Compactness and Continuity on Neutrosophic Soft Metric Space

2018·0.22 MB·English

by  Tuhin Bera, Nirmal Kumar Mahapatra;

InternationalJournalofAdvancesinMathematics
Volume2018,Number4,Pages1-24,2018
eISSN2456-6098
(cid:13)c adv-math.com
CompactnessandContinuityOnNeutrosophicSoftMetricSpace
TuhinBera1andNirmalKumarMahapatra∗
1DepartmentofMathematics,BororS.S.HighSchool,Bagnan,
Howrah-711312,W.B,India.
∗DepartmentofMathematics,PanskuraBanamaliCollege,
PanskuraRS-721152,W.B,India.
Email:nirmal [email protected]
ABSTRACT. Inthispaper,thenotionofcompactneutrosophicsoftmetricspaceisintroduced. Theconceptofneutrosophic
softfunctionandthecompositionoffunctionsinaneutrosophicsoftmetricspacealongwithsuitableexamplesalsohavebeen
brought.Thecontinuityanduniformcontinuityofaneutrosophicsoftfunctioninthisspacehavebeendefinedandverifiedby
properexamples.Severalrelatedproperties,theoremsandstructuralcharacteristicsofthesehavebeeninvestigatedhere.
1 Introduction
ThetheoryofNeutrosophicset(NS)introducedbySmarandache[19,20]isthegeneralizationofmanytheories
e.g.,fuzzyset,intuitionisticfuzzysetetcpracticedtohandlethevariousuncertaintiesinmanyrealapplication
over the past many years. The neutrosophic logic includes the information about the percentage of truth, in-
determinacy and falsity grade in several real world problem like in law, medicine, engineering, management,
industrial,ITsectoretcwhichisnotavailableinfuzzysettheoryandintuitionisticfuzzysettheory.
Molodtsovhasshownthateachoftheabovetopicsdealingwithuncertaintiessufferfrominherentdifficulties
possibly due to inadequacy of their parametrization tool. So, Molodtsov [1] proposed the concept of ‘soft set
*CorrespondingAuthor.
ReceivedMarch05,2018;acceptedJune17,2018.
2010MathematicsSubjectClassification:47H10,54H25.
Keywordsandphrases: Compactneutrosophicsoftmetricspace;Neutrosophicsoftfunctions;Compositionofneutrosophic
softfunction;Continuity;Uniformcontinuity.
ThisisanopenaccessarticleundertheCCBYlicensehttp://creativecommons.org/licenses/by/3.0/.
1
TuhinBeraandNirmalKumarMahapatra 2
theory’ for modeling vagueness and uncertainties. It is completely free from the parametrization inadequacy
syndrome. Thismakesthetheoryveryconvenient,efficientandeasytoapplyinpractice. Inaccordanceofthis,
Majietal.[2-4]studiedtheseveralbasicoperationsinsoftsetstheoryoverfuzzysetsandintuitionisticfuzzysets.
Thenotionsoffuzzymetricspacewerestudiedin[5-13]fromdifferentpointofview.RoyandSamanta[14]have
definedopenandclosedsetsonfuzzytopologicalspaces. Park[15]andAlacaetal. [16]definedtheconceptof
intuitionisticfuzzymetricspaceintermofcontinuoust-normsandcontinuoust-conormsasageneralisationof
fuzzymetricspace.Usingalltheseconcepts,Beaulaetal.[17,18]proposedthenotionoffuzzysoftmetricspaces
intermsoffuzzysoftpoints.
After introduction of NS theory, Maji [21] has brought a combined notion Neutrosophic soft set (NSS). In
continuation,severalmathematicianshavepresentedtheirresearchworksindifferentmathematicalstructures.
DeliandBroumi[22],CetkinandAygun[24-26],BeraandMahapatra[27-34]studiedsomefundamentalalgebraic
structuresinNSStheorycontext.DeliandBroumi[23]havealsomodifiedsomeoperationsrelatedtoindetermin-
isticfunctionofNSSsgivenbyMaji. Broumietal. [35,36]havedonesomeconsecutiveworksingraphtheory
overNSS.
Themotivationofthepresentpaperistoextendtheconceptneutrosophicsoftmetricspace(NSMS)proposed
in [32]. The current article presents the notion of compact NSMS, the continuity and uniform continuity of a
neutrosophicsoftfunctioninanNSMSalongwithinvestigationofsomerelatedpropertiesandtheorems. The
contentofthepresentpaperisdesignedasfollows:
Section2givessomepreliminaryusefuldefinitions,examplesandtheoremswhichwillbeusedthroughout
thepaper.Insection3,compactnessofNSMSisdefinedandillustratedbyexamples.Somerelatedbasicproperties
havebeenstudiedhere,also.Section4dealswiththecontinuityofneutrosophicsoftfunctionandthecomposition
ofneutrosophicsoftfunctionsinanNSMSalongwiththestudyoftheirstructuralcharacteristics. Theconcept
ofuniformcontinuityofaneutrosophicsoftfunctioninanNSMShasbeenintroducedinsection5. Finally,the
conclusionofthepresentworkisstatedinsection6.
2 Preliminaries
Werecallsomebasicdefinitionsandtheoremsrelatedtofuzzyset,softset,NS,NSS,NSMSforthesakeofcom-
pleteness.
2.1 DefinitionsrelatedtoFuzzySetandSoftset
ThissectiongivessomeimportantdefinitionsrelatedtoFuzzyset,SoftSet[1,28]:
1.Abinaryoperation∗:[0,1]×[0,1]→[0,1]iscontinuoust-normif∗satisfiesthefollowingconditions:
(i) ∗iscommutativeandassociative.
(ii) ∗iscontinuous.
(iii)a∗1=1∗a=a, ∀a∈[0,1].
(iv) a∗b≤c∗d if a≤c, b≤d with a,b,c,d∈[0,1].
TuhinBeraandNirmalKumarMahapatra 3
Afewexamplesofcontinuoust-normarea∗b=ab,a∗b=min{a,b},a∗b=max{a+b−1,0}.
2. A binary operation (cid:5) : [0,1]×[0,1] → [0,1] is continuous t - conorm (s - norm) if (cid:5) satisfies the following
conditions:
(i) (cid:5)iscommutativeandassociative.
(ii) (cid:5)iscontinuous.
(iii)a(cid:5)0=0(cid:5)a=a, ∀a∈[0,1].
(iv) a(cid:5)b≤c(cid:5)d if a≤c, b≤d with a,b,c,d∈[0,1].
Afewexamplesofcontinuouss-normarea(cid:5)b=a+b−ab,a(cid:5)b=max{a,b},a(cid:5)b=min{a+b,1}.
3. LetU beaninitialuniversesetand Ebeasetofparameters. Let P(U)denotethepowersetofU. Thenfor
A⊆E,apair(F,A)iscalledasoftsetoverU,whereF: A→P(U)isamapping.
2.2 DefinitionsrelatedtoNSandNSS
Fewrelevantdefinitionsaregivenbelow[19,21,23,33]:
1. Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X
is characterized by a truth-membership function T , an indeterminacy-membership function I and a falsity-
A A
membershipfunctionF . T (x), I (x)andF (x)arerealstandardornon-standardsubsetsof]−0,1+[. Thatis
A A A A
T ,I ,F : X →]−0,1+[. Thereisnorestrictiononthesumof T (x),I (x), F (x) andso, −0 ≤ supT (x)+
A A A A A A A
supI (x)+supF (x)≤3+.
A A
2.LetUbeaninitialuniversesetandEbeasetofparameters.LetNS(U)denotethesetofallNSsofU.Thenfor
A⊆E,apair(F,A)iscalledanNSSoverU,whereF: A→ NS(U)isamapping.
ThisconcepthasbeenmodifiedbyDeliandBroumiasgivenbelow:
3. Let U be an initial universe set and E be a set of parameters. Let NS(U) denote the set of all NSs of U.
Then, a neutrosophic soft set N over U is a set defined by a set valued function f representing a mapping
N
f : E → NS(U) where f iscalledapproximatefunctionoftheneutrosophicsoftset N. Inotherwords, the
N N
neutrosophicsoftsetisaparameterizedfamilyofsomeelementsofthesetNS(U)andthereforeitcanbewritten
asasetoforderedpairs,
N = {(e,f (e)):e∈E}
N
= {(e,{<x,TfN(e)(x),IfN(e)(x),FfN(e)(x)>:x∈U}):e∈E}
whereTfN(e)(x),IfN(e)(x),FfN(e)(x)∈[0,1]andtheyarerespectivelycalledthetruth-membership,indeterminacy-
membership, falsity-membership function of f (e). Since supremum of each T,I,F is 1 so the inequality 0 ≤
N
TfN(e)(x)+IfN(e)(x)+FfN(e)(x)≤3isobvious.
4.ThecomplementofaneutrosophicsoftsetNisdenotedbyNcandisdefinedby:
Nc ={(e,{<x,FfN(e)(x),1−IfN(e)(x),TfN(e)(x)>:x∈U}):e∈E}
5. Let N and N be two NSSs over the common universe (U,E). Then N is said to be the neutrosophic soft
1 2 1
TuhinBeraandNirmalKumarMahapatra 4
subsetofN if∀e∈E,∀x∈U,
2
TfN1(e)(x)≤TfN2(e)(x);IfN1(e)(x)≥ IfN2(e)(x);FfN1(e)(x)≥FfN2(e)(x).
WewriteN ⊆ N andthenN istheneutrosophicsoftsupersetofN .
1 2 2 1
6. LetN andN betwoNSSsoverthecommonuniverse(U,E). ThentheirunionisdenotedbyN ∪N = N
1 2 1 2 3
andisdefinedby:
N3 ={(e,{<x,TfN3(e)(x),IfN3(e)(x),FfN3(e)(x)>:x∈U}):e∈E}
where TfN3(e)(x)=TfN1(e)(x)(cid:5)TfN2(e)(x), IfN3(e)(x)= IfN1(e)(x)∗IfN2(e)(x)and
FfN3(e)(x)=FfN1(e)(x)∗FfN2(e)(x);
7.LetN andN betwoNSSsoverthecommonuniverse(U,E).ThentheirintersectionisdenotedbyN ∩N =
1 2 1 2
N andisdefinedby:
3
N3 ={(e,{<x,TfN3(e)(x),IfN3(e)(x),FfN3(e)(x)>:x∈U}):e∈E}
where TfN3(e)(x)=TfN1(e)(x)∗TfN2(e)(x), IfN3(e)(x)= IfN1(e)(x)(cid:5)IfN2(e)(x)and
FfN3(e)(x)=FfN1(e)(x)(cid:5)FfN2(e)(x);
8. A neutrosophic soft set N over (U,E) is said to be null neutrosophic soft set if TfN(e)(x) = 0,IfN(e)(x) =
1,FfN(e)(x)=1;∀e∈E,∀x∈U.Itisdenotedbyφu.
A neutrosophic soft set N over (U,E) is said to be absolute neutrosophic soft set if TfN(e)(x) = 1,IfN(e)(x) =
0,FfN(e)(x)=0;∀e∈E,∀x∈U.Itisdenotedby1u.
Clearly,φuc =1uand1cu =φu.
9.AneutrosophicsoftpointinanNSSNisdefinedasanelement(e,f (e))ofN,fore∈Eandisdenotedbye ,
N N
if fN(e)∈/φuand fN(e(cid:48))∈φu,∀e(cid:48) ∈E−{e}.
The complement of a neutrosophic soft point e is another neutrosophic soft point ec such that fc(e) =
N N N
(f (e))c.
N
AneutrosophicsoftpointeN ∈ M, MbeinganNSSiffore∈E, fN(e)≤ fM(e)i.e., TfN(e)(x)≤TfM(e)(x), IfN(e)(x)≥
IfM(e)(x), FfN(e)(x)≥FfM(e)(x), ∀x∈U.
Example:LetU={x ,x ,x }andE={e ,e }.Then,
1 2 3 1 2
e ={<x ,(0.6,0.4,0.8)>,<x ,(0.8,0.3,0.5)>,<x ,(0.3,0.7,0.6)>}
1N 1 2 3
isaneutrosophicsoftpointwhosecomplementis:
ec ={<x ,(0.8,0.6,0.6)>,<x ,(0.5,0.7,0.8)>,<x ,(0.6,0.3,0.3)>}.
1N 1 2 3
ForanotherNSSMdefinedonsame(U,E),let
f (e )={<x ,(0.7,0.4,0.7)>,<x ,(0.8,0.2,0.4)>,<x ,(0.5,0.6,0.5)>}.
M 1 1 2 3
Then f (e )≤ f (e ) i.e., e ∈ M.
N 1 M 1 1N
2.3 Definitionsrelatedtoneutrosophicsoftmetricspace
Followingnecessarydefinitionsareprovidedhere[32]:
1. LetNS(U )bethecollectionofallneutrosophicsoftpointsover(U,E). Thentheneutrosophicsoftmetricin
E
TuhinBeraandNirmalKumarMahapatra 5
termsofneutrosophicsoftpointsisdefinedbyamappingd: NS(U )×NS(U )→[0,3]satisfyingthefollowing
E E
conditions:
NSM1:d(e ,e )≥0, ∀e ,e ∈ NS(U ).
M N M N E
NSM2:d(e ,e )=0⇔e =e .
M N M N
NSM3:d(e ,e )=d(e ,e ).
M N N M
NSM4:d(e ,e )≤d(e ,e )+d(e ,e ), ∀e ,e ,e ∈ NS(U ).
M N M P P N M P N E
Then NS(U ) is said to form an NSMS with respect to the neutrosophic soft metric ‘d’ over (U,E) and
E
is denoted by (NS(UE),d). Here eM = eN in the sense that 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)1k},k(≥1)beinganyrealnumber.This‘d’ satisfiesallthemetricaxiomsandso,itisaneutrosophicsoft
metricover(U,E).
(ii)Let‘d’ beaneutrosophicsoftmetriconNS(U ). Supposed (e ,e ) = d(eM,eN) ;Then‘d ’ satisfiesallthe
E 1 M N 1+d(eM,eN) 1
metricaxioms.So,(NS(U ),d )isanNSMSwithrespecttotheneutrosophicsoftmetricd .
E 1 1
3.Let(NS(U ),d)beaneutrosophicsoftmetricspaceandt∈(0,3].Anopenballhavingcenterate ∈ NS(U )
E N E
andradius‘t’ isdefinedbyasetB(e ,t)={e ∈ NS(U ):d(e ,e )<t}.
N iN E N iN
Theneutrosophicsoftclosedballisdefinedas: B[e ,t]={e ∈ NS(U ):d(e ,e )≤t}.
N iN E N iN
Aneighbourhoodofe ∈ NS(U )isdefinedbyanopenballB(e ,t)withcenterate andradiust∈(0,3].
N E N N
4.InanNSMS(NS(U ),d)over(U,E),aneutrosophicsoftpointe iscalledaninteriorpointofNS(U )ifthere
E N E
existanopenballB(e ,t)suchthatB(e ,t)⊂ NS(U ).
N N E
ForanNSMS(NS(U ),d)over(U,E),anNSSMiscalledopenifeachofit’spointsisaninteriorpoint.
E
5. Aneutrosophicsoftpointe inanNSMS(NS(U ),d)iscalledalimitpoint/accumulationpointofanNSS
N E
M⊂ NS(U )ifforeveryt∈(0,3], B(e ,t)containsatleastoneneutrosophicsoftpointofMdistinctfrome .
E N N
CollectionofalllimitpointsofMiscalledderivedNSSofMandisdenotedbyD(M).AnNSSM⊂ NS(U )
E
inanNSMS(NS(U ),d)over(U,E)isclosedNSSifD(M)⊂ MorMhasnolimitpoint.
E
6. Asequenceofneutrosophicsoftpoints{e }inanNSMS(NS(U ),d)issaidtoconvergein(NS(U ),d)if
nN E E
thereexistsaneutrosophicsoftpointe ∈ NS(U )suchthatd(e ,e ) → 0asn → ∞ore → e asn → ∞.
N E nN N nN N
Analytically,forevery(cid:101)>0thereexistsanaturalnumbern suchthatd(e ,e )<(cid:101) ∀n≥n .
0 nN N 0
7.Asequence{e }ofneutrosophicsoftpointinanNSMS(NS(U ),d)issaidtobeaCauchysequenceiftoevery
nN E
(cid:101)>0thereexistsann ∈N(setofnaturalnumbers)suchthatd(e ,e )<(cid:101) ∀m,n≥n i.e.,d(e ,e )→0
0 mN nN 0 mN nN
asm,n→∞.
8. AnNSMS(NS(U ),d)issaidtobecompleteifeveryCauchysequencein(NS(U ),d)convergestoaneutro-
E E
sophicsoftpointofNS(U ).
E
9.Let(NS(U ),d)beanNSMS.ThenthediameterofNS(U )isdefinedas:
E E
δ(NS(U ))=sup{d(e ,e ):e ,e ∈ NS(U )}.
E 1N 2N 1N 2N E
TuhinBeraandNirmalKumarMahapatra 6
An NSS M ⊂ NS(U ) is bounded if it has a finite diameter i.e., if d(e ,e ) ≤ r, for r ∈ (0,3] and
E 1M 2M
∀e ,e ∈ M.
1M 2M
2.4 Theoremsrelatedtoneutrosophicsoftmetricspace
Somenecessarytheoremsarestatedforthesakeofcompleteness[32]:
1.InanNSMS(NS(U ),d),everyneutrosophicsoftopenballB(e ,t)isopenandeveryneutrosophicsoftclosed
E N
ballB[e ,t]isclosed.
N
2.Let(NS(U ),d)beanNSMSover(U,E).Then,
E
(i) theintersectionoffinitenumberofopenNSSsin(NS(U ),d)isopen.
E
(ii) theintersectionofanyfamilyofclosedNSSsin(NS(U ),d)isclosed.
E
3.EveryfiniteneutrosophicsoftsubsetofanNSMSisclosed.
3 Compactness of NSMS
Inthissection,thecompactNSMShasbeendefinedandillustratedbyexamples.Somerelatedtheoremsalsohave
beendevelopedhere.
3.1 Definition
AnNSMS(NS(U ),d)issaidtobecompactifeverysequenceofneutrosophicsoftpoints{e }ofthespacehas
E nM
asubsequence{e }convergingtoaneutrosophicsoftpointofNS(U ).
nkM E
AnNSSM⊂ NS(U )issaidtobecompactifeverysequenceofneutrosophicsoftpointschosenfromMhas
E
asubsequenceconvergingtoapointofM.IfthelimitofthesubsequencebelongstoNS(U )andnotnecessarily
E
toM,thenMissaidtobecompactin(NS(U ),d).
E
3.1.1 Example
(1)LetE={e}andU={x,y,z}.DefineadistancefunctiononNS(U )as:
E

 1 if e (cid:54)=e
M N
d(e ,e )=
M N
 0 if e =e .
M N
Then‘d’ isaneutrosophicsoftmetriconNS(U )andiscalleddiscreteneutrosophicsoftmetric.Thus(NS(U ),d)
E E
isadiscreteNSMS.ItisacompactNSMS.
(2)ConsidertheNSMS(NS(U ),d)whereE = N(thesetofnaturalnumber)betheparametricset,U = Z(the
E
setofallintegers)betheuniversalsetanddisdefinedasin(2)(i)of[2.3];SinceT, I, F∈[0,1],everysequenceof
neutrosophicsoftpointsofthespacehasaconvergentsubsequenceandso(NS(U ),d)iscompact.
E
(3)TaketheNSMS(NS(U ),d)whereE=N(thesetofnaturalnumber),U=Z(thesetofallintegers)and‘d’ is
E
definedasin(2)(i)of[2.3];Considerasequenceofneutrosophicsoftpoints{e }as,∀x∈Z:
nM
TuhinBeraandNirmalKumarMahapatra 7
TenM(x)= 21n, IenM(x)=1− 21n, FenM(x)= 1+n2n forTenM,IenM,FenM ∈(0,1)
ThenMisnotcompactitselfbutiscompactonNS(U ).
E
(4)LetE = {e ,e ,e ,e ,e ,e ,e ,e }andU =Z. Define‘d’ asin(2)(i)of[2.3];Then(NS(U ),d)isnotcompact.
1 2 3 4 5 6 7 8 E
WeshallverifyitbytakingasequenceofneutrosophicsoftpointsasgiveninTable1.
Table1:Tabularformofneutrosophicsoftsequence
e e e e e e e e
1M 2M 3M 4M 5M 6M 7M 8M
x (1,0,0) (0,1,0) (0,0,1) (1,1,0) (1,0,1) (0,1,1) (0,0,0) (1,1,1)
1
x (1,0,0) (0,1,0) (0,0,1) (1,1,0) (1,0,1) (0,1,1) (0,0,0) (1,1,1)
2
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Thend(e ,e )(cid:54)=0fori(cid:54)= j.So,neitherthesequencenoranyofit’ssubsequenceisconvergent.
iM jM
3.2 Theorem
AcompactNSMSiscomplete.
Proof. Let (NS(U ),d) be a compact NSMS and {e } be a Cauchy sequence of neutrosophic soft points in
E nM
NS(U ). Thentoevery(cid:101) > 0thereexistsann ∈ N(setofnaturalnumbers)suchthatd(e ,e ) < (cid:101), ∀n >
E 0 mM nM
m≥n .
0
Since (NS(UE),d) be compact, ∃ a subsequence {enkM} such that limn→∞enkM = eP, say. Then d(enkM,eP) <
(cid:101), ∀n ≥n .Alsod(e ,e )<(cid:101), ∀n >m≥n .
k 0 mM nkM k 0
Nowforn > m, d(e ,e ) ≤ d(e ,e )+d(e ,e )+d(e ,e ) < 3(cid:101). Thus{e }beingaCauchy
nM P nM mM mM nkM nkM P nM
sequenceconvergestoapointinNS(U )andso(NS(U ),d)beacompleteNSMS.
E E
3.3 Theorem
EverycompactsetinanNSMSisclosedandbounded.
Proof. LetMbeacompactNSSinanNSMS(NS(U ),d). SupposeMisnotclosed. Thenthereexistsasequence
E
{e }ofneutrosophicsoftpointsinMconvergingtoapointe (say)notbelongtoM.Theneverysubsequence
nM M
of{e }alsoconvergestoe notbelongto M. Thusthereisnosubsequenceof{e }convergingtoapointof
nM M nM
MwhichcontradictsthecompactnessofM.HenceMisclosed.
NextsupposeMisnotboundedande befixedneutrosophicsoftpoint. Then∃apointe ∈ Msuchthat
M 1M
d(e ,e ) > 3. By similar argument ∃ a point e ∈ M such that d(e ,e ) > d(e ,e )+3. Continuing
M 1M 2M M 2M M 1M
thisprocess, wegetasequenceofneutrosophicsoftpoints e ,e ,··· ,e ,··· ∈ M suchthat d(e ,e ) >
1M 2M nM M nM
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(e ,e ) ≤ d(e ,e )+d(e ,e ) and so d(e e ) > 3 whenever n > m. This shows that neither
nM M nM mM mM M nM mM
thesequence{e }noranyofit’ssubsequencecanconverge,contradictingthefactthatMiscompact.HenceM
nM
isbounded.
TuhinBeraandNirmalKumarMahapatra 8
3.3.1 Remark
Converseofabovemaynotbetrue.Thefactisshownbytheexample(4)of[3.1.1];
Here,d(e ,e )<3foralli(cid:54)= jandD(M)=φ⊂ M.So,Misboundedandclosed.ButMisnotcompact.
iM jM
4 Continuity on NSMS
Here,theconceptofneutrosophicsoftfunction,it’scontinuityonanNSMS,thecompositionofneutrosophicsoft
functionshavebeenintroducedandillustratedbysuitableexamples.Severalproperties,structuralcharacteristics
andtheoremsrelatedtothesealsohavebeenpresentedhere.
4.1 Definition
Let (NS(UE),d)and (NS(VE(cid:48)),d(cid:48))betwoNSMSsand (ϕ,ψ) : (NS(UE),d) → (NS(VE(cid:48)),d(cid:48))beaneutrosophic
softfunctionwhere ϕ : U → V andψ : E → E(cid:48) betwocrispfunctions. Considertwoneutrosophicsoftpoints
e , e(cid:48) as:
M N
eM ={<x,(TeM(x),IeM(x),FeM(x))>:x∈U}∈ NS(UE), e∈E and
e(cid:48)N ={<y,(Te(cid:48)N(y),Ie(cid:48)N(y),Fe(cid:48)N(y))>:y∈ ϕ(U)}∈ NS(VE(cid:48)), e(cid:48) ∈ψ(E)
(1)Thentheimageof e under (ϕ,ψ) isdenotedby (ϕ,ψ)(e ). Itisalsoaneutrosophicsoftpoint e(cid:48) (say) ∈
M M N
NS(VE(cid:48))definedasfollows:

Te(cid:48) (y)= maxϕ(x)=y maxψ(e)=e(cid:48)[TeM(x)], ifx∈ ϕ−1(y)
N  0 , otherwise.

Ie(cid:48) (y)= minϕ(x)=y minψ(e)=e(cid:48)[IeM(x)], ifx∈ ϕ−1(y)
N  1 , otherwise.

Fe(cid:48) (y)= minϕ(x)=y minψ(e)=e(cid:48)[FeM(x)], ifx∈ ϕ−1(y)
N  1 , otherwise.
(2)Thepre-imageofe(cid:48) under(ϕ,ψ),denotedby(ϕ,ψ)−1(e(cid:48) ),isaneutrosophicsoftpointe (say) ∈ NS(U )
N N M E
andisdefinedasfollows,∀x∈U, ∀e∈ψ−1(E(cid:48)):
TeM(x) = T[ψ(e)]N(ϕ(x))=Te(cid:48)N(ϕ(x))
IeM(x) = I[ψ(e)]N(ϕ(x))= Ie(cid:48)N(ϕ(x))
FeM(x) = F[ψ(e)]N(ϕ(x))=Fe(cid:48)N(ϕ(x))
Ifψandϕareinjective(surjective),then(ϕ,ψ)isinjective(surjective).
TuhinBeraandNirmalKumarMahapatra 9
4.1.1 Example
LetE=N(thesetofnaturalnumbers)betheparametricsetandU =Z(thesetofintegers)betheuniversalset.
Consideraneutrosophicsoftpointn ∈ NS(Z )asfollows,foranyn∈Nandx∈Z:
M N

 0 if x=2k−1, k∈Z
TnM(x)= 1 if x=2k, k∈Z.
n

 1 if x=2k−1, k∈Z
InM(x)= 2n
 0 if x=2k, k∈Z.

 1− 1 if x=2k−1, k∈Z
FnM(x)= n
 0 if x=2k, k∈Z.
Then(NS(Z ),d)formsanNSMSwhere‘d’isdefinedin(2)(i)of[2.3]. Now,let ϕ : Z → Zandψ : N → Nbe
N
twocrispfunctionsdefinedas ϕ(x) = 2x+3 = y (say)and ψ(n) = 2n−1 = m (say), respectively. Thenthe
neutrosophicsoftfunction(ϕ,ψ) : (NS(Z ),d) → (NS(Z ),d)isgivenby(ϕ,ψ)(n ) = m , m ∈ Nanditis
N N M P
definedas:

 0 if y=4k+1, k∈Z
TmP(y)= 1+2m if y=4k+3, k∈Z
 0 if y=otherwise.

 1+1m if y=4k+1, k∈Z
ImP(y)= 0 if y=4k+3, k∈Z
 1 if y=otherwise.

 mm−+11 if y=4k+1, k∈Z
FmP(y)= 0 if y=4k+3, k∈Z
 1 if y=otherwise.
4.2 Proposition
Let(ϕ,ψ):(NS(UE),d)→(NS(VE(cid:48)),d(cid:48))beaneutrosophicsoftfunction. Thentheimageset{(φ,ψ)(eM):eM ∈
NS(U )}formsanNSMSwithrespectto‘d(cid:48)’.
E
TuhinBeraandNirmalKumarMahapatra 10
Proof.Letusconsiderthreeneutrosophicsoftpointse ,e ,e ∈ NS(U ).Now,
M N P E
(1) e (cid:54)=e ⇒ d(e ,e )>0
M N M N
i.e., (TeM(x),IeM(x),FeM(x))(cid:54)=(TeN(x),IeN(x),FeN(x)) ⇒
d[(TeM(x),IeM(x),FeM(x)),(TeN(x),IeN(x),FeN(x))]>0,∀x∈U
i.e., (mϕ(axx)mψ(ae)x[TeM(x)],mϕ(ixn)mψ(ien)[IeM(x)],mϕ(ixn)mψ(ien)[FeM(x)])(cid:54)=
(mϕ(axx)mψ(ae)x[TeN(x)],mϕ(ixn)mψ(ien)[IeN(x)],mϕ(ixn)mψ(ien)[FeN(x)]) ⇒
d(cid:48)[(mϕ(axx)mψ(ae)x[TeM(x)],mϕ(ixn)mψ(ien)[IeM(x)],mϕ(ixn)mψ(ien)[FeM(x)]),
(mϕ(axx)mψ(ae)x[TeN(x)],mϕ(ixn)mψ(ien)[IeN(x)],mϕ(ixn)mψ(ien)[FeN(x)])]>0
i.e., (φ,ψ)(e )(cid:54)=(φ,ψ)(e ) ⇒ d(cid:48)[(φ,ψ)(e ),(φ,ψ)(e )]>0
M N M N
(2) e =e ⇔ d(e ,e )=0
M N M N
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., mϕ(axx)mψ(ae)x[TeM(x)]=mϕ(axx)mψ(ae)x[TeN(x)],mϕ(ixn)mψ(ien)[IeM(x)]=
mϕ(ixn)mψ(ien)[IeN(x)],mϕ(ixn)mψ(ien)[FeM(x)]=mϕ(ixn)mψ(ien)[FeN(x)] ⇔
d(cid:48)[(mϕ(axx)mψ(ae)x[TeM(x)],mϕ(ixn)mψ(ien)[IeM(x)],mϕ(ixn)mψ(ien)[FeM(x)]),
(mϕ(axx)mψ(ae)x[TeN(x)],mϕ(ixn)mψ(ien)[IeN(x)],mϕ(ixn)mψ(ien)[FeN(x)])]=0
i.e., (φ,ψ)(e )=(φ,ψ)(e ) ⇔ d(cid:48)[(φ,ψ)(e ),(φ,ψ)(e )]=0
M N M N
(3) d(e ,e )=d(e ,e )
M N N M
⇒ 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(cid:48)[(mϕ(axx)mψ(ae)x[TeM(x)],mϕ(ixn)mψ(ien)[IeM(x)],mϕ(ixn)mψ(ien)[FeM(x)]),
(mϕ(axx)mψ(ae)x[TeN(x)],mϕ(ixn)mψ(ien)[IeN(x)],mϕ(ixn)mψ(ien)[FeN(x)])]
=d(cid:48)[(mϕ(axx)mψ(ae)x[TeN(x)],mϕ(ixn)mψ(ien)[IeN(x)],mϕ(ixn)mψ(ien)[FeN(x)]),
(mϕ(axx)mψ(ae)x[TeM(x)],mϕ(ixn)mψ(ien)[IeM(x)],mϕ(ixn)mψ(ien)[FeM(x)])]
⇒ d(cid:48)[(φ,ψ)(e ),(φ,ψ)(e )]=d(cid:48)[(φ,ψ)(e ),(φ,ψ)(e )]
M N N M