Single valued neutrosophic multisets

AnnalsofFuzzyMathematicsandInformaticsVolumex,No. x,(Month201y), pp. 1xxISSN:20939310(printversion)ISSN:22876235(electronicversion)http://www.afmi.or.kr@FMIc KyungMoonSaCo.http://www.kyungmoon.comSinglevaluedneutrosophicmultisetsRajashi Chatterjee, P.Majumdar, S.K.SamantaReceived2December2014;Revised17January2015;Accepted25March2015Abstract. In this paper, we have investigated single valued neutrosophicmultisets in detail. Several operations have been dened on them and theirimportant algebraic properties are studied.We have further introduced thenotionofdistanceandsimilaritymeasuresbetweentwosinglevaluedneu-trosophic multisets. An application of single valued neutrosophic multisetsinmedicaldiagnosishasbeendiscussed.2010AMSClassication: 03E72,03E75Keywords: Neutrosophicsets,SingleValuedNeutrosophicSets,Multisets,Simi-larity,SingleValuedNeutrosophicMultisets.CorrespondingAuthor: S.K.Samanta([email protected] )1. IntroductionTheconceptof multisetsstemmedfromtheviolationof oneof thebasicprop-ertiesof classical settheory, whichstatesthatanelementcanoccurinasetonlyonce. The term multiset was rst introduced by N. G. Bruijn [8]. Multisets, oftenreferredtoasbags, arecollectionsof objectsthatmaycontainanitenumberofduplicates. Multisetsaremathematicalstructuresthatcomehandyinareaslikedatabase enquiries relatedtocomputer science. Also, indealingwiththe prob-lemsofconstructingmathematicalmodelsforreal-lifesituations, thedataathandaremainlyimpreciseandindeterministic. In1965, Zadeh[31] cameupwithhisremarkabletheoryof fuzzysetswhereheintroducedthenotionof partial belong-ingness of anelement inaset. Later thesetwoconceptshavebeencombinedtogenerateFuzzyMultisets[18]andwereappliedinmanyareasofcomputerscience.Aftertheintroductionof IntuitionisticfuzzysetsbyAtanassov[4, 5] in1986, thetheoryof Intuitionistic FuzzyMultisets [21] have beendeveloped. Onthe otherhand, in1995, FlorentinSmarandache incorporatedthe concept of NeutrosophicLogic [22, 23] which sprouted from a branch of philosophy, known as Neutrosophy,meaningthestudyof neutralitiesandgavebirthtothetheoryof NeutrosophicRajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxxSets. Unlike intuitionistic fuzzysets whichassociate toeachmember of the setadegreeof membershipandadegreeof non-membership; , [0, 1], neutro-sophic sets characterize each member x of the set with a truth-membership functionTA(x), an indeterminacy-membership function IA(x) and a falsity-membership func-tionFA(x),eachofwhichbelongstothenon-standardunitinterval]0, 1+[. Thus,although in some cases intuitionistic sets consider a particular indeterminacy or hes-itation margin, = 1 , neutrosophic sets are capable of handling uncertaintyinabetter waysinceincaseof neutrosophicsets indeterminacyis takencareofseparately. Further, in2005, Wanget al, introducedthenotionof Single-ValuedNeutrosophicSets(SVNS)[25],whichdierfromneutrosophicsetsonlyinthefactthat inthe formers case, TA(x), IA(x), FA(x)[0, 1] andcanbe appliedtosolvemanypractical problems. Researchinvolvingsinglevaluedneutrosophicsetsandinterval neutrosophicsets together withtheir applications arenowonfull swing.Notionsof similarity, entropy, subsethoodmeasureetc. of neutrosophicsetshavebeenintroducedandtheirapplicationsinseveralareas[7,17,20,28,29]arebeingexecutedbymanyauthors.Again the theory ofsoft sets was initiated byD. Molodstov[19] in 1999 for mod-elling uncertainty present in real life. Roughly speaking, a soft set is a parameterizedclassicationoftheobjectsoftheuniverse. Molodstovhadshownseveral applica-tionsof softsetsindierentareaslikeintegration, gametheory, decisionmakingetc. LaterMaji etal. [11] denedseveral operationsonsoftsets. Perhapsthisisthe only theory available with a parameterization tool for modelling uncertainty. H.AktasandN. Cagman[1] haveshownthatfuzzysetsarespecial casesofsoftsets.Latermanyauthors[12,13,16]havecombinedsoftsetswithothersetstogeneratehybridstructureslikefuzzysoftsets,intuitionisticfuzzysoftsets,generalizedfuzzysoftsets,vaguesoftsetsetc. andappliedtheminmanyareaslikedecisionmaking,medical diagnosis, similaritymeasureetc. Fewauthors[2, 3, 6, 14, 15] havealsodened the notions of soft multisets,fuzzysoft multisets,neutrosophic soft sets etc.VeryrecentlyS. YeandJ. Ye[30] combinedtheconceptsof singlevaluedneutro-sophicsetsalongwiththetheoryofmultisetsandproposedanewtheoryofSingleValuedNeutrosophicMultisets(SVNMSinshort).InthispaperwehaveslightlymodiedthedenitionofSVNMSandstudieditsproperties. The initial contributions of this paper involve the introduction of variousnewset-theoreticoperatorsonSVNMSandtheirproperties. Later, thenotionofsingle valued neutrosophic sets has been applied in solving a decision making problemregardingmedicaldiagnosis.2. PreliminariesIn this section we give the denition and some important results regarding singlevaluedneutrosophicsets[25]andmultisets.Denition2.1([25]). Let Xbeaspaceof points, withagenericelement inXdenotedbyx. AsinglevaluedneutrosophicsetAinXischaracterizedbyatruth-membershipfunctionTA, anindeterminacy-membershipfunctionIAandafalsity-membershipfunctionFA. ForeachpointxinX,TA(x), IA(x), FA(x)[0, 1].2RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxxWhenXiscontinuous,aSVNSAcanbewrittenasA =_XT(x), I(x), F(x) x, x XWhenXisdiscrete,aSVNSAcanbewrittenasA =n

i=1T(xi), I(xi), F(xi) xi, xi XDenition2.2([25]). Thecomplement of aSVNSAisdenotedbyc(A) andisdenedbyTc(A)(x) = FA(x),Ic(A)(x) = 1 IA(x)andFc(A)(x) = TA(x),x X.Denition2.3([25]). ASVNSAis containedinanother SVNSBi.e.ABiTA(x) TB(x),IA(x) IB(x)andFA(x) FB(x), x X.Denition2.4([25]). TwoSVNSAandBaresaidtobeequal iA BandB A.Denition 2.5 ([25]). The union of two SVNMS A and B is a SVNMS C, written as,C= A B,whosetruth-membership,indeterminacymembershipandfalsitymem-bershipfunctionsarerelatedtothoseofAandBasTC(x)=max(TA(x), TB(x)),IC(x) = max(IA(x), IB(x))andFC(x) = min(FA(x), FB(x)), x X.Denition2.6([25]). The intersection of two SVNMS A and Bis a SVNS C, writ-tenas,C= AB,whosetruth-membership,indeterminacymembershipandfalsitymembership functions are related to those of Aand B as TC(x) = min(TA(x), TB(x)),IC(x) = min(IA(x), IB(x))andFC(x) = max(FA(x), FB(x)), x X.Denition2.7([25]). ThedierencebetweentwoSVNMSAandBis aSVNSC, writtenas, C=AB, whosetruth-membership, indeterminacymembershipandfalsitymembershipfunctions are relatedto those of AandBas TC(x) =min(TA(x), FB(x)), IC(x) = min(IA(x), 1 IB(x)) and FC(x) = max(FA(x), TB(x)),x X.Denition2.8([25]). Thetruth-favoriteof aSVNSA, denotedby A, whosetruth-membership, indeterminacy-membership and falsity-membership functions aredenedas TA(x) =min(TA(x) +IA(x), 1), IA(x) =0andFA(x) =FA(x),x X.Denition2.9([25]). Thefalsity-favoriteof aSVNSA, is aSVNSdenotedbyA, whosetruth-membership, indeterminacy-membershipandfalsity-membershipfunctionsaredenedasTA(x) = TA(x),IA(x) = 0andFA(x) = min(FA(x) +IA(x), 1), x X.Denition 2.10 ([24]). Let U= {x1, x2, ..., xn} be the universal set. A crisp bag ormultiset Mof Uis characterized by a function CM (.), (CM: U N) correspondingtoeachx U , knownasthecountfunction. AmultisetMisexpressedasM=_k1x1,k2x2, ...,knxn_suchthatxiappearskitimesinM.3RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxx3. SingleValuedNeutrosophicMultisetsThe notion of single valued neutrosophic multisets were rst dened in [30]. Herethe three count functions are real valued but in this paper we have redened the no-tion of SVNMS with positive integer valued count functions for the sake of practicaluse.Denition3.1. ASingleValuedNeutrosophicMultiset(SVNMS)A, denedonauniverseXissuchthatcorrespondingtoeachelementof thesetthereexistsafunction, namely, the count functionCf: XN, whichdenotes the numberof timesthatparticularelementoccursintheset, suchthateachx Xischarac-terizedbythreesequencesoflengthsCf(x),namely,atruth-membershipsequence_T1A(x), T2A(x), ..., TkA(x)_, an indeterminacy-membership sequence_I1A(x), I2A(x), ..., IkA(x)_andafalsity-membershipsequence_F1A(x), F2A(x), ..., FkA(x)_.Whentheuniverseunder considerationX= {x1, x2, ..., xn}, is discrete, aSingleValuedNeutrosophicMultisetAoverXisrepresentedasA=n

i=1__T1A(xi), T2A(xi), ..., TkiA(xi)_,_I1A(xi), I2A(xi), ..., IkiA(xi)_,_F1A(xi), F2A(xi), ..., FkiA(xi)__wheretheelementxi Xisrepeatedki= Cf(xi)timesinX.Remark3.2. Thetruth-membershipsequenceisalwaysadecreasingsequenceofmembershipvalueswhereastheindeterminacy-membershipandfalsity-membershipsequencesaresuchthattheycanassumethemembershipvaluesinanyorderand0 TrA(xi) + IrA(xi) + FrA(xi) 3, xi X, i=1, 2, ..., nandr=1, 2, ..., ki. ThishasbeendonetokeepparitywiththedenitionsofFuzzyandIntuitionisticfuzzymultisets.Denition3.3. Thelengthof anelement xi Xof asingle-valuedneutrosophicmultisetA,denedonthesetX,isdenedasl(xi: A) = Cf(xi), i = 1, 2, ..., n.Denition 3.4. The cardinality of a single-valued neutrosophic multiset A is denedasCard(A) =

ni=1

l(xi:A)j=1_TjA(xi) +IjA(xi)_,xi X,i = 1, 2, ..., n.Example3.5. Suppose X= {x1, x2, x3} denotes three shirts displayed for sale in aparticularshop. Wenowsetforthtoregistertheopinionofadomainofcustomersabout the quality of shirts based on whether the shirts are made up of good fabric,a level of indeterminacy on the part of the customers and whether they feel that theshirtismadeupofanotsogoodfabric. Basedontheopinionofthedomainofcustomersconcerned,asingle-valuedneutrosophicmultisetcanbedenedonXasfollows:A = (0.6, 0.4) , (0.5, 0.3) , (0.2, 0.3) /x1 +0.2, 0.4, 0.7 /x2+(0.8, 0.6, 0.5) , (0.2, 0.2, 0.3) , (0.1, 0.3, 0.4) /x3Also,l (x1: A) = 2,l (x2: A) = 1,l(x3: A) = 3andCard(A) = 5.Denition3.6. Anabsolutesingle-valuedneutrosophicmultisetAis aSVNMSwhereTi(x) = 1,Ii(x) = 1andFi(x) = 0, x X,i = 1, 2, ..., l(x :A).Denition3.7. Anull single-valuedneutrosophicmultisetisaSVNMSwhereTi(x) = 0,Ii(x) = 0andFi(x) = 1, x X,i = 1, 2, ..., l(x :).4RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxx4. OperationsoverSingleValuedNeutrosophicMultisetsInthissectionwehavedenedseveral settheoreticandalgebraicoperationsonSVNMS.Theseoperationsaredierentfromtheoperationsthathavebeendenedin [30]. Moreover, in our case the behaviour of indeterminacy membership is similarto the behaviour of truth membership whereas in [30] the indeterminacy membershipissimilartothebehaviouroffalsitymembership.4.1. Set-theoreticoperationsoverSinglevaluedneutrosophicmultisets.Denition4.1. ASVNMSAis saidtobe containedinanother SVNMSBixi X,i = 1, 2, ..., n,(i)l(xi: A) l(xi: B)(ii)TrA(xi) TrB(xi)(iii)IrA(xi) IrB(xi)(iv)FrA(xi) FrB(xi), xi X, i = 1, 2, ..., nandr = 1, 2, ..., l(xi: A).Denition4.2. TwoSVNMSAandBaresaidtobeequalitheyaresubsetsofoneanother.Remark4.3. LetAandBbetwoSVNMSovertheuniverseX. Inordertocarryoutanyoperation(set-theoreticoralgebraic)betweenAandBitisveriedatrstwhetherl(xi: A)=l(xi: B), xi X. If l(xm: A) =l(xm: B)foranyxm Xthenwithoutanylossof generality, asucientnumberof 0sareappendedwiththetruth-membershipandtheindeterminacymembershipvalues andasucientnumberof1sareappendedwiththefalsity-membershipvaluesrespectivelytothesequences of smaller lengththerebymakingthelengths equal andfaciltatingtheexecutionofoperations.Example4.4. SupposetwoSVNMSAandBaregivenby,A = 0.5, 0.4, 0.4 /x1 +(0.3, 0.2) , (0.5, 0.4) , (0.6, 0.7) /x2 +0.8, 0.3, 0.2 /x3B= (0.8, 0.6) , (0.4, 0.5) , (0.1, 0.3) /x1 +(0.5, 0.4) , (0.7, 0.7) , (0.5, 0.3) /x2 +0.9, 0.5, 0.2 /x3HereA B.Denition4.5. TheunionoftwoSVNMSAandBoverX, denotedbyAB, isa SVNMS over Xwhose truth-membership,indeterminacy-membership and falsity-membershipvaluesaregivenbyTrAB(xi) = max(TrA(xi), TrB(xi))IrAB(xi) = max(IrA(xi), IrB(xi))FrAB(xi) = min(FrA(xi), FrB(xi))xi X,i = 1, 2, ..., n;r = 1, 2, ..., lwherel = max{l(xi: A), l(xi: B)}.Denition4.6. The intersectionof twoSVNMSAandBover X, denotedbyA B, isaSVNMSoverXwhosetruth-membership, indeterminacy-membershipandfalsity-membershipvaluesaregivenbyTrAB(xi) = min(TrA(xi), TrB(xi))IrAB(xi) = min(IrA(xi), IrB(xi))FrAB(xi) = max(FrA(xi), FrB(xi))xi X,i = 1, 2, ..., n;r = 1, 2, ..., lwherel = max{l(xi: A), l(xi: B)}.5RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxxExample 4.7. Let two SVNMS A and B, dened over the universe X= {x1, x2, x3}begivenby,A = (0.5, 0.3) , (0.1, 0.1) , (0.7, 0.8) /x1 +(0.7, 0.68, 0.62) , (0.3, 0.45, 0.5) , (0.34, 0.28, 0.49) /x2+(0.67, 0.5, 0.3) , (0.2, 0.3, 0.4) , (0.4, 0.5, 0.7) /x3B= 0.75, 0.2, 0.15 /x1 +(0.43, 0.37, 0.28, (0.5, 0.2, 0.3) , (0.7, 0.8, 0.9)) /x2+(1.0, 0.86, 0.79) , (0.01, 0.1, 0.2) , (0.0, 0.3, 0.2) /x3 A B= (0.75, 0.3) , (0.2, 0.1) , (0.15, 0.8) /x1+(0.7, 0.68, 0.62) , (0.5, 0.45, 0.5) , (0.34, 0.28, 0.49) /x3+(1.0, 0.86, 0.79) , (0.2, 0.3, 0.4) , (0.0, 0.3, 0.2) /x3Denition 4.8. The truth-favorite of a SVNMS A, denoted by A, is a SVNMS andischaracterizedbythetruth-membership, indeterminacy-membershipandfalsity-membershipvalueswhicharerespectivelydenedasTrA(xi) = min(TrA(xi) +IrA(xi), 1)IrA(xi) = 0FrA(xi) = FrA(xi)xi X, i = 1, 2, ..., n; r = 1, 2, ..., lwherel = max{l(xi: A), l(xi: B)}.Denition4.9. Thefalsity-favoriteofaSVNMSA,denotedby A,isaSVNMSand is characterized by the truth-membership, indeterminacy-membership and falsity-membershipvalueswhicharerespectivelydenedasTrA(xi) = TrA(xi)IrA(xi) = 0FrA(xi) = min(FrA(xi) +IrA(xi), 1)xi X,i = 1, 2, ..., n;r = 1, 2, ..., lwherel = max{l(xi: A), l(xi: B)}.Example4.10. ConsideringtheSVNMSAofexample4.7wehave,A = (0.6, 0.4) , (0.0, 0.0) , (0.7, 0.8) /x1+(1.0, 1.0, 1.0) , (0.0, 0.0, 0.0) , (0.34, 0.28, 0.49) /x2+(0.87, 0.8, 0.7) ,(0.0, 0.0, 0.0) ,(0.4, 0.5, 0.7) /x3A = (0.5, 0.3) , (0.0, 0.0) , (0.8, 0.9) /x1+(0.7, 0.68, 0.62) , (0.0, 0.0, 0.0) , (0.64, 0.73, 0.99) /x2+(0.67, 0.5, 0.3) , (0.0, 0.0, 0.0) , (0.6, 0.8, 1.0) /x3Proposition4.11. It hasbeenobservedthat single-valuedneutrosophicmultisetssatisfythefollowingpropertiesunderset-theoreticoperations:1. CommutativeProperty(i)A B= B A(ii)A B= B A2. AssociativeProperty(i)A (B C) = (A B) C(ii)A (B C) = (A B) C6RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxx3. IdempotentProperty(i)A A = A(ii)A A = A(iii) A = A(iv) A = A4. AbsorptiveProperty(i)A (A B) = A(ii)A (A B) = A5. (i)AA =A(ii)AA = A(iii)A = A(iv)A =Proof. Theproofsarestraight-forward. 4.2. AlgebraicOperationsoverSVNMS.Denition4.12. The addition between two SVNMS A and Bover the universe X,denotedbyA B, isaSVNMSoverX, whosetruth-membership, indeterminacy-membershipandfalsity-membershipvaluesaredenedasTrAB(xi) = TrA(xi) +TrB(xi) TrA(xi).TrB(xi)IrAB(xi) = IrA(xi) +IrB(xi) IrA(xi).IrB(xi)FrAB(xi) = FrA(xi).FrB(xi)xi X,i = 1, 2, ..., n;r = 1, 2, ..., l.Denition4.13. ThemultiplicationbetweentwoSVNMSAandBovertheuni-verse X, denoted by AB, is a SVNMS over X, whose truth-membership, indeterminacy-membershipandfalsity-membershipvaluesaredenedasTrAB(xi) = TrA(xi).TrB(xi)IrAB(xi) = IrA(xi).IrB(xi)FrAB(xi) = FrA(xi) +FrB(xi) FrA(xi).FrB(xi)xi X,i = 1, 2, ..., n;r = 1, 2, ..., l.Example4.14. LettheSVNMSunderconsiderationbethosestatedinexample4.7ThenwehaveAB= (0.875, 0.3) , (0.28, 0.1) , (0.105, 0.0) /x1+(0.829, 0.7894, 0.7264) , (0.65, 0.56, 0.65) , (0.238, 0.224, 0.441) /x2+(1.0, 0.93, 0.853) , (0.208, 0.37, 0.52) , (0.0, 0.15, 0.14) /x3AB= (0.375, 0.0) , (0.02, 0.0) , (0.745, 0.8) /x1+(0.301, 0.252, 0.174) , (0.15, 0.09, 0.15) , (0.802, 0.856, 0.949) /x2+(0.67, 0.43, 0.237) , (0.002, 0.03, 0.08) , (0.4, 0.65, 0.76) /x35. ProposeddistancemeasurebetweentwoSVNMSInthissection, thenotionof distance, denotedbyd(A, B), ingeneral, betweentwoSVNMSAandBdenedovertheuniverseX= {x1, x2, ..., xn}hasbeenpro-posedinthesenseofHamming, NormalizedHamming, EuclideanandNormalizedEuclidean.7RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxxSupposethat,li= maxi{l(xi: A), l(xi: B)}andL = maxi{li},i = 1, 2, ..., n.Denition5.1. TheHammingdistancebetweenAandBisgivenby,dN(A, B) =n

i=1li

r=1(|TrA(xi) TrB(xi)| +|IrA(xi) IrB(xi)| +|FrA(xi) FrB(xi)|)Denition5.2. TheNormalizedHammingdistancebetweenAandBisgivenby,lN(A, B) =13nLn

i=1li

r=1(|TrA(xi) TrB(xi)| +|IrA(xi) IrB(xi)| +|FrA(xi) FrB(xi)|)Denition5.3. TheEuclideandistancebetweenAandBisgivenby,eN(A, B) =_n

i=1li

r=1_TrA(xi) TrB(xi)2+IrA(xi) IrB(xi)2+FrA(xi) FrB(xi)2_Denition5.4. TheNormalizedEuclideandistancebetweenAandBisgivenby,qN(A, B) =_13nLn

i=1li

r=1_TrA(xi) TrB(xi)2+IrA(xi) IrB(xi)2+FrA(xi) FrB(xi)2_Remark5.5. Ithasbeenobservedthattheproposeddistancemeasuresasstatedabovesatisesthefollowingproperties:(i)dN (A, B)[0, 3nL](ii)lN (A, B)[0, 1](iii)eN (A, B)_0,3nL_(iv)qN (A, B)[0, 1]Proof. Theproofsarestraight-forward. Proposition 5.6. It has been observed that in general, whatever might be the notioninwhichthe distance betweenany twoSVNMSbe dened, the distance measured (A, B)betweenanytwoSVNMSAandBsatisesthefollowingproperties:(i)d (A, B) 0andtheequalityholdsiA = B.(ii)d (A, B) = d (B, A)(iii)d (A, B) d (A, C) +d (B, C),whereCisaSVNMSoverX.Proof. Theproofs of (i) and(ii) arestraightforward. Wegivetheoutlineof theproofsof(iii)only.ConsiderthreearbitrarySVNMSA, BandC. Thenforr = 1, 2, ..., liwehaveTrA(xi) TrB(xi) =TrA(xi) TrC(xi) +TrC(xi) TrB(xi)TrA(xi) TrC(xi)+TrB(xi) TrC(xi)Similarlyitcanbeshownthat,IrA(xi) IrB(xi)IrA(xi) IrC(xi)+IrB(xi) IrC(xi)FrA(xi) FrB(xi)FrA(xi) FrC(xi)+FrB(xi) FrC(xi)8RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxxHenceitfollowsthat,

ni=1

lir=1_TrA(xi) TrB(xi)+IrA(xi) IrB(xi)+FrA(xi) FrB(xi)_

ni=1

lir=1_TrA(xi) TrC(xi)+IrA(xi) IrC(xi)+FrA(xi) FrC(xi)_+

ni=1

lir=1_TrB(xi) TrC(xi)+IrB(xi) IrC(xi)+FrB(xi) FrC(xi)_Again,weseeTrA(xi) TrB(xi)2=TrA(xi) TrC(xi) +TrC(xi) TrB(xi)2TrA(xi) TrC(xi)2+TrB(xi) TrC(xi)2SimilarlywehaveIrA(xi) IrB(xi)IrA(xi) IrC(xi)+IrB(xi) IrC(xi)FrA(xi) FrB(xi)FrA(xi) FrC(xi)+FrB(xi) FrC(xi)andhence,

ni=1

lir=1_TrA(xi) TrB(xi)2+IrA(xi) IrB(xi)2+FrA(xi) FrB(xi)2_

ni=1

lir=1_TrA(xi) TrC(xi)2+IrA(xi) IrC(xi)2+FrA(xi) FrC(xi)2_+

ni=1

lir=1_TrB(xi) TrC(xi)2+IrB(xi) IrC(xi)2+FrB(xi) FrC(xi)2_Thustheproofsfollowautomaticallyfromtheaboveresults. Remark5.7. Fromtheaforementionedobservationsitcanbeconcludedthattheproposed notion of distance measures actually dene metrics over the set of all single-valuedneutrosophicmultisetsandhenceifNmdenotesthecollectionofallSVNMSoverauniverseXthen(Nm, d)denesametricspaceoverX.Example 5.8. Let us consider the SVNMS A and Bas stated in example 4.7. ThenwehavedN (A, B) = 7.83,lN (A, B) = 0.29,eN (A, B) = 1.772andqN (A, B) = 0.3418.6. SimilarityMeasurebetweentwoSVNMSInthissectionthenotionof similaritymeasurebetweentwoSVNMShasbeenstated. ThevarioustypesofsimilaritymeasuresbetweentwoSVNMSareproposedasfollows:Denition6.1. TheDistanceBasedSimilarityMeasureisdenedas,Sd (A, B) =11 + d (A, B)wherethedistancemeasurecanbetakeninanyof themethodsasmentionedinSection5.Denition6.2. TheSimilarityMeasurebasedonmembershipdegreesisdenedas,Sm (A, B)=

ni=1

lir=1{min(TrA(xi), TrB(xi)) + min(IrA(xi), IrB(xi)) + min(FrA(xi), FrB(xi))}

ni=1

lir=1_max_TrA(xi), TrB(xi)_+ max_IrA(xi), IrB(xi)_+ max_FrA(xi), FrB(xi)__9RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxxRemark6.3. Insomecases, especiallywhiledealingwithreal lifeproblems, theelements of the universe under consideration are associated with weights in order tospecifythevaryingdegreesofimportanceoftheelementsathand. Inthiscasewehave considered this possibility and accordingly the similarity measure between twoSVNMShavebeendenedwheretheuniverseoverwhichthesetsaredenedhaveweightsassociatedtoitsconstituentelements.Denition6.4. Theweightedsimilaritymeasureinthesenseof Majumdar andSamanta[17]isdenedas,Sw (A, B)=

ni=1 wi_

lir=1{TrA(xi).TrB(xi)+IrA(xi).IrB(xi)+FrA(xi).FrB(xi)}_

ni=1 wi[(TrA(xi)2+IrA(xi)2+FrA(xi)2).(TrB(xi)2+IrB(xi)2+FrB(xi)2)]._1

ni=1 wi_where0 wi 1,i=1, 2, ..., nandwiaretheweightsassociatedwiththexisrespectivelywherexi X.Denition 6.5. The proposed weighted similarity measure in the sense of DengfengandChuntian[9]isdenedasfollows:LetAbeaSVNMSoverauniverseX. WedeneafunctionA: X [0, 1]as,A(x) =

lr=1{TrA(x) + IrA(x) + (1 FrA(x))}3lwherel = l(x : A)isthelengthofanelementx X.Dene the weightedsimilaritymeasure betweentwoSVNMSAandBover thesameuniverseXas,S

w (A, B) = 1 p

ni=1 wi |B(xi) A(xi)|p

ni=1 wiwhere0 wi 1, i =1, 2, ..., nandwiaretheweightsassociatedwiththexisrespectivelywherexi X.Herepisapositiveintegercalledthesimilaritydegree.Example6.6. WeconsidertheSVNMSAandBasstatedinexample4.7. ThentakingintoconsiderationtheresultlN (A, B) = 0.29fromexample5.8wehave,Sd (A, B) = 0.775andSm (A, B) = 0.485.Proposition6.7. 1. Thedistancebasedsimilaritymeasuresatisesthefollowingproperties:(i)0 Sd (A, B) 1andequalityoccursiA = B(ii)Sd (A, B) = Sd (B, A)(iii)ForA B C,Sd (A, C) Sd (A, B) Sd (B, C)2. Thesimilaritymeasurebasedonmembershipdegreessatisesthefollowingprop-erties:(i)0 Sm (A, B) 1andequalityoccursiA = B(ii)Sm (A, B) = Sm (B, A)10RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxx(iii)ForA B C,Sd (A, C) Sd (A, B) Sd (B, C)3. Theweightedsimilaritymeasuressatisfythefollowingproperties:(a).(i)0 Sw (A, B) 1andequalityoccursiA = B(ii)Sw (A, B) = Sw (B, A)(b).(i)0 Sw(A, B) 1andequalityoccursiA = B(ii)Sw(A, B) = Sw(B, A)(iii)ForA B C,Sw(A, C) Sw(A, B) Sw(B, C)(iv)CorrespondingtoanytwoSVNMS,foranascendingsequenceofintegralvaluesofpadecreasingsequenceofsimilaritymeasuresisobtained.Proof. Weonlyprove3(b)(iii)and3(b)(iv)sincetheremainingproofsarestraight-forward.LetA B Cthenitiseasytoprovethat{C(xi) A(xi)} {B(xi) A(xi)}sinceTrC(xi) TrB(xi),IrC(xi) IrB(xi)andFrC(xi) FrB(xi), xiX,i = 1, 2, ..., nandr= 1, 2, ..., max{l (xi: B) , l (xi: C)}.Thus,

l(xi:C)r=1_TrC(xi) +IrC(xi) +_1 FrC(xi)__

l(xi:B)r=1_TrB(xi) +IrB(xi) +_1 FrB(xi)__andhence,{C(xi) A(xi)} {B(xi) A(xi)} = {C(xi) B(xi)} 0.So,|{C(xi) A(xi)}|p |{B(xi) A(xi)}p|

ni=1wi|{C(xi) A(xi)}|p

ni=1wi|{B(xi) A(xi)}|pSw (A, C) Sw (A, B)Similarly,itcanbeshownthatSw (A, C) Sw (B, C).Next, suppose that for aparticular pN, ai= |B(xi) A(xi)| andqi=wi,i = 1, 2, ..., n.Thus,p

ni=1 wi |B(xi) A(xi)|p

ni=1 wi=

ni=1 qi (ai)p

ni=1 qi1p= f(p), say.Hence, f(p) is a power meanfunction[10] therebybeing strictlymonotonicallyincreasinginnaturei.e. forp1, p2 Nwithp1< p2, f(p1) < f(p2)._ni=1qi (ai)p1

ni=1qi_1p1 1 _ni=1qi (ai)p2

ni=1qi_1p211RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxxThis proves the fact that as p increases, the value of the similarity measureSw (A, B)decreasesandhenceitmaybeconcludedthatforasequenceofascendingintegralvalues for pweobtainadecreasingsequenceof similaritymeasures betweentwoSVNMSAandB,whichcompletestheproof. Remark6.8. Fromthe aforementionedproperties it is clear that the weightedsimilarity measure Sw (A, B) in the sense of [17] is not an actual similarity measure inthe actual sense of the term since it does not satisfy all the properties of the axiomaticdenitionof asimilaritymeasure [26]. Wethus termthis particular measure ofsimilarityaweightedquasi-similaritymeasure.7. AnapplicationofSinglevaluedneutrosophicmultisetsinmedicaldiagnosisSince real-life situations involve uncertainties, while constructing a mathematicalmodel ofpractical importanceweneedtoincorporatevariablesthatcandeal withuncertainties. Inthis sectionwestateanexamplewithaviewtoshowhowthetheoryofsingle-valuedneutrosophicmultisetscanbeusedindiagnosingamedicalcondition.Afeverisoneof themostcommonmedical signsanditischaracterizedbyanelevationofbodytemperatureabovethenormal rangeof97.7F 98.5F. Itcanbecausedbymanymedical conditionsrangingfrombenigntopotentiallyserious.Besidesanelevatedbodytemperaturethereareadditionalsymptomssuchasshiv-ering, sweating, loss of appetiteetc. whichareassociatedwithfever. Moreover,there are specic patterns of temperature changes during a fever according to whichafever maybeclassiedas Continuous fever, Pel-Ebsteinfever, Remittent feverand Intermittent fever, which in turn may be classied into Quotidian fever, TertianfeverandQuartanfever. Thesefeverpatternsalongwiththeassociatedsymptomsaidinthediagnosisofaparticulardisease.Consider the case of apersonprimarilydiagnosedwithfever associatedwithshivering, headache, muscleandjointache, cough, runningnoseaccompaniedwithsneezing, lossof appetite, chestpainandfatigue. Theassociatedsymptomsthatare prominent in the person hint at the fact that the person might be suering fromTuberculosis,InuenzaorCommonCold.Tuberculosisisaninfectiousdisease, typicallyof thelungsandischaracterizedbyremittent fever accompaniedbybadcough, at times accompaniedbyblood,paininthe chest, fatigue, loss of appetite, chills andnight sweating. Inuenzaor Fluis characterizedbyremittent fever, extremechills, shivering, cough, nasalcongestionandrunnynoseaccompaniedbysneezing,bodyache,particularlyinthejoints,fatigueandheadache. Ontheotherhand,CommonColdischaracterizedbyremittentfeveraccompaniedbyrunnynose,shivering,cough,bodyache,headacheandsneezing.Forthesakeofdiagnosis,thepatientiskeptundersupervisionforadayandhisfeverpatternalongwiththeothersymptomsaremonitoredthrice, atintervalsof8 hrs, starting from 6:00 hrs in the morning, then at 14:00 hrs and nally at 22:00 hrs.12RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxxThemedicalndingsofthepatientarerepresentedinatabularformasfollows:SymptomsTimings T Sh Sw H MJ C LA CP F RNS6:00hrs 97.5F - - - m m m - m m14:00hrs 100F m - m m m m - m m22:00hrs 101.2F h - m m m m - m mTable7.1. Tablerepresentingthemedicalndingsofthepatient.HereT, Sh, Sw, H, MJ, C, LA, CP, FandRNSdenotebodytemprature, shiv-ering, sweating, headache, muscle and joint pain, cough, loss of appetite, chest pain,fatigueandrunningnosewithsneezingrespectivelyandthesesymptomsaltogetherconstitute the univrsal set. On the other hand, the symbols m and h are abbrevi-ations for moderate and high respectively which denote the qualitative intensityoftheelementsoftheuniversalset.The above ndings of the patient can be summarized and represented with the helpof a SVNMS, denoted by Pf(symbolic representation for the patients ndings) overtheabovementioneduniverseasfollows:A = (0, 0.5, 0.7) , (0.2, 0.1, 0) , (0.9, 0.1, 0) /T+(0, 0.5, 0.9) , (0.1, 0.1, 0) , (1, 0.3, 0) /Sh+(0, 0, 0) , (0.1, 0, 0) , (0.9, 0.9, 1) /Sw +(0, 0.5, 0.4) , (0.1, 0.2, 0.2) , (0.8, 0.3, 0.3) /H+(0.5, 0.5, 0.5) , (0, 0.1, 0) , (0.2, 0.3, 0.4) /MJ+(0.55, 0.5, 0.4) , (0.2, 0, 0) , (0.5, 0.4, 0.5) /C+(0.5, 0.5, 0.45) , (0.2, 0.2, 0.1) , (0.4, 0.3, 0.4) /LA+(0, 0, 0) , (0.1, 0.1, 0) , (0.8, 0.9, 0.9) /CP+(0.58, 0.5, 0.5) , (0.3, 0.2, 0.1) , (0.3, 0.3, 0.4) /F+(0.5, 0.4, 0.4) , (0.1, 0.1, 0) , (0.2, 0.3, 0.4) /RNSSupposethestandardsymptomaticcharacteristicsof thediseasesarerepresentedbythefollowingSVNMSas,TB= (0, 0.5, 0.6) , (0, 0.1, 0) , (0.9, 0.4, 0.2) /T+(0.6, 0.6, 0.5) , (0.1, 0.1, 0) , (0.2, 0.1, 0.1) /Sh+(0.5, 0.5, 0.7) , (0.1, 0.2, 0.1) , (0.4, 0.3, 0.1) /Sw +(0, 0, 0) , (0, 0.1, 0) , (0.8, 0.8, 0.9) /H+(0, 0, 0) , (0.1, 0.1, 0.1) , (0.9, 1, 0.9) /MJ+(1, 0.9, 0.9) , (0.1, 0, 0.1) , (0, 0, 0.1) /C+(0.6, 0.6, 0.5) , (0.3, 0.2, 0.2) , (0.4, 0.5, 0.4) /LA+(0.7, 0.6, 0.6) , (0.2, 0.1, 0.1) , (0.2, 0.3, 0.1) /CP+(0.5, 0.4, 0.3) , (0.4, 0.4, 0.3) , (0.4, 0.5, 0.3) /F+(0.4, 0.3, 0.3) , (0.3, 0.3, 0.2) , (0.5, 0.4, 0.5) /RNSInf= (0.8, 0.7, 0.7) , (0, 0.1, 0.1) , (0.1, 0.1, 0.2) /T+(0.9, 0.8, 0.8) , (0, 0, 0.1) , (0.1, 0.2, 0.1) /Sh+(0, 0, 0) , (0, 0, 0) , (0.9, 0.9, 0.1) /Sw +(0.6, 0.6, 0.5) , (0.2, 0.1, 0.1) , (0.3, 0.3, 0.2) /H+(0.9, 0.8, 0.8) , (0.1, 0.1, 0) , (0.1, 0, 0) /MJ+(0.6, 0.6, 0.5) , (0.1, 0.2, 0.1) , (0.3, 0.4, 0.2) /C+(0.6, 0.5, 0.5) , (0.2, 0.2, 0.1) , (0.4, 0.3, 0.3) /LA+(0, 0, 0) , (0.1, 0.2, 0.1) , (0.8, 0.9, 0.8) /CP+(0.5, 0.4, 0.3) , (0.4, 0.4, 0.3) , (0.4, 0.5, 0.3) /F+(0.4, 0.3, 0.3) , (0.3, 0.3, 0.2) , (0.5, 0.4, 0.5) /RNSCC= (0.1, 0.5, 0.7) , (0.1, 0.1, 0.2) , (0.9, 0.4, 0.3) /T+(0.9, 0.8, 0.8) , (0.1, 0, 0.1) , (0, 0.2, 0.1) /Sh+(0, 0, 0) , (0.2, 0.1, 0.1) , (0.9, 0.4, 0.3) /Sw +(0.6, 0.6, 0.5) , (0.2, 0.1, 0.1) , (0.3, 0.2, 0.2) /H+(0.7, 0.6, 0.6) , (0.3, 0.1, 0.1) , (0.2, 0.3, 0.3) /MJ +(0.6, 0.6, 0.5) , (0.1, 0.2, 0.1) , (0.3, 0.4, 0.2) /C+(0.6, 0.5, 0.5) , (0.2, 0.2, 0.1) , (0.4, 0.3, 0.3) /LA+(0.5, 0.5, 0.4) , (0.1, 0.1, 0) , (0.4, 0.3, 0.3) /CP+(0.5, 0.4, 0.3) , (0.4, 0.4, 0.3) , (0.4, 0.5, 0.3) /F+(0.7, 0.7, 0.6) , (0.2, 0.2, 0.1) , (0.3, 0.3, 0.4) /RNS13RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxx.The disease of the patient is diagnosed using the weighted similarity measure (Def-inition6.5.). Suppose, forthesakeofdiagnosis, thehighestpriorityisassignedtoheadacheandmuscleandjointacheandconsequentlysymptomssuchas,shivering,sweating,coughing,runningnoseandshivering,chestpain,fatigue,lossofappetiteandbodytemperaturehave beenprioritized. Letthe respectiveweights assignedtoT, Sh, Sw, H, MJ, C, LA, CP, LA, CP, F, RNSbe0.2, 0.7, 0.7, 0.9, 0.9, 0.6, 0.3,0.4,0.3and0.6.Thedecisionmakingprocess involves calculatingtheweightedsimilaritymea-sures between the SVNMS Pfand the respective SVNMS representing the diseases.ThesetbearingthehighestmeasureofsimilaritywithrespecttoPfisthediseasethathasaectedtheperson. Inordertoconrmtheobtainedresult,theprocessisrepeatedformorethanoneintegralvaluesofpi.e. thesimilaritydegree.Thecalculationshavebeenrepresentedinatabularformasfollows:DiseaseSetsSimilarityDegree(p) PatientData TB Inf CCp = 1 Pf0.69 0.89 0.87p = 2 Pf0.68 0.86 0.85p = 3 Pf0.66 0.85 0.83Table 7.2. Tableshowingthe similaritymeasures betweenthediseasesetsandPf.Thus, fromtheabovendingsitisclearthatthepatienthasbeensueringfromInuenza.8. ConclusionIn this paper a new hybridized concept, namely Single Valued Neutrosophic Mul-tisethasbeenstudied. Varioussettheoreticandalgebraicoperatorshavebeende-ned and their properties have been discussed. The notions of distance and similaritymeasureshavealsobeenincorporated. Finallyanexamplecitingtheapplicabilityofsinglevaluedneutrosophicmultisetsinproblemspertainingtomedicaldiagnosishas been stated. Being characterized by an indeterminacy membership value, singlevalued neutrosophic sets provide a far more generalized tool in handling uncertaintyascomparedtofuzzysetsorintuitionisticfuzzysets. Sincesinglevaluedneutro-sophicmultisetshaveresultedbymergingtogethertheconceptsof multisetsandsinglevaluedneutrosophicsets,theformerareafurthergeneralizationofthelatterin the sense that in this case multiple occurrances of an element with varying degreesof membershipvaluesaretakenintoconsiderationandthushavemoredegreesoffreedomcomparedtothelatter. Moreover, thenotionsof distanceandsimilaritymeasuresarestatedwithaviewtoaidinthewidespreadapplicabilityof SVNMSin elds like medical diagnosis,data retrieval on the web or in multicriteria decisionmakingproblems.14RajashiChatterjeeetal./Ann. FuzzyMath. Inform. x(201y),No. x,xxxxAcknowledgments. Theauthorsexpresstheirsincerethankstotheanonymousreferees for their valuable and constructive suggestions which have improved the pre-sentation. 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