NEUTROSOPHIC VAGUE SOFT EXPERT SET THEORY

Transcription

1 NEUTROSOPHIC VAGUE SOFT EXPERT SET THEORY Ashraf Al-Quran a Nasruddin Hassan a1 and Florentin Smarandache b a School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia UKM Bangi Selangor DE Malaysia b Department of Mathematics University of New Mexico 705 Gurley Avenue Gallup NM USA Abstract In this paper we first introduce the concept of neutrosophic vague soft expert sets (NVSESs for short) which combines neutrosophic vague sets and soft expert sets to be more effective and useful We also define its basic operation namely complement union intersection AND and OR along with illustrative examples and study some related properties supporting proofs In our model the user can know the opinion of all experts in one model Keywords neutrosophic soft expert set; neutrosophic vague set; neutrosophic vague soft set; soft expert set 1 Introduction In reality the limitation of precise research is increasingly being recognized in many fields such as economics social science and management science etc It is well known that the real world is full of uncertainty imprecision and vagueness so researches on these areas are of great importance In recent years uncertain theories such as probability theory fuzzy set theory [1] intuitionistic fuzzy set theory [2] vague set theory [3] rough set theory [4] and interval mathematics have developed greatly and achievements have been widely applied in lots of social fields But as we know in some real life problems for proper description of an object in uncertain and ambiguous environment we need to handle the indeterminate and incomplete information But these theories do not handle the indeterminate and inconsistent information Thus neutrosophic set (NS in short) is defined by Smarandache [5] as a new mathematical tool for dealing with problems involving incomplete indeterminacy inconsistent knowledge In NS the indeterminacy is quantified explicitly and truth-membership indeterminacy membership and false-membership are completely independent Furthermore many research and applications in the literature based on neutrosophic set are undertaken such as some aggregation operators of interval neutrosophic linguistic numbers for multiple attribute decision making [6] similarity measures between interval neutrosophic sets and their applications in 1 Corresponding author Nasruddin Hassan School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia UKM Bangi Selangor DE Malaysia Tel: ; nas@ukmedumy

2 multicriteria decision-making [7] multicriteria decision-making method using aggregation operators for simplified neutrosophic sets [8] and improved correlation coefficients of single valued neutrosophic sets and interval neutrosophic sets for multiple attribute decision making [9] However all of these theories have their inherent difficulties in order to overcome these disadvantages Molodtsov [10] firstly proposed a new mathematical tool called soft set theory to deal with uncertainty and imprecision and it has been demonstrated that this new theory brings about a rich potential for applications in decision making measurement theory game theory etc Presently research on the theoretical and application aspects of soft sets and its various generalizations are progressing rapidly Thus After Molodtsovs work some operations and application of soft sets were studied by Chen et al[11] and Maji et al [1213] Also Maji et al[14] have introduced the concept of fuzzy soft set a more general concept which is a combination of fuzzy set and soft set and studied its properties and also Roy and Maji [15] used this theory to solve some decision-making problems Vague soft set theory was provided by Xu [16] later on Alhazaymeh and Hassan [17] introduced the concept of generalized vague soft set and its application followed by possibility vague soft set along with a few applications in decision making [18] and interval-valued vague soft sets and its application [19] they also introduced the concept of Possibility Interval- Valued Vague Soft Set [20] By the concept of neutrosophic set and soft set first time Maji [21] introduced neutrosophic soft set As a combination of neutrosophic set and vague set Alkhazaleh [22] introduced the concept of neutrosophic vague set Among the significant milestones in the development of the theory of soft sets and its generalizations is the element of expert sets which enables the users to know the opinion of all the experts in one model without the need for any operations This new aspect has further improved the theory of soft sets and made it better suited to be used in solving decision making problems especially when used with the more accurate generalizations of soft sets such as fuzzy soft sets intuitionistic fuzzy soft sets neutrosophic soft set and neutrosophic vague soft set this aspect was established by Alkhazaleh and Salleh (see [23]) The duo of Alkhazaleh and Salleh then proceeded to introduce the notion of fuzzy soft expert sets [24] and gave the application of this concept in decision making and medical diagnosis problems Hassan and Alhazaymeh introduced the theory of vague soft expert sets (see [25] ) and used it to solve decision making and medical diagnosis problems They also introduced the mapping on generalized vague soft expert set (see [26] ) In this paper we introduce the concept of neutrosophic vague soft expert set which is a combination of neutrosophic vague set and soft expert set and then define its basic operation namely complement union intersection AND and OR and study their properties 2 Preliminaries In this section we recall some basic notions in neutrosophic vague set neutrosophic vague soft set soft expert set and neutrosophic soft expert set DEFINITION 21 (see [22]) A neutrosophic vague set A NV (NVS in short) on the universe of discourse X written as A NV = < x; T ANV (x);î ANV (x); F ANV (x) >;x X}

3 whose truth-membership indeterminacy - membership and falsity-membership functions is defined as T ANV (x) = [T T + ] Î ANV (x) = [I I + ] and F ANV (x) = [F F + ] where (1) T + = 1 F (2) F + = 1 T and (3) 0 T + I + F 2 + DEFINITION 22 (see [22]) Let Ψ NV be a NVS of the universe U where u i U T ΨNV (x) = [11] Î ΨNV (x) = [00] F ΨNV (x) = [00] then Ψ NV is called a unit NVS where 1 i n Let Φ NV be a NVS of the universe U where u i U T ΦNV (x) = [00] Î ΦNV (x) = [11] F ΦNV (x) = [11] then Φ NV is called a zero NVS where 1 i n DEFINITION 23 (see [22]) Let A NV and B NV be two NVSs of the universe U If u i U (1) T ANV (u i ) = T BNV (u i ) (2) Î ANV (u i ) = Î BNV (u i ) and (3) F ANV (u i ) = F BNV (u i ) then the NVS A NV is equal to B NV denoted by A NV = B NV where 1 i n DEFINITION 24 (see [22]) Let A NV and B NV be two NVSs of the universe U If u i U (1) T ANV (u i ) T BNV (u i ) (2) Î ANV (u i ) Î BNV (u i ) and (3) F ANV (u i ) F BNV (u i ) then the NVS A NV is included by B NV denoted by A NV B NV where 1 i n DEFINITION 25 (see [22]) The complement of a NVS A NV is denoted by A c and is defined by T A c NV (x) = [1 T + 1 T ] ÎA c NV (x) = [1 I + 1 I ] and F A c NV (x) = [1 F + 1 F ] DEFINITION 26 (see [22] The union of two NVSs A NV and B NV is a NVS C NV written as C NV = A NV B NV whose truth-membership indeterminacy-membership and false-membership functions are related to those of A NV and B NV given by [ ) )] T CNV (x) = max (T ANV x T BNV x max (T + ANV x T + BNV x [ ) )] (I ANV x I BNV x (I + ANV x I + BNV x Î CNV (x) = F CNV (x) = min [ min min ) (F ANV x F BNV x min and )] (F + ANV x F + BNV x DEFINITION 27 (see [22]) The intersection of two NVSs A NV and B NV is a NVS C NV written as H NV = A NV B NV whose truth-membership indeterminacy-membership and false-membership functions are related to those of A NV and B NV given by [ ) )] T HNV (x) = min (T ANV x T BNV x min (T + ANV x T + BNV x [ ) )] (I ANV x I BNV x (I + ANV x I + BNV x Î HNV (x) = F HNV (x) = max [ max max ) (F ANV x F BNV x max and )] (F + ANV x F + BNV x DEFINITION 28 (see [22]) Let U be an initial universal set and let E be a set of parameters Let NV (U) denote the power set of all neutrosophic vague subsets of U and

4 let A E A collection of pairs ( FE) is called a neutrosophic vague soft set NV SS} over U where F is a mapping given by F : A NV (U) Let Ube a universe E a set of parameters X a set of experts (agents) and O a set of opinions Let Z = E X O and A Z DEFINITION 29 (see [23]) A pair ( F A ) is called a soft expert set over U where F is a mapping given by F : A P(U) where P(U ) denotes the power set of U Let Ube a universe E a set of parameters X a set of experts (agents) and O = 1 = agree0 = disagree}a set of opinions Let Z = E X O and A Z DEFINITION 210 (see [27]) A pair ( F A ) is called a neutrosophic soft expert set (NSES in short) over U where F is a mapping given by F : A P(U) where P(U ) denotes the power neutrosophic set of U DEFINITION 211 (see [27]) Let (F A) and (G B) be two NSESs over the common universe U (F A) is said to be neutrosophic soft expert subset of (G B) if A B and T F(e) (X) T G(e) (X) I F(e) (X) I G(e) (X) F F(e) (X) F G(e) (X) e A X U We denote it by (FA) (GB) (F A) is said to be neutrosophic soft expert superset of (G B) if (G B) is a neutrosophic soft expert subset of (F A) We denote by (FA) (GB) DEFINITION 212 (see [27]) Two ( NSESs ) (F A) and (G B) over the common universe U are said to be equal if (F A) is neutrosophic soft expert subset of (G B) and (G B) is neutrosophic soft expert subset of (F A) We denote it by (FA) = (GB) DEFINITION 213 (see [27]) NOT set of set parameters Let E = e 1 e 2 e n } be a set of parameters The NOT set of E is denoted by E = e 1 e 2 e n } where e i = not e i i DEFINITION 214 (see [27]) The complement of a NSES (F A) denoted by (FA) c and is defined as (FA) c = (F c A) where F c : A P(U) is mapping given by F c (x) = neutrosophic soft expert complement with T F c (X) = F F(X) I F c (X) = I F(X) F F c (X) = T F(X) DEFINITION 215 (see [27]) An agree-nses (FA) 1 over U is a neutrosophic soft expert subset of (F A) defined as follow

5 (FA) 1 = F 1 (m) : m E X 1}} DEFINITION 216 (see [27]) A disagree-nses (FA) 0 over U is a neutrosophic soft expert subset of (F A) defined as follow (FA) 0 = F 0 (m) : m E X 0}} DEFINITION 217 (see [27]) Let (H A) and (G B) be two NSESs over the common universe U Then the union of (H A) and (G B) is denoted by (HA) (GB) and is defined by (HA) (GB) = (KC) where C = A B and the truthmembership indeterminacy-membership and falsity-membership of (KC) are as follows: T H(e) (m) i f e A B T H(e) (m) = T G(e) (m) i f e B A max (T H(e) (m)t G(e) (m)) i f e A B I H(e) (m) i f e A B I H(e) (m) = I G(e) (m) i f e B A I H(e) (m)+i G(e) (m) 2 i f e A B F H(e) (m) i f e A B F H(e) (m) = F G(e) (m) i f e B A min (F H(e) (m)f G(e) (m)) i f e A B DEFINITION 218 (see [27]) Let (H A) and (G B) be two NSESs over the common universe U Then the intersection of (H A) and (G B) is denoted by (HA) (GB) and is defined by (HA) (GB) = (KC) where C = A B and the truth membership indeterminacy-membership and falsity-membership of (KC) are as follows: T K(e) (m) = min (T H(e) (m)t G(e) (m)) I K(e) (m) = I H(e)(m) + I G(e) (m) 2 F K(e) (m) = max (F H(e) (m)f G(e) (m)) i f e A B DEFINITION 219 (see [27]) AND operation on two NSESs Let (H A) and (G B) be two NSESs over the common universe U Then AND operation on them is denoted by (H A) (G B) and is defined by (H A) (G B) =(K A B) where the truth-membership indeterminacy-membership and falsity-membership of (KA B) are as follows: T K(αβ) (m) = min (T H(α) (m)t G(β) (m)) I K(αβ) (m) = I H(α)(m) + I G(β) (m) 2 F K(αβ) (m) = max (F H(α) (m)f G(β) (m)) α A β B

6 DEFINITION 220 (see [27]) OR operation on two NSESs Let (H A) and (G B) be two NSESs over the common universe U Then OR operation on them is denoted by (H A) (G B) and is defined by (H A) (G B) =(O A B) where the truth-membership indeterminacy-membership and falsity-membership of (OA B) are as follows: T O(αβ) (m) = max (T H(α) (m)t G(β) (m)) I O(αβ) (m) = I H(α)(m) + I G(β) (m) 2 F O(αβ) (m) = min (F H(α) (m)f G(β) (m)) α A β B 3 Neutrosophic Vague Soft Expert Set In this section we introduce the definition of a neutrosophic vague soft expert set and give basic properties of this concept Let Ube a universe E a set of parameters X a set of experts (agents) and O = 1 = agree0 = disagree}a set of opinions Let Z = E X O and A Z DEFINITION 31 A pair ( F A ) is called a neutrosophic vague soft expert set over U where F is a mapping given by F : A NV U where NV U denotes the power neutrosophic vague set of U Example 32 Suppose that a company produced new types of its products and wishes to take the opinion of some experts about concerning these products Let U = u 2 u 4 } be a set of products E = e 1 e 2 } a set of decision parameters where e i (i = 12) denotes the decision easy to use and quality respectively and let X = pq} be a set of experts Suppose that the company has distributed a questionnaire to three experts to make decisions on the companys products and we get the following: F (e 1 p1) = F (e 1 q1) = F (e 2 p1) = F (e 2 q1) = [0208];[0103];[0208] u 2 [0107];[0205];[0309] [0506];[0307];[0405] u 4 [081];[0102];[002] [0809];[0304];[0102] u 2 [0204];[0204];[0608] [005];[0507];[051] u 4 [0607];[0204];[0304] [0309];[0103];[0107] u 2 [0205];[0205];[0508] } } [0609];[0107];[0104] u 4 [0204];[0202];[0608] [0406];[0104];[0406] u 2 [0103];[0204];[0709] }

7 } [0105];[0507];[0509] u 4 [0207];[0204];[0308] u F (e 1 p0) = 1 [0107];[0203];[0309] u 2 [0709];[0505];[0103] [0102];[0304];[0809] u 4 [0203];[0204];[0708] u F (e 1 q0) = 1 [0203];[0305];[0708] u 2 [0507];[0304];[0305] F (e 2 p0) = [0508];[0607];[0205] u 4 [0102];[0405];[0809] [0102];[0303];[0809] u 2 [0708];[0405];[0203] [0104];[0507];[0609] u 4 [0508];[0205];[0205] u F (e 2 q0) = 1 [0708];[0304];[0203] u 2 [0506];[0809];[0405] [0607];[0607];[0304] u 4 [0809];[0204];[0102] The neutrosophic vague soft expert set (F Z ) is a parameterized family F(e i )i = 123} of all neutrosophic vague sets of U and describes a collection of approximation of an object DEFINITION 33 Let (F A) and (G B) be two neutrosophic vague soft expert sets over the common universe U (F A) is said to be neutrosophic vague soft expert subset of (G B) if 1 A B 2 ε A F(ε) is a neutrosophic vague subset of G(ε) This relationship is denoted by (FA) (GB) In this case (G B) is called a neutrosophic vague soft expert superset of (F A) DEFINITION 34 Two neutrosophic vague soft expert sets (F A) and (G B) over U are said to be equal if (F A) is a neutrosophic vague soft expert subset of (G B) and (G B) is a neutrosophic vague soft expert subset of (F A) Example 35Consider Example 32 Suppose that the company takes the opinion of the experts once again after a month of using the products Let A = (e 1 p1)(e 1 q0)} and B = (e 1 p1)(e 1 q0)(e 2 p1)} Clearly A B Let (F A) and (G B) be defined as follows: } } } }

8 (FA) = (e 1 p1) (e 1 q0) [0208];[0103];[0208] u 2 [0107];[0205];[0309] [0102];[0607];[0809] u 4 [0405];[0102];[0506] [0609];[0204];[0104] u 2 [0708];[0305];[0203] [0809];[0304];[0102] u 4 [0506];[0506];[0405] } (GB) = (e 1 p1) ( (e 1 q0) [0309];[0102];[0107] u 2 [0507];[0205];[0305] [0506];[0307];[0405] u 4 [081];[0102];[002] (e 2 p1) Therefore (F A) (G B) [0709];[0203];[0103] u 2 [0809];[0204];[0102] [0909];[0103];[0101] u 4 [0809];[0304];[0102] [0102];[0204];[0809] u 2 [0708];[0305];[0203] [0808];[0104];[0202] u 4 [0506];[0506];[0405] } DEFINITION 36 An agree-neutrosophic vague soft expert set (FA) 1 over U is a neutrosophic vague soft expert subset of (F A) defined as follow (FA) 1 = F 1 (m) : m E X 1}} Example 37 Consider Example 32 Then the agree-neutrosophic vague soft expert set (FA) 1 over U is (FA) 1 = (e 1 p1) ( (e 1 q1) [0208];[0103];[0208] u 2 [0107];[0205];[0309] [0506];[0307];[0405] u 4 [081];[0102];[002] (e 2 p1) (e 2 q1) [0809];[0304];[0102] u 2 [0204];[0204];[0608] [005];[0507];[051] u 4 [0607];[0204];[0304] [0309];[0103];[0107] u 2 [0205];[0205];[0508] [0609];[0107];[0104] u 4 [0204];[0202];[0608] [0406];[0104];[0406] u 2 [0103];[0204];[0709] [0105];[0507];[0509] u 4 [0207];[0204];[0308] }

9 DEFINITION 38 A disagree-neutrosophic vague soft expert set(fa) 0 over U is a neutrosophic vague soft expert subset of (F A) defined as follows: (FA) 0 = F 0 (m) : m E X 0}} Example 39 Consider Example 32 Then the disagree-neutrosophic vague soft expert set (FA) 0 over U is (FA) 0 = (e 1 p0) ( (e 1 q0) [0107];[0203];[0309] u 2 [0709];[0505];[0103] [0102];[0304];[0809] u 4 [0203];[0204];[0708] (e 2 p0) (e 2 q0) [0203];[0305];[0708] u 2 [0507];[0304];[0305] [0508];[0607];[0205] u 4 [0102];[0405];[0809] [0102];[0303];[0809] u 2 [0708];[0405];[0203] [0104];[0507];[0609] u 4 [0508];[0205];[0205] [0708];[0304];[0203] u 2 [0506];[0809];[0405] [0607];[0607];[0304] u 4 [0809];[0204];[0102] } 4 Basic Operations on Neutrosophic Vague Soft Expert Sets In this section we introduce some basic operations on neutrosophic vague soft expert sets namely the complement union and intersection of neutrosophic vague soft expert sets derive their properties and give some examples We define the complement operation for neutrosophic vague soft expert set and give an illustrative example and proved proposition DEFINITION 41 The complement of a neutrosophic vague soft expert set (FA) is denoted by (FA) c and is defined by (FA) c = (F c A) where F c : A NV U is a mapping given by F c (α) = c(f(α)) α A where c is a neutrosophic vague complement Example 42 Consider Example 32 By using the basic neutrosophic vague complement we have

10 (FZ) c = (e 1 p1) ( (e 1 q1) [0208];[0709];[0208] u 2 [0309];[0508];[0107] [0405];[0307];[0506] u 4 [002];[0809];[081] (e 2 p1) (e 2 q1) (e 1 p0) (e 1 q0) (e 2 p0) (e 2 q0) [0102];[0607];[0809] u 2 [0608];[0608];[0204] [051];[0305];[005] u 4 [0304];[0608];[0607] [0107];[0709];[0309] u 2 [0508];[0508];[0205] [0104];[0309];[0609] u 4 [0608];[0808];[0204] [0406];[0609];[0406] u 2 [0709];[0608];[0103] [0509];[0305];[0105] u 4 [0308];[0608];[0207] [0309];[0708];[0107] u 2 [0103];[0505];[0709] [0809];[0607];[0102] u 4 [0708];[0607];[0203] [0708];[0507];[0203] u 2 [0305];[0607];[0507] [0205];[0304];[0508] u 4 [0809];[0506];[0102] [0809];[0707];[0102] u 2 [0203];[0506];[0708] [0609];[0305];[0104] u 4 [0205];[0508];[0508] [0203];[0607];[0708] u 2 [0405];[0102];[0506] [0304];[0304];[0607] u 4 [0102];[0608];[0809] } Proposition 43 If (F A) is a neutrosophic vague soft expert set over U then ((FA) c ) c = (FA) Proof From Definition 41 we have (FA) c = (F c A) where F c (α) = 1 F(α) α A Now ((FA) c ) c = ((F c ) c A) where (F c ) c (α) = 1 (1 F(α)) α A = F(α) α A We define the union of two neutrosophic vague soft expert sets and give an illustrative example DEFINITION 44 The union of two neutrosophic vague soft expert sets (F A) and (G B) over U denoted by(fa) (GB) is a neutrosophic vague soft expert set (HC) where C = A B and ε C

11 F (ε) i f ε A B (HC) = G(ε) i f ε B A F (ε) G(ε) i f ε A B where denote the neutrosophic vague set union Example 45 Consider Example 32 Let A = (e 1 p1)(e 1 q0)(e 1 p0)} and B = (e 1 p1)(e 1 q0)(e 2 p1)} Suppose (F A) and (G B) are two neutrosophic vague soft expert sets over U such that: (FA) = (e 1 p1) (e 1 q0) (e 1 p0) [0708];[0103];[0203] u 2 [0407];[0205];[0306] [0102];[0607];[0809] u 4 [0405];[0102];[0506] [0109];[0204];[0109] u 2 [0408];[0305];[0206] [0809];[0304];[0102] u 4 [0506];[0506];[0405] [0508];[0104];[0205] u 2 } [0409];[0809];[0106] [0203];[0406];[0708] u 4 [0507];[0406];[0305] } (GB) = (e 1 p1) ( (e 1 q0) [0306];[0104];[0407] u 2 [0507];[0205];[0305] [0508];[0307];[0205] u 4 [081];[0102];[002] (e 2 p1) [0109];[0203];[0109] u 2 [0809];[0204];[0102] [0909];[0103];[0101] u 4 [0809];[0304];[0102] [0104];[0204];[0609] u 2 [0508];[0305];[0205] [0808];[0104];[0202] u 4 [0506];[0506];[0405] } By using basic neutrosophic vague union we have(fa) (GB) = (HC) where (HC) = (e 1 p1) [0708];[0103];[0203] u 2 [0507];[0205];[0305] [0508];[0307];[0205] u 4 [081];[0102];[002]

12 (e 1 q0) (e 1 p0) (e 2 p1) [0109];[0203];[0109] u 2 [0809];[0204];[0102] [0909];[0103];[0101] u 4 [0809];[0304];[0102] [0508];[0104];[0205] u 2 [0409];[0809];[0106] [0203];[0406];[0708] u 4 [0507];[0406];[0305] [0104];[0204];[0609] u 2 } [0508];[0305];[0205] [0808];[0104];[0202] u 4 [0506];[0506];[0405] } We define the intersection of two neutrosophic vague soft expert sets and give an illustrative example DEFINITION 46 The intersection of two neutrosophic vague soft expert sets (FA) and (GB) over a universe U is a neutrosophic vague soft expert set (HC) denoted by (FA) (GB) such that C = A B and e c F (e) i f e A B (HC) = G(e) i f e B A F (e) G(e) i f e A B where denoted the neutrosophic vague set intersection Example 47 Consider Example 45By using basic neutrosophic vague intersection we have (FA) (GB) = (HC) where (HC) = (e 1 p1) ( (e 1 q0) [0306];[0104];[0407] u 2 [0407];[0205];[0306] [0102];[0607];[0809] u 4 [0405];[0102];[0506] (e 1 p0) (e 2 p1) [0109];[0204];[0109] u 2 [0408];[0305];[0206] [0809];[0304];[0102] u 4 [0507];[0406];[0305] [0508];[0104];[0205] u 2 [0409];[0809];[0106] [0203];[0406];[0708] u 4 [0507];[0406];[0305] [0104];[0204];[0609] u 2 } [0508];[0305];[0205] [0808];[0104];[0202] u 4 [0506];[0506];[0405] } 5 AND and OR Operations In this section we introduce the definitions of AND and OR operations for neutrosophic vague soft expert set and derive their properties

13 DEFINITION 51 Let ( F A ) and ( G B ) be any two neutrosophic vague soft expert sets over a soft universe ( U Z ) Then ( F A ) AND ( G B ) denoted (FA) (GB) is defined by : (FA) (GB) = (HA B) where (HA B) = H(αβ) such that H(αβ) = F(α) G(β) f orall(αβ) A B And represent the basic intersection DEFINITION 52 Let ( F A ) and ( G B ) be any two neutrosophic vague soft expert sets over a soft universe ( U Z ) Then ( F A ) OR ( G B ) denoted (FA) (GB) is defined by : (FA) (GB) = (HA B) where (HA B) = H(αβ) such that H(αβ) = F(α) G(β) f orall(αβ) A B And represent the basic union Proposition 53 If ( F A)and( G B) are two neutrosophic vague soft expert sets over a soft universe ( U Z ) Then 1 ((FA) (GB)) c = (FA) c (GB) c 2 ((FA) (GB)) c = (FA) c (GB) c Proof (1) Suppose that (F A) and (G B) are two neutrosophic vague soft expert sets over a soft universe (U Z) defined as: (F A) = F(α) f orall α A Z and(gb) = G(β) f orall β B Z Then by definitions 48 and 49 it follows that: ((FA) (GB)) c = ((F(α) G(β)) c = ((F(α) G(β)) c = ( c(f(α) G(β))) = ( c(f(α) cg(β))) = (F(α)) c (G(β)) c = (FA) c (GB) c (2) The proof is similar to that in part(1)and therefore is omitted

14 6 Conclusion In this paper we reviewed the basic concepts of neutrosophic vague set and neutrosophic soft expert set and gave some basic operations on both neutrosophic vague set and neutrosophic soft expert set the concept of neutrosophic vague soft expert set was establishedthe basic operations on neutrosophic vague soft expert set namely complement union intersection AND and OR operations were defined Subsequently the basic properties of these operations such as De Morgans laws and other relevant laws pertaining to the concept of neutrosophic vague soft expert set are proved Finally we intend to further explore the applications of neutrosophic vague soft expert set approach to solve certain decision making problems This new extension will provide a significant addition to existing theories for handling uncertainties and lead to potential areas of further research and pertinent applications 7 Acknowledgements The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant IP References [1] LAZadeh Fuzzy set Information and Control 8 (3) (1965) [2] K Atanassov Intuitionistic fuzzy sets Fuzzy Sets Syst 20 (1) (1986) [3] WLGau DJ Buehrer Vague sets IEEE Transactions on Systems Man and Cybernetics 23 (2) (1993) [4] ZPawlak (1982) Rough Sets International Journal of Information and Computer Sciences 11 (1982) [5] F Smarandache Neutrosophic set a generalisation of the intuitionistic fuzzy sets International Journal of Pure and Applied Mathematics 24 (3) (2005) [6] JYe Some aggregation operators of interval neutrosophic linguistic numbers for multiple attribute decision making Journal of Intelligent and Fuzzy Systems 27 (2014) [7] JYe Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making Journal of Intelligent and Fuzzy Systems 26 (2014) [8] JYe A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets Journal of Intelligent and Fuzzy Systems 26 (2014) [9] JYe Improved correlation coefficients of single valued neutrosophic sets and interval neutrosophic sets for multiple attribute decision making Journal of Intelligent and Fuzzy Systems 27 (2014) [10] D Molodtsov Soft set theory first results Computers and Mathematics with Applications 37(2)(1999) [11] D Chen E C C Tsang D S Yeung and XWang The parameterization reduction of soft sets and its applications Computers and Mathematics with Applications 49 (2005) [12] P K Maji R Biswas and A R Roy Soft set theory Computers and Mathematics with Applications 45 (2003) [13] P K Maji A R Roy and R Biswas An application of soft sets in a decision making problem Computers and Mathematics with Applications 44 (2002) [14] P K Maji R Biswas and A R Roy Fuzzy soft sets Journal of Fuzzy Mathematics 9 (3) (2001) [15] A R Roy and P K Maji A fuzzy soft set theoretic approach to decision making problems Journal of Computational and Applied Mathematics 203(2) (2007) [16] W Xu J Ma S Wang and G Hao Vague soft sets and their properties Computers and Mathematics with Applications 59 (2) (2010)

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