29 Critical Review. Volume X, 2015 Neutrosophic Vague Set Theory Shawkat Alkhazaleh1 1 Department of Mathematics, Faculty of Science and Art Shaqra University, Saudi Arabiashmk79@gmail.com Abstract In 1993, Gau and Buehrer proposed the theory of vague sets as an extension of fuzzy set theory. Vague sets are regarded as a special case of context-dependent fuzzy sets. In 1995, Smarandache talked for the first time about neutrosophy, and he dened the neutrosophic set theory as a new mathematical tool for handling problems involving imprecise, indeterminacy, and inconsistent data. In this paper, we define the concept of a neutrosophic vague set as a combination of neutrosophic set and vague set. We alsodefineandstudytheoperationsandpropertiesofneutrosophicvaguesetand give some examples. Keywords Vague set, Neutrosophy, Neutrosophic set, Neutrosophic vague set. Acknowledgement We would like to acknowledge the financial support received from Shaqra University. With our sincere thanks and appreciation to Professor Smarandache for his support and his comments. 1Introduction Manyscientistswishtondappropriatesolutionstosomemathematical problems that cannot be solved by traditional methods. These problems lie in the fact that traditional methods cannot solve the problems of uncertainty in economy,engineering,medicine,problemsofdecision-making,andothers. There have been a great amount of research and applications in the literature concerningsomespecialtoolslikeprobabilitytheory,fuzzysettheory[13], rough set theory [19], vague set theory [18], intuitionistic fuzzy set theory [10, 12] and interval mathematics [11, 14]. 30 Shawkat Alkhazaleh Neutrosophic Vague Set Theory Critical Review. Volume X, 2015 SinceZadehpublishedhisclassicalpaperalmostftyyearsago,fuzzyset theory has received more and more attention from researchers in a wide range of scientic areas, especially in the past few years.Thedifferencebetweenabinarysetandafuzzysetisthatinanormalset every element is either a member or a non-member of the set; it either has to be A or not A. In a fuzzy set, an element can be a member of a set to some degree and at the same time a non-member of the same set to some degree. In classical set theory, the membership of elements in a set is assessed in binary terms: according to a bivalent condition, an element either belongs or does not belong to the set.Bycontrast,fuzzysettheorypermitsthegradualassessmentofthe membership of elements in a set; this is described with the aid of a member-ship function valued in the closed unit interval [0, 1]. Fuzzysetsgeneraliseclassicalsets,sincetheindicatorfunctionsofclassical setsarespecialcasesofthemembershipfunctionsoffuzzysets,ifthelater only take values 0 or 1. Therefore, a fuzzy set A in an universe of discourse X is afunction: [0, 1] ,andusuallythisfunctionisreferredtoasthe membership function and denoted by (). ThetheoryofvaguesetswasfirstproposedbyGauandBuehrer[18]asan extension of fuzzy set theory and vague sets are regarded as a special case of context-dependent fuzzy sets.Avaguesetisdefinedbyatruth-membershipfunction

andafalse-membershipfunction

,where

()isalowerboundonthegradeof membership of derived from the evidence for , and

() is a lower bound on the negation of derived from the evidenceagainst . The values of

() and

() arebothdefinedontheclosedinterval[0, 1]witheachpointina basic setX , where

() +

() 1.For more information, see [1, 2, 3, 7, 15, 16, 19]. In 1995, Smarandache talked for the first time about neutrosophy, and in 1999 and 2005 [4, 6] dened the neutrosophic set theory, one of the most important newmathematicaltoolsforhandlingproblemsinvolvingimprecise, indeterminacy, and inconsistent data.Inthispaper,wedefinetheconceptofaneutrosophicvaguesetasa combination of neutrosophic set and vague set. We also define and study the operations and properties of neutrosophic vague set and give examples. 31 Critical Review. Volume X, 2015 Shawkat Alkhazaleh Neutrosophic Vague Set Theory 2Preliminaries Inthissection,werecallsomebasicnotionsinvaguesettheoryand neutrosophicsettheory.GauandBuehrerhaveintroducedthefollowing definitions concerning its operations, which will be useful tounderstand the subsequent discussion. Definition2.1([18]).Let beavaguevalue, [ ,1 ]x xx t f = ,where | | 0,1 ,xt e| | 0,1xf e,and 0 11x xt f s s s .If 1xt = and 0xf= (i.e.,| | 1,1 x =),then is called a unit vague value. If 0xt = and 1xf = (i.e.,| | 0, 0 x =), then is called a zero vague value. Definition 2.2 ([18]). Let andybe two vague values, where [ ,1 ]x xx t f = and ,1 .y yy t f( = If x yt t = andx yf f = , then vague values andyare called equal (i.e.[ ,1 ] ,1x x y yt f t f( = ).Definition2.3([18]).LetA beavaguesetoftheuniverse.If iu U e , ( ) 1A it u =and ( ) 0A if u =,thenAiscalledaunitvagueset,where1i n s s .If iu U e , ( ) 0A it u =and ( ) 1A if u =,thenAiscalledazerovagueset,where 1. i n s sDefinition 2.4 ([18]). The complement of a vague setAis denoted by cA and is defined by ,11 .ccAAAAt ff t= = Definition2.5([18]).LetAandBbetwovaguesetsoftheuniverse .If ,iu U e ( ) ( ) ( ) ( ) ,1 ,1 ,A i A i B i B it u f u t u f u = (( thenthevaguesetAandB are called equal, where1. i n s sDefinition2.6([18]).LetAandBbetwovaguesetsoftheuniverse. UIf ,iu U e( ) ( )A i B it u t u sand ( ) ( ) 11 ,A i B if u f u s thenthevaguesetAare included byB , denoted by A B _ , where1. i n s sDefinition 2.7 ([18]). The union of two vague setsAandB is a vague setC , writtenas C A B = ,whosetruth-membershipandfalse-membership functions are related to those of A and B by ( ) , ,C A Bt max t t =( ) ( ) 1 1 ,1 1 , .C A B A Bf max f f min f f = = 32 Shawkat Alkhazaleh Neutrosophic Vague Set Theory Critical Review. Volume X, 2015 Definition 2.8 ([18]). The intersection of two vague setsAandB is a vague setC , written as C A B = , whose truth-membership and false-membership functions are related to those of A and B by ( ) , ,C A Bt min t t =( ) ( ) 1 1 ,1 1 , .C A B A Bf min f f max f f = = In the following, we recall some definitions related to neutrosophic set given by Smarandache. Smarandache defined neutrosophic set in the following way: Definition2.9[6]AneutrosophicsetAontheuniverseofdiscourse is defined as = {< , ( ), ( ), ( ) >, }A A AA x T x I x F x x X

where , , : ] 0,1[and0 ( ) ( ) ( ) 3.A A AT x I x F x Smarandache explained his concept as it follows: "For example, neutrosophic logic is a generalization of the fuzzy logic. In neutrosophic logic a proposition isT true ,I indeterminate ,andF false .Forexample,letsanalyzethe following proposition: Pakistan will win against India in the next soccer game. This proposition can be(0.6, 0.3, 0.1) , which means that there is a possibility of 60%thatPakistanwins,30%thatPakistanhasatiegame,and10%that Pakistan looses in the next game vs. India."Now we give a brief overview of concepts of neutrosophic set dened in [8, 5, 17]. Let 1S and 2Sbe two real standard or non-standard subsets, then1 2 1 2 1 1 2 2 { | , }, S S x x s s s S and s S = = + e e{ }2 2 2 21 { |1 , }, S x x s s S+ + = = + e1 2 1 2 1 1 2 2 { | , }, S S x x s s s S and s S = = e e1 2 1 2 1 1 2 2 { | . , }, S S x x s s s S and s S = = e e{ }2 2 2 21 { |1 , }. S x x s s S+ + = = eDefinition 2.10 (Containment) A neutrosophic setA is contained in the other neutrosophic set B ,A B _ , if and only if ( ) ( ) ( ) ( ) infinf,sup sup ,A B A BT x T x T x T x s s( ) ( ) ( ) ( ) infinf, sup sup ,A B A BI x I x I x I x > >( ) ( ) ( ) ( ) infinf, sup sup ,A B A BF x F x F x F x > >for allx X e . 33 Critical Review. Volume X, 2015 Shawkat Alkhazaleh Neutrosophic Vague Set Theory Definition 2.11 The complement of a neutrosophic setA is denoted by Aand is dened by ( ) { } ( ) 1 , AT x T x+= ( ) { } ( ) 1 , AI x I x+= ( ) { } ( ) 1 , AF x F x+= for all x X e . Definition 2.12 (Intersection) The intersection of two neutrosophic sets and isaneutrosophicset,writtenas = ,whosetruth-membership, indeterminacy-membershipandfalsity-membershipfunctionsarerelatedto those ofA andB by ( ) ( ) ( ),C A BT x T x T x =( ) ( ) ( ),C A BI x I x I x =( ) ( ) ( ),C A BF x F x F x =for allx X e . Definition2.11(Union)Theunionoftwoneutrosophicsets and isa neutrosophicsetwrittenas = ,whosetruth-membership, indeterminacy-membershipandfalsity-membershipfunctionsarerelatedto those ofA andB by ( ) ( ) ( ) ( ) ( ),C A B A BT x T x T x T x T x = ( ) ( ) ( ) ( ) ( ),C A B A BI x I x I x I x I x = ( ) ( ) ( ) ( ) ( ),C A B A BF x F x F x F x F x = for allx X e . 3Neutrosophic Vague Set Avaguesetover ischaracterizedbyatruth-membershipfunction vt anda false-membershipfunctionvf , | | : 0,1vt U and | | : 0,1vf U respectively where ( )v it uisalowerboundonthegradeofmembershipof iu whichis derived from the evidence foriu , ( )v if uis a lower bound on the negation of iuderivedfromtheevidenceagainst iuand( ) ( ) 1v i v it u f u + s.Thegradeof membership of iu in the vague set is bounded to a subinterval( ) ( ) ,1v i v it u f u ( of | | 0,1.Thevaguevalue ( ) ( ) ,1v i v it u f u ( indicatesthattheexactgradeof 34 Shawkat Alkhazaleh Neutrosophic Vague Set Theory Critical Review. Volume X, 2015 membership( )v i uof iumaybeunknown,butitisboundedby ( ) ( ) ( )v i v i v it u u f u s swhere ( ) ( ) 1.v i v it u f u + s LetUbeaspaceofpoints (objects),withagenericelementin denotedbyu .Aneutrosophicsets(N-sets)inischaracterizedbyatruth-membershipfunctionAT ,an indeterminacy-membership function AI and a falsity-membership functionAF .( )AT u; ( )AI uand ( )AF uare real standard or nonstandard subsets of | | 0,1. It can be written as: ( ) | | { }, ( ), ( ), ( ) : , ( ), ( ), ( ) 0,1 .A A A A A AA u Tu I u F u u U Tu I u F u = < > e e There is no restriction on the sum of( )AT u; ( )AI uand( )AF u, so: 0 sup ( )sup ( )sup ( ) 3A A ATu I u Fu s + + s .By using the above information and by addingthe restriction of vague set to neutrosophic set, we define the concept of neutrosophic vague set as it follows.Definition 3.1 A neutrosophic vague set NVA (NVS in short) on the universe of discourseXwritten as

= {< ,

(),

(),

() >, } whosetruth-membership,indeterminacy-membershipandfalsity-member-ship functions is defined as( ) , , ( ) , , ( ) , ,NV NV NVA A AT x T T I x I I F x F F where , 1 , 1 T F F Tand0 2 T I F , when is continuous, a NVS

can be written as ( ) ( ) ( ) , , , / , .NV NV NVNV A A AXA x T x I x F x x x X = < > e} When is discrete, a NVS

can be written as ( ) ( ) ( )1, , , / , .NV NV NVnNV A i A i A i i iiA x T x I x F x x x X== < > e

Inneutrosophiclogic,apropositionisT true ,I indeterminate ,and F false such that: 0 sup ( )sup ( )sup ( ) 3.N N NA A AT u I u F u s + + s 35 Critical Review. Volume X, 2015 Shawkat Alkhazaleh Neutrosophic Vague Set Theory Also,vaguelogicisageneralizationofthefuzzylogicwhereapropositionis T trueandF false ,suchthat:( ) ( ) 1,v i v it u f u + s heexactgradeof membership( )v i uof iumaybe unknown, but it is bounded by( ) ( ) ( ).v i v i v it u u f u s s Forexample,letsanalyzetheSmarandache'spropositionusingournew concept:PakistanwillwinagainstIndiainthenextsoccergame.This proposition can be as it follows: 0.6, 0.9 , 0.3, 0.4 and 0.4, 0.6 ,NV NV NVA A AT I F whichmeansthatthereispossibilityof 60% 90% to thatPakistanwins, 30% 40% to thatPakistanhasatiegame,and 40% 60% to thatPakistan looses in the next game vs. India. Example3.1Let { }1 2 3, , U u u u = beasetofuniversewedefinetheNVS NVA as follows: | | | | | || | | | | || | | | | |123,0.3, 0.5 , 0.5, 0.5 , 0.5, 0.7,0.4, 0.7 , 0.6, 0.6 , 0.3, 0.6.0.1, 0.5 , 0.5, 0.5 , 0.5, 0.9NVuAuu= `) Definition 3.2 Let NV+ be a NVS of the universeU whereiu U e , ( ) 1,1 , ( ) 0, 0 , ( ) 0, 0 ,NV NV NVT x I x F x then NV+is called a unit NVS, where1i n s s .Let NVu be a NVS of the universeU where iu U e , ( ) 0, 0 , ( ) 1,1 , ( ) 1,1 ,NV NV NVT x I x F x then NVuis called a zero NVS, where1. i n s sDefinition 3.3 The complement of aNVSNVAis denoted by cA and is defined by( ) 1 , 1 ,( ) 1 , 1 ,( ) 1 , 1 ,NVNVNVAAAcccT x T TI x I IF x F F 36 Shawkat Alkhazaleh Neutrosophic Vague Set Theory Critical Review. Volume X, 2015 Example 3.2 Considering Example 3.1, we have: | | | | | || | | | | || | | | | |123,0.5, 0.7 , 0.5, 0.5 , 0.3, 0.5,0.3, 0.6 , 0.4, 0.4 , 0.4, 0.7.0.5, 0.9 , 0.5, 0.5 , 0.1, 0.5cNVuAuu= `) Definition 3.5 Let NVAand NVBbe two NVSs of the universeU . If,iu U e ( ) ( ) ( ) ( ) ( ) ( ) , and =,NV NV NV NV NV NVA i B i A i B i A i B iT u T u I u I u F u F u = =then the NVS NVAand NVB are called equal, where1. i n s sDefinition 3.6 Let NVAand NVBbe two NVSs of the universe. UIf,iu U e( ) ( ) ( ) ( )( ) ( ), and,NV NV NV NVNV NVA i B i A i B iA i B iT u T u I u I uF u F us >>

then the NVS NVAare included by NVB , denoted byNV NVA B _ , where1. i n s s Definition3.7TheunionoftwoNVSs NVAand NVB isaNVS NVC ,writtenas NV NV NVC A B = ,whosetruth-membership,indeterminacy-membershipand false-membership functions are related to those of NVAand NVB by

() = [(

,

), (

+,

+ )],

() = [(

,

), (

+,

+ )],

() = [(

,

), (

+,

+ )]. Definition 3.8 The intersection of two NVSs NVAand NVB is a NVS NVH , written as NV NV NVH A B = ,whosetruth-membership,indeterminacy-membership and false-membership functions are related to those of NVAand NVB by

() = [(

,

), (

+,

+ )],

() = [(

,

), (

+,

+ )],

() = [(

,

), (

+,

+ )]. Example3.3Let { }1 2 3, , U u u u = beasetofuniverseandletNVSNVA and NVBdefine as follows: 37 Critical Review. Volume X, 2015 Shawkat Alkhazaleh Neutrosophic Vague Set Theory | | | | | || | | | | || | | | | |123,0.3, 0.5 , 0.7, 0.8 , 0.5, 0.7,0.4, 0.7 , 0.6, 0.8 , 0.3, 0.6.0.1, 0.5 , 0.3, 0.6 , 0.5, 0.9NVuAuu= `) | | | | | || | | | | || | | | | |123,0.7, 0.8 , 0.3, 0.5 , 0.2, 0.3,0.2, 0.4 , 0.2, 0.4 , 0.6, 0.8.0.9,1 , 0.6, 0.7 , 0, 0.1NVuBuu= `) Then we haveNV NV NVC A B =where | | | | | || | | | | || | | | | |123,0.7, 0.8 , 0.3, 0.5 , 0.2, 0.3,0.4, 0.7 , 0.2, 0.4 , 0.3, 0.6.0.9,1 , 0.3, 0.6 , 0, 0.1NVuCuu= `) Moreover, we haveNV NV NVH A B =where | | | | | || | | | | || | | | | |123,0.3, 0.5 , 0.7, 0.8 , 0.5, 0.7,0.2, 0.4 , 0.6, 0.8 , 0.6, 0.8.0.1, 0.5 , 0.6, 0.7 , 0.5, 0.9NVuHuu= `) Theorem 3.1 LetP be the power set of all NVS dened in the universe X . Then ; ,NV NVP is a distributive lattice. Proof Let , , be the arbitrary NVSs dened on . It is easy to verify that , A A A A A A = =(idempotency), , A B B A A B B A = =(commutativity), ( ) ( ), ( ) ( ) A B C A B C A B C A B C = =(associativity), and ( )( ) ( ), ( ) ( ) ( ) A B C A B A CA B C A B A C = =(distributivity). 38 Shawkat Alkhazaleh Neutrosophic Vague Set Theory Critical Review. Volume X, 2015 4Conclusion In this paper, we have defined and studied the concept of a neutrosophic vague set, as well as its properties, and its operations, giving some examples. 5References [1]A.Kumar,S.P.Yadav,S.Kumar,Fuzzysystemreliabilityanalysisusing basedarithmeticoperationsonLRtypeintervalvaluedvaguesets,in InternationalJournalofQuality&ReliabilityManagement,24(8) (2007), 846860. [2]D.H.Hong,C.H.Choi,Multicriteriafuzzydecision-makingproblems basedonvaguesettheory,inFuzzySetsandSystems,114(2000), 103113. [3]F.Smarandache,Neutrosophicset,ageneralisationoftheintuitionistic fuzzy sets, in Inter. J. Pure Appl. Math., 24 (2005), 287-297. [4]F. Smarandache, Neutrosophy: Neutrosophic Probability, Set, and Logic, Amer. Res. Press, Rehoboth, USA, 105 p., 1998. [5]FlorentinSmarandache,Aunifyingeldinlogics.Neutrosophy: Neutrosophicprobability,setandlogic,AmericanResearchPress, Rehoboth, 1999. [6]H. Bustince, P. Burillo, Vague sets are intuitionistic fuzzy sets, in Fuzzy Sets and Systems, 79 (1996), 403405. [7]H.Wang,F.Smarandache,Y.Q.Zhang,andR.Sunderraman,Interval NeutrosophicSetsandLogic:TheoryandApplicationsinComputing, Hexis, Phoenix, AZ, 2005. [8]J.Wang,S.Y.Liu,J.Zhang,S.Y.Wang,OntheparameterizedOWA operatorsforfuzzyMCDMbasedonvaguesettheory,inFuzzy Optimization and Decision Making, 5 (2006), 520. [9]K.Atanassov,Intuitionisticfuzzysets,inFuzzySetsandSystems,20 (1986), 8796. [10]K. Atanassov,Operators over interval valued intuitionistic fuzzy sets, in Fuzzy Sets and Systems, 64 (1994), 159174. [11]L.Zhou,W.Z.Wu,Ongeneralizedintuitionisticfuzzyrough approximationoperators,inInformationScience,178(11)(2008), 24482465. [12]L.A. Zadeh, Fuzzy sets, in Information and Control, 8 (1965), 338353. [13]M. B. Gorzalzany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, in Fuzzy Sets and Systems, 21 (1987), 117. 39 Critical Review. Volume X, 2015 Shawkat Alkhazaleh Neutrosophic Vague Set Theory [14]S. M. Chen, Analyzing fuzzy system reliability using vague set theory, in InternationalJournalofAppliedScienceandEngineering,1(1) (2003), 8288. [15]S.M.Chen,Similaritymeasuresbetweenvaguesetsandbetween elements,IEEETransactionsonSystems,Man andCybernetics,27(1) (1997), 153158. [16]F. Smarandache, Neutrosophic set - A generalization of the intuitionistic fuzzyset,IEEEInternationalConferenceonGranularComputing, (2006), 38-42.[17]W. L. Gau, D.J. Buehrer, Vague sets, IEEE Transactions on Systems. Man and Cybernetics, 23 (2) (1993), 610614. [18]Z.Pawlak,Roughsets,inInternationalJournalofInformationand Computer Sciences, 11 (1982), 341356.