Compactness and Continuity on Neutrosophic Soft Metric Space

InternationalJournalofAdvancesinMathematics

Volume2018,Number4,Pages1-24,2018 eISSN2456-6098 c adv-math.com

CompactnessandContinuityOnNeutrosophicSoftMetricSpace

TuhinBera1 andNirmalKumarMahapatra∗ 1 DepartmentofMathematics,BororS.S.HighSchool,Bagnan, Howrah-711312,W.B,India. ∗ DepartmentofMathematics,PanskuraBanamaliCollege, PanskuraRS-721152,W.B,India. Email:nirmal hridoy@yahoo.co.in

ABSTRACT.Inthispaper,thenotionofcompactneutrosophicsoftmetricspaceisintroduced.Theconceptofneutrosophic softfunctionandthecompositionoffunctionsinaneutrosophicsoftmetricspacealongwithsuitableexamplesalsohavebeen brought.Thecontinuityanduniformcontinuityofaneutrosophicsoftfunctioninthisspacehavebeendefinedandverifiedby properexamples.Severalrelatedproperties,theoremsandstructuralcharacteristicsofthesehavebeeninvestigatedhere.

1Introduction

ThetheoryofNeutrosophicset(NS)introducedbySmarandache[19,20]isthegeneralizationofmanytheories e.g.,fuzzyset,intuitionisticfuzzysetetcpracticedtohandlethevariousuncertaintiesinmanyrealapplication overthepastmanyyears.Theneutrosophiclogicincludestheinformationaboutthepercentageoftruth,indeterminacyandfalsitygradeinseveralrealworldproblemlikeinlaw,medicine,engineering,management, industrial,ITsectoretcwhichisnotavailableinfuzzysettheoryandintuitionisticfuzzysettheory.

Molodtsovhasshownthateachoftheabovetopicsdealingwithuncertaintiessufferfrominherentdifficulties possiblyduetoinadequacyoftheirparametrizationtool.So,Molodtsov[1]proposedtheconceptof‘softset

*CorrespondingAuthor. ReceivedMarch05,2018;acceptedJune17,2018.

2010MathematicsSubjectClassification:47H10,54H25.

Keywordsandphrases:Compactneutrosophicsoftmetricspace;Neutrosophicsoftfunctions;Compositionofneutrosophic softfunction;Continuity;Uniformcontinuity.

ThisisanopenaccessarticleundertheCCBYlicense http://creativecommons.org/licenses/by/3.0/

TuhinBeraandNirmalKumarMahapatra2

theory’formodelingvaguenessanduncertainties.Itiscompletelyfreefromtheparametrizationinadequacy syndrome.Thismakesthetheoryveryconvenient,efficientandeasytoapplyinpractice.Inaccordanceofthis, Majietal.[2-4]studiedtheseveralbasicoperationsinsoftsetstheoryoverfuzzysetsandintuitionisticfuzzysets. Thenotionsoffuzzymetricspacewerestudiedin[5-13]fromdifferentpointofview.RoyandSamanta[14]have definedopenandclosedsetsonfuzzytopologicalspaces.Park[15]andAlacaetal.[16]definedtheconceptof intuitionisticfuzzymetricspaceintermofcontinuous t-normsandcontinuous t-conormsasageneralisationof fuzzymetricspace.Usingalltheseconcepts,Beaulaetal.[17,18]proposedthenotionoffuzzysoftmetricspaces intermsoffuzzysoftpoints.

AfterintroductionofNStheory,Maji[21]hasbroughtacombinednotionNeutrosophicsoftset(NSS).In continuation,severalmathematicianshavepresentedtheirresearchworksindifferentmathematicalstructures. DeliandBroumi[22],CetkinandAygun[24-26],BeraandMahapatra[27-34]studiedsomefundamentalalgebraic structuresinNSStheorycontext.DeliandBroumi[23]havealsomodifiedsomeoperationsrelatedtoindeterministicfunctionofNSSsgivenbyMaji.Broumietal.[35,36]havedonesomeconsecutiveworksingraphtheory overNSS.

Themotivationofthepresentpaperistoextendtheconceptneutrosophicsoftmetricspace(NSMS)proposed in[32].ThecurrentarticlepresentsthenotionofcompactNSMS,thecontinuityanduniformcontinuityofa neutrosophicsoftfunctioninanNSMSalongwithinvestigationofsomerelatedpropertiesandtheorems.The contentofthepresentpaperisdesignedasfollows:

Section2givessomepreliminaryusefuldefinitions,examplesandtheoremswhichwillbeusedthroughout thepaper.Insection3,compactnessofNSMSisdefinedandillustratedbyexamples.Somerelatedbasicproperties havebeenstudiedhere,also.Section4dealswiththecontinuityofneutrosophicsoftfunctionandthecomposition ofneutrosophicsoftfunctionsinanNSMSalongwiththestudyoftheirstructuralcharacteristics.Theconcept ofuniformcontinuityofaneutrosophicsoftfunctioninanNSMShasbeenintroducedinsection5.Finally,the conclusionofthepresentworkisstatedinsection6.

Afewexamplesofcontinuous t-normare a ∗ b = ab, a ∗ b = min{a, b}, a ∗ b = max{a + b 1,0}.

2. Abinaryoperation : [0,1] × [0,1] → [0,1] iscontinuous t -conorm(s -norm)if satisfiesthefollowing conditions:

(i) iscommutativeandassociative. (ii) iscontinuous. (iii) a 0 = 0 a = a, ∀a ∈ [0,1] (iv) a b ≤ c d if a ≤ c, b ≤ d with a, b, c, d ∈ [0,1]

Afewexamplesofcontinuous s-normare a b = a + b ab, a b = max{a, b}, a b = min{a + b,1}.

3. Let U beaninitialuniversesetand E beasetofparameters.Let P(U) denotethepowersetof U.Thenfor A ⊆ E,apair (F, A) iscalledasoftsetover U,where F : A → P(U) isamapping.

2.2DefinitionsrelatedtoNSandNSS

Fewrelevantdefinitionsaregivenbelow[19,21,23,33]:

1. Let X beaspaceofpoints(objects),withagenericelementin X denotedby x.Aneutrosophicset A in X ischaracterizedbyatruth-membershipfunction TA,anindeterminacy-membershipfunction IA andafalsitymembershipfunction FA. TA (x), IA (x) and FA (x) arerealstandardornon-standardsubsetsof ] 0,1+ [.Thatis TA, IA, FA : X →] 0,1+ [.Thereisnorestrictiononthesumof TA (x), IA (x), FA (x) andso, 0 ≤ sup TA (x)+ sup IA (x)+ sup FA (x) ≤ 3+ .

2. Let U beaninitialuniversesetand E beasetofparameters.Let NS(U) denotethesetofallNSsof U.Thenfor A ⊆ E,apair (F, A) iscalledanNSSover U,where F : A → NS(U) isamapping. ThisconcepthasbeenmodifiedbyDeliandBroumiasgivenbelow: 3. Let U beaninitialuniversesetand E beasetofparameters.Let NS(U) denotethesetofallNSsof U Then,aneutrosophicsoftset N over U isasetdefinedbyasetvaluedfunction f N representingamapping f N : E → NS(U) where f N iscalledapproximatefunctionoftheneutrosophicsoftset N.Inotherwords,the neutrosophicsoftsetisaparameterizedfamilyofsomeelementsoftheset NS(U) andthereforeitcanbewritten asasetoforderedpairs,

subsetof N2 if ∀e ∈ E, ∀x ∈ U, Tf N1 (e) (x) ≤ Tf N2 (e) (x); I f N1 (e) (x) ≥ I f N2 (e) (x); Ff N1 (e) (x) ≥ Ff N2 (e) (x). Wewrite N1 ⊆ N2 andthen N2 istheneutrosophicsoftsupersetof N1.

6. Let N1 and N2 betwoNSSsoverthecommonuniverse (U, E).Thentheirunionisdenotedby N1 ∪ N2 = N3 andisdefinedby: N3 = {(e, {< x, Tf N3 (e) (x), I f N3 (e) (x), Ff N3 (e) (x) >: x ∈ U}) : e ∈ E} where Tf N3 (e) (x)= Tf N1 (e) (x) Tf N2 (e) (x), I f N3 (e) (x)= I f N1 (e) (x) ∗ I f N2 (e) (x) and Ff N3 (e) (x)= Ff N1 (e) (x) ∗ Ff N2 (e) (x);

7. Let N1 and N2 betwoNSSsoverthecommonuniverse (U, E).Thentheirintersectionisdenotedby N1 ∩ N2 = N3 andisdefinedby: N3 = {(e, {< x, Tf N3 (e) (x), I f N3 (e) (x), Ff N3 (e) (x) >: x ∈ U}) : e ∈ E} where Tf N3 (e) (x)= Tf N1 (e) (x) ∗ Tf N2 (e) (x), I f N3 (e) (x)= I f N1 (e) (x) I f N2 (e) (x) and Ff N3 (e) (x)= Ff N1 (e) (x) Ff N2 (e) (x);

8. Aneutrosophicsoftset N over (U, E) issaidtobenullneutrosophicsoftsetif Tf N (e) (x)= 0, I f N (e) (x)= 1, Ff N (e) (x)= 1; ∀e ∈ E, ∀x ∈ U.Itisdenotedby φu. Aneutrosophicsoftset N over (U, E) issaidtobeabsoluteneutrosophicsoftsetif Tf N (e) (x)= 1, I f N (e) (x)= 0, Ff N (e) (x)= 0; ∀e ∈ E, ∀x ∈ U.Itisdenotedby1u. Clearly, φc u = 1u and1c u = φu.

9. AneutrosophicsoftpointinanNSS N isdefinedasanelement (e, f N (e)) of N,for e ∈ E andisdenotedby eN , if f N (e) / ∈ φu and f N (e ) ∈ φu, ∀e ∈ E −{e} Thecomplementofaneutrosophicsoftpoint eN isanotherneutrosophicsoftpoint ec N suchthat f c N (e)= ( f N (e))c Aneutrosophicsoftpoint eN ∈ M, M beinganNSSiffor e ∈ E, f N (e) ≤ f M (e) i.e., Tf N (e) (x) ≤ Tf M (e) (x), I f N (e) (x) ≥ I f M (e) (x), Ff N (e) (x) ≥ Ff M (e) (x), ∀x ∈ U Example: Let U = {x1, x2, x3} and E = {e1, e2}.Then, e1N = {< x1, (0.6,0.4,0.8) >, < x2, (0.8,0.3,0.5) >, < x3, (0.3,0.7,0.6) >} isaneutrosophicsoftpointwhosecomplementis: ec 1N = {< x1, (0.8,0.6,0.6) >, < x2, (0.5,0.7,0.8) >, < x3, (0.6,0.3,0.3) >} ForanotherNSS M definedonsame (U, E),let f M (e1)= {< x1, (0.7,0.4,0.7) >, < x2, (0.8,0.2,0.4) >, < x3, (0.5,0.6,0.5) >} Then f N (e1) ≤ f M (e1) i.e., e1N ∈ M

termsofneutrosophicsoftpointsisdefinedbyamapping d : NS(UE ) × NS(UE ) → [0,3] satisfyingthefollowing conditions:

NSM1: d(eM, eN ) ≥ 0, ∀eM, eN ∈ NS(UE )

NSM2: d(eM, eN )= 0 ⇔ eM = eN

NSM3: d(eM, eN )= d(eN , eM )

NSM4: d(eM, eN ) ≤ d(eM, eP )+ d(eP, eN ), ∀eM, eP, eN ∈ NS(UE ) Then NS(UE ) issaidtoformanNSMSwithrespecttotheneutrosophicsoftmetric‘d’over (U, E) and isdenotedby (NS(UE ), d).Here eM = eN inthesensethat TeM (xi )= TeN (xi ), IeM (xi )= IeN (xi ), FeM (xi )= FeN (xi ), ∀xi ∈ U 2. Example (i)On NS(UE ) define d(eM, eN )= minxi {(|TeM (xi ) TeN (xi )|k + |IeM (xi ) IeN (xi )|k + |FeM (xi ) FeN (xi )|k ) 1 k }, k(≥ 1) beinganyrealnumber.This‘d’satisfiesallthemetricaxiomsandso,itisaneutrosophicsoft metricover (U, E) (ii)Let‘d’beaneutrosophicsoftmetricon NS(UE ).Suppose d1(eM, eN )= d(eM ,eN ) 1+d(eM ,eN ) ;Then‘d1’satisfiesallthe metricaxioms.So, (NS(UE ), d1) isanNSMSwithrespecttotheneutrosophicsoftmetric d1

3. Let (NS(UE ), d) beaneutrosophicsoftmetricspaceand t ∈ (0,3].Anopenballhavingcenterat eN ∈ NS(UE ) andradius‘t’isdefinedbyaset B(eN , t)= {eiN ∈ NS(UE ) : d(eN , eiN ) < t}.

Theneutrosophicsoftclosedballisdefinedas: B[eN , t]= {eiN ∈ NS(UE ) : d(eN , eiN ) ≤ t}.

Aneighbourhoodof eN ∈ NS(UE ) isdefinedbyanopenball B(eN , t) withcenterat eN andradius t ∈ (0,3].

4. InanNSMS (NS(UE ), d) over (U, E),aneutrosophicsoftpoint eN iscalledaninteriorpointof NS(UE ) ifthere existanopenball B(eN , t) suchthat B(eN , t) ⊂ NS(UE )

ForanNSMS (NS(UE ), d) over (U, E),anNSS M iscalledopenifeachofit’spointsisaninteriorpoint.

5. Aneutrosophicsoftpoint eN inanNSMS (NS(UE ), d) iscalledalimitpoint/accumulationpointofanNSS M ⊂ NS(UE ) ifforevery t ∈ (0,3], B(eN , t) containsatleastoneneutrosophicsoftpointof M distinctfrom eN Collectionofalllimitpointsof M iscalledderivedNSSof M andisdenotedby D(M).AnNSS M ⊂ NS(UE ) inanNSMS (NS(UE ), d) over (U, E) isclosedNSSif D(M) ⊂ M or M hasnolimitpoint.

6. Asequenceofneutrosophicsoftpoints {enN } inanNSMS (NS(UE ), d) issaidtoconvergein (NS(UE ), d) if thereexistsaneutrosophicsoftpoint eN ∈ NS(UE ) suchthat d(enN , eN ) → 0as n → ∞ or enN → eN as n → ∞. Analytically,forevery > 0thereexistsanaturalnumber n0 suchthat d(enN , eN ) < ∀n ≥ n0. 7. Asequence {enN } ofneutrosophicsoftpointinanNSMS (NS(UE ), d) issaidtobeaCauchysequenceiftoevery > 0thereexistsan n0 ∈ N (setofnaturalnumbers)suchthat d(emN , enN ) < ∀m, n ≥ n0 i.e., d(emN , enN ) → 0 as m, n → ∞ 8. AnNSMS (NS(UE ), d) issaidtobecompleteifeveryCauchysequencein (NS(UE ), d) convergestoaneutrosophicsoftpointof NS(UE ) 9. Let (NS(UE ), d) beanNSMS.Thenthediameterof NS(UE ) isdefinedas: δ(NS(UE ))= sup {d(e1N , e2N ) : e1N , e2N ∈ NS(UE )}

AnNSS M ⊂ NS(UE ) isboundedifithasafinitediameteri.e.,if d(e1M, e2M ) ≤ r,for r ∈ (0,3] and ∀e1M, e2M ∈ M.

2.4Theoremsrelatedtoneutrosophicsoftmetricspace

Somenecessarytheoremsarestatedforthesakeofcompleteness[32]:

1. InanNSMS (NS(UE ), d),everyneutrosophicsoftopenball B(eN , t) isopenandeveryneutrosophicsoftclosed ball B[eN , t] isclosed.

2. Let (NS(UE ), d) beanNSMSover (U, E).Then, (i)theintersectionoffinitenumberofopenNSSsin (NS(UE ), d) isopen. (ii)theintersectionofanyfamilyofclosedNSSsin (NS(UE ), d) isclosed.

3. EveryfiniteneutrosophicsoftsubsetofanNSMSisclosed.

3CompactnessofNSMS

Inthissection,thecompactNSMShasbeendefinedandillustratedbyexamples.Somerelatedtheoremsalsohave beendevelopedhere.

3.1Definition

AnNSMS (NS(UE ), d) issaidtobecompactifeverysequenceofneutrosophicsoftpoints {enM } ofthespacehas asubsequence {enk M } convergingtoaneutrosophicsoftpointof NS(UE )

AnNSS M ⊂ NS(UE ) issaidtobecompactifeverysequenceofneutrosophicsoftpointschosenfrom M has asubsequenceconvergingtoapointof M.Ifthelimitofthesubsequencebelongsto NS(UE ) andnotnecessarily to M,then M issaidtobecompactin (NS(UE ), d)

3.1.1 Example

(1)Let E = {e} and U = {x, y, z}.Defineadistancefunctionon NS(UE ) as: d(eM, eN )=    1if eM = eN 0if eM = eN

Then‘d’isaneutrosophicsoftmetricon NS(UE ) andiscalleddiscreteneutrosophicsoftmetric.Thus (NS(UE ), d) isadiscreteNSMS.ItisacompactNSMS.

(2)ConsidertheNSMS (NS(UE ), d) where E = N (thesetofnaturalnumber)betheparametricset, U = Z (the setofallintegers)betheuniversalsetand d isdefinedasin(2)(i)of[2.3];Since T, I, F ∈ [0,1],everysequenceof neutrosophicsoftpointsofthespacehasaconvergentsubsequenceandso (NS(UE ), d) iscompact.

(3)TaketheNSMS (NS(UE ), d) where E = N (thesetofnaturalnumber), U = Z (thesetofallintegers)and‘d’is definedasin(2)(i)of[2.3];Considerasequenceofneutrosophicsoftpoints {enM } as, ∀x ∈ Z :

TenM (x)= 1 2n , IenM (x)= 1 1 2n , FenM (x)= n 1+2n for TenM , IenM , FenM ∈ (0,1)

Then M isnotcompactitselfbutiscompacton NS(UE ). (4)Let E = {e1, e2, e3, e4, e5, e6, e7, e8} and U = Z.Define‘d’asin(2)(i)of[2.3];Then (NS(UE ), d) isnotcompact. WeshallverifyitbytakingasequenceofneutrosophicsoftpointsasgiveninTable1.

Table1:Tabularformofneutrosophicsoftsequence e1M e2M e3M e4M e5M e6M e7M e8M x1 (1,0,0)(0,1,0)(0,0,1)(1,1,0)(1,0,1)(0,1,1)(0,0,0)(1,1,1) x2 (1,0,0)(0,1,0)(0,0,1)(1,1,0)(1,0,1)(0,1,1)(0,0,0)(1,1,1) . . . . . . . . .

Then d(eiM, ejM ) = 0for i = j.So,neitherthesequencenoranyofit’ssubsequenceisconvergent.

3.2Theorem

AcompactNSMSiscomplete.

Proof. Let (NS(UE ), d) beacompactNSMSand {enM } beaCauchysequenceofneutrosophicsoftpointsin NS(UE ).Thentoevery > 0thereexistsan n0 ∈ N (setofnaturalnumbers)suchthat d(emM, enM ) < , ∀n > m ≥ n0.

Since (NS(UE ), d) becompact, ∃ asubsequence {enk M } suchthat limn→∞ enk M = eP,say.Then d(enk M, eP ) < , ∀nk ≥ n0.Also d(emM, enk M ) < , ∀nk > m ≥ n0. Nowfor n > m, d(enM, eP ) ≤ d(enM, emM )+ d(emM, enk M )+ d(enk M, eP ) < 3 .Thus {enM } beingaCauchy sequenceconvergestoapointin NS(UE ) andso (NS(UE ), d) beacompleteNSMS.

3.3Theorem

EverycompactsetinanNSMSisclosedandbounded.

Proof. Let M beacompactNSSinanNSMS (NS(UE ), d).Suppose M isnotclosed.Thenthereexistsasequence {enM } ofneutrosophicsoftpointsin M convergingtoapoint eM (say)notbelongto M.Theneverysubsequence of {enM } alsoconvergesto eM notbelongto M.Thusthereisnosubsequenceof {enM } convergingtoapointof M whichcontradictsthecompactnessof M.Hence M isclosed. Nextsuppose M isnotboundedand eM befixedneutrosophicsoftpoint.Then ∃ apoint e1M ∈ M suchthat d(eM, e1M ) > 3.Bysimilarargument ∃ apoint e2M ∈ M suchthat d(eM, e2M ) > d(eM, e1M )+ 3.Continuing thisprocess,wegetasequenceofneutrosophicsoftpoints e1M, e2M, , enM, ···∈ M suchthat d(eM, enM ) > d(eM, e1M )+ d(eM, e2M )+ + d(eM, e(n 1)M )+ 3.So,for n > m, d(eM, enM ) > d(eM, emM )+ 3.Now, d(enM, eM ) ≤ d(enM, emM )+ d(emM, eM ) andso d(enM emM ) > 3whenever n > m.Thisshowsthatneither thesequence {enM } noranyofit’ssubsequencecanconverge,contradictingthefactthat M iscompact.Hence M isbounded.

3.3.1 Remark

Converseofabovemaynotbetrue.Thefactisshownbytheexample(4)of[3.1.1]; Here, d(eiM, ejM ) < 3forall i = j and D(M)= φ ⊂ M.So, M isboundedandclosed.But M isnotcompact.

4ContinuityonNSMS

Here,theconceptofneutrosophicsoftfunction,it’scontinuityonanNSMS,thecompositionofneutrosophicsoft functionshavebeenintroducedandillustratedbysuitableexamples.Severalproperties,structuralcharacteristics andtheoremsrelatedtothesealsohavebeenpresentedhere.

4.1Definition

Let (NS(UE ), d) and (NS(VE ), d ) betwoNSMSsand ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) beaneutrosophic softfunctionwhere ϕ : U → V and ψ : E → E betwocrispfunctions.Considertwoneutrosophicsoftpoints eM, eN as: eM = {< x, (TeM (x), IeM (x), FeM (x)) >: x ∈ U}∈ NS(UE ), e ∈ E and eN = {< y, (TeN (y), IeN (y), FeN (y)) >: y ∈ ϕ(U)}∈ NS(VE ), e ∈ ψ(E)

(1) Thentheimageof eM under ( ϕ, ψ) isdenotedby ( ϕ, ψ)(eM ).Itisalsoaneutrosophicsoftpoint eN (say) ∈ NS(VE ) definedasfollows:

TeN (y)=    maxϕ(x)=y maxψ(e)=e [TeM (x)],if x ∈ ϕ 1(y) 0,otherwise.

IeN (y)=    minϕ(x)=y minψ(e)=e [IeM (x)],if x ∈ ϕ 1(y) 1,otherwise.

FeN (y)=    minϕ(x)=y minψ(e)=e [FeM (x)],if x ∈ ϕ 1(y) 1,otherwise.

(2) Thepre-imageof eN under ( ϕ, ψ),denotedby ( ϕ, ψ) 1(eN ),isaneutrosophicsoftpoint eM (say) ∈ NS(UE ) andisdefinedasfollows, ∀x ∈ U, ∀e ∈ ψ 1(E ) :

TeM (x)= T[ψ(e)]N ( ϕ(x))= TeN ( ϕ(x))

IeM (x)= I[ψ(e)]N ( ϕ(x))= IeN ( ϕ(x))

FeM (x)= F[ψ(e)]N ( ϕ(x))= FeN ( ϕ(x))

If ψ and ϕ areinjective(surjective),then ( ϕ, ψ) isinjective(surjective).

4.1.1 Example

Let E = N (thesetofnaturalnumbers)betheparametricsetand U = Z (thesetofintegers)betheuniversalset. Consideraneutrosophicsoftpoint nM ∈ NS(ZN) asfollows,forany n ∈ N and x ∈ Z :

TnM (x)=    0if x = 2k 1, k ∈ Z 1 n if x = 2k, k ∈ Z

InM (x)=    1 2n if x = 2k 1, k ∈ Z 0if x = 2k, k ∈ Z

FnM (x)=    1 1 n if x = 2k 1, k ∈ Z 0if x = 2k, k ∈ Z

Then (NS(ZN), d) formsanNSMSwhere‘d’isdefinedin(2)(i)of[2.3].Now,let ϕ : Z → Z and ψ : N → N be twocrispfunctionsdefinedas ϕ(x)= 2x + 3 = y (say)and ψ(n)= 2n 1 = m (say),respectively.Thenthe neutrosophicsoftfunction ( ϕ, ψ) : (NS(ZN), d) → (NS(ZN), d) isgivenby ( ϕ, ψ)(nM )= mP, m ∈ N anditis definedas:



TmP (y)=

4.2Proposition



 0if y = 4k + 1, k ∈ Z 2 1+m if y = 4k + 3, k ∈ Z 0if y = otherwise.





    

FmP (y)=  

 m 1 m+1 if y = 4k + 1, k ∈ Z 0if y = 4k + 3, k ∈ Z 1if y = otherwise.

(NS(VE ), d ) beaneutrosophicsoftfunction.Thentheimageset

Proof. Letusconsiderthreeneutrosophicsoftpoints eM, eN , eP ∈ NS(UE ).Now,

(1) eM = eN ⇒ d(eM, eN ) > 0

i e., (TeM (x), IeM (x), FeM (x)) =(TeN (x), IeN (x), FeN (x)) ⇒

d[(TeM (x), IeM (x), FeM (x)), (TeN (x), IeN (x), FeN (x))] > 0, ∀x ∈ U

i e., (max ϕ(x) max ψ(e) [TeM (x)],min ϕ(x) min ψ(e) [IeM (x)],min ϕ(x) min ψ(e) [FeM (x)]) =

(max ϕ(x) max ψ(e) [TeN (x)],min ϕ(x) min ψ(e) [IeN (x)],min ϕ(x) min ψ(e) [FeN (x)]) ⇒

d [(max ϕ(x) max ψ(e) [TeM (x)],min ϕ(x) min ψ(e) [IeM (x)],min ϕ(x) min ψ(e) [FeM (x)]),

(max ϕ(x) max ψ(e) [TeN (x)],min ϕ(x) min ψ(e) [IeN (x)],min ϕ(x) min ψ(e) [FeN (x)])] > 0

i e., (φ, ψ)(eM ) =(φ, ψ)(eN ) ⇒ d [(φ, ψ)(eM ), (φ, ψ)(eN )] > 0

(2) eM = eN ⇔ d(eM, eN )= 0

i e., TeM (x)= TeN (x), IeM (x)= IeN (x), FeM (x)= FeN (x), ∀x ∈ U ⇔

d[(TeM (x), IeM (x), FeM (x)), (TeN (x), IeN (x), FeN (x))]= 0, ∀x ∈ U

i e.,max ϕ(x) max ψ(e) [TeM (x)]= max ϕ(x) max ψ(e) [TeN (x)],min ϕ(x) min ψ(e) [IeM (x)]= min ϕ(x) min ψ(e) [IeN (x)],min ϕ(x) min ψ(e) [FeM (x)]= min ϕ(x) min ψ(e) [FeN (x)] ⇔ d [(max ϕ(x) max ψ(e) [TeM (x)],min ϕ(x) min ψ(e) [IeM (x)],min ϕ(x) min ψ(e) [FeM (x)]), (max ϕ(x) max ψ(e) [TeN (x)],min ϕ(x) min ψ(e) [IeN (x)],min ϕ(x) min ψ(e) [FeN (x)])]= 0 i e., (φ, ψ)(eM )=(φ, ψ)(eN ) ⇔ d [(φ, ψ)(eM ), (φ, ψ)(eN )]= 0

(3) d(eM, eN )= d(eN , eM )

⇒ d[(TeM (x), IeM (x), FeM (x)), (TeN (x), IeN (x), FeN (x))] = d[(TeN (x), IeN (x), FeN (x)), (TeM (x), IeM (x), FeM (x))]

⇒ d [(max ϕ(x) max ψ(e) [TeM (x)],min ϕ(x) min ψ(e) [IeM (x)],min ϕ(x) min ψ(e) [FeM (x)]),

(max ϕ(x) max ψ(e) [TeN (x)],min ϕ(x) min ψ(e) [IeN (x)],min ϕ(x) min ψ(e) [FeN (x)])]

= d [(max ϕ(x) max ψ(e) [TeN (x)],min ϕ(x) min ψ(e) [IeN (x)],min ϕ(x) min ψ(e) [FeN (x)]),

(max ϕ(x) max ψ(e) [TeM (x)],min ϕ(x) min ψ(e) [IeM (x)],min ϕ(x) min ψ(e) [FeM (x)])]

⇒ d [(φ, ψ)(eM ), (φ, ψ)(eN )]= d [(φ, ψ)(eN ), (φ, ψ)(eM )]

(4) d(eM, eN ) ≤ d(eM, eP )+ d(eP, eN )

⇒ d[(TeM (x), IeM (x), FeM (x)), (TeN (x), IeN (x), FeN (x))]

≤ d[(TeM (x), IeM (x), FeM (x)), (TeP (x), IeP (x), FeP (x))]+ d[(TeP (x), IeP (x), FeP (x)), (TeN (x), IeN (x), FeN (x))], ∀x ∈ U

⇒ d [(max ϕ(x) max ψ(e) [TeM (x)],min ϕ(x) min ψ(e) [IeM (x)],min ϕ(x) min ψ(e) [FeM (x)]),

(max ϕ(x) max ψ(e) [TeN (x)],min ϕ(x) min ψ(e) [IeN (x)],min ϕ(x) min ψ(e) [FeN (x)])]

≤ d [(max ϕ(x) max ψ(e) [TeM (x)],min ϕ(x) min ψ(e) [IeM (x)],min ϕ(x) min ψ(e) [FeM (x)]),

(max ϕ(x) max ψ(e) [TeP (x)],min ϕ(x) min ψ(e) [IeP (x)],min ϕ(x) min ψ(e) [FeP (x)])]

+d [(max ϕ(x) max ψ(e) [TeP (x)],min ϕ(x) min ψ(e) [IeP (x)],min ϕ(x) min ψ(e) [FeP (x)])

(max ϕ(x) max ψ(e) [TeN (x)],min ϕ(x) min ψ(e) [IeN (x)],min ϕ(x) min ψ(e) [FeN (x)])] ⇒ d [(φ, ψ)(eM ), (φ, ψ)(eN )] ≤ d [(φ, ψ)(eM ), (φ, ψ)(eP )]+ d [(φ, ψ)(eP ), (φ, ψ)(eN )]

Thiscompletestheproof.

4.3Proposition

Let ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) beanontoneutrosophicsoftfunction.Thenthepre-imageset {( ϕ, ψ) 1(eQ ) : eQ ∈ NS(VE )} formsalsoanNSMSwithrespectto‘d’. [ Notethat ( ϕ, ψ) 1 istheinverseimageof NS(VE ) under themapping ( ϕ, ψ).Here ( ϕ, ψ) 1 maynotbeamapping.]

Proof. Let eM, eN , eP ∈ NS(UE ) and eQ, eR, eS ∈ NS(VE ) suchthat ( ϕ, ψ) 1(eQ )= eM, ( ϕ, ψ) 1(eR )= eN , ( ϕ, ψ) 1(eS )= eP and ϕ(x)= y for x ∈ U, y ∈ V.Now, (1) eQ = eR ⇒ d (eQ, eR ) > 0 i.e., (TeQ (y), IeQ (y), FeQ (y)) =(TeR (y), IeR (y), FeR (y)) ⇒ d [(TeQ (y), IeQ (y), FeQ (y)), (TeR (y), IeR (y), FeR (y))] > 0, ∀y ∈ V

i e., (TeM (x), IeM (x), FeM (x)) =(TeN (x), IeN (x), FeN (x)) ⇒ d[(TeM (x), IeM (x), FeM (x)), (TeN (x), IeN (x), FeN (x))] > 0

i e., eM = eN ⇒ d(eM, eN ) > 0 i e., ( ϕ, ψ) 1(eQ ) =( ϕ, ψ) 1(eR ) ⇒ d[( ϕ, ψ) 1(eQ ), ( ϕ, ψ) 1(eR )] > 0

(2) eQ = eR ⇔ d(eQ, eR )= 0

i e., TeQ (y)= TeR (y), IeQ (y)= IeR (y), FeQ (y)= FeR (y), ∀y ∈ V ⇔ d[(TeQ (y), IeQ (y), FeQ (y)), (TeR (y), IeR (y), FeR (y))]= 0, ∀y ∈ V

i e., TeM (x)= TeN (x), IeM (x)= IeN (x), FeM (x)= FeN (x) ⇔ d[(TeM (x), IeM (x), FeM (x)), (TeN (x), IeN (x), FeN (x))]= 0

i e., eM = eN ⇔ d(eM, eN )= 0

i e., ( ϕ, ψ) 1(eQ )=( ϕ, ψ) 1(eR ) ⇔ d[( ϕ, ψ) 1(eQ ), ( ϕ, ψ) 1(eR )]= 0

(3) d (eQ, eR )= d (eR, eQ )

⇒ d [(TeQ (y), IeQ (y), FeQ (y)), (TeR (y), IeR (y), FeR (y))]= d [(TeR (y), IeR (y), FeR (y)), (TeQ (y), IeQ (y), FeQ (y))]

⇒ d[(TeM (x), IeM (x), FeM (x)), (TeN (x), IeN (x), FeN (x))]= d[(TeN (x), IeN (x), FeN (x)), (TeM (x), IeM (x), FeM (x))]

⇒ d(eM, eN )= d(eN , eM ) i e., d[( ϕ, ψ) 1(eQ ), ( ϕ, ψ) 1(eR )]= d[( ϕ, ψ) 1(eR ), ( ϕ, ψ) 1(eQ )]

(4) d (eQ, eS ) ≤ d (eQ, eR )+ d (eR, eS )

⇒ d [(TeQ (y), IeQ (y), FeQ (y)), (TeS (y), IeS (y), FeS (y))]

≤ d [(TeQ (y), IeQ (y), FeQ (y)), (TeR (y), IeR (y), FeR (y))]+ d [(TeR (y), IeR (y), FeR (y)), (TeS (y), IeS (y), FeS (y))], ∀y ∈ V

⇒ d[(TeM (x), IeM (x), FeM (x)), (TeP (x), IeP (x), FeP (x))] ≤ d[(TeM (x), IeM (x), FeM (x)), (TeN (x), IeN (x), FeN (x))]+ d[(TeN (x), IeN (x), FeN (x)), (TeP (x), IeP (x), FeP (x))]

⇒ d(eM, eP ) ≤ d(eM, eN )+ d(eN , eP ) i e., d[( ϕ, ψ) 1(eQ ), ( ϕ, ψ) 1(eS )] ≤ d[( ϕ, ψ) 1(eQ ), ( ϕ, ψ) 1(eR )] +d[( ϕ, ψ) 1(eR ), ( ϕ, ψ) 1(eS )]

Thiscompletestheproof.

4.4Proposition

Let P, Q ⊂ NS(UE ) and M, N ⊂ NS(VE ).Thenforaneutrosophicsoftfunction ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ),thefollowingshold.

(1) ( ϕ, ψ) 1(M) ⊆ ( ϕ, ψ) 1(N) ⇔ M ⊆ N (2) P ⊆ Q ⇔ ( ϕ, ψ)(P) ⊆ ( ϕ, ψ)(Q)

(3) M ⊆ ( ϕ, ψ)(P) ⇔ ( ϕ, ψ) 1(M) ⊆ P. (4) ( ϕ, ψ)(Q) ⊆ N ⇔ Q ⊆ ( ϕ, ψ) 1(N).

Proof. Let ϕ(x)= y and ψ(e)= e for x ∈ U, y ∈ V and e ∈ E, e ∈ E .Then, (1) M ⊆ N

⇔ TeM (y) ≤ TeN (y), IeM (y) ≥ IeN (y), FeM (y) ≥ FeN (y), ∀e , ∀y; ⇔ T[ψ(e)]M ( ϕ(x)) ≤ T[ψ(e)]N ( ϕ(x)), I[ψ(e)]M ( ϕ(x)) ≥ I[ψ(e)]N ( ϕ(x)),

F[ψ(e)]M ( ϕ(x)) ≥ F[ψ(e)]N ( ϕ(x)), ∀e, ∀x; ⇔ Te( ϕ,ψ) 1 (M) (x) ≤ Te( ϕ,ψ) 1 (N) (x), Ie( ϕ,ψ) 1 (M) (x) ≥ Ie( ϕ,ψ) 1 (N) (x),

Fe( ϕ,ψ) 1 (M) (x) ≥ Fe( ϕ,ψ) 1 (N) (x), ∀e, ∀x; ⇔ ( ϕ, ψ) 1(M) ⊆ ( ϕ, ψ) 1(N)

(2)( ϕ, ψ)(P) ⊆ ( ϕ, ψ)(Q)

⇔ max ϕ(x) max ψ(e) [TeP (x)] ≤ max ϕ(x) max ψ(e) [TeQ (x)],min ϕ(x) min ψ(e) [IeP (x)] ≥ min ϕ(x) min ψ(e) [IeQ (x)],min ϕ(x) min ψ(e) [FeP (x)] ≥ min ϕ(x) min ψ(e) [FeQ (x)]

⇔ TeP (x) ≤ TeQ (x), IeP (x) ≥ IeQ (x), FeP (x) ≥ FeQ (x), ∀e, ∀x

⇔ P ⊆ Q

(3) M ⊆ ( ϕ, ψ)(P)

⇔ TeM (y) ≤ max ϕ(x) max ψ(e) [TeP (x)], IeM (y) ≥ min ϕ(x) min ψ(e) [IeP (x)], FeM (y) ≥ min ϕ(x) min ψ(e) [FeP (x)]

⇔ T[ψ(e)]M ( ϕ(x)) ≤ max ϕ(x) max ψ(e) [TeP (x)], I[ψ(e)]M ( ϕ(x)) ≥ min ϕ(x) min ψ(e) [IeP (x)], F[ψ(e)]M ( ϕ(x)) ≥ min ϕ(x) min ψ(e) [FeP (x)]

⇔ Te( ϕ ψ) 1 (M) (x) ≤ TeP (x), Ie( ϕ ψ) 1 (M) (x) ≥ IeP (x), Fe( ϕ ψ) 1 (M) (x) ≥ FeP (x), ∀e, ∀x

⇔ ( ϕ, ψ) 1(M) ⊆ P

(4)( ϕ, ψ)(Q) ⊆ N

⇔ max ϕ(x) max ψ(e) [TeQ (x)] ≤ TeN (y),min ϕ(x) min ψ(e) [IeP (x)] ≥ IeN (y), min ϕ(x) min ψ(e) [FeQ (x)] ≥ FeN (y) ⇔ max ϕ(x) max ψ(e) [TeQ (x)] ≤ T[ψ(e)]N ( ϕ(x)),min ϕ(x) min ψ(e) [IeQ (x)] ≥ I[ψ(e)]N ( ϕ(x)),min ϕ(x) min ψ(e) [FeQ (x)] ≥ F[ψ(e)]N ( ϕ(x)) ⇔ TeQ (x) ≤ Te( ϕ ψ) 1 (N) (x), IeQ (x) ≥ Ie( ϕ ψ) 1 (N) (x), FeQ (x) ≥ Fe( ϕ ψ) 1 (N) (x), ∀e, ∀x ⇔ Q ⊆ ( ϕ, ψ) 1(N)

4.5Definition

Let (NS(UE ), d) and (NS(VE ), d ) betwoNSMSs.Thenaneutrosophicsoftfunction ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) issaidtobecontinuousat e0N ∈ NS(UE ) ifforeach > 0thereexistsa δ > 0suchthat d [( ϕ, ψ)(eM ), ( ϕ, ψ)(e0N )] < whenever d(eM, e0N ) < δ, eM ∈ NS(UE ) i.e.,if ( ϕ, ψ)[Bu (e0N , δ)] ⊂ Bv (( ϕ, ψ)(e0N ), ) holds. ( ϕ, ψ) iscalledneutrosophicsoftcontinuousfunctionifitis continuousateverypointin NS(UE )

4.6Theorem

Let ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) beaneutrosophicsoftfunction.

(1)If e0N isalimitpointof NS(UE ),then ( ϕ, ψ) isneutrosophicsoftcontinuousat e0N ifflimeM →e0N ( ϕ, ψ)(eM )= ( ϕ, ψ)(e0N ). (2) ( ϕ, ψ) iscontinuousat e0N ∈ NS(UE ) iffforeverysequence {enN } ofneutrosophicsoftpointsin NS(UE ) convergingto e0N ,wehavelimn→∞ ( ϕ, ψ)(enN )=( ϕ, ψ)(e0N ) Proof. (1)Itisstraightforward.

(2)Firstsupposethat ( ϕ, ψ) iscontinuousat e0N ∈ NS(UE ) andlimn→∞ enN = e0N .Thengiven > 0,there existsa δ > 0suchthat d [( ϕ, ψ)(eM ), ( ϕ, ψ)(e0N )] < whenever d(eM, e0N ) < δ, eM ∈ NS(UE ) Sincelimn→∞ enN = e0N ,thereexistsanaturalnumber n0 suchthat d(enN , e0N ) < δ, ∀n ≥ n0.Putting eM = enN ,wehave d [( ϕ, ψ)(enN ), ( ϕ, ψ)(e0N )] < whenever d(enN , e0N ) < δ, ∀n ≥ n0 Thus d [( ϕ, ψ)(enN ), ( ϕ, ψ)(e0N )] < , ∀n ≥ n0 andthiscompletesthe‘if’part. Conversely,lettheconditionbeholdbut ( ϕ, ψ) isnotcontinuousat e0N ∈ NS(UE ).Thengiven > 0,there existsa δ > 0suchthat d [( ϕ, ψ)(eM ), ( ϕ, ψ)(e0N )] ≥ whenever d(eM, e0N ) < δ, eM ∈ NS(UE ); (1)

Butbyhypothesis,thereexistsanaturalnumber n0 suchthat d [( ϕ, ψ)(enN ), ( ϕ, ψ)(e0N )] < whenever d(enN , e0N ) < δ, ∀n ≥ n0 Putting eM = enN in(1),wehave d [( ϕ, ψ)(enN ), ( ϕ, ψ)(e0N )] ≥ whenever d(enN , e0N ) < δ Thiscontradictsthehypothesisandso ( ϕ, ψ) iscontinuousat e0N ∈ NS(UE )

4.6.1 Example

1. Let E = N (thesetofnaturalnumbers), E = I (unitinterval [0,1])and U = V = Q (thesetofnonzerorational numbers).Consideraneutrosophicsoftsequence {nM } in NS(QN) asfollowing,forany n ∈ N : TnM (x)= n n+1 , InM (x)= 1 2n , FnM (x)= 1 3n , ∀x ∈ Q Then (NS(QN), d) formsanNSMSwhere‘d’isdefinedin(2)(i)of[2.3].Now,let ϕ : Q → Q and ψ : N → I betwocrispfunctionsdefinedas ϕ(x)= 1 x = y (say)and ψ(n)= 1 1 n = m (say),respectively.Thenthe

neutrosophicsoftfunction ( ϕ, ψ) : NS(QN) → NS(QI ) isgivenby ( ϕ, ψ)(nM )= mP, m ∈ I andisdefinedas: TmP (y)= 1 2 m , ImP (y)= 1 m 2 , FmP (y)= 1 m 3 , ∀y = 1 x ∈ Q . Wenowdefineaneutrosophicsoftpoint aS ∈ NS(QN), a ∈ N givenas: TaS (x)= 1, IaS (x)= 0, FaS (x)= 0, ∀x ∈ Q Weshallestimatethedistancefunction‘d’herefor k = 1only.Similarconclusioncanbedrawnfordifferent valuesof k d(nM, aS )= |TnM (x) TaS (x)| + |InM (x) IaS (x)| + |FnM (x) FaS (x)| = | n n + 1 1| + | 1 2n 0| + | 1 3n 0| = 1 n + 1 + 1 2n + 1 3n = 11n + 5 6n2 + 6n = 11 + 5 n n(6 + 6 n )

Hence, d(nM, aS ) −→ 0as n −→ ∞ i.e., {nM } convergesto aS

Totestthecontinuityof ( ϕ, ψ) at aS,weshallusethetheorem(2)of[4.6].

Clearly, ( ϕ, ψ)(aS )= {< y, (1,0,0) >: y ∈ ϕ(Q )};Forsame‘d’statedabove, d[( ϕ, ψ)(nM ), ( ϕ, ψ)(aS )]= | 1 2 m 1| + | 1 m 2 0| + | 1 m 3 0| = 1 m 2 m + 1 m 2 + 1 m 3 = (1 m)(16 5m) 6(2 m)

Thisshows d[( ϕ, ψ)(nM ), ( ϕ, ψ)(aS )] −→ 0as n −→ ∞ (i.e.,as m → 1).Hence {( ϕ, ψ)(nM )} convergesto ( ϕ, ψ)(aS ) andso ( ϕ, ψ) iscontinuousat (aS ).

2. Consideraneutrosophicsoftsequence {nM } in NS(ZN) (Z beingthesetofintegersand N beingthesetof naturalnumbers)asfollowing,forany n ∈ N : TnM (x)= 1 1 n , InM (x)= 1 7n , FnM (x)= 1 n+1 , ∀x ∈ Z Then (NS(ZN), d) formsanNSMSwhere‘d’isdefinedin(2)(i)of[2.3].Now,letaneutrosophicsoftfunction ( ϕ, ψ) : NS(ZN) → NS(ZN∪{0}) begivenby ( ϕ, ψ)(nM )= mP, m ∈ N where ϕ : Z → Z and ψ : N → N ∪{0} betwocrispfunctionsdefinedas ϕ(x)= 3x = y (say)and ψ(n)= n 1 = m (say).Then ( ϕ, ψ)(nM )= mP is definedas:

TmP (y)=    1 1 m+1 if y = ϕ(x) 0otherwise.

ImP (y)=    1 7(m+1) if y = ϕ(x) 1otherwise.

FmP (y)=    1 m+2 if y = ϕ(x) 1otherwise.

Wenowdefineaneutrosophicsoftpoint aS ∈ NS(ZN), a ∈ N givenas:

TaS (x)= 1, IaS (x)= 0, FaS (x)= 0, ∀x ∈ Z.Thenfor k = 1, d(nM, aS )= |TnM (x) TaS (x)| + |InM (x) IaS (x)| + |FnM (x) FaS (x)|

= |1 1 n 1| + | 1 7n 0| + | 1 n + 1 0|

= 1 n + 1 7n + 1 n + 1

= 15n + 8 7n2 + 7n = 15 + 8 n n(7 + 7 n )

Similarconclusioncanbedrawnfordifferentchoiceof k.Hence, d(nM, aS ) −→ 0as n −→ ∞ i.e., {nM } convergesto aS.But {( ϕ, ψ)(nM )} doesnotconvergeto ( ϕ, ψ)(aS ) clearlyandhence, ( ϕ, ψ) isnotcontinuousat aS

4.7Theorem

Let ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) beaneutrosophicsoftfunction.Then ( ϕ, ψ) iscontinuouson NS(UE ) iff ( ϕ, ψ) 1(P) isopenin NS(UE ) whenever P ⊂ NS(VE ) isopen.

Proof. Firstsuppose ( ϕ, ψ) becontinuouson NS(UE ) and P ⊂ NS(VE ) beanopenNSS.Let e0M ∈ ( ϕ, ψ) 1(P) Then ( ϕ, ψ)(e0M ) ∈ P.Since P isopenNSS,thereexistsanopenball Bv (( ϕ, ψ)(e0M ), ) ⊂ P.Againas ( ϕ, ψ) is continuousat e0M,thereexists δ > 0suchthat d [( ϕ, ψ)(eN ), ( ϕ, ψ)(e0M )] < whenever d(eN , e0M ) < δ for eN ∈ NS(UE ).Itimplies ( ϕ, ψ)(eN ) ∈ Bv (( ϕ, ψ)(e0M ), ), ∀eN ∈ Bu (e0M, δ).But ( ϕ, ψ)(eN ) ∈ Bv (( ϕ, ψ)(e0M ), ) ⊂ P ⇒ eN ∈ ( ϕ, ψ) 1(P).Thus Bu (e0M, δ) ⊂ ( ϕ, ψ) 1(P) whenever e0M ∈ ( ϕ, ψ) 1(P).Hence e0M isaninterior pointof ( ϕ, ψ) 1(P).Since e0M isarbitrary, ( ϕ, ψ) 1(P) isopenin NS(UE ). Conversely,assumethat ( ϕ, ψ) 1(P) isopenin NS(UE ) foreveryopenNSS P ⊂ NS(VE ) and e0M ∈ NS(UE ) bearbitrarybutfixed.Then ( ϕ, ψ)(e0M ) ∈ NS(VE ) and Bv (( ϕ, ψ)(e0M ), ) beinganopenballisanopensetin NS(VE ).Sobyhypothesis, ( ϕ, ψ) 1[Bv (( ϕ, ψ)(e0M ), )] isopenin NS(UE ).Now ( ϕ, ψ)(e0M ) ∈ Bv (( ϕ, ψ)(e0M ), ), clearlyandso e0M ∈ ( ϕ, ψ) 1[Bv (( ϕ, ψ)(e0M ), )].Since ( ϕ, ψ) 1[Bv (( ϕ, ψ)(e0M ), )] isopenin NS(UE ),so Bu (e0M, δ) ⊂ ( ϕ, ψ) 1[Bv (( ϕ, ψ)(e0M ), )].Let eN ∈ Bu (e0M, δ) ⊂ ( ϕ, ψ) 1[Bv (( ϕ, ψ)(e0M ), )].Then eN ∈ Bu (e0M, δ) and ( ϕ, ψ)(eN ) ∈ Bv (( ϕ, ψ)(e0M ), ).Thisshowsthat d [( ϕ, ψ)(eN ), ( ϕ, ψ)(e0M )] < whenever d(eN , e0M ) < δ i.e., ( ϕ, ψ) iscontinuousat e0M.Since e0M ∈ NS(UE ) isarbitrary,so ( ϕ, ψ) iscontinuouson NS(UE )

4.8Theorem

Let ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) beaninjectiveandcontinuousneutrosophicsoftfunction.Then ( ϕ, ψ) 1(Q) isclosedin NS(UE ) whenever Q ⊂ NS(VE ) isclosed. Proof. Let e0M ∈ NS(UE ) bealimitpointof ( ϕ, ψ) 1(Q) ⊂ NS(UE ) and eN ∈ Bu (e0M, δ) ∩ ( ϕ, ψ) 1(Q), eN = e0M Thenbysenseof[2.2](9), eN ∈ Bu (e0M, δ) and eN ∈ ( ϕ, ψ) 1(Q) ⇒ eN ∈ Bu (e0M, δ) and ( ϕ, ψ)(eN ) ∈ Q.Again as ( ϕ, ψ) iscontinuousat e0M,thereexists > 0suchthat ( ϕ, ψ)(eN ) ∈ Bv (( ϕ, ψ)(e0M ), ) whenever eN ∈ Bu (e0M, δ) for eN ∈ NS(UE ).Thus ( ϕ, ψ)(eN ) ∈ Bv (( ϕ, ψ)(e0M ), ) ∩ Q with ( ϕ, ψ)(eN ) =( ϕ, ψ)(e0M ),as ( ϕ, ψ)

isinjective.Thisshowsthat ( ϕ, ψ)(e0M ) isalimitpointof Q.Since Q isclosedin NS(VE ),so ( ϕ, ψ)(e0M ) ∈ Q i.e., e0M ∈ ( ϕ, ψ) 1(Q).Hence e0M beingalimitpointof ( ϕ, ψ) 1(Q) belongsto ( ϕ, ψ) 1(Q).Since e0M isarbitrary, so ( ϕ, ψ) 1(Q) isclosedin NS(UE ).

4.9Definition

Let (NS(UE ), d) beanNSMSover (U, E) and M ⊂ NS(UE ) beanarbitraryNSS.Thentheclosureof M isdenoted by M andisdefinedasfollows: M = ∩{N ⊂ NS(UE ) : N isneutrosophicsoftclosedand N ⊃ M} i.e.,itistheintersectionofallclosedneutrosophicsoftsupersetsof M

4.9.1 Example

Let (NS(UE ), d) beanNSMSwithrespectto‘d’definedin(2)(i)of[2.3]where U = {x1, x2, x3} and E = {e1, e2} TheneveryNSSdefinedover (U, E) isfinite.AlsoeveryfiniteNSSonanNSMSisclosedby[2.4](3).Nowconsider fourNSSs M, N, P,1u ⊂ NS(UE ) suchthat M ⊂ N, P,1u onlyandtheyaregivenasfollowing:

f M (e1)= {< x1, (0.6,0.7,0.8) >, < x2, (0.5,0.3,0.7) >, < x3, (0.4,0.4,0.5) >}

f M (e2)= {< x1, (0.4,0.5,0.7) >, < x2, (0.3,0.4,0.8) >, < x3, (0.6,0.4,0.6) >}

f N (e1)= {< x1, (0.6,0.5,0.8) >, < x2, (0.6,0.3,0.5) >, < x3, (0.5,0.3,0.4) >}

f N (e2)= {< x1, (0.5,0.4,0.7) >, < x2, (0.4,0.2,0.6) >, < x3, (0.6,0.2,0.5) >} fP (e1)= {< x1, (0.7,0.4,0.6) >, < x2, (0.8,0.2,0.4) >, < x3, (0.6,0.2,0.3) >} fP (e2)= {< x1, (0.6,0.2,0.5) >, < x2, (0.5,0.1,0.5) >, < x3, (0.7,0.1,0.2) >} f1u (e1)= {< x1, (1,0,0) >, < x2, (1,0,0) >, < x3, (1,0,0) >} f1u (e2)= {< x1, (1,0,0) >, < x2, (1,0,0) >, < x3, (1,0,0) >} Then M = 1u ∩ N ∩ P = N.Thecorresponding t-norm (∗) and s-norm ( ) are: a ∗ b = min{a, b} and a b = max{a, b}.

(2)If M = M,then M isclosedby(1). Conversely,let M beclosed.By(1), M ⊂ M.Hence,weshallonlyshow M ⊂ M. M = ∩{P ⊂ NS(UE ) : P isneutrosophicsoftclosedand P ⊃ M} ⊂{M ⊂ NS(UE ) : M isneutrosophicsoftclosedand M ⊃ M} = M

(3) M ⊂ M and P ⊂ P ⇒ M ⊂ P ⊂ P ⇒ M ⊂ P But M isthesmallestclosedsetcontaining M i.e., M ⊂ M ⊂ P.Hence, M ⊂ P (4)If N isclosedthen N = N.Since M isclosed,replacing N by M,weget M = M (5) M ⊂ M ∪ P and P ⊂ M ∪ P ⇒ M ⊂ M ∪ P and P ⊂ M ∪ P ⇒ M ∪ P ⊂ M ∪ P Also, M ⊂ M and P ⊂ P ⇒ M ∪ P ⊂ M ∪ P.Butwehave, M ∪ P ⊂ M ∪ P ⊂ M ∪ P Thus, M ∪ P = M ∪ P (6) M ∩ P ⊂ M and M ∩ P ⊂ P ⇒ M ∩ P ⊂ M and M ∩ P ⊂ P ⇒ M ∩ P ⊂ M ∩ P

4.10Theorem

Let ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) beaninjectiveaswellascontinuousneutrosophicsoftfunction.Then, (1) ( ϕ, ψ)(N) ⊂ ( ϕ, ψ)(N) in NS(VE ) forevery N ⊂ NS(UE ) (2) ( ϕ, ψ) 1(M) ⊂ ( ϕ, ψ) 1(M) in NS(UE ) forevery M ⊂ NS(VE )

Proof. (1)Here ( ϕ, ψ)(N) ∈ NS(VE ) andso ( ϕ, ψ)(N) isclosedin NS(VE ).Since ( ϕ, ψ) iscontinuous,so ( ϕ, ψ) 1[( ϕ, ψ)(N)] isclosedin NS(UE ) by[4.8].Then ( ϕ, ψ) 1[( ϕ, ψ)(N)]=( ϕ, ψ) 1[( ϕ, ψ)(N)] by[4.9.2](2). Now ( ϕ, ψ)(N) istheclosureof ( ϕ, ψ)(N).So, ( ϕ, ψ)(N) ⊂ ( ϕ, ψ)(N) ⇒ N ⊂ ( ϕ, ψ) 1[( ϕ, ψ)(N)] ⇒ N ⊂ ( ϕ, ψ) 1[( ϕ, ψ)(N)]=( ϕ, ψ) 1[( ϕ, ψ)(N)].Thus ( ϕ, ψ)(N) ⊂ ( ϕ, ψ)(N) (2)Here M isclosedin NS(VE ) andsois ( ϕ, ψ) 1(M) in NS(UE ) by[4.8].But M ⊂ M ⇒ ( ϕ, ψ) 1(M) ⊂ ( ϕ, ψ) 1(M) ⇒ ( ϕ, ψ) 1(M) ⊂ ( ϕ, ψ) 1(M)=( ϕ, ψ) 1(M), as ( ϕ, ψ) 1(M) isclosed.Thus ( ϕ, ψ) 1(M) ⊂ ( ϕ, ψ) 1(M)

4.11Definition

where eM ∈ NS(UE ), eN ∈ NS(VE ), eR ∈ NS(WE ) andfor x ∈ U, z ∈ W

TeR (z)=    max( ϕ2 ◦ ϕ1 )(x)=z max(ψ2 ◦ψ1 )(e)=e [TeM (x)],if x ∈ ( ϕ2 ◦ ϕ1) 1(z) 0otherwise.

IeR (z)=    min( ϕ2 ◦ ϕ1 )(x)=z min(ψ2 ◦ψ1 )(e)=e [IeM (x)],if x ∈ ( ϕ2 ◦ ϕ1) 1(z) 1otherwise.

FeR (z)=    min( ϕ2 ◦ ϕ1 )(x)=z min(ψ2 ◦ψ1 )(e)=e [FeM (x)],if x ∈ ( ϕ2 ◦ ϕ1) 1(z) 1otherwise.

4.11.1 Example

Let ( ϕ1, ψ1) : (NS(UE ), d) → (NS(VE ), d), ( ϕ2, ψ2) : (NS(VE ), d) → (NS(WE ), d) betwoneutrosophicsoft functionswhere d isdefinedin(2)(i)of[2.3].Let U = {x1, x2} and E = {e1, e2}.Weconsiderthe NS(UE ) asgiven bytheTable2.

Table2:Tabularformof NS(UE )

e1A e2A e1B e2B e1C e2C x1 (0.5,0.6,0.3)(0.6,0.3,0.5)(0.7,0.4,0.3)(0.6,0.2,0.3)(0.8,0.6,0.2)(0.7,0.2,0.5) x2 (0.4,0.7,0.6)(0.7,0.4,0.3)(0.6,0.7,0.2)(0.4,0.3,0.5)(0.5,0.7,0.4)(0.1,0.5,0.8)

Nowlet ϕ1(x1)= y1, ϕ1(x2)= y1 and ψ1(e1)= e2, ψ1(e2)= e1.Suppose, ( ϕ1, ψ1)(e1A )= e2D, ( ϕ1, ψ1)(e2A )= e1G, ( ϕ1, ψ1)(e1B )= e2G ( ϕ1, ψ1)(e2B )= e1H , ( ϕ1, ψ1)(e1C )= e2H , ( ϕ1, ψ1)(e2C )= e1D

Thenthefollowingtable(Table3)represents ( ϕ1, ψ1)(NS(UE )) :

Table3:Tabularformof ( ϕ1, ψ1)(NS(UE )) e1D e2D e1G e2G e1H e2H y1 (0.7,0.2,0.5)(0.5,0.6,0.3)(0.7,0.3,0.3)(0.7,0.4,0.2)(0.6,0.2,0.3)(0.8,0.6,0.2) y2 (0,1,1)(0,1,1)(0,1,1)(0,1,1)(0,1,1)(0,1,1)

OnecalculationisprovidedheretomakeouttheTable3.

Te1G (y1)= max { ϕ1 (x1 ), ϕ1 (x2 )} max ψ1 (e2 ) [Te2A (x)],as x1, x2 ∈ ϕ 1 1 (y1) = max (0.6,0.7)= 0.7

Ie1G (y1)= min { ϕ1 (x1 ), ϕ1 (x2 )} min ψ1 (e2 ) [Ie2A (x)],as x1, x2 ∈ ϕ 1 1 (y1) = min (0.3,0.4)= 0.3

Fe1G (y1)= min { ϕ1 (x1 ), ϕ1 (x2 )} min ψ1 (e2 ) [Fe2A (x)],as x1, x2 ∈ ϕ 1 1 (y1) = min (0.5,0.3)= 0.3

Further Te1G (y2)= 0, Ie1G (y2)= 1, Fe1G (y2)= 1as x1, x2 / ∈ ϕ 1 1 (y2)

Nowassume ϕ2(y1)= z2, ϕ2(y2)= z1 and ψ2(e1)= e1, ψ2(e2)= e2.Suppose, ( ϕ2, ψ2)(e1D )= e1L, ( ϕ2, ψ2)(e1G )= e1Q, ( ϕ2, ψ2)(e1H )= e1M ( ϕ2, ψ2)(e2D )= e2M, ( ϕ2, ψ2)(e2G )= e2L, ( ϕ2, ψ2)(e2H )= e2Q

Then ( ϕ2, ψ2)[( ϕ1, ψ1)(NS(UE ))] isgivenbytheTable4.

Table4:Tabularformof ( ϕ2, ψ2)[( ϕ1, ψ1)(NS(UE ))] e1L e2L e1M e2M e1Q e2Q z1 (0,1,1)(0,1,1)(0,1,1)(0,1,1)(0,1,1)(0,1,1) z2 (0.7,0.2,0.5)(0.7,0.4,0.2)(0.6,0.2,0.3)(0.5,0.6,0.3)(0.7,0.3,0.3)(0.8,0.6,0.2)

ThustheTable4gives ( ϕ2, ψ2)[( ϕ1, ψ1)(NS(UE ))]=[( ϕ2, ψ2) ◦ ( ϕ1, ψ1)](NS(UE )).Forconvenience, [( ϕ2, ψ2) ◦ ( ϕ1, ψ1)](e1A )=( ϕ2, ψ2)[( ϕ1, ψ1)(e1A )]=( ϕ2, ψ2)(e2D )= e2M andsoon.

4.12Theorem

Let ( ϕ1, ψ1) : (NS(UE ), d1) → (NS(VE ), d2), ( ϕ2, ψ2) : (NS(VE ), d2) → (NS(WE ), d3) betwoneutrosophic softfunctionswhere (NS(UE ), d1), (NS(VE ), d2), (NS(WE ), d3) arethreeNSMSs.If ( ϕ1, ψ1) iscontinuousat e0N ∈ NS(UE ) and ( ϕ2, ψ2) iscontinuousatthecorrespondingpoint ( ϕ1, ψ1)(e0N ) ∈ NS(VE ),thenthecomposite function ( ϕ2, ψ2) ◦ ( ϕ1, ψ1) : (NS(UE ), d1) → (NS(WE ), d3) iscontinuousat e0N ∈ NS(UE )

Proof. Let {enN } beasequenceofneutrosophicsoftpointsin NS(UE ) suchthatlimn→∞ enN = e0N ∈ NS(UE ) Since ( ϕ1, ψ1) iscontinuousat e0N ,so ( ϕ1, ψ1)(enN ) → ( ϕ1, ψ1)(e0N ) ∈ NS(VE ) as n → ∞.Againsince ( ϕ2, ψ2) iscontinuousat ( ϕ1, ψ1)(e0N ),so ( ϕ2, ψ2)[( ϕ1, ψ1)(enN )] → ( ϕ2, ψ2)[( ϕ1, ψ1)(e0N )] ∈ NS(WE ) as n → ∞.This implies [( ϕ2, ψ2) ◦ ( ϕ1, ψ1)](enN ) → [( ϕ2, ψ2) ◦ ( ϕ1, ψ1)](e0N ) ∈ NS(WE ) as n → ∞ Hence ( ϕ2, ψ2) ◦ ( ϕ1, ψ1) iscontinuousat e0N ∈ NS(UE )

4.13Theorem

ContinuousimageofacompactNSMSiscompact. Proof. Let ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) beacontinuousneutrosophicsoftfunctionand NS(UE ) bea compactNSMS.Wearetoshowthat ( ϕ, ψ)(NS(UE ))= NS(VE ) ⊆ NS(VE ) (say)iscompact.Let {enN } beasoft sequencein NS(VE ).Thenforeach enN thereexists enM ∈ NS(UE ) suchthat ( ϕ, ψ)(enM )= enN , n = 1,2,3, ; Since NS(UE ) iscompact,thesoftsequence {enM } hasasubsequence {enk M } suchthatlimk→∞ enk M = e0P ∈ NS(UE ) (say).Again ( ϕ, ψ) iscontinuouson (NS(UE ), d),soitiscontinuousat e0P.Thenby(2)oftheorem [4.6],limk→∞ ( ϕ, ψ)(enk M )=( ϕ, ψ)(e0P );But, ( ϕ, ψ)(enk M )= enk N andsolimk→∞ enk N =( ϕ, ψ)(e0P );Thusasoft sequence {enN } in NS(VE ) hasasubsequence {enk N } convergingtoasoftpointin NS(VE ).Thisfollowsthe theorem.

5UniformcontinuityonNSMS

ThissectiongivestheconceptofuniformcontinuityofaneutrosophicsoftfunctiononanNSMSandit’scharacteristicsonNSMS.

5.1Definition

Let (NS(UE ), d) and (NS(VE ), d ) betwoNSMSs.Thenaneutrosophicsoftfunction ( ϕ, ψ) : NS(UE ) → NS(VE ) issaidtobeuniformlycontinuouson NS(UE ) ifforeach > 0thereexistsa δ > 0dependingonlyon ,noton thepointsuchthat d [( ϕ, ψ)(eM ), ( ϕ, ψ)(eN )] < whenever d(eM, eN ) < δ ∀eM, eN ∈ NS(UE )

5.1.1 Example

Consideraneutrosophicsoftfunction ( ϕ, ψ) : (NS(ZE ), d) → (NS(ZE ), d) where‘d’isdefinedin(2)(i)of[2.3] and Z bethesetofintegers.Thefunctionisdefinedas ( ϕ, ψ)(eM )= ρ(eM, P) foranyNSS P ⊂ NS(ZE ) and eM ∈ NS(ZE ),where ρ(eM, P)= {< x,min eP ∈P {|TeM (x) TeP (x)|},max eP ∈P {|IeM (x) IeP (x)|}, max eP ∈P {|FeM (x) FeP (x)|} >: x ∈ Z}

Nowforanytwopoints eM, eN ∈ NS(ZE ) andfor P ⊂ NS(ZE ) ,wehave d[( ϕ, ψ)(eM ), ( ϕ, ψ)(eN )]

= d[ρ(eM, P), ρ(eN , P)]

= min x | min eP ∈P {|TeM (x) TeP (x)|}− min eP ∈P {|TeN (x) TeP (x)|}|

+ | max eP ∈P {|IeM (x) IeP (x)|}− max eP ∈P {|IeN (x) IeP (x)|}|

+ | max eP ∈P {|FeM (x) FeP (x)|}− max eP ∈P {|FeN (x) FeP (x)|}|

< min x | min eP ∈P {TeM (x)+ TeP (x)}− min eP ∈P {TeN (x)+ TeP (x)}|

+ | max eP ∈P {IeM (x)+ IeP (x)}− max eP ∈P {IeN (x)+ IeP (x)}|

+ | max eP ∈P {FeM (x)+ FeP (x)}− max eP ∈P {FeN (x)+ FeP (x)}|

= min x | min eP ∈P {TeM (x)+ TeP (x) TeN (x) TeP (x)}| + | max eP ∈P {IeM (x)+ IeP (x) IeN (x) IeP (x)}|

+ | max eP ∈P {FeM (x)+ FeP (x) FeN (x) FeP (x)}| = min x |TeM (x) TeN (x)| + |IeM (x) IeN (x)| + |FeM (x) FeN (x)|

= d(eM, eN ) < δ =

Hence ( ϕ, ψ) isuniformlycontinuouson NS(ZE )

5.2Theorem

TheimageofaCauchysequenceinanNSMSunderauniformlycontinuousneutrosophicsoftfunctionisagaina Cauchysequence.

Proof. Let ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) beauniformlycontinuousneutrosophicsoftfunction.Thenfor each > 0thereexistsa δ > 0dependingonlyon ,notonthepointsuchthat d [( ϕ, ψ)(eM ), ( ϕ, ψ)(eN )] < whenever d(eM, eN ) < δ ∀eM, eN ∈ NS(UE ) Let {enP } beaCauchysequencein NS(UE ).Thentoevery δ > 0thereexistsan n0 ∈ N (setofnaturalnumbers) suchthat d(emP, enP ) < δ ∀m, n ≥ n0 Thisshows d [( ϕ, ψ)(emP ), ( ϕ, ψ)(enP )] < ∀m, n ≥ n0 andthatendsthetheorem.

5.3Theorem

EveryuniformlycontinuousneutrosophicsoftfunctiononanNSMSiscontinuous. Proof. Replacing eN by e0N ,anarbitrarybutfixedneutrosophicsoftpoint,itdirectlyfollowsfromdefinition[5.1];

5.4Theorem

UniformcontinuousimageofacompleteNSSinanNSMSiscomplete.

Proof. Let ( ϕ, ψ) : (NS(UE ), d) → (NS(VE ), d ) beauniformlycontinuousneutrosophicsoftfunctionand M ⊂ NS(UE ) beacompleteNSS.Wearetoshowthat ( ϕ, ψ)(M)= P (say)iscomplete.Let {enM } beaneutrosophic softCauchysequencein M suchthatlimn→∞ enM = e0M ∈ M.Then {( ϕ, ψ)(enM )} isaCauchysequencein P by theorem[5.2];Again ( ϕ, ψ) beinguniformlycontinuousneutrosophicsoftfunctioniscontinuousbytheorem[5.3] andso,limn→∞ ( ϕ, ψ)(enM )=( ϕ, ψ)(e0M ) ∈ P bytheorem(2)of[4.6];Thusacauchysequence {( ϕ, ψ)(enM )} in P convergestoapointin P andthiscompletestheproof.

5.5Theorem

UniformcontinuousimageofacompactNSMSiscompact.

Proof. Itisthecombinationoftheorem[5.3]andthetheorem[4.13];

6Conclusion

Inthispaper,thenotionofcompactNSMShasbeenintroducedandisillustratedbysuitableexamples.The continuityanduniformcontinuityofaneutrosophicsoftfunctioninanNSMShavebeendefinedandverifiedby properexamples.Severalrelatedproperties,theoremsandstructuralcharacteristicsoftheseinanNSMShave beeninvestigated.Somearejustifiedbysuitableexamplesalso.Themotivationofthepresentpaperistoput

forwardtheconceptintroducedin[32].Weexpect,theseconceptswillbringanopportunityoffurtherresearch worktodeveloptheNSStheory.

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