, y   R x, y ∈

U, (x, y) ∈ R then (y, x) ∈ R. •

Transitive:

if

for

all

x,y,z  U,  x, y   R,  y, z   R x, y, z ∈ U, (x, y) ∈

R, (y, z) ∈ R then (x, z) ∈ R • Similarity: if R is reflexive and symmetric • Preorder: if R is reflexive and transitive • Equivalence: if R is reflexive and symmetric, transitive.

Aαθ = Aα,0 θ Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

83

Neutrosophic Sets and Systems, Vol. 14, 2016

A crisp approximation space is a pair (U, R). For an arbitrary crisp relation R on U, we can define a set-valued mapping R s : U ď&#x201A;Ž P ď&#x20AC;¨ U ď&#x20AC;Š by:

R s ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď ťy ď&#x192;&#x17D; U| ď&#x20AC;¨ x, y ď&#x20AC;Š ď&#x192;&#x17D; Rď ˝ , x ď&#x192;&#x17D; U. Then, R s (x) is called the successor neighborhood of x x with respect to (w.r.t) R .

Definition 4.[9]. Let (U, R) be a crisp approximation space. For each crisp set A â&#x160;&#x2020; U , we define the upper and Ě&#x2026;(A) lower approximations of A (w.r.t) (U, R) denoted by R and R(A), respectively, are defined as follows:

A rough standard neutrosophic set is the approximation of a standard neutrosophic set w. r. t a crisp approximation space. Here, we consider the upper and lower approximations of a standard neutrosophic set in the crisp approximation spaces together with their membership functions, respectively.

Definition 5: Let (U, R) be a crisp approximation space. For A â&#x2C6;&#x2C6; PFS(U) , the upper and lower approximations of A (w.r.t) (U, R) denoted by RP ď&#x20AC;¨ A ď&#x20AC;Š Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026; RP(A) and RP(A) , respectively, are defined as follows: Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) = {(x, ÎźRP RP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x), ΡRP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x), ÎłRP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x))|x â&#x2C6;&#x2C6; U}

Ě&#x2026;(A) = {x â&#x2C6;&#x2C6; U: R s (x) â&#x2C6;Š A â&#x2030; â&#x2C6;&#x2026;}, R

R ď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;˝ ď ťx ď&#x192;&#x17D; U : R s ď&#x20AC;¨ x ď&#x20AC;Š ď&#x192;? Aď ˝

3. Rough standard neutrosophic set

R(A) = {x â&#x2C6;&#x2C6;

U: R s (x) â&#x160;&#x2020; A}.

Remark 2.1. Let (U, R) be a Pawlak approximation space, i.e. R is an equivalence relation. Then R s (x) = [x]R holds.

For each crisp set A â&#x160;&#x2020; U , the upper and lower Ě&#x2026;(A) and approximations of A (w.r.t) (U, R) denoted by R R(A), respectively, are defined as follows: Ě&#x2026;(A) = {x â&#x2C6;&#x2C6; U: [x]R â&#x2C6;Š A â&#x2030; â&#x2C6;&#x2026;}R(A) = {x â&#x2C6;&#x2C6; U: [x]R â&#x160;&#x2020; R

ď&#x20AC;¨

where

Îź RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; Îź A ď&#x20AC;¨ y ď&#x20AC;Š , ΡRPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; ΡA ď&#x20AC;¨ y ď&#x20AC;Š , yď&#x192;&#x17D;R s ď&#x20AC;¨ x ď&#x20AC;Š yď&#x192;&#x17D;R s ď&#x20AC;¨ x ď&#x20AC;Š RP(A) = {(x, ÎźRP(A) (x), ÎłRP(A) (x), ΡRP(A) (x))|x â&#x2C6;&#x2C6; U}; and RP(A) = {(x, ÎźRP(A) (x), ÎłRP(A) (x), ΡRP(A) (x))|x â&#x2C6;&#x2C6; U}

ΡRPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝

A} Definition 5. [16] Let (U, R) be a crisp approximation space. For each fuzzy set A â&#x160;&#x2020; U, we define the upper and lower approximations of A (w.r.t) (U, R) denoted by R ď&#x20AC;¨ A ď&#x20AC;Š and R(A), respectively, are defined as follows: Ě&#x2026;(A) = {x â&#x2C6;&#x2C6; U: R s (x) â&#x2C6;Š A â&#x2030; â&#x2C6;&#x2026;}, R

ď&#x20AC;Š

RP ď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;˝ { x, Îź RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š , ΡRPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š , Îł RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š | x ď&#x192;&#x17D; U}

,

ď&#x192;&#x2122; ΡA ď&#x20AC;¨ y ď&#x20AC;Š , Îł RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; Îł A ď&#x20AC;¨ y ď&#x20AC;Š .

yď&#x192;&#x17D;R s ď&#x20AC;¨ x ď&#x20AC;Š

yď&#x192;&#x17D;R s ď&#x20AC;¨ x ď&#x20AC;Š

RP(A) = {(x, ÎźRP(A) (x), ÎłRP(A) (x), ΡRP(A) (x))|x â&#x2C6;&#x2C6; U} We have Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026; RP(A) and RP ď&#x20AC;¨ A ď&#x20AC;Š , two standard neutrosophic sets in U. Indeed, for each x â&#x2C6;&#x2C6; U, for all Ďľ > 0 , it exists y0 ď&#x192;&#x17D; U y0 â&#x2C6;&#x2C6; U such that ÎźRP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x)-Ďľ â&#x2030;¤

R ď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;˝ ď ťx ď&#x192;&#x17D; U : R s ď&#x20AC;¨ x ď&#x20AC;Š ď&#x192;? Aď ˝

ÎźA (y0 ) â&#x2030;¤ ÎźĚ&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026; RP(A) (x) , ΡĚ&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026; RP(A) (x) â&#x2030;¤ ΡA (y0 ) , ÎłĚ&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026; RP(A) (x) â&#x2030;¤ ÎłA (y0 )

where

so that Îź RP

ÎźRĚ&#x2026;(A) (x) = max{ÎźA (y)|y â&#x2C6;&#x2C6; R s (x)},

Îź RA ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ min{ď ­ A ď&#x20AC;¨ y ď&#x20AC;Š | y ď&#x192;&#x17D; Rs ď&#x20AC;¨ x ď&#x20AC;Š}

ď&#x20AC;¨Aď&#x20AC;Š

ď&#x20AC;¨xď&#x20AC;Š ď&#x20AC;­

ď&#x20AC;Ť ΡRPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;Ť Îł RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š

ď&#x201A;Ł Îź A ď&#x20AC;¨ y0 ď&#x20AC;Š ď&#x20AC;Ť ΡA ď&#x20AC;¨ y0 ď&#x20AC;Š ď&#x20AC;Ť ď § A  ď&#x20AC;¨ y0 ď&#x20AC;Š ď&#x201A;Ł 1 ÎźRP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x)-Ďľ + ΡRP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x)+ÎłRP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x) â&#x2030;¤.

Remark 2.2. Let (U, R) be a Pawlak approximation space, i.e. đ?&#x2018;&#x2026; is an equivalence relation. Then R s (x) = [x]R holds.

For each fuzzy set A â&#x160;&#x2020; U , the upper and lower Ě&#x2026;(A) and approximations of A (w.r.t) (U, R) denoted by R R(A), respectively, are defined as follows: Ě&#x2026;(A) = {x â&#x2C6;&#x2C6; U: [x]R â&#x2C6;Š A â&#x2030; â&#x2C6;&#x2026;}, R R(A) = {x â&#x2C6;&#x2C6; U: [x]R â&#x160;&#x2020; A}

Hence ÎźRP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x) + ΡRP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x)+ÎłRP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) (x) â&#x2030;¤ 1 + Ďľ , for all Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026; Ďľ > 0. It means that RP(A) is a standard neutrosophic set.

By the same way, we obtain RP(A) a standard neutrosophic Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A). set. Moreover, RP(A) â&#x160;&#x201A; RP

This is the rough fuzzy set in [6].

PR( A) ď&#x20AC;˝ ( PR( A), RP ď&#x20AC;¨ A ď&#x20AC;Š) is called the rough standard

Thus, the standard neutrosophic mappings Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026; RP, RP: PFS(U) â&#x2020;&#x2019; PFS(U)are referred to as the upper and lower PF approximation operators, respectively, and the pair

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

84

Neutrosophic Sets and Systems, Vol. 14, 2016

neutrosophic set of A w.r.t the approximation space. The picture fuzzy set denoted by ~RP(A) and is defined by ~RP(A) = PR( A) ď&#x20AC;˝ ( PR( A), RP ď&#x20AC;¨ A ď&#x20AC;Š) Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A)) where ~RP(A) and ~RP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) are the (~RP(A), ~RP Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(A) and RP(A) respectively. complements of the PF sets RP Example 2. We consider the universe set U = {u1 , u2 , u3 , u4 , u5 } and a binary relation R on U in Table 1.

Here, if ui Ruj then cell (i, j) takes a value of 1, cell (i, j) takes a value of 0 (i, j = 1, 2, 3, 4, 5). A standard neutrosophic

A ď&#x20AC;˝ {ď&#x20AC;¨ u1 , 0.7, 0.1, 0.2 ď&#x20AC;Š , ď&#x20AC;¨ u2 , 0.6, 0.2, 0.1ď&#x20AC;Š , ď&#x20AC;¨ u3 , 0.6, 0.2, 0.05ď&#x20AC;Š ,

ď&#x20AC;¨ u2 , 0.6, 0.2, 0.1ď&#x20AC;Š , ď&#x20AC;¨ u3 , 0.6, 0.2, 0.05ď&#x20AC;Š}

ď § RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ u1 ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;R ď&#x20AC;¨ u ď&#x20AC;Šď § A ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ min ď ťď § A ď&#x20AC;¨ u1 ď&#x20AC;Š , ď § A ď&#x20AC;¨ u3 ď&#x20AC;Šď ˝ s

1

ÎłRP(A) (u1 ) = â&#x2039;&#x20AC;yâ&#x2C6;&#x2C6;Rs(u1) ÎłA (y) = min{ÎłA (u1 ), ÎłA (u3 )} = max{0.7,0.6} = 0.7 min{0.2,0.05} = 0.05 Similar calculations for other elements of U, we have upper approximations of A RP ď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;˝ {(u1 , 0.7, 0.1, 0.05), (u2 , 0.6, 0.2, 0.1),

ď&#x20AC;¨ u3 ,0.7,0.1, 0.05ď&#x20AC;Š ,ď&#x20AC;¨ u4 ,0.6, 0.2, 0.1ď&#x20AC;Š , ď&#x20AC;¨ u5 ,0.6,0.2, 0.05ď&#x20AC;Š} and lower approximations of A is RP ď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;˝ {(u1 ,0.6,0.1,0.2),(u2 ,0.4,0.2,0.2),

ď&#x20AC;¨ u3 ,0.4,0.1, 0.2ď&#x20AC;Š , ď&#x20AC;¨ u4 ,0.5, 0.2, 0.15ď&#x20AC;Š , ď&#x20AC;¨u5 ,0.4,0.2, 0.2ď&#x20AC;Š} .

Some basic properties of rough standard neutrosophic set operators are presented in the following theorem:

Table 1: Binary relation đ?&#x2018;&#x2026; on đ?&#x2018;&#x2C6; R

u1

u2

u3

u4

u5

u1

1

0

1

0

0

u2

0

1

0

1

1

Theorem 1. Let (U, R) be a crisp approximation space, then the upper and lower rough standard neutrosophic approximation operators satisfy the following properties: â&#x2C6;&#x20AC;A, B, Aj â&#x2C6;&#x2C6; PFS(U), j â&#x2C6;&#x2C6; J, J is an index set, (PL1) PR( A) = (PL2)

RP ď&#x20AC;¨ A ď&#x20AC;Š

ď&#x20AC;Š

ď&#x20AC;¨

RP A ď&#x192;&#x2C6; ď&#x20AC;¨ Îą,β,θ ď&#x20AC;Š ď&#x20AC;˝  RP ď&#x20AC;¨ A ď&#x20AC;Š ď&#x192;&#x2C6; ď&#x20AC;¨ Îą,β,θ ď&#x20AC;Š

u3

1

0

1

0

1

RP(A â&#x2C6;Ş (Îą,Ě&#x201A; β, θ)) = RP(A) â&#x2C6;Ş (Îą,Ě&#x201A; β, θ)

u4

0

1

0

1

0

(PL3) RP ď&#x20AC;¨ U ď&#x20AC;Š ď&#x20AC;˝ U RP(U) = U

u5

0

0

1

1

1

ΡRP(A) (x) = â&#x2039;&#x20AC;yâ&#x2C6;&#x2C6;Rs(x) ΡA (y)

We have R s (u1 ) = {u1 , u3 }, R s (u2 ) = {u2 , u4 , u5 }, R s (u3 ) = {u1 , u3 , u5 }, R s (u4 ) = {u2 , u4 }, R s ď&#x20AC;¨ u 5 ď&#x20AC;Š ď&#x20AC;˝ ď ťu3 , u4 , u5 ď ˝ R s (u5 ) = {u3 , u4 , u5 }.

Therefore, we obtain the results

ÎźĚ&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026; RP(A) (u1 ) = â&#x2039; yâ&#x2C6;&#x2C6;Rs (u1 ) ÎźA (y)

(PL5) RP ď&#x20AC;¨ A ď&#x192;&#x2C6; Bď&#x20AC;Š ď&#x192;&#x160; RP ď&#x20AC;¨ A ď&#x20AC;Š ď&#x192;&#x2C6; RP ď&#x20AC;¨ B ď&#x20AC;Š (PL6) A â&#x160;&#x2020; B â&#x2021;&#x2019; RP(A) â&#x160;&#x2020; RP(B) (PU1)

Ě&#x2026;Ě&#x2026;Ě&#x2026;Ě&#x2026;(~A) = ~RP(A) RP ď&#x20AC;¨ A ď&#x20AC;Š RP

ď&#x20AC;˝ PR( A)

Îź RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ u1 ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; yď&#x192;&#x17D;R ď&#x20AC;¨ u ď&#x20AC;ŠÎź A ď&#x20AC;¨ y ď&#x20AC;Š = max{ÎźA (u1 ), ÎźA (u3 )}

(PU2) PR(A â&#x2C6;Š (Îą,Ě&#x201A; β, θ)) = PR(A) â&#x2C6;Š (Îą,Ě&#x201A; β, θ)

= max{0.7,0.6} = 0.7,

(PU3) PR(â&#x2C6;&#x2026;) = â&#x2C6;&#x2026;

ΡRPď&#x20AC;¨ Aď&#x20AC;Š ď&#x20AC;¨ u1 ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;R ď&#x20AC;¨ u ď&#x20AC;ŠÎˇA ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ min ď ťď ¨ A ď&#x20AC;¨ u1 ď&#x20AC;Š ,ď ¨ A ď&#x20AC;¨ u3 ď&#x20AC;Šď ˝

(PU4) RP(â&#x2039;&#x192;jâ&#x2C6;&#x2C6;J Aj ) = â&#x2039;&#x192;jâ&#x2C6;&#x2C6;J RP(Aj )

s

1

s

=max{0.7,0.6} = 0.7,

1

(PU5) RP(A â&#x2C6;Š B) â&#x160;&#x2020; RP(A) â&#x2C6;Š RP(B) (PU6) A â&#x160;&#x2020; B â&#x2021;&#x2019; RP(A) â&#x160;&#x2020; RP(B)

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

85

Neutrosophic Sets and Systems, Vol. 14, 2016

Proof.

(PL1).

ď&#x20AC;¨

ď&#x20AC;Š

RP ď&#x20AC;¨ ~ A ď&#x20AC;Š ď&#x20AC;˝ { x, Îź RPď&#x20AC;¨ ~A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š , ΡRPď&#x20AC;¨ ~A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ,  γ RPď&#x20AC;¨ ~A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š | x ď&#x192;&#x17D; U}

The results (PL4), (PL5), (PL6) were proved by using the definition of lower and upper approximation spaces (definition 5) and lemma 1. ΟΟ (x) Ě&#x201A; RP((Îą,β,θ))

in which,

Similarly, we have (PU1), (PU2), (PU3), (PU4), (PU5), PU(6). â&#x2013;Ą

Îź RPď&#x20AC;¨ ~A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; yď&#x192;&#x17D;Rs ď&#x20AC;¨ x ď&#x20AC;Š ď ­~ A ď&#x20AC;¨ y ď&#x20AC;Š = ď&#x192;&#x161; yď&#x192;&#x17D;Rs ď&#x20AC;¨ x ď&#x20AC;Š ď § A ď&#x20AC;¨ y ď&#x20AC;Š =

Theorem 2. Let (U, R) be a crisp approximation space.

Then

ď § RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ;

a) RP(U) = U = RP(U) and RP ď&#x20AC;¨ ď&#x192;&#x2020; ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2020; ď&#x20AC;˝ RP ď&#x20AC;¨ ď&#x192;&#x2020; ď&#x20AC;Š RP(â&#x2C6;&#x2026;) = â&#x2C6;&#x2026; = RP(â&#x2C6;&#x2026;).

ď ¨RPď&#x20AC;¨ ~A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;R ď&#x20AC;¨ x ď&#x20AC;Šď ¨~ A ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;R ď&#x20AC;¨ x ď&#x20AC;Šď ¨ A ď&#x20AC;¨ y ď&#x20AC;Š = s

b) RP(A) â&#x160;&#x2020; RP(A) forall A â&#x2C6;&#x2C6; PFS(U).â&#x2013;Ą

s

ď ¨ RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š

Proof.

Îł RPď&#x20AC;¨ ~A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ Â ď&#x192;&#x2122; yď&#x192;&#x17D;Rs ď&#x20AC;¨ x ď&#x20AC;Š ď § ~ A ď&#x20AC;¨ y ď&#x20AC;Š = ď&#x192;&#x2122; yď&#x192;&#x17D;Rs ď&#x20AC;¨ x ď&#x20AC;Š ď ­ A ď&#x20AC;¨ y ď&#x20AC;Š =

(a) Using (PL3), (PL6), (PU3), (PU6), we easy prove

ď ­ RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š

(b) Based on definition 5, we have

From that and lemma 1, we have

RP(U) = U = RP(U) and RP(â&#x2C6;&#x2026;) = â&#x2C6;&#x2026; = RP(â&#x2C6;&#x2026;).

PR( A) = RP ď&#x20AC;¨ A ď&#x20AC;Š .

(PL2) Because (Îą,Ě&#x201A; β, θ) = {(x, Îą, β, θ)|x â&#x2C6;&#x2C6; U}, we have

ď ­RP Aď&#x192;&#x2C6; Îą,β,θ

ď&#x20AC;¨

ď&#x20AC;¨

ď&#x20AC;¨ x ď&#x20AC;Š = ď&#x192;&#x161; yď&#x192;&#x17D;R ď&#x20AC;¨ x ď&#x20AC;Šď ­RPď&#x20AC;¨ Aď&#x192;&#x2C6;ď&#x20AC;¨Îą,β,θ ď&#x20AC;Šď&#x20AC;Š ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;Šď&#x20AC;Š

Îź RP ď&#x20AC;¨ Aď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;R

ď&#x201A;Ł Îź RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; yď&#x192;&#x17D;R

s

ď&#x192;&#x161;

ď ť

max ď ­ RPď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ y ď&#x20AC;Š , ď Ą

yď&#x192;&#x17D;Rs ď&#x20AC;¨ x ď&#x20AC;Š

and

ď ˝

max{â&#x2039; yâ&#x2C6;&#x2C6;Rs(x) ÎźRP(A) (y), â&#x2039; yâ&#x2C6;&#x2C6;Rs(x) Îą} ( x ),

ď ­ ď&#x20AC;¨ď&#x20AC;¨

Îą ,β ,θ ď&#x20AC;Š

ď&#x20AC;Š ( x )} = ď ­ RP ď&#x20AC;¨ Aď&#x20AC;Šď&#x192;&#x2C6;ď&#x20AC;¨Îą,β,θď&#x20AC;Š ( x ) .

By the same way, we have

ď ¨ RP

x ď&#x20AC;˝ ď ¨ RP Aď&#x192;&#x2C6;ď&#x20AC;¨ Îą,β,θ ď&#x20AC;Š ( x ) ď&#x20AC;¨ A ď&#x192;&#x2C6;ď&#x20AC;¨ Îą,β,θ ď&#x20AC;Š ď&#x20AC;Š ď&#x20AC;¨ ď&#x20AC;Š

and

ď § RP A ď&#x192;&#x2C6; Îą,β,θ

ď&#x20AC;¨

ď&#x20AC;¨

ď&#x20AC;Šď&#x20AC;Š

ÎźA ď&#x20AC;¨ yď&#x20AC;Š ,

ď&#x20AC;¨xď&#x20AC;Š

s

Îł RP ď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; yď&#x192;&#x17D;R

ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď § RPAď&#x192;&#x2C6;ď&#x20AC;¨ Îą,β,θ ď&#x20AC;Š ( x) .

s

ÎłA ď&#x20AC;¨ y ď&#x20AC;Š ď&#x201A;ł

ď&#x20AC;¨xď&#x20AC;Š

ď&#x192;&#x2122;

s

= RP ď&#x20AC;¨ Aď&#x20AC;Š

s

ď ¨RPď&#x20AC;¨ Aď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;R ď&#x20AC;¨ x ď&#x20AC;ŠÎź A ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ ΡRP ď&#x20AC;¨ A ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ,

= max{ď&#x192;&#x161; ď ­ ď&#x20AC;¨ y ď&#x20AC;Š , ď&#x192;&#x161; yď&#x192;&#x17D;R ď&#x20AC;¨ xď&#x20AC;Šď Ą } yď&#x192;&#x17D;R ď&#x20AC;¨ x ď&#x20AC;Š RPď&#x20AC;¨ A ď&#x20AC;Š

ď ­

ÎźA ď&#x20AC;¨ yď&#x20AC;Š

ď&#x20AC;¨xď&#x20AC;Š

s

â&#x2039; yâ&#x2C6;&#x2C6;Rs(x) ÎźRP(Aâ&#x2C6;Ş(Îą,β,θ) Ě&#x201A; ) (y)=

max{

s

ď §

yď&#x192;&#x17D;R s ď&#x20AC;¨ x ď&#x20AC;Š A

ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ ď § RPď&#x20AC;¨ Aď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š

So RP(A) â&#x160;&#x2020; RP(A) for all A â&#x2C6;&#x2C6; PFS(U).â&#x2013;Ą In the case of connections between special types of crisp relation on U , and properties of rough standard neutrosophic approximation operators, we have the following: Lemma 2. If R is a symmetric crisp binary relation on U, then for all A, B â&#x2C6;&#x2C6; PFS(U),

RP( A) ď&#x192;? B ď&#x192;&#x203A; A ď&#x192;? RP( B) Proof.

It means RP(A â&#x2C6;Ş (Îą,Ě&#x201A; β, θ)) = RP(A) â&#x2C6;Ş (Îą,Ě&#x201A; β, θ). Ě&#x201A; = {(x, 1,0,0)|x â&#x2C6;&#x2C6; U} , then (PL3) Since U = 1U = (1,0,0) we can obtain (PL3) RP(U) = U by using definition 5.

Let R be a symmetric crisp binary relation on U, i.e. y â&#x2C6;&#x2C6; R s (x) â&#x;ş x â&#x2C6;&#x2C6; R s (y), â&#x2C6;&#x20AC;x, y â&#x2C6;&#x2C6; U. We assume contradiction that RP( A) ď&#x192;? B but A ď&#x192;&#x2039; RP( B) .

For each đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2C6;, we consider all the cases:

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

86

Neutrosophic Sets and Systems, Vol. 14, 2016

+ if  A ( x)  μ RP B  x    μ  y  then it exists y0 ∈ yR  x  B

It means A  RP  A  , A  PFS  U  , i.e. (a2) is

s

R s (x) such that

  y  zR s

A

 A ( x)   B ( y0 )  RP ( A)  y0  

satisfied.

Now, assume that (a1) RP  A   A , A  PFS  U  ; we

( z)   A ( x) (because y0 ∈ R s (x) then

0

show that R is reflexive. Indeed, we assume contradiction that R is not reflexive, i.e. x  R  x  . s

x  R s  y0  . This is not true. + the cases



( x)  

A

RP ( B )

( x) or  A ( x)   RP ( B ) ( x) are

A = 1U {x}

also not true. □

We consider

Theorem 3. Let (U, R) be a crisp approximation space, and ̅​̅​̅​̅, the upper and lower PF approximation operators.

RP

1

U {x }

Then: (a) R is reflexive if and only if at least one of the following conditions are satisfied

, i.e.

0 if y  x , μ1U {x}  y    1 if y  x

0 if y  x 1 if y  x , 1 . y     U {x } 0 if y  x 0 if y  x

 y  

Then γ RP A   x  



γ

yR s  x  A

 y  0   A  x  1 .

This is not true. It implies R is reflexive.

(a1) (PLR)RP(A) ⊆ A∀A ∈ PFS(U)

Similarly, we assume that (a2) A  RP  A  , A  PFS  U  ;

(a2) (PUR)A ⊆ RP(A)∀A ∈ PFS(U) (b) R is symmetric if and only if at least one of the following conditions are satisfied

we show that R is reflexive. Indeed, we assume contradiction that R is not reflexive, i.e., x  R  x  . s

1 if y  x A = 1x , i.e., μ1x  y    , 0 if y  x (b2) (PUR)A ⊆ RP (RP(A)) ∀A ∈ PFS(U) 0 if y  x 0 if y  x (c) R is transitive if and only if at least one of the 1x  y   0 if y  x ,  1x  y   1 if y  x .   (b1) (PLR)RP(RP(A)) ⊆ A∀A ∈ PFS(U)

We consider

following conditions are satisfied

μ RP A   x    yR  x μ A  y   0  μ A  x   1 .

(c1) (PLT)RP(A) ⊆ RP(RP(A))∀A ∈ PFS(U)

Then

(c2) (PUT)RP(A) ⊆ RP (RP(A)) ∀A ∈ PFS(U)

This is not true. It implies R is reflexive.

s

(b).

Proof. (a). We assume that R is reflexive, i.e., x  RS ( x) , so that

We verify case (b1).

A  PFS  U 

We assume that R is symmetric, i.e., if

we

have

μ RP A  x    yR  x μ A  y   μ A  x  s

RP A  x    yR  x μ A  y   ηA  x  , s

and γ RP  A   x  



yR s

γ y   A  x  . It means x A  

that RP  A   A , A  PFS  U  , i.e. (a1) was verified.

Similarly, we consider upper approximation of:

μ RP A   x    yR  x μ A  y   μ A  x  , ηRP A   x  = s

 

μ

yR s  x  A yR s 

 y   ηA  x  ,

  y   A  x  . x A

and  RP A   x  =

,

x  RS ( y )

y  RS ( x) . For all A  PFS  U  , because then

 zR s  y  μ A  z   μ A  x 

,

x  RS then ( y)

 zR s  y  μ A  z 

 μ A  x  ,  zR  y   A  z    A  x  for all y  RS ( x) , s

μ RP (RP A ) 



we have

x

 yR s  x  (  zR s  y  μ A  z  )  μ A  x  ,

 RP (RP A )  x    yR  x  (  zR  y  A  z  )  A  x  ; and s

s

 RP (RP A )  x    yR  x  (  zR  y   A  z  )   A  x  . s

s



Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

87

Neutrosophic Sets and Systems, Vol. 14, 2017

It means that RP  RP  A    A A  PFS  U  .

Theorem 4. Let R be a similarity crisp binary relation

We assume contradiction that RP  RP  A    A A  PFS  U  but R is not symmetric, i.e., if x  RS ( y ) then y  RS ( x)

on

lower PF approximation operators. Then, for all A ∈

PFS(U) A  RP  A  –

and if y  RS ( x) then x  RS ( y ) .

We consider

–

A = 1U {x} . Then, μ RP (RP A )  x  

 z  ) =1 > μ A  x   0

 yR s  x  (  zR s  y  μ A

. It

̅​̅​̅​̅, RP: PFS(U) → PFS(U) the upper and U and RP

RP  A   A

~ A  RP  ~ A   RP  ~ A  ~ A .

4. The standard neutrosophic information systems In this section, we introduce a new concept: standard

μ RP (RP A  )  x    A ( x), for all neutrosophic information system.

is not true, because

x U . So that R is symmetric.

Let (U, A, F) be a classical information system. Here U is the (nonempty) set of objects, i.e. U = {u1 , u2 , … , un },

By the same way, it yields (b2).

A = {a1 , a2 , … , am } is the attribute set, and F is the relation set of U and A, i.e. F = {fj : U → Vj , j = 1,2, … , m} ,

(c). R is transitive, i.e., if for all

x, y , z  U :

z  RS ( y ), y  RS ( x) then z  RS ( x) . It means that RS ( y )  RS ( x) , so that for all A  PFS (U ) we have  zR s  x  μ A  z    zR s  y  μ A  z  .

Hence .

.

We call (U, A, F, D, G) an information system or decision table, where U, A, F) is the classical information system, A is the condition attribute set and D is the decision at-

 RP ( RP ( A)) ( x)   yR  x  ( zR  y  μ A  z ) . s

s

So RP( A) ( x)  RP( RP( A)) ( x) , for all x  U , A  PFS (U ) .

It mean that (c1) was varified. Now, we assume contradiction that (c1): RP  A   RP  RP  A   A  PFS  U  , R

is

not

transitive,

i.e.,

x, y , z  U

set of U an D, i.e. G = {g j : U → Vj' , j = 1,2, … , p} where

Vj' is the domain of the attribute d j , j  1, 2,..., p .

Because  RP ( A) ( x)   yR  x  ( zR  x  μ A  z ) s s

but

a j , j  1, 2,..., m

tribute set, i.e. D = {d1 , d2 , … , dp } and G is the relation

 yR s  x  ( zR s  x  μ A  z )   yR s  x  ( zR s  y  μ A  z )

and

where Vj is the domain of the attribute

:

z  RS ( y ), y  RS ( x) then z  RS ( x) . We consider

Let (U, A, F, D, G) be the information system. For B ⊆ A ∪ D, we define a relation, denoted R B = IND(B), as follows, ∀x, y ∈ U: xIND(B)y ⟺ fj (x) = fj (y) for all j ∈ {j: aj ∈ B}.

The equivalence class of x ∈ U based on R B is [x]B = {y ∈ U: yR B x}.

Here, we consider R A = IND(A), R D = IND(D). If RA  RD R A ⊆ R D , i.e., for any [x]A , x ∈ U there exists

[x]D such that [x]A ⊆ [x]D , then the information system is called a consistent information system, other called an in-

A = 1U {x} , then  RP ( A) ( x)   zR s  x  μ A  z   1 , but consistent information system.

 RP ( RP ( A)) ( x)   yR  x  ( zR  y  μ A  z )  0 . s

s

It is false. By same way, we show that (c2) is true. Hence, (c) was verified.⧠ Now, according to Theorem 1, Lemma 1 and Theorem 3, we obtain the following results:

Let (U, A, F, D, G) be the information system, where (U, A, F) is a classical information system.

If D = {Dk |k = 1,2, … , q}, where Dk is a fuzzy subset of U, then (U, A, F, D, G) is the fuzzy information system.

If D = {Dk |k = 1,2, … , q} where Dk is an intutionistic fuzzy subset of U, then (U, A, F, D, G) is an intuitionistic fuzzy information system.

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

88

Neutrosophic Sets and Systems, Vol. 14, 2016

Definition 6. Let (U, A, F, D, G) be the information system or decision table, where (U, A, F) is a classical information system. If D = {Dk |k = 1,2, â&#x20AC;Ś , q}, where Dk is a standard neutrosophic subset of U, and G is the relation set of U and D, then (U, A, F, D, G) is called a standard neutrosophic information system.

Example 2. The following Table 2 gives a standard neutrosophic information system, where the objects set U = {u1 , u2 , â&#x20AC;Ś , u10 }, , the condition attribute set is A = {a1 , a2 , a3 } , and the decision attribute set is D = {D1 , D2 , D3 } , where Dk (k = 1,2,3) is the standard neutrosophic subsets of đ?&#x2018;&#x2C6;.

Table 2: A standard neutrosophic information system

U

a1

a2

a3

D1

D2

D3

(0.15,0.6,0.2)

(0.4,0.05,0.5)

= RPB (Di )(x) > RPB (Dj )(x)(i â&#x2030; j), β(x),0

ď &#x203A; x ď ?B ď&#x192;&#x2021; ď&#x20AC;¨ ď ž D j ď&#x20AC;Šď Ą ď&#x20AC;¨ x ď&#x20AC;Š

ď ą ď&#x20AC;¨ x ď&#x20AC;Š,0

ď &#x203A; x ď ?B

β(x),0

[x]B â&#x2C6;Š (â&#x2C6;ź Dj )Îą(x) â&#x2030; â&#x2C6;&#x2026;

then

θ(x),0

[x]B â&#x2C6;Š (â&#x2C6;ź Dj )Îą(x) â&#x2030; â&#x2C6;&#x2026;

ď&#x201A;šď&#x192;&#x2020;

ι(x),θ(x)

and

β(x),0

ď Ą ď&#x20AC;¨ x ď&#x20AC;Š, ď ˘ ď&#x20AC;¨ x ď&#x20AC;Š

ď&#x192;? ď&#x20AC;¨ Di ď&#x20AC;Šď ą ď&#x20AC;¨ x ď&#x20AC;Š

(Di )β(x)

[x]B â&#x2C6;Š (â&#x2C6;ź Dj )Îą(x) â&#x2030; â&#x2C6;&#x2026;

[x]B â&#x2C6;Š (â&#x2C6;ź Dj )Îą(x) â&#x2030; â&#x2C6;&#x2026;[x]B â&#x160;&#x2020; β(x),0

[x]B â&#x2C6;Š (â&#x2C6;ź Dj )Îą(x) â&#x2030; â&#x2C6;&#x2026;

where (Îą(x), β(x), θ(x)) â&#x2C6;&#x2C6; D* .

Proof.

We have

ď&#x20AC;¨ Di ď&#x20AC;Šď ą ď&#x20AC;¨ď&#x20AC;¨x ď&#x20AC;Šď&#x20AC;Š

ď Ą x ,ď ˘ ď&#x20AC;¨ xď&#x20AC;Š

ď&#x20AC;¨

ď&#x20AC;˝ { y ď&#x192;&#x17D; U : ď ­ Di ď&#x20AC;¨ y ď&#x20AC;Š ,ď ¨ Di ď&#x20AC;¨ y ď&#x20AC;Š , ď § Di ď&#x20AC;¨ y ď&#x20AC;Š

ď&#x20AC;Š

â&#x2030;Ľ (Îą(x), β(x), θ(x))}.

u1

3

2

1

(0.2,0,3,0.5)

u2

1

3

2

(0.3,0.1,0.5)

(0.3,0.3,0.3)

(0.35,0.1,0.4)

Since (ι(x), β(x), θ(x)) = RPB (Di )(x),

u3

3

2

1

(0.6,0,0.4)

(0.3,0.05,0.6)

(0.1,0.45,0.4)

we have ď Ą ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;ď &#x203A; xď ? ď ­D ď&#x20AC;¨ y ď&#x20AC;Š , ď ˘ ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;ď &#x203A; xď ? ď ¨D ď&#x20AC;¨ y ď&#x20AC;Š , i i

u4

3

3

1

(0.15,0.1,0.7)

(0.1,0.05,0.8)

(0.2,0.4,0.3)

and ď ą ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; yď&#x192;&#x17D;ď &#x203A; xď ? ď § D ď&#x20AC;¨ y ď&#x20AC;Š . So that, for any x â&#x2C6;&#x2C6; U, y â&#x2C6;&#x2C6; [x]B i

u5

2

2

4

(0.05,0,2,0.7)

(0.2,0.4,0.3)

(0.05,0.4,0.5)

then ÎźDi (y) â&#x2030;Ľ Îą(x) , ď ¨D ď&#x20AC;¨ y ď&#x20AC;Š ď&#x201A;łď ą ď&#x20AC;¨ x ď&#x20AC;Š ÎłDi (y) â&#x2030;¤ θ(x) and i

u6

2

3

4

(0.1,0.3,0.5)

(0.2,0.3,0.4)

(1,0,0)

ΡDi (y) â&#x2030;Ľ θ(x) . It means that

u7

1

3

2

(0.25,0.3,0.4)

(1,0,0)

(0.3,0.3,0.4)

u8

2

2

4

(0.1,0.6,0.2)

(0.25,0.3,0.4)

(0.4,0,0.6)

u9

3

2

1

(0.45,0,1,0.45)

(0.25,0.4,0.3)

(0.2,0.5,0.3)

u10

1

3

2

(0.05,0.05,0.9)

(0.4,0.2,0.3)

(0.05,0.7,0.2)

B

B

B

ď Ą ď&#x20AC;¨ x ď&#x20AC;Š, ď ˘ ď&#x20AC;¨ x ď&#x20AC;Š

[ x]B ď&#x192;? ď&#x20AC;¨ Di ď&#x20AC;Šď ą ď&#x20AC;¨ x ď&#x20AC;Š

ď Ą ď&#x20AC;¨ x ď&#x20AC;Š, ď ˘ ď&#x20AC;¨ x ď&#x20AC;Š

y ď&#x192;&#x17D; ď&#x20AC;¨ Di ď&#x20AC;Šď ą ď&#x20AC;¨ x ď&#x20AC;Š

, i.e.,

ι(x),β(x)

[x]B â&#x160;&#x2020; (Di )θ(x)

Now, since

ď&#x20AC;¨ď Ą ď&#x20AC;¨ x ď&#x20AC;Š , ď ˘ ď&#x20AC;¨ x ď&#x20AC;Š ,ď ą ď&#x20AC;¨ x ď&#x20AC;Šď&#x20AC;Š ď&#x20AC;˝ RP ď&#x20AC;¨ D ď&#x20AC;Šď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;ž RP ď&#x20AC;¨ D ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Šď&#x20AC;¨i ď&#x201A;š j ď&#x20AC;Š B

i

B

j

then there exists y â&#x2C6;&#x2C6; [x]B such that

5. The knowledge discovery in the standard neutrosophic information systems In this section, we will give some results about the knowledge discovery for a standard neutrosophic information systems by using the basic theory of rough standard neutrosophic set in Section 3. Throughout this paper, let (U, A, F, D, G) be the standard neutrosophic information system and by B â&#x160;&#x2020; A, we denote RPB (Dj ) the lower rough standard neutrosophic approximation of Dj â&#x2C6;&#x2C6; PFS(U) on approximation space (U, R B ).

Theorem 5. Let (U, A, F, D, G) be the standard neutrosophic information system and B â&#x160;&#x2020; A. If for any đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2C6;:

ď&#x20AC;¨ ď ­ ď&#x20AC;¨ x ď&#x20AC;Š ,ď ¨ ď&#x20AC;¨ x ď&#x20AC;Š ,ď § ď&#x20AC;¨ x ď&#x20AC;Šď&#x20AC;Š ď&#x201A;ł ď&#x20AC;¨ď Ą ď&#x20AC;¨ x ď&#x20AC;Š , ď ˘ ď&#x20AC;¨ x ď&#x20AC;Š ,ď ą ď&#x20AC;¨ x ď&#x20AC;Šď&#x20AC;Š Di

Di

Di

ď&#x20AC;¨ ď ­ ď&#x20AC;¨ y ď&#x20AC;Š ,ď ¨ ď&#x20AC;¨ y ď&#x20AC;Š ,ď § ď&#x20AC;¨ y ď&#x20AC;Šď&#x20AC;Š ď&#x20AC;ź ď&#x20AC;¨ď Ą ď&#x20AC;¨ x ď&#x20AC;Š , ď ˘ ď&#x20AC;¨ x ď&#x20AC;Š ,ď ą ď&#x20AC;¨ x ď&#x20AC;Šď&#x20AC;Š Di

Di

Di

(ΟDi (y), ΡDi (y), γDi (y)) < (ι(x), β(x), θ(x))

,i.e.,

or

(ÎźDi (y) < Îą(x) , ÎłDi (y) â&#x2030;Ľ θ(x)) or (ÎźDi (y) = Îą(x) , ÎłDi (y) > θ(x)) or (ÎźDi (y) = Îą(x) , ÎłDi (y) > θ(x)) and ΡDi (y) < β(x)). It means that here exists y â&#x2C6;&#x2C6; [x]B such that

ď&#x20AC;¨ď § ď&#x20AC;¨ y ď&#x20AC;Š ,ď ¨ ď&#x20AC;¨ y ď&#x20AC;Š , ď ­ ď&#x20AC;¨ y ď&#x20AC;Šď&#x20AC;Š ď&#x201A;ł ď&#x20AC;¨ď ą ď&#x20AC;¨ x ď&#x20AC;Š ,0,ď Ą ď&#x20AC;¨ x ď&#x20AC;Šď&#x20AC;Š , Di

θ(x),0

Di

Di

i.e.

y â&#x2C6;&#x2C6; (â&#x2C6;ź

θ(x),0

Dj )Îą(x) . So that [x]B â&#x2C6;Š (â&#x2C6;ź Dj )Îą(x) â&#x2030; â&#x2C6;&#x2026;.â&#x2013;Ą Let (U, A, F, D, G) be the standard neutrosophic information system, R A the equivalence classes which are induced by the condition attribute set đ??´, and the universe is divided by R A as following: Uâ &#x201E;R A = {X1 , X2 â&#x20AC;Ś , Xk }. Then the approximation of the standard neutrosophic decision denoted as, for all i = 1,2, â&#x20AC;Ś , k

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

89

Neutrosophic Sets and Systems, Vol. 14, 2016

ď&#x20AC;¨

RP A ď&#x20AC;¨ D ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š ď&#x20AC;˝ RP A ď&#x20AC;¨ D1 ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š , RP A ď&#x20AC;¨ D2 ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š ,ď&#x201A;ź, RP A ď&#x20AC;¨ Dq ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š

ď&#x20AC;Š

Example 3. We consider the standard neutrosophic information system in Table 2. The equivalent classes

U / RA ď&#x20AC;˝ { X1 ď&#x20AC;˝ ď ťu1 , u3 , u9 ď ˝ , X 2 ď&#x20AC;˝ ď ťu2 , u7 , u10 ď ˝ ,

maxi={1,2,3} RPA (Di )(x) = RPA (D2 )(x) = (0.3,0.3,0.1), and X2 = {u2 , u7 , u10 } â&#x160;&#x2020; (D2 )0.3,0.1 = {u2 , u7 , u10 }.

0.3 For X3 = {u4 }, we have â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; X2 , maxi={1,2,3} RPA (Di )(x) = RPA (D3 )(x) = (0.2,0.3,0.4),

đ?&#x2018;&#x2039;3 = {đ?&#x2018;˘4 }, đ?&#x2018;&#x2039;4 = {đ?&#x2018;˘5 , đ?&#x2018;˘8 }, đ?&#x2018;&#x2039;5 = {đ?&#x2018;˘6 }}

and

X 3 ď&#x20AC;˝ ď ťu4 ď ˝ ď&#x192;? ď&#x20AC;¨ D2 ď&#x20AC;Š0.3

0.3,0.1

The approximation of the standard neutrosophic decision is as follows:

(D2 )0.3,0.1 = {u4 , u6 , u9 }.

0.3

Table 3:

For X3 = {u4 }, we have â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; X2

The approximation of the picture fuzzy decision

U / RA

RP A ď&#x20AC;¨ D1 ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š (0.2,0,0.5)

X1

RP A ď&#x20AC;¨ D2 ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š

RP A ď&#x20AC;¨ D3 ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š

maxi={1,2,3} RPA (Di )(x) = RPA (D3 )(x) = (0.2,0.3,0.4)

(0.15,0.05,0.6)

(0.1,0.05,0.5)

and

(0.3,0.1,0.3)

X2

(0.05,0.05,0.9)

X3

(0.15, 0.1,0.7)

(0.1,0.05,0.8)

(0.2,0.4,0.3)

X4

(0.05,0.2,0.7)

(0.2,0.3,0.4)

(0.05,0,0.6)

X5

(0.1,0.3,0.5)

(0.2,0.3,0.4)

(1,0,0)

ď&#x20AC;¨ D1 ď&#x20AC;Š

1

1

1

1

A

ď&#x20AC;¨ D2 ď&#x20AC;Š

ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;X

1

6 The knowledge reduction and extension of standard neutrosophic information systems

1

1

ď ­ D ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ min ď ť0.15,0.3,0.25ď ˝ ď&#x20AC;˝ 0.15 ,

A ď&#x20AC;¨ D2 ď&#x20AC;Š

ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; yď&#x192;&#x17D;X ď § D ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ max ď ť0.2, 0.6, 0.3ď ˝ ď&#x20AC;˝ 0.6 1

2

ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;X ď ¨D ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ min ď ť0.05,0.45,0.5ď ˝ ď&#x20AC;˝ 0.05 ď § RP ď&#x20AC;¨ D ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; yď&#x192;&#x17D;X ď § D ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ max ď ť0.5,0.2,03ď ˝ ď&#x20AC;˝ 0.5 A

A

ď&#x20AC;¨ D3 ď&#x20AC;Š 3

mation system and B â&#x160;&#x2020; A. B is called the standard neutrosophic reduction of the classical information system (U, A, F), if đ??ľ is the minimum set which satisfies the following relations: for any X â&#x2C6;&#x2C6; PFS(U), x â&#x2C6;&#x2C6; U. RP A ď&#x20AC;¨ X ď&#x20AC;Š ď&#x20AC;˝ RP B ď&#x20AC;¨ X ď&#x20AC;Š , RP A ď&#x20AC;¨ X ď&#x20AC;Š ď&#x20AC;˝ RP B ď&#x20AC;¨ X ď&#x20AC;Š

so RPA (D2 )(x) = (0.15,0.6,0.05) and ÎźRPA (D3) (x) =â&#x2C6;§yâ&#x2C6;&#x2C6;X1 ÎźD3 (y) = min{0.4,0.1,0.2} = 0.1,

ď ¨RP

Definition 7. (i) Let ď&#x20AC;¨U , A, F ď&#x20AC;Š (U, A, F) be the classical infor-

2

ΡRPA(D2) (x) =â&#x2C6;§yâ&#x2C6;&#x2C6;X1 ΡD2 (y) = min{0.6,0.05,0.4} = 0.05 ,

ď § RP

1

3

1

,

3

so that RPA (D3 )(x) = (0.1,0.5,0.05).

Hence, for X1 = {u1 , u3 , u9 } maxi ď&#x20AC;˝ď ť1,2,3ď ˝ RP A ď&#x20AC;¨ Di ď&#x20AC;Šď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝

,

â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; X2

RP A ď&#x20AC;¨ D1 ď&#x20AC;Šď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x20AC;¨ 0.2,0.5,0 ď&#x20AC;Š ,maxi={1,2,3} RPA (Di )(x) =

and X1 = {u1 , u3 , u9 } â&#x160;&#x2020; (D1 )0.2,0 0.5 = {u1 , u2 , u3 , u7 , u9 }; For X2 = {u2 , u7 , u10 }. We have â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; X2 ,

= {u2 , u5 , u8 , u9 , u10 }.

1,0

β(x),0

A

ď&#x20AC;˝ ď ťu2 , u5 , u8 , u9 , u10 ď ˝

X 5 ď&#x20AC;˝ ď ťu6 ď ˝ ď&#x192;? ď&#x20AC;¨ D2 ď&#x20AC;Š0 ď&#x20AC;˝ ď ťu6 ď ˝ .

y â&#x2C6;&#x2C6; (â&#x2C6;ź Dj )Îą(x) , so that RPA (D1 )(x) = (0.2,0.5,0). And

ď ­ RP

(D2 )0.2,0.3 0.4

maxi={1,2,3} RPA (Di )(x) = RPA (D3 )(x) = (0.2,0.3,0.4), and

1

1

A

0.2,0.3

For X3 = {u4 }, we have â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; X2 ,

ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;X ď ­D ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ min ď ť0.2, 0.6, 0.45ď ˝ ď&#x20AC;˝ 0.2 , ď ¨ RP ď&#x20AC;¨ D ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x2122; yď&#x192;&#x17D;X ď ¨ D ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ min ď ť0.3, 0, 0.1ď ˝ ď&#x20AC;˝ 0 ď § RP ď&#x20AC;¨ D ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;&#x161; yď&#x192;&#x17D;X ď § D ď&#x20AC;¨ y ď&#x20AC;Š ď&#x20AC;˝ max ď ť0.5, 0.4, 0.45ď ˝ ď&#x20AC;˝ 0.5 , A

X 4 ď&#x20AC;˝ ď ťu5 , u8 ď ˝ ď&#x192;? ď&#x20AC;¨ D2 ď&#x20AC;Š0.4

X4 = {u5 , u8 } â&#x160;&#x2020;

(0.05,0.1,0.4)

Indeed, for X1 = {u1 , u3 , u9 }.

We have â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; X1 ,

ď ­ RP

ď&#x20AC;˝ ď ťu4 , u6 , u9 ď ˝ X3 = {u4 } â&#x160;&#x2020;

,

(ii) B is called the standard neutrosophic lower approximation reduction of the classical information system (U, A, F), if B is the minimum set which satisfies the following relations: for any X â&#x2C6;&#x2C6; PFS(U), x â&#x2C6;&#x2C6; U

RPA (X) = RPB (X), (iii) B is called the standard neutrosophic upper approximation reduction of the classical information system (U, A, F), if B is the minimum set which satisfies the following relations: for any X â&#x2C6;&#x2C6; PFS(U), x â&#x2C6;&#x2C6; U RP A ď&#x20AC;¨ X ď&#x20AC;Š ď&#x20AC;˝ RP B ď&#x20AC;¨ X ď&#x20AC;Š

where

RP A ď&#x20AC;¨ X ď&#x20AC;Š , RP B ď&#x20AC;¨ X ď&#x20AC;Š , RP A ď&#x20AC;¨ X ď&#x20AC;Š , RP B ď&#x20AC;¨ X ď&#x20AC;Š

RPA (X), RPB (X), RPA (X), RPB (X) are standard neutro-

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

90

Neutrosophic Sets and Systems, Vol. 14, 2016

sophic lower and standard neutrosophic upper approximation sets of standard neutrosophic set X â&#x2C6;&#x2C6; PFS(U) based on RA , RB R A , R B , respectively.

Now, we express the knowledge of the reduction of standard neutrosophic information system by introducing the discernibility matrix.

Definition 8. Let (U, A, F, D, G) be the standard

M ď&#x20AC;˝ [ Dij ]kď&#x201A;´k

neutrosophic information system. Then where

ď ť

ď ˝

ď&#x192;Ź al ď&#x192;&#x17D; A : f l ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x201A;š f l ď&#x20AC;¨ X j ď&#x20AC;Š ; ď&#x192;Ż Dij ď&#x20AC;˝ ď&#x192;­ A ď&#x192;Ż ď&#x192;Ž

g X i ď&#x20AC;¨ Dt ď&#x20AC;Š ď&#x201A;š g X j ď&#x20AC;¨ Dt ď&#x20AC;Š

ď&#x20AC;¨

=

ď&#x20AC;Š

ď ť

ď ˝

max RP A ď&#x20AC;¨ Dz ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š , z ď&#x20AC;˝ 1, 2,ď&#x201A;ź, q ) g Xi (Dk ) =

RPA (Dk (X i )) = max{RPA (Dt (X i )), t = 1,2, â&#x20AC;Ś , q}).

Definition 9. Let (U, A, F, D, G) be the standard neutrosophic information system, for any B â&#x160;&#x2020; A, if the following relations holds, for any x â&#x2C6;&#x2C6; U: RP B ď&#x20AC;¨ Di ď&#x20AC;Šď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;ž RP B ď&#x20AC;¨ D j ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Š â&#x20AC;&#x201C;

of A. Proof. If B â&#x2C6;Š DCij = â&#x2C6;&#x2026;, then B â&#x160;&#x2020; Dij . According to Theorem 6, B is the consistent set of A.â&#x2013;Ą The extension of a standard neutrosophic information system suggested the following definition: Definition 11. (i) Let (U, A, F) be the classical information system and A

â&#x160;&#x2020; B. B is called the standard neutrosophic extension of the classical information system (U, A, F), if B satisfies the

; g X i ď&#x20AC;¨ Dt ď&#x20AC;Š ď&#x20AC;˝ g X j ď&#x20AC;¨ Dt ď&#x20AC;Š following relations:

is called the discernibility matrix of (U, A, F, D, G) (where g Xi (Dk ) is the maximum of RPA (D(Xi )) obtained at Dt Dk , i.e., g X ď&#x20AC;¨ Dt ď&#x20AC;Š ď&#x20AC;˝ RP A Dt ď&#x20AC;¨ X i ď&#x20AC;Š i

B â&#x160;&#x2020; A such that B â&#x2C6;Š DCij = â&#x2C6;&#x2026;, then B is the consistent set

RP A ď&#x20AC;¨ Di ď&#x20AC;Šď&#x20AC;¨ x ď&#x20AC;Š ď&#x20AC;ž RP A ď&#x20AC;¨ D j ď&#x20AC;Š ď&#x20AC;¨ x ď&#x20AC;Šď&#x20AC;¨i ď&#x201A;š j ď&#x20AC;Š

for any X â&#x2C6;&#x2C6; PFS(U), x â&#x2C6;&#x2C6; U RP A ď&#x20AC;¨ X ď&#x20AC;Š ď&#x20AC;˝ RP B ď&#x20AC;¨ X ď&#x20AC;Š , RP A ď&#x20AC;¨ X ď&#x20AC;Š ď&#x20AC;˝ RP B ď&#x20AC;¨ X ď&#x20AC;Š

(ii) B is called the standard neutrosophic lower approximation extension of the classical information system (U, A, F), if B B satisfies the following relations: for any X â&#x2C6;&#x2C6; PFS(U), x â&#x2C6;&#x2C6; U

RP A ď&#x20AC;¨ X ď&#x20AC;Š ď&#x20AC;˝ RP B ď&#x20AC;¨ X ď&#x20AC;Š (iii) B is called the standard neutrosophic upper approximation extension of the classical information system (U, A, F), if B satisfies the following relations: for any X â&#x2C6;&#x2C6; PFS(U), x â&#x2C6;&#x2C6; U

then B is called the consistent set of A. RP A ď&#x20AC;¨ X ď&#x20AC;Š ď&#x20AC;˝ RP B ď&#x20AC;¨ X ď&#x20AC;Š Theorem 6. Let (U, A, F, D, G) be the standard where RPA (X), RPB (X), RPA (X), RPB (X) are picture neutrosophic information system. If there exists a subset B fuzzy lower and upper approximation sets of standard neuâ&#x160;&#x2020; A such that B â&#x2C6;Š Dij â&#x2030; â&#x2C6;&#x2026;, then B is the consistent set of trosophic set X â&#x2C6;&#x2C6; PFS(U) based on R A , R B , respectively.

A. We can easily obtain the following results: Definition 10. Let (U, A, F, D, G) be the standard Definition 12. Let (U, A, F) be the classical information neutrosophic information system system, for any hyper set B, such that đ??´ â&#x160;&#x2020; đ??ľ , if đ??´ is the ď&#x192;Ź al ď&#x192;&#x17D; A : fl ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;˝ fl ď&#x20AC;¨ X j ď&#x20AC;Š ; g D ď&#x201A;š g D ď&#x20AC;¨ ď&#x20AC;Š ď&#x20AC;¨ ď&#x20AC;Š X t X t ď&#x192;Ż standard neutrosophic reduction of the classical information DiCj ď&#x20AC;˝ ď&#x192;­ ď&#x192;&#x2020; ; g D ď&#x20AC;˝ g D ď&#x20AC;¨ ď&#x20AC;Š ď&#x20AC;¨ ď&#x20AC;Š system (U, B, F) , then (U, B, F) is the standard neutroď&#x192;Żď&#x192;Ž X t X t is called the discernibility matrix of (U, A, F, D, G) (where sophic extension of (U, A, F), but not conversely necessary.

Example 4. In the approximation of the standard neutrog Xi (Dk ) is the maximum of RPA (D(X i )) obtained at Dk , sophic decision in Table 2, Table 3. Let B = {a1 , a2 }, then i.e. we obtain the family of all equivalent classes of đ?&#x2018;&#x2C6; based on g Xi ď&#x20AC;¨ Dt ď&#x20AC;Š ď&#x20AC;˝ RP A ď&#x20AC;¨ Dt ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š ď&#x20AC;˝ max RP A ď&#x20AC;¨ Dz ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š , z ď&#x20AC;˝ 1, 2,ď&#x201A;ź, q ). the equivalent relation R B = IND(B) as follows:

ď ť

ď ˝

i

j

i

j

ď ť

ď ˝

g Xi (Dk ) = RPA (Dk (X i )) = max{RPA (Dt (X i )), t = 1,2, â&#x20AC;Ś , q}).

Theorem 7. Let (U, A, F, D, G) be the standard neutrosophic information system. If there exists a subset

U / RB ď&#x20AC;˝ ď ť X1 ď&#x20AC;˝ ď ťu1 , u3 , u9 ď ˝ , X 2 ď&#x20AC;˝ ď ťu2 , u7 , u10 ď ˝, X 3 ď&#x20AC;˝ ď ťu4 ď ˝, X 4 ď&#x20AC;˝ ď ťu5 , u8 ď ˝, X 5 ď&#x20AC;˝ ď ťu6 ď ˝ď ˝

We can get the approximation value given in Table 4.

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

91

Neutrosophic Sets and Systems, Vol. 14, 2016

Table 4: decision

The approximation of the standard neutrosophic

RP B ď&#x20AC;¨ D1 ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š

U / RB

(0.2,0,0.5)

X1

RP B ď&#x20AC;¨ D2 ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š

RP B ď&#x20AC;¨ D3 ď&#x20AC;¨ X i ď&#x20AC;Š ď&#x20AC;Š

(0.15,0.05,0.6)

(0.1,0.05,0.5)

(0.3,0.1,0.3)

(0.05,0.1,0.4)

X2

(0.05,0.05,0.9)

X3

(0.15, 0.1,0.7)

(0.1,0.05,0.8)

(0.2,0.4,0.3)

X4

(0.05,0.2,0.7)

(0.2,0.3,0.4)

(0.05,0,0.6)

X5

(0.1,0.3,0.5)

(0.2,0.3,0.4)

(1,0,0)

It is easy to see that đ??ľ satisfies Definition 7 (ii), i.e., đ??ľ is the standard neutrosophic lower reduction of the classical information system (đ?&#x2018;&#x2C6;, đ??´, đ??š).

The discernibility matrix of the standard neutrosophic information system (đ?&#x2018;&#x2C6;, đ??´, đ??š, đ??ˇ, đ??ş) will be presented in Table 5.

Table 5: The discernibility matrix of the standard neutrosophic information system đ?&#x2018;&#x2C6;â &#x201E;đ?&#x2018;&#x2026;đ??ľ

X1

X2

X3

X4

X1

đ??´

X2

đ??´

đ??´

X3

{đ?&#x2018;&#x17D;2 }

{đ?&#x2018;&#x17D;1 , đ?&#x2018;&#x17D;3 }

đ??´

X4

{đ?&#x2018;&#x17D;1 , đ?&#x2018;&#x17D;3 }

đ??´

đ??´

đ??´

X5

{đ?&#x2018;&#x17D;1 , đ?&#x2018;&#x17D;3 }

đ??´

đ??´

{đ?&#x2018;&#x17D;2 }

X5

đ??´

7 Conclusion In this paper, we introduced the concept of standard neutrosophic information system, and studied the knowledge discovery of standard neutrosophic information system based on rough standard neutrosophic sets. We investigated some problems of the knowledge discovery of standard neutrosophic information system: the knowledge reduction and extension of the standard neutrosophic information systems.

Acknowledgment This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.01-2017.02 .

References [1] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, vol. 11, no.5 , pp 341 â&#x20AC;&#x201C; 356, 1982.

[2] L. A. Zadeh, Fuzzy Sets, Information and Control, Vol. 8, No. 3 (1965), p 338-353 [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy set and systems, vol.20, pp.87-96, 1986.

[4] B.C. Cuong, V. Kreinovick, Picture fuzzy sets â&#x20AC;&#x201C; a new concept for computational intelligence problems, in the Proceedings of the 3rd World congress on information and communication technologies WICTâ&#x20AC;&#x2122;2013, Hanoi, Vietnam, December 15-18, pp 16, 2013.

[5] B.C. Cuong, Picture Fuzzy Sets, Journal of Computer Science and Cybernetics, Vol.30, n.4, 2014, 409-420.

[6] B.C. Cuong, P.H.Phong and F. Smarandache, Standard Neutrosophic Soft Theory: Some First Results, Neutrosophic Sets and Systems, vol.12, 2016, pp.80-91.

[7] L.H. Son, DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets, Expert systems with applications 42, pp 51-66, 2015.

[8] P.H. Thong and L.H.Son, Picture Fuzzy Clustering : A New Computational Intelligence Method, Soft Computing, v.20 (9) 3544-3562, 2016.

[9] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International journal of general systems, Vol. 17, p 191-209, 1990.

[10] Y.Y. Yao, Combination of rough and fuzzy sets based on Îąlevel sets, Rough sets and Data mining: analysis for imprecise data, Kluwer Academic Publisher, Boston, p 301 â&#x20AC;&#x201C; 321, 1997.

[11] W. Z. Wu, J. S. Mi, W. X. Zhang, Generalized fuzzy rough sets, Information Sciences 151, p. 263-282, 2003.

[12] W. Z. Wu, Y. H. Xu, On fuzzy topological structures of rough fuzzy sets, Transactions on rough sets XVI, LNCS 7736, Springer â&#x20AC;&#x201C; Verlag Berlin Heidelberg, p 125-143, 2013.

[13] Y.H. Xu, W.Z. Wu, Intuitionistic fuzzy topologies in crisp approximation spaces, RSKT 2012, LNAI 7414, Š Springer â&#x20AC;&#x201C; Verlag Berlin Heidelberg, pp 496-503, 2012.

[14] B. Davvaz, M. Jafarzadeh, Rough intuitionistic fuzzy information systems, Fuzzy Inform., vol.4, pp 445-458, 2013.

[15] N.X. Thao, N.V. Dinh, Rough picture fuzzy set and picture fuzzy topologies, Science computer and Cybernetics, Vol 31, No 3 (2015), pp 245-254.

[16] B. Sun, Z. Gong, Rough fuzzy set in generalized approximation space, Fifth Int. Conf. on Fuzzy Systems and Knowledge Discovery, IEEE computer society 2008, pp 416-420.

[17] F. Smarandache, A unifying field in logics. Neutrosophy: Neutrosophic probability, set and logic, ARP, Rehoboth 1999.

[18] H. Wang, F. Smarandache, Y.Q. Zhang et al., Interval neutrosophic sets and logic: Theory and applications in computing, Hexis, Phoenix, AZ 2005.

[19] H. Wang, F. Smarandache, Y.Q. Zhang, et al., Single valued neutrosophic sets, Multispace & Multistructure 4 (2010), 410-413.

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems

92

[20] J. Ye, A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets, Journal of Intelligent & Fuzzy Systems 26 (2014) 2459-2466.

[21] P. Majumdar, Neutrosophic sets and its applications to decision making, Computation intelligentce for big data analysis (2015), V.19, pp 97-115.

Neutrosophic Sets and Systems, Vol. 14, 2016

[22] J. Peng, J. Q. Wang, J. Wang, H. Zhang, X. Chen, Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems, International journal of systems science (2016), V.47, issue 10, pp 2342-2358.

Received: December 7, 2016. Accepted: December 21, 2016

Nguyen Xuan Thao, Bui Cong Cuong, Florentin Smarandache, Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems