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University of New Mexico
Single Valued Neutrosophic Soft Approach to Rough Sets, Theory and Application Emad Marei Department of Mathematics, Faculty of Science and Art, Sager, Shaqra University, Saudi Arabia. E-mail: via_marei@yahoo.com
Abstract. This paper aims to introduce a single valued neutrosophic soft approach to rough sets based on neutrosophic right minimal structure. Some of its properties are deduced and proved. A comparison between traditional rough model and suggested model, by using their properties is concluded to show that Pawlak’s approach
to rough sets can be viewed as a special case of single valued neutrosophic soft approach to rough sets. Some of rough concepts are redefined and then some properties of these concepts are deduced, proved and illustrated by several examples. Finally, suggested model is applied in a decision making problem, supported with an algorithm.
Keywords: Neutrosophic set, soft set, rough set approximations, neutrosophic soft set, single valued neutrosophic soft set.
1 Introduction
a collected data. This model has been successfully used in the decision making problems and it has been modified in Set theory is a basic branch of a classical mathematics, many papers such as [13-17]. In 2011, F. Feng et al.[18] which requires that all input data must be precise, but introduced a soft rough set model and proved its properties. almost, real life problems in biology, engineering, E.A. Marei generalized this model in [19]. In 2013, P.K. economics, environmental science, social science, medical Maji [20] introduced neutrosophic soft set, which can be science and many other fields, involve imprecise data. In viewed as a new path of thinking to engineers, 1965, L.A. Zadeh [1] introduced the concept of fuzzy logic mathematicians, computer scientists and many others in which extends classical logic by assigning a membership various tests. In 2014, Broumi et al. [21] introuduced the function ranging in degree between 0 and 1 to variables. concept of rough neutrosophic sets. It is generalized and As a generalization of fuzzy logic, F. Smarandache in 1995, applied in many papers such as [22-31]. In 2015, E.A. initiated a neutrosophic logic which introduces a new Marei [32] introduced the notion of neutrosophic soft component called indeterminacy and carries more rough sets and its modification. information than fuzzy logic. In it, each proposition is estimated to have three components: the percentage of This paper aims to introduce a new approach to soft truth (t %), the percentage of indeterminacy (i %) and the rough sets based on the neutrosophic logic, named single percentage of falsity (f %), his work was published in [2]. valued neutrosophic soft (VNS in short) rough set From scientific or engineering point of view, neutrosophic approximations. Properties of VNS-lower and VNS-upper set’s operators need to be specified. Otherwise, it will be approximations are included along with supported proofs difficult to apply in the real applications. Therefore, Wang and illustrated examples. A comparison between traditional et al.[3] defined a single valued neutrosophic set and rough and single valued neutrosophic soft rough various properties of it. This thinking is further extended to approaches is concluded to show that Pawlak’s approach to many applications in decision making problems such as [4, rough sets can be viewed as a special case of single valued 5]. neutrosophic soft approach to rough sets. This paper delves Rough set theory, proposed by Z. Pawlak [6], is an into single valued neutrosophic soft rough set by defining effective tool in solving many real life problems, based on some concepts on it as a generalization of rough concepts. imprecise data, as it does not need any additional data to Single valued neutrosophic soft rough concepts (NRdiscover a knowledge hidden in uncertain data. Recently, concepts in short) include NR-definability, NRmany papers have been appeared to development rough set membership function, NR-membership relations, NRmodel and then apply it in many real life applications such inclusion relations and NR-equality relations. Properties of as [7-11]. In 1999, D. Molodtsov [12], suggested a soft set these concepts are deduced, proved and illustrated by model. By using it, he created an information system from
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several examples. Finally, suggested model is applied in a decision making problem, supported with an algorithm. 2 Preliminaries In this section, we recall some definitions and properties regarding rough set approximations, neutrosophic set, soft set and neutrosophic soft set required in this paper. Definition 2.1 [6] Lower, upper and boundary approximations of a subset X U , with respect to an equivalence relation, are defined as E ( X ) {[ x]
E
: [ x]
E
, X }, E ( X ) {[ x]E : [ x]E X },
BNDE ( X ) E( X ) E( X ), where
[ x]E {x, U : E ( x) E ( x, )}.
Definition 2.2 [6] Pawlak determined the degree of crispness of any subset X U by a mathematical tool, named the accuracy measure of it, which is defined as ( X ) E ( X ) / E ( X ), E ( X ) . E
Obviously, 0 E ( X ) 1 . If E( X ) E ( X ) , then X is crisp (exact) set, with respect to E , otherwise X is rough set. Properties of Pawlak’s approximations are listed in the following proposition. Proposition 2.1 [6] Let (U , E ) be a Pawlak proximation space and let X , Y U . Then, (a) E( X ) X E( X ) . (b) E( ) = = E( ) and E(U ) = U = E(U ) . (c) E( X Y ) = E( X ) E(Y ) . (d) E( X Y ) = E( X ) E(Y ) . (e) X Y , then E ( X ) E (Y ) and E ( X ) E(Y ) . (f) E ( X Y ) E ( X ) E (Y ) . (g) E( X Y ) E( X ) E(Y ) . (h) E ( X c ) = [ E ( X )]c , X is the complement of X . (i) E ( X c ) = [ E ( X )]c . (j) E( E( X )) = E( E( X )) = E( X ) . (k) E( E( X )) = E( E( X )) = E( X ) . C
Definition 2.3 [33] An information system is a quadruple IS = (U , A, V , f ) , where U is a non-empty finite set of objects, A is a non-empty finite set of attributes, V = {V , e A} , V is the value set of attribute e , e
e
f : U A V is called an information (knowledge) function. Definition 2.4 [12] Let U be an initial universe set, E be a set of parameters, A E and let P (U ) denotes the
power set of U . Then, a pair S = ( F , A) is called a soft set over U , where F is a mapping given by F : A P (U ) . In other words, a soft set over U is a parameterized family of subsets of U . For e A, F (e) may be considered as the set of e -approximate elements of S . Definition 2.5 [2] A neutrosophic set A on the universe of discourse U is defined as A = { x, T ( x), I ( x), F ( x) : x U }, where A A A 0 T ( x) I ( x) F ( x) 3 , andT , I , F 0,1 A A A
Definition 2.6 [20] Let U be an initial universe set and E be a set of parameters. Consider A E , and let P (U ) denotes the set of all neutrosophic sets of U . The collection ( F , A) is termed to be the neutrosophic soft set over U , where F is a mapping given by F : A P (U ). Definition 2.7 [3] Let X be a space of points (objects), with a generic element in X denoted by x . A single valued neutrosophic set A in X is characterized by truth-embership function T A , indeterminacy-membership function I A and falsity-membership function FA . For each point x in X , TA(X),I A(X),F A(X) 0,1 . When X is continuous, a single valued neutrosophic set A can be written as A X (T(x),I(x), F(x)) /x,x X . When X is discrete, A can be written as A in1 (T(xi ),I(x i ),F(x i )) /xi ,xi X . 3 Single valued neutrosophic soft rough set approximations In this section, we give a definition of a single valued neutrosophic soft (VNS in short) set. VNS-lower and VNS-upper approximations are introduced and their properties are deduced, proved and illustrated by many counter examples. Definition 3.1 Let U be an initial universe set and E be a set of parameters. Consider A E , and let P (U ) denotes the set of all single valued neutrosophic sets of U . The collection (G,A) is termed to be VNS set over U , where G is a mapping given by G : A P (U ) . For more illustration the meaning of VNS set, we consider the following example Example 3.1 Let U be a set of cars under consideration and E is the set of parameters (or qualities). Each parameter is a neutrosophic word. Consider E = {elegant, trustworthy, sporty, comfortable, modern}. In this case, to define a VNS means to point out elegant cars, trustworthy cars and so on. Suppose that, there are five cars in the universe U , given by U {h1 , h2 , h3 , h4 , h5} and the set of parameters A {e1 , e2 , e3 , e4 } , where A E and each ei is a specific criterion for cars: e1 stands for elegant, e 2 stands
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for trustworthy, e 3 stands for sporty and e 4 stands for comfortable. A VNS set can be represented in a tabular form as shown in Table 1. In this table, the entries are c ij corresponding to the car hi and the parameter e j , where Cij = (true membership value of hi , indeterminacy-membership value of hi , falsity membership value of hi ) in G ( e i ) . e1
e2
e3
e4
h1
(.6, .6, .2)
(.8, .4, .3)
(.7, .4, .3)
(.8, .6, .4)
h3
(.4, .6, .6) (.6, .4, .2)
(.6, .2, .4)
(.6, .4, .3)
(.8, .1, .3)
(.7, .6, .6)
(.7, .2, .5)
Proof Let h , T (h ), I (h ), F (h ) , h , T (h ), I (h ), F (h ) 2 e 2 e 2 and h , T (h 1), e I 1(h e), 1F (eh 1) G (2A) e. Then, 3
h5
(.7, .6, .4)
(.6, .3, .3)
(.8, .2, .2)
(.5, .2, .6)
(.7, .5, .6)
(.8, .2, .3)
(.8, .3, .2)
(.7, .3, .4)
(.9, .5, .7)
Table1: Tabular representation of (G, A) of Example 3.1. Definition 3.2 Let (G , A) be a VNS set on a universe U . For any element h U , a neutrosophic right neighborhood, with respect to e A is defined as follows he = {hi U : Te (hi ) Te (h), I e (hi ) I e (h), Fe (hi ) Fe (h)}.
Definition 3.3 Let (G,A) be a VNS set on U. Neutrosophic right minimal structure is defined as follows {U , , h e : h U , e A}
Illustration of Definitions 3.2 and 3.3 is introduced in the following example Example 3.2 According Example 3.1, we can deduce the following results: h1e h1e h1e h1e {h } , h2 e h2e 1
2
3
4
1
1
3
{h , h } , h2e {h , h , h , h } , h2e {h , h , h } , h3 e h3e {h , h } , 1 2 1 2 4 5 1 2 3 1 3 2
4
1
3
e
3
e
3
e
1
e
1
e
1
e
1
e
1
= F (h ) . For every e A , h1 h1e . Then h1 R e h1 and e
1
(b) Let h R h and h R h , then h h and h
4
h3e2 {h , h , h , h } , h3e3 {h , h , h } , h4e1 {h , h3 , h } , h4e2 {h , h } , 1 3 4 5 1 3 5 4 5 1 4
e
1
2
h . Hence, T ( h ) e
2e
e
3
1
e
3
1
e
2
e
2
3
T (h ) , I (h ) e
e
1
2
T (h ) , I (h ) e
2
e
3
1e
3
I (h ) , F (h ) e
e
1
e
2
3
I ( h ) and F ( h ) e
1
e
3
2
I ( h ) and e
2
F ( h ) . Consequently, we have T ( h )
T (h ) , I (h ) e
2
F (h ) , T (h )
F (h ) e
h4
e
(a) Obviously, T (h ) = T (h ) , I (h ) = I (h ) and F (h )
then R e is reflexive relation.
U
h2
(c) R e may be not symmetric relation.
e
3
F (h ) . It e
1
follows that, h h . Then h R h and then R is 3
1e
1
e
3
e
transitive relation. The following example proves (c) of Proposition 3.1. Example 3.3 From Example 3.2, we have, h1e {h1} and 1 h3 e1 {h , h } . Hence, (h2 , h1 ) Re1 but (h , h ) Re1 . 1 3 1 3 Then, R e isn’t symmetric relation. Definition 3.4 Let (G,A) be a VNS set on U , and let be a neutrosophic right minimal structure on it. Then, VNSlower and VNS-upper approximations of any subset X based on , respectively, are
S X {Y : Y X }, S X {Y : Y X }.
Remark 3.1 For any considered set X in a VNS set (G,A), the sets c PNR X S X , N NR X [ S X ] , b NR X S X P NR X
are called single valued neutrosophic positive, single valued neutrosophic negative and single valued neutrosophic boundary regions of a considered set X , It follows that, respectively. The real meaning of single valued {{h1 }, {h5 }, {h1 , h2 }, {h1 , h3 }, {h1 , h5 }, {h4 , h5 }, neutrosophic positive of X is the set of all elements which are surely belonging to X, single valued neutrosophic {h1 , h2 , h3 }, {h1 , h3 , h4 }, {h1 , h3 , h5 }, {h1 , h2 , h3 , h4 } negative of X is the set of all elements which are surely not , {h1 , h2 , h4 , h5 }, {h1 , h3 , h4 , h5 }, U , } belonging to X and single valued neutrosophic boundary of Proposition 3.1 Let (G , A) be a VNS set on a universe U , X is the elements of X which are not determined by (G,A). is the family of all neutrosophic right neighborhoods on Consequently, the single valued neutrosophic boundary region of any considered set is the initial problem of any it, and let real life application. Re : U , Re (h) = he VNS rough set approximations properties are introduced in Then, the following proposition. (a) R e is reflexive relation. Proposition 3.2 Let (G,A) be a VNS set on U, and let (b) R e is transitive relation. X , Z U . Then the following properties hold h4 e U , h4e {h , h , h , h } , h5 e h5e h5e {h } , h5e {h , h } . 1 2 3 4 5 1 5 3
4
1
2
4
3
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(a) S X X S X .
S( X Z ) S X S Z .
(b) S = S = .
(i) Let h S ( X Z ) . But S ( X Z ) {Y : Y X Z } . Then, there exists Y such that Y X Z
(c) S U = S U = U . (d) X Z S X S Z . (e) X Z S X S Z .
(f) S ( X Z ) S X SZ . (g) S ( X Z ) S X SZ . (h) S ( X Z ) S X S Z (i) S ( X Z ) S X S Z . Proof (a) From Definition 3.3, obviously, we can deduce that, S X X S X .
(b) From Definition 3.4, we can deduce that S and S {Y : Y } . (c) From Property (a), we have U S U but U is the universe set, then S U U . Also, from Definition 3.4, we have SU {Y : Y U } , but U . Then, SU U (d) Let X Z and h S X , then there exists Y such that h Y X . But X Z , then h Y Z . Hence, h SZ . Consequently S X S Z . (e) Let X Z and h S Z . But S Z {Y : Y .
h Y and Y Z such that U there exists Then. Z } But X Z , then Y X and h Y . Hence h S Z . Thus S X S Z . (f) Let h S ( X Z ) {Y : Y X Z} . So, there exists Y such that, h Y X Z , then h Y X and h Y Z . Consequently, h S X and h S Z , then h S X SZ . Thus S ( X Z ) S X S Z . (g) Let h S ( X Z ) {Y : Y X Z } . So, for all Y , h Y , we have Y X Z , then Y X and
Y Z . Consequently, h S X and h S Z . So
and
h Y . Then, Y X , h Y and Y Z , h Y .
It follows that, h S X S Z . Thus S ( X Z ) S X S Z . The following example illustrates that the converse of Property (a) doesn’t hold Example 3.4 From Example 3.1, if X {h3 } , then S X . X S X and S X X Hence. S X {h1 , h3 } and The following example illustrates that the converse of Property (d) doesn’t hold Example 3.5 From Example 3.1, if X {h2 } and Z {h1, h2} , then S X , S Z {h1 , h2 } . Thus S X SZ . The following example illustrates that the converse of Property (e) doesn’t hold Example 3.6 From Example 3.1, if X {h5 } and Z {h2 , h5 } , then, S X {h5 } and S Z {h1 , h2 , h4 , h5 } . Hence, S X S Z .
The following example illustrates that the converse of Property (f) doesn’t hold Example 3.7 From Example 3.1, If X {h1 , h3 , h4 } and Z {h1 , h4 , h5 } , then S X {h1 , h3 , h4 } , S Z {h1 ,
S ( X Z ) S X Hence. S ( X Z ) {h1} and h4 , h5 } . S Z The following example illustrates that the converse of Property (g) doesn’t hold Example 3.8 From Example 3.1, if X {h1 } and Z {h2 } then S X {h1} , S Z and S ( X Z ) {h1, h2} . Hence S ( X Z ) S X S Z . The following example illustrates that the converse of Property (h) doesn’t hold Example 3.9 From Example 3.1, if X {h1 , h2 , h4 } and Z {h1 , h2 , h5 } then S X {h1 , h2 , h4 } , S Z {h1 , h2 , h4 , h5 } and S ( X Z ) {h1 , h2 } . Hence
h S X S Z . Thus S ( X Z ) S X S Z .
S ( X Z ) S X S Z
(h) Let h S X S Z . Then, h S X or h S Z and then there exists Y such that Y X , h Y or Y X ,
The following example illustrates that the converse of Property (i) doesn’t hold
h Y . Consequently
Example 3.10 From Example 3.1, if X {h2 , h3 } and
h S ( X Z ) . Thus
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Z {h5 } then S X {h1 , h2 , h3 } , S Z {h5 } and S ( X Z ) U . Hence S ( X Z ) S X S Z .
Proposition 3.3 Let (G , A) be a neutrosophic soft set on a unverse U , and let X , Z U . Then the following properties hold. (a) S S X S X (b) S S X S X
and h Y Z . Consequently, h S X and h S Z , then h S X S Z . Therefore S ( X Z ) S X S Z . The following example illustrates that the converse of Proposition 3.4 doesn’t hold. Example 3.13 From Example 3.1, if X {h1 , h3 , h5 } and Z {h1, h5} , then S X {h1 , h3 , h5 } , S Z {h1 , h5 } , S ( X Z ) and S X S Z {h3 } . Hence, S ( X Z ) S X S Z
(d) S S X S X
Proposition 3.5 Let (G , A) be a VNS set on U and let X , Z U . Then the following properties don’t hold (a) S X c [ S X ]c
Proof
(b) S X c [ S X ]c
(c) S S X S X
(a) Let W = S X and some
h W {Y : Y X } . Then, for
e A , we have h Y W . So h SW . Hence W
SW . Thus, SW S S W . Also, from Property (a) of Proposition 3.2, we have S X X and by using Property (d) of Proposition 3.2, we get S S X S X . Consequently. S X = S S X (b) Let W S X and h W , from Definition 3.4, we have W {Y : Y X }. Then there exists Y , such that Y X and h Y . Hence, there exists Y , such that Y W and h Y , it follows that h S W . Consequently W S W . Also, by using Property (a) of Proposition 3.2, we have W S W . Thus S S W S W Properties (c) and (d) can be proved directly from Proposition 3.2. The following example illustrates that the converse of Property (c) doesn’t hold. Example 3.11 From Example 3.1, if X {h4 } . Then S X {h4 } and S S X . Hence, S S X S X . The following example illustrates that the converse of Property (c) doesn’t hold. Example 3.12 From Example 3.1, if X {h1 , h2 , h5 } , then S X {h1 , h2 , h5 } and S S X {h1 , h2 , h4 , h5 } . Hence
S S X S X
Proposition 3.4 Let (G , A) be a VNS set on X , Z U . Then
U and let
S ( X Z ) S X S Z Proof Let h S ( X Z ) {Y : Y ( X Z )} . So, there exists Y such that h Y ( X Z ) , then h Y X
(c) S ( X Z ) S X S Z The following example proves Properties (a) and (b) of Proposition 3.5. Example 3.14 From Example 3.1, if X {h1} . Then, c S X S X {h1 } , S X c {h4 , h5 }and S X U . Thus c c c c S X [ S X ] and S X [ S X ] The following example proves Property (c) of Proposition 3.5. Example 3.15 From Example 3.1, if X {h1 , h2 } and Z {h1 } . Then S X {h1 , h2 } , S Z {h1} , S ( X Z ) {h1 , h2 } . Hence S ( X Z ) S X S Z . Remark 3.2 A comparison between traditional rough and single valued neutrosophic soft rough approaches, by using their properties, is concluded in Table 2, as follows 4 Single valued neutrosophic soft rough concepts In this section, some of single valued neutrosophic soft rough concepts (NR-concepts in short) are defined as a generalization of traditional rough concepts. Definition 4.1 Let (G , A) be a VNS set on U . A subset X U is called (a) NR-definable (NR-exact) set if S X S X X (b) Internally NR-definable set if S X X and S X X (c) Externally NR-definable set if S X X and S X X (d) NR-rough set if S X X and S X X The following example illustrates Definition 4.1. Example 4.1 From Example 3.1, we can deduce that {h1 } , {h5 }, {h1 , h2 }, {h1 , h3 }, {h1 , h5 }, {h4 , h5 } , {h1 , h2 , h3},{h1 , h3 , h4 },{h1 , h3 , h5},{h1 , h4 , h5},{h1 , h2 , h3 , h4 } , {h1 , h2 , h4 , h5 }, {h1 , h3 , h4 , h5 } are NR-definable sets, {h1 , h2 , h5 }, {h1 , h2 , h3 , h5 } are internally NR-definable sets, {h4 }, {h1 , h4 }, {h1 , h2 , h4 } are externally NR-definable sets and the rest of proper subsets of U are
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NR-rough sets. We can determine the degree of single valued neutrosophic soft-crispness (exactness) of any subset X U by using NR-accuracy measure, denoted by C X , which is defined as follows Definition 4.2 Let (G,A) be a VNS on U , and let X U . Then C X S X S X , X Remark 4.1 Let (G,A) be a VNS on U . A subset X U is NR-definable (NR-exact) if and only if C X 1 . Definition 4.3 Let (G,A) be a VNS on U and let X U , x X . NR-membership function of an element x to a set X denoted by X x is defined as follows: X x | xA X | / | xA |, where x A {xe : e A} and xe is a neutrosophic right neighborhood, defined in Definition 3.2. Proposition 4.1 Let (G,A) be a VNS on U , X U and let X x be the membership function defined in Definition 4.3. Then X x [0,1] Proof Where x A X x A then 0 x A X x A and then 0 X x 1. Proposition 4.2 Let (G,A) be a VNS on U and let X U , then X x 1 x X Proof Let X x 1, then x A X x A . Consequantly x A X . From Proposition 3.1, we have Re is a reflexive relation for all e A . Hence x xe e A . It follows that x x A . Thus x X The following example illustrates that the converse of Proposition 4.2 doesn’t hold. Example 4.2 From Example 3.2, we get h3 A {h1 , h3 } . If X {h , h , h } , then X h3 1 2 . Although h3 X 2 3 5 Proposition 4.3 Let (G,A) be a VNS on U and let X , Z U . If X Z , then the following properties hold (a)
X x Z x
(b) S X x S Z x (c) S X x S Z x Proof (a) Where X U , for any x U we can deduce that X x Z x . Thus x A X x A Z then xA Z , x A X
We get the proof of Properties (b) and (c) of Proposition 4.3, directly from property (a) of Proposition 4.3 and properties (d) and (e) of Proposition 3.2. Traditional rough properties VNS rough properties S ( X Z ) S X S Z E( X Z ) E X EZ E ( X Y ) = E ( X ) E (Y ) S ( X Z ) S X S Z E ( E ( X )) = E ( X ) E ( E ( X )) = E ( X ) E ( X c ) = [ E ( X )]c c E ( X c ) = [ E ( X )]
S S X S X
S S X S X
S X c [ S X ]c
S X c [ S X ]c
Table 2: Comparison between traditional, VNS rough Proposition 4.4 Let (G,A) be a VNS on U and let XU, then the following properties hold (a) S X x X x
(b) X x x S X
(c) S X x x S X
Proof can be obtained directly from Propositions 3.2 and property (a) of Proposition 4.3. Definition 4.4 Let (G,A) be a VNS set on U , and let x U , X U . NR-membership relations, denoted by and are defined as follows x X if x S X and x X if x S X Proposition 4.5 Let (G,A) be a VNS set on U , and let x U , X U . Then (a) x X x X (b) x X x X Proof (a) Let x X , hence by using Definition 4.4, we get x S X . But from Proposition 3.2, we have S X X , then x X . (b) Let x X , according to Proposition 3.2, we have X S X , then x S X , by using Definition 4.4, we can deduce that x X . Consequently x X x X . The following example illustrates that the converse of Proposition 4.5 doesn’t hold. Example 4.3 From Example 3.1, if X {h2 , h5 } , then S X {h5 } and S X {h1 , h2 , h4 , h5 } . Hence, h2 X , although h2 X and h4 X , although h4 X . Proposition 4.6 Let (G,A) be a VNS on Then the following properties hold
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(a) x X X x 1 (b) X x 1 x X
Proposition 4.9 Let (G,A) be a VNS on U . Then
Proof can be obtained directly from Definition 4.4 and Propositions 4.2 and 4.5.
Proof comes directly From Proposition 3.2.
The following example illustrates that the converse of property (a) does not hold. Example 4.4 From Example 3.1, if X {h1 , h4 } then S X {h1} and h4 A {h4 } , it follows that X h4 1 . Although h4 X
The following example illustrates that, the converse of Proposition 4.9 doesn’t hold. Example 4.9 In Example 3.1, if X {h1 , h4 } and Z {h1 , h2 , h5 } , then S X {h1}, S Z {h1 , h2 , h5 } , S X {h1 , h4 } and S Z {h1 , h2 , h4 , h5 } . Hence, X Z and X Z . Although X Z
U and let X , Z
X Z X Z X Z
The following example illustrates that the converse of property (b) does not hold. Example 4.5 From Example 3.1, if X {h2 } , then S X {h1 , h2 } and h2 A {h1 , h2 } , it follows that h2 X , although X h2 1
From Definition 4.5 and Proposition 4.3, the following remarks can be deduced Remark 4.2 Let (G,A) be a VNS on U and let X , Z U . If X Z , then the following properties hold (a) S X x S Z x
Proposition 4.7 Let (G,A) be a VNS on X U . Then (a) X x 0 x X
U and let
(b) X x 0 x X Proof is straightforward and therefore is omitted. The following example illustrates that the converse of property (a), does not hold. Example 4.6 From Example 3.1, if X {h1 , h3 , h4 } and from Example 3.2, we get h2 A {h1 , h2 } , then X h2 0 , although h2 X The following example illustrates that the converse of property (b), does not hold. Example 4.7 From Example 3.1, if X {h1 , h4 , h5 } , then S X {h1 , h4 , h5 } , from Example 3.2, we get h2 A {h1 , h2 } , it follows that X h2 0 , although h2 X Proposition 4.8 Let (G,A) be a VNS on U and let XU. The following property does not hold X x 0 x X
The following example proves Proposition 4.8. Example 4.8 From Example 3.1, if X {h2 } then S X {h1, h2} , from Example 3.2, we get h1 A {h1} , it follows that h1 X , although X h1 0 Definition 4.5 Let (G,A) be a VNS on U and let X , Z U . NR-inclusion relations, denoted by and which are defined as follows X Z If S X S Z
X Z If S X S Z
(b) S X x Z x
(c) S X x x S Z
Remark 4.3 Let (G,A) be a VNS on U and let X , Z U . If X Z , then the following properties hold (a) x x SX
S Z
(b) X x x S Z
(c) S X x x S Z
Definition 4.6 Let (G,A) be a VNS on U and let X , Z U . NR-equality relations are defined as follows
X Z If
X Z If If X Z
S X SZ
S X S Z
X Z X Z
The following example illustrates Definition 4.6. Example 4.10 According to Example 3.1. Let A {e1} , then {U , ,{h1}, {h5 }, {h1 , h2 }, {h1 , h3}, {h1 , h3 , h4 }} . If X1 {h2}, X 2 {h3}, X 3 {h1 , h2 }, X 4 {h2 , h3} and X 5 {h2 , h4 } , then S X1 S X 2 , S X 1 S X 3 {h1 , h2 } , S X 4 S X 5 and S X 4 S X 5 U . Consequently X 1 X 2 , X 1 X 3 and X 4 X 5 Proposition 4.10 Let (G,A) be a VNS set on X , Z U . Then (a) X S X
U and let
(b) X S X (c) X Z X Z (d) X Z , Z X
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(e) X Z , X U Z U (f) X Z , Z X (g) X Z , X U Z U Proof. From Definition 4.6 and Propositions 3.2 and 3.3 we get the proof, directly. From Definition 4.6 and Proposition 4.3, the following remarks can be deduced Remark 4.4 Let (G,A) be a VNS on U and let X , Z U . If X Z , then the following properties hold (a) S X x S Z x
parameters of the person X. To solve this problem, we need the following definitions Definition 5.1 Let (G,A) be a VNS set on U {h1, h2 ,..., hn } as the objects and A {e1 , e2 ,.., em } is the set of parameters. The value matrix is a matrix whose rows are labeled by the objects, its columns are labeled by the parameters and the entries Cij are calculated by Cij (Tej (hi ) I ej (hi ) Fej (hi )),
1 i n,1 j m
Definition 5.2 Let (G,A) be a VNS set on U {h1, h2 ,..., hn } , where A {e1 , e2 ,.., em } . The score of an object h j is defined as follows S ( hi ) m j 1 Cij
(b) S X x Z x
(c) S X x x S Z
Remark 4.5 Let (G,A) be a VNS on U and let X , Z U . If X Z , then the following properties hold (a) x x SX
U and
(b) m S (hi ) 2m, hi U
S Z
(b) X x x S Z
(c) S X x x S Z
Remark 5.1 Let (G, A) be a VNS set on A {e1 , e2 , then is the set of parameters. .., em } (a) 1 Cij 2, 1 i n,1 j m
The real meaning of C A is the degree of crispness of A . Hence, if C A 1 , then A is NR-definable set. It means that the collected data are sufficient to determine the set A . Also, from the meaning of the neutrosophic right neighborhood, we can deduce the most suitable choice by using the following algorithm.
The following remark is introduced to show that Pawlak’s approach to rough sets can be viewed as a special case of proposed model. Remark 4.6 Let (G,A) be a VNS on U and let X , Z U . Algorithm If we consider the following case 1. Input VNS set (G,A) ( If Te (hi ) 0.5 , then e(h) 1 , otherwise e(h) 0 ) 2. Compute the accuracy measures of all singleton sets and the neutrosophic right neighborhood of an element h is 3. Consider the objects of NR-definable singleton sets replaced by the following equivalence class 4. Compute the value matrix of the considered objects [h] e {hi U : e(hi ) e(h), e A}. 5. Compute the score of all considered objects in a tabular form Then VNS-lower and VNS-upper approximations will be traditional Pawlak’s approximations. It follows that NR- 6. Find the maximum score of the considered objects concepts will be Pawlak’s concepts. Therefor Pawlak’s 7. If there are more than one object has the maximum approach to rough sets can be viewed as a special case of scare, then any object of them could be the suitable suggested single valued neutrosophic soft approach to choice rough sets. 8. If there is no NR-definable singleton set, then we consider the objects of all NR-definable sets consisting 5 A decision making problem two elements and then repeat steps (4-7), else, consider In this section, suggested single valued neutrosophic the objects of all NR-definable sets consisting three soft rough model is applied in a decision making problem. elements and then repeat steps (4-7),and so on... We consider the problem to select the most suitable car which a person X is going to choose from n cars (h1, h2 ,..., For illustration the previous technique, the following hn ) by using m parameters ( e1 , e2 ,.., em ). Since these data are not crisp but neutrosophic, the example is introduced. selection is not straightforward. Hence our problem in this Example 5.1 According to Example 3.1, we can create section is to select the most suitable car with the choice Tables 3, as follows
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Singleton sets
{h1}
{h2 }
{h3 } {h4 } {h5 }
1 0 0 0 1 C X Table 3: Accuracy measures of all singleton sets. Hence C {h1 } C {h5 } 1 . It follows that h1 and h5 are the NR-definable singleton sets. Consequently h1 and h5 are concidered objects. Therefore Table 4 can be created as follows Object h1
e1
e3
e2
e4
(.6,.6,.2) (.8,.4,.3) (.7,.4,.3) (.8,.6,.4) (.8,.2,.3) (.8,.3,.2) (.7,.3,.4) (.9,.5,.7) h5 Table 4: Tabular representation of considered objects. The value matrix of considered objects can be viewed as Table 5. Object h1
e1
e2
e3
e4
1 0.9 0.8 1 0.7 0.9 0.6 0.7 h5 Table 5: Value matrix of considered objects.
Finally, the scores of considered objects are concluded in Table 6, as follows Object
Score of the object 3.7 2.9 Table 6: The scores of considered objects.
h1 h5
Clearly, the maximum score is 3.7, which is scored by the car h1 . Hence, our decision in this case study is that a car h1 is the most suitable car for a person X , under his choice parameters. Also, the second suitable car for him is a car h5 .
Obviously, the selection is dependent on the choice parameters of the buyer. Consequently, the most suitable car for a person X need not be suitable car for another person Y . Conclusion This paper introduces the notion of single valued neutrosophic soft rough set approximations by using a new neighborhood named neutrosophic right neighborhood. Suggested model is more realistic than the other traditional models, as each proposition is estimated to have three components: the percentage of truth, the percentage of indeterminacy and the percentage of falsity. Several properties of single valued neutrosophic soft rough sets have been defined and propositions and illustrative examples have been presented. It has been shown that Pawlak’s approach to rough sets can be viewed as a special case of single valued neutrosophic soft approach to rough
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Received : April 11, 2018. Accepted : April 25, 2018.
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Emad Marei, Single valued neutrosophic soft approach to rough sets, theory and application