Generalized inverse of fuzzy neutrosophic soft matrix

Generalizedinverseoffuzzyneutrosophic softmatrix

R.Uma∗ ,P.Murugadas†andS.Sriram‡

*† DepartmentofMathematics,AnnamalaiUniversity,Annamalainagar-608002,India. ‡ DepartmentofMathematics,Mathematicswing,D.D.E,AnnamalaiUniversity, Annamalainagar-608002,India.

Abstract

Aimofthisarticleistofindthemaximumandminimumsolutionofthe fuzzyneutrosophicsoftrelationalequation 𝑥𝐴 = 𝑏 and 𝐴𝑥 = 𝑏, where 𝑥 and 𝑏 arefuzzyneutrosophicsoftvectorand 𝐴 isafuzzyneutrosophicsoftmatrix. Whenever 𝐴 issingularwecannotfind 𝐴 1 Inthatcasewecanuseginversetogetthesolutionoftheaboverelationalequation.Further,usingthis conceptmaximumandminimumg-inverseoffuzzyneutrosophicsoftmatrix areobtained.

AMSSubjectclassificationcode:03E72,15B15. Keywords: FuzzyNeutrosophicSoftSet(FNSS),FuzzyNeutrosophicSoftMatrix (FNSM),g-inverse.

1Introduction

MostofourreallifeproblemsinMedicalScience,Engineering,Management, EnvironmentandSocialSciencesofteninvolvedatawhicharenotnecessarilycrisp, preciseanddeterministicincharacterduetovariousuncertaintiesassociatedwith theseproblems.Suchuncertaintiesareusuallybeinghandledwiththehelpofthe

∗ uma83bala@gmail.com

† bodi muruga@yahoo.com ‡ ssm 3096@yahoo.co.in

topicslikeprobability,fuzzysets,intervalMathematicsandroughsetsetc.,IntuitionisticfuzzysetsintroducedbyAtanassov[3]isappropriateforsuchasituation. Theintuitionisticfuzzysetscanonlyhandletheincompleteinformationconsidering boththetruthmembershipandfalsitymembership.Itdoesnothandletheindeterminateandinconsistentinformationwhichexistsinbeliefsystem.Smarandache [13]announcedandevincedtheconceptofneutrosophicsetwhichisaMathematical toolforhandlingproblemsinvolvingimprecise,indeterminacyandinconsistentdata. TheneutrosophiccomponentsT,I,Fwhichrepresentsthemembership,indeterminancy,andnon-membershipvaluesrespectively,where] 0, 1+[isthenon-standard unitinterval,andthusonedefinestheneutrosophicset.

ForexampletheSchrodinger’scattheorysaysthatbasicallythequantumstateof aphotoncanbasicallybeinmorethanoneplaceatthesametime,whichtranslated totheneutrosophicsetwhichmeansanelement(quantumstate)belongsanddoes notbelongtoaset(oneplace)atthesametime;oranelement(quantumstate) belongstotwodifferentsets(twodifferentplaces)inthesametime.Diletheismis theviewthatsomestatementscanbebothtrueandfalsesimultaneously.Moreprecisely,itisbeliefthattherecanbetruestatementwhosenegationisalsotrue.Such statementarecalledtruecontradiction,diletheiaornondualism.”Allstatementsare true”isafalsestatement.Theaboveexampleoftruecontradictionsthatdialetheists accept.Neutrosophicset,likedialetheism,candescribeparadoxistelements,Neutrosophicset(paradoxistelement)=(1,1,1),whileintuitionisticfuzzylogiccannot describeaparadoxbecausethesumofcomponentsshouldbe1inintuitionisticfuzzy set.

InneutrosophicsetthereisnorestrictiononT,I,Fotherthantheyaresubsetsof

Neutrosophicsetsandlogicarethefoundationsformanytheorieswhicharemore generalthantheirclassicalcounterpartsinfuzzy,intuitionisticfuzzy,paraconsistent set,dialetheistset,paradoxistsetandtautologicalset.

In1999,Molodtsov[9]initiatedthenovelconceptofsoftsettheorywhichisa completelynewapproachformodelingvaguenessanduncertainty.In[7]Majiet al.,initiatedtheconceptoffuzzysoftsetswithsomepropertiesregardingfuzzysoft union,intersection,complementoffuzzysoftset.Moreoverin[8,11]Majietal., extendedsoftsetstointuitionisticfuzzysoftsetsandneutrosophicsoftsets.

OneoftheimportanttheoryofMathematicswhichhasavastapplicationin ScienceandEngineeringisthetheoryofmatrices.Let 𝐴 beasquarematrixoffull rank.Then,thereexistsamatrix 𝑋 suchthat 𝐴𝑋 = 𝑋𝐴 = 𝐼. This 𝑋 iscalledthe

inverseof 𝐴 andisdenotedby 𝐴 1 . Suppose 𝐴 isnotamatrixoffullrankoritis arectangularmatrix,insuchacaseinversedoesnotexists.Needfeltinnumerous areasofappliedMathematicsforsomekindofpartialinverseofamatrixwhich issingularorevenrectangular,suchinversearecalledgeneralizedinverse.Solving fuzzymatrixequationofthetype 𝑥𝐴 = 𝑏 where 𝑥 =(𝑥11,𝑥12,𝑥1𝑚),𝑏 =(𝑏11,𝑏12,𝑏1𝑛) and 𝐴 isafuzzymatrixoforder 𝑚 × 𝑛 isofgreatinterestinvariousfields.We say 𝑥𝐴 = 𝑏 iscomptiable,ifthereexistsasolutionfor 𝑥𝐴 = 𝑏 andinthiscasewe writemax 𝑗 𝑚𝑖𝑛(𝑥1𝑗 ,𝑎𝑗𝑘)= 𝑏1𝑘 forall 𝑗 ∈ 𝐼𝑚 and 𝑘 ∈ 𝐼𝑛, where 𝐼𝑛 isanindexset, 𝑖 =1, 2,...,𝑛. Ω1(𝐴,𝑏)representsthesetofallsolutionsof 𝑥𝐴 = 𝑏. Theauthorsextendthisconceptintofuzzyneutrosophicsoftmatrix.Thefuzzy neutrosophicsoftmatrixequationisoftheform 𝑥𝐴 = 𝑏,.....(1) where 𝑥

,𝑏

𝑘⟩ forall 𝑗 ∈ 𝐼𝑚 and 𝑘 ∈ 𝐼𝑛. Denote Ω1(𝐴,𝑏)= {𝑥∣𝑥𝐴 = 𝑏} representsthesetofallsolutionsof 𝑥𝐴 = 𝑏. Severalauthors [4,6,12]havestudiedaboutthemaximumsolutionˆ 𝑥 andtheminimumsolutionˇ 𝑥 of 𝑥𝐴 = 𝑏 forfuzzymatrixaswellasIFMs. LiJian-Xin[6]andKatarinaCechlarova[5]discussedthesolvabilityofmaxmin fuzzyequation 𝑥𝐴 = 𝑏 and 𝐴𝑥 = 𝑏. Inboththecasesthemaximumsolutionis uniqueandtheminimumsolutionneednotbeunique.LetΩ2(𝐴,𝑏)bethesetof allsolutionsfor 𝐴𝑥 = 𝑏. Murugadas[10]introducedamethodtofindmaximumginverseaswellasminimumg-inverseoffuzzymatrixandintuitionisticfuzzymatrix. Letusrestrictourfurtherdiscussioninthissectiontofuzzyneutrosophicsoftmatrix equationoftheform 𝐴𝑥 = 𝑏 with 𝑥 =[⟨𝑥𝑇 𝑖1,𝑥𝐼 𝑖1,𝑥𝐹 𝑖1⟩∣𝑖 ∈ 𝐼𝑛],𝑏 =[⟨𝑏𝑇 𝑘1,𝑏𝐼 𝑘1,𝑏𝐹 𝑘1⟩∈ 𝐼𝑚] where 𝐴 ∈

findsthemaximumandminimumsolutionoftherelationalequation 𝑥𝐴 = 𝑏 when 𝐴 isaFNSM.Furtherthisconcepthasbeenextendedinfindingg-inverseofFNSM.

dardornon-standardsubsetsof

scientificandEngineeringproblemsitisdifficulttouseneutrosophicsetwithvalue fromrealstandardornon-standardsubsetof ] 0, 1+[ .Henceweconsidertheneutrosophicsetwhichtakesthevaluefromthesubsetof [0, 1] Thereforewecanrewritetheequation(1)as 0 ≤ 𝑇𝐴(𝑥)+ 𝐼𝐴(𝑥)+ 𝐹𝐴(𝑥) ≤ 3. Inshortanelement 𝑎 intheneutrosophicset 𝐴, canbewrittenas 𝑎 = ⟨𝑎𝑇 ,𝑎𝐼 ,𝑎𝐹 ⟩, where 𝑎𝑇 denotesdegreeoftruth, 𝑎𝐼 denotesdegreeofindeterminacy, 𝑎𝐹 denotes degreeoffalsitysuchthat 0 ≤ 𝑎𝑇 + 𝑎𝐼 + 𝑎𝐹 ≤ 3

Example2.2. Assumethattheuniverseofdiscourse 𝑋 = {𝑥1,𝑥2,𝑥3},where 𝑥1,𝑥2, and 𝑥3 characterisesthequality,relaibility,andthepriceoftheobjects.Itmaybe furtherassumedthatthevaluesof {𝑥1,𝑥2,𝑥3} arein [0, 1] andtheyareobtainedfrom someinvestigationsofsomeexperts.Theexpertsmayimposetheiropinioninthree componentsviz;thedegreeofgoodness,thedegreeofindeterminacyandthedegree ofpoornesstoexplainthecharacteristicsoftheobjects.Suppose 𝐴 isaNeutrosophic Set(NS)of 𝑋,suchthat 𝐴 = {⟨𝑥1, 0.4, 0.5, 0.3⟩, ⟨𝑥2, 0.7, 0.2, 0.4⟩, ⟨𝑥3, 0.8, 0.3, 0.4⟩}, wherefor 𝑥1 thedegreeofgoodnessofqualityis0.4,degreeofindeterminacyof qualityis0.5anddegreeoffalsityofqualityis0.3etc,.

Definition2.3. [9]LetUbeaninitialuniversesetand 𝐸 beasetofparameters. LetP(U)denotesthepowersetofU.Consideranonemptyset 𝐴,𝐴 ⊂ 𝐸.Apair (F,A)iscalledasoftsetoverU,whereFisamappinggivenby 𝐹 : 𝐴 → 𝑃 (𝑈 )

Definition2.4. [1]Let 𝑈 beaninitialuniversesetand 𝐸 beasetofparameters. Consideranonemptyset 𝐴,𝐴 ⊂ 𝐸.Let 𝑃 (𝑈 ) denotesthesetofallfuzzyneutrosophicsetsof 𝑈 .Thecollection (𝐹,𝐴) istermedtobetheFuzzyNeutrosophicSoft Set(FNSS)over 𝑈, whereFisamappinggivenby 𝐹 : 𝐴 → 𝑃 (𝑈 ).Hereafterwe simplyconsider 𝐴 asFNSSover 𝑈 insteadof (𝐹,𝐴)

Definition2.5. [2]Let 𝑈 = {𝑐1,𝑐2,...𝑐𝑚} betheuniversalsetand 𝐸 bethesetof parametersgivenby 𝐸 = {𝑒1,𝑒2,...𝑒𝑛}.Let 𝐴 ⊆ 𝐸 .Apair (𝐹,𝐴) beaFNSSover 𝑈 .Thenthesubsetof 𝑈 × 𝐸 isdefinedby 𝑅𝐴 = {(𝑢,𝑒); 𝑒 ∈ 𝐴,𝑢 ∈ 𝐹𝐴(𝑒)} which iscalledarelationformof (𝐹𝐴,𝐸).Themembershipfunction,indeterminacymembershipfunctionandnonmembershipfunctionarewrittenby 𝑇𝑅𝐴 : 𝑈 × 𝐸 → [0, 1], 𝐼

𝑚𝑛,𝐼𝑚𝑛,𝐹𝑚𝑛⟩ ⎤ ⎥ ⎦ whichiscalledan 𝑚 × 𝑛 FNSMoftheFNSS(𝐹𝐴,𝐸)over 𝑈. Definition2.6. Let 𝑈 = {𝑐1,𝑐2...𝑐𝑚} betheuniversalsetand 𝐸 bethesetofparametersgivenby 𝐸 = {𝑒1,𝑒2,...𝑒𝑛}.Let 𝐴 ⊆ 𝐸.Apair (𝐹,𝐴) beafuzzyneutrosophic softset.Thenfuzzyneutrosophicsoftset (𝐹,𝐴) inamatrixformas 𝐴𝑚×𝑛 =(𝑎𝑖𝑗 )𝑚×𝑛 or 𝐴 =(𝑎𝑖𝑗 ),𝑖 =1, 2,...𝑚,𝑗 =1, 2,...𝑛 where (𝑎𝑖𝑗 )= {(𝑇 (𝑐𝑖,𝑒𝑗 ),𝐼(𝑐𝑖,𝑒𝑗 ),𝐹 (𝑐𝑖,𝑒𝑗 )) if 𝑒𝑗 ∈ 𝐴 ⟨0, 0, 1⟩ if 𝑒𝑗 / ∈ 𝐴 where 𝑇𝑗 (𝑐𝑖) representthemembershipof 𝑐𝑖,𝐼𝑗 (𝑐𝑖) representtheindeterminacyof 𝑐𝑖 and 𝐹𝑗 (𝑐𝑖) representthenon-membershipof 𝑐𝑖 intheFNSS (𝐹,𝐴) Ifwereplacetheidentityelement ⟨0, 0, 1⟩ by ⟨0, 1, 1⟩ intheaboveformwegetFNSM oftype-II.

FNSMofType-I[14] Let 𝒩𝑚×𝑛 denotesFNSMoforder 𝑚 × 𝑛 and 𝒩𝑛 denotesFNSMoforder 𝑛 × 𝑛. Definition2.7. Let 𝐴 =(〈𝑎𝑇 𝑖𝑗 ,𝑎𝐼 𝑖𝑗 ,𝑎𝐹 𝑖𝑗 〉),𝐵 =(〈𝑏𝑇 𝑖𝑗 ,𝑏𝐼 𝑖𝑗 ,𝑏𝐹 𝑖𝑗 〉) ∈𝒩𝑚×𝑛 thecomponentwiseadditionandcomponentwisemultiplicationisdefinedas 𝐴 ⊕ 𝐵 =(𝑠𝑢𝑝 {𝑎𝑇 𝑖𝑗 ,𝑏𝑇 𝑖𝑗 } ,𝑠𝑢𝑝 {𝑎𝐼 𝑖𝑗 ,𝑏𝐼 𝑖𝑗 } ,𝑖𝑛𝑓 {𝑎𝐹 𝑖𝑗 ,𝑏𝐹 𝑖𝑗 }) 𝐴 ⊙ 𝐵 =(𝑖𝑛𝑓 {𝑎𝑇 𝑖𝑗 ,𝑏𝑇 𝑖𝑗 },𝑖𝑛𝑓 {𝑎𝐼 𝑖𝑗 ,𝑏𝐼 𝑖𝑗 },𝑠𝑢𝑝{𝑎𝐹 𝑖𝑗 ,𝑏𝐹 𝑖𝑗 }). Definition2.8. Let 𝐴 ∈𝒩𝑚×𝑛,𝐵 ∈𝒩

Definition2.9. Let 𝐴 =(⟨𝑎𝑇 𝑖𝑗 ,𝑎𝐼 𝑖𝑗 ,𝑎𝐹 𝑖𝑗 ⟩),𝐵 =(⟨𝑏𝑇 𝑖𝑗 ,𝑏𝐼 𝑖𝑗 ,𝑏𝐹 𝑖𝑗 ⟩) ∈𝒩𝑚×𝑛, thecomponent wiseadditionandcomponentwisemultiplicationisdefinedas 𝐴 ⊕ 𝐵 =(⟨𝑠𝑢𝑝 {𝑎𝑇 𝑖𝑗 ,𝑏𝑇 𝑖𝑗 } ,𝑖𝑛𝑓 {𝑎𝐼 𝑖𝑗 ,𝑏𝐼 𝑖𝑗 } ,𝑖𝑛𝑓 {𝑎𝐹 𝑖𝑗 ,𝑏𝐹 𝑖𝑗 }⟩) 𝐴 ⊙ 𝐵

theproduct 𝐴 ∗ 𝐵 isdefinedifandonlyifthenumberofcolumnsofAissameasthe numberofrowsofB.AandBaresaidtobeconformableformultiplication.

3Mainresults

Definition3.1. 𝐴 ∈ 𝑁𝑚×𝑛 issaidtoberegularifthereexists 𝑋 ∈ 𝑁𝑛×𝑚 suchthat 𝐴𝑋𝐴 = 𝐴.

Definition3.2. If 𝐴 and 𝑋 aretwoFNSMoforder 𝑚 × 𝑛 satisfiestherelation 𝐴𝑋𝐴 = 𝐴, then 𝑋 iscalledageneralizedinverse(g-inverse)of 𝐴 whichisdenoted by 𝐴 Theg-inverseofanFNSMisnotnecessarilyunique.Wedenotethesetofall g-inverseby 𝐴{1}

Definition3.3. Anyelementˆ 𝑥 ∈ Ω1(𝐴,𝑏)iscalledamaximalsolutionifforall

IfΩ1(𝐴,𝑏)

𝜙, thenˆ 𝑥 isasolutionof 𝑥𝐴 = 𝑏. Forifˆ 𝑥 isnotasolution, thenˆ𝑥𝐴

andtherefore

,𝑎

,𝑏

𝑘

,𝑏𝐹

𝑘

⟩

isthemaximumsolution. Theconversepartistrivial.

.....(2)

isanfuzzyneutrosophicsoftmatrixoforder

)betheset allsolutionoftherelationalequation 𝐴𝑥 = 𝑏.

Definition3.7. Anyelementˆ 𝑥 ∈ Ω2(𝐴,𝑏)iscalledamaximalsolutionifforall 𝑥 ∈ Ω2(𝐴,𝑏),𝑥 ≥

implies 𝑥 =ˆ𝑥. Thatiselements 𝑥, ˆ 𝑥 arecomponentwiseequal. Definition3.8. Anyelementˇ 𝑥 ∈ Ω2(𝐴,𝑏)iscalledaminimalsolutionifforall

arecomponentwiseequal.

Lemma3.9. Let 𝐴𝑥 = 𝑏 asdefinedin(2). If ⟨max

,𝑏

,𝑏

⟩ forsome 𝑘 ∈ 𝐼

, then Ω2(𝐴,𝑏)= 𝜙.

Theorem3.10. Fortheequation 𝐴𝑥 = 𝑏, Ω2(𝐴,𝑏) = 𝜙 ifonlyif

otherwise

⟨min

(

2𝑘,𝑏𝑇 1𝑘), min 𝑘 𝜎 ′ (𝑎 𝐼 2𝑘,𝑏𝐼 1𝑘), max 𝑘 𝜎 ′′ (𝑎 𝐹 2𝑘,𝑏𝐹 1𝑘)⟩ = ⟨min 𝑘 (1, 0.5), min 𝑘 (1, 0.3), max 𝑘 (0, 0.1)⟩ = ⟨0.5, 0.3, 0.1⟩

Thenclearly (⟨0 2, 0 3, 0 5⟩⟨0 5, 0 3, 0 1⟩) [⟨0.50.60.2⟩⟨0.7, 0.5, 0.1⟩ ⟨0 20 30 5⟩⟨0 6, 0 4, 0⟩ ] =(⟨0 2, 0 3, 0 5⟩⟨0 5, 0 3, 0 1⟩) Togettheminimalsolution ˇ 𝑥 of 𝑥𝐴 = 𝑏 wefollowtheprocedureasfollowed forfuzzyneutrosophicsoftmatrixequation.

Step.1 Determinethesets 𝐽𝑘(ˆ 𝑥)= {𝑗 ∈ 𝐼𝑚∣𝑚𝑖𝑛(⟨ˆ 𝑥𝑇 1𝑗 , ˆ 𝑥𝐼 1𝑗 , ˆ 𝑥𝐹 1𝑗 ⟩, ⟨𝑎𝑇 𝑗𝑘,𝑎𝐼 𝑗𝑘,𝑎𝐹 𝑗𝑘⟩)= 𝑏𝑇 1𝑘} forall 𝑘 ∈ 𝐼𝑛. Constructtheircartesianproduct 𝐽(ˆ 𝑥)= 𝐽1(ˆ 𝑥) × 𝐽2(ˆ 𝑥) × ... × 𝐽𝑛(ˆ 𝑥).

Step.2 Denotetheelementsof 𝐽(ˆ 𝑥), by 𝛽 =[𝛽𝑘/𝑘 ∈ 𝐼𝑛] Foreach 𝛽 ∈ 𝐽(ˆ 𝑥)andeach 𝑗 ∈ 𝐼𝑚, determinetheset 𝑘(𝛽,𝑗)= {𝑘 ∈ 𝐼𝑚∣𝛽𝑘 = 𝑗}.

Step.3 Foreach 𝛽 ∈ 𝐽(ˆ 𝑥)generatethen-tuple 𝑔(𝛽)= 𝑔𝑗 (𝛽)∣𝑗 ∈ 𝐼𝑚}, where 𝑔𝑗 (𝛽)= ⎧ ⎨ ⎩ max 𝑘(𝛽,𝑗)⟨𝑏𝑇 1𝑘,𝑏𝐼 1𝑘,𝑏𝐹 1𝑘⟩ if 𝑘(𝛽,𝑗) =0 ⟨0, 0, 1⟩ otherwise Step.4 Fromallthem-tuples 𝑔(𝛽)generatedinstep.3,selectonlytheminimalone bypairwisecomparison.Theresultingsetofn-tuplesistheminimalsolutionofthe reducedformofequation 𝑥𝐴 = 𝑏.

Example3.12. LetusfindtheminimalsolutiontothelinearequationgiveninExample3.11usingthemaximalsolution ˆ 𝑥 Step1. Todetermine 𝐽𝑘(ˆ 𝑥) for 𝑘 =1, 2.

1(ˆ 𝑥)= {𝑗 =1, 2∣𝑚𝑖𝑛(⟨𝑥

1𝑗 ,𝑥

𝑗 ,𝑥

1

⟩, ⟨

,𝑎

,𝑎

⟩)= ⟨𝑏𝑇 1𝑘,𝑏𝐼 1𝑘,𝑏𝐹 1𝑘⟩}

{𝑚𝑖𝑛{⟨0 2, 0 3, 0 5⟩⟨0 5, 0 6, 0 2⟩},𝑚𝑖𝑛{⟨0 5, 0 3, 0 1⟩⟨0 2, 0 3, 0 5⟩}} = ⟨0 2, 0 3, 0 5⟩ = {1, 2} 𝐽2(ˆ 𝑥)= {𝑚𝑖𝑛{⟨0 2, 0 3, 0 5⟩⟨0 7, 0 5, 0 1⟩},𝑚𝑖𝑛{⟨0 5, 0 3, 0 1⟩⟨0 6, 0 4, 0⟩}} = ⟨0 5, 0 3, 0 1⟩ = {2} Therefore 𝐽𝑘(ˆ 𝑥)= 𝐽1(ˆ 𝑥) × 𝐽2(ˆ 𝑥)= {1, 2}×{2} = {(1, 2, (2, 2))} = 𝛽

Step2: Todeterminethesets 𝐾(𝛽,𝑗)foreach 𝛽 = 𝐽𝑘(ˆ 𝑥)andforeachj=1,2.

For 𝛽 =(1, 2) 𝐾(𝛽, 1)= {𝑘 =1, 2∣𝛽𝑘 =1} = {1} 𝐾(𝛽, 2)= {𝑘 =1, 2∣𝛽𝑘 =2} = {2}

For 𝛽 =(2, 2) 𝐾(𝛽, 1)= {𝑘 =1, 2∣𝛽𝑘 =1} = {𝜑} 𝐾(𝛽, 2)= {𝑘 =1, 2∣𝛽𝑘 =2} = {1, 2}

Thusthesets 𝐾(𝛽,𝑗)foreach 𝛽 ∈ 𝐽(ˆ 𝑥)and 𝑗 =1, 2arelistedinthefollowingtable.

𝐾{𝛽,𝑗} 1 𝐾 (1, 2) {1} {2} (2, 2) {𝜙} {1, 2}

Step3. Foreach 𝛽 ∈ 𝐽(ˆ 𝑥)wegeneratethetuples 𝑔(𝛽)

For 𝛽 =(1, 2) 𝑔1(𝛽)= ⎧ ⎨ ⎩ max 𝑘∈𝑘(𝛽,1)⟨0.2, 0.3, 0.5⟩ if 𝑘(𝛽, 1) = 𝜙 ⟨0, 0, 1⟩ otherwise = ⟨0.2, 0.3, 0.5⟩ 𝑔2(𝛽)= ⟨0.5, 0.3, 0.1⟩ For 𝛽 =(2, 2) 𝑔1(𝛽)= ⟨0, 0, 1⟩ 𝑔2(𝛽)= ⟨0.5, 0.3, 0.1⟩

Thereforewecangetthefollowingtablefor 𝛽 𝛽 𝑔(𝛽) (1, 2) ⟨0 2, 0 3, 0 5⟩, ⟨0 5, 0 3, 0 1⟩ (2, 2) ⟨0, 0, 1⟩, ⟨0 5, 0 3, 0 1⟩

Outofwhich(⟨0, 0, 1⟩, ⟨0.5, 0.3, 0.1⟩)istheminimalone.Andalsoitsatisfy 𝑥𝐴 = 𝑏 thatisˆ 𝑥 =(⟨0, 0, 1⟩, ⟨0.5, 0.3, 0.1⟩) 10

Usingthesamemethodwehavefollowed,onecanfindtheg-inverseofafuzzyneutrosophicsoftmatrixifitexits.

(𝑎 ′ 2𝑘,𝑏′ 2𝑘), max 𝑘 𝜎 ′′ (𝑎 ′′ 2𝑘,𝑏′′ 2𝑘)⟩ = ⟨1, 1, 0⟩ Take (⟨𝑏𝑇 21,𝑏𝐼 21,𝑏𝐹 21⟩, ⟨𝑏𝑇 22,𝑏𝐼 22,𝑏𝐹 22⟩) [⟨1, 1, 0⟩⟨1, 1, 0⟩ ⟨1, 1, 0⟩⟨0, 0, 1⟩] =(⟨1, 1, 0⟩, ⟨0, 0, 1⟩) ⟨𝑏𝑇 21,𝑏𝐼 21,𝑏𝐹 21⟩ = ⟨min 𝑘 𝜎1𝑘,𝑏2𝑘), min 𝑘 (𝜎 ′ 1𝑘,𝑏′ 2𝑘), max 𝑘 (𝜎 ′′ 1𝑘,𝑏′′ 2𝑘)⟩ = ⟨0, 0, 1⟩ ⟨𝑏𝑇 22,𝑏𝐼 22,𝑏𝐹 22⟩ = ⟨min 𝑘 (𝜎2𝑘,𝑏2𝑘), min 𝑘 (𝜎 ′ 2𝑘,𝑏′ 2𝑘), max 𝑘 (𝜎 ′′ 2𝑘,𝑏′′ 2𝑘)⟩ = ⟨1, 1, 0⟩ Therefore ˆ 𝐵 = [⟨1, 1, 0⟩⟨1, 1, 0⟩ ⟨0, 0, 1⟩⟨1, 1, 0⟩] whichsatisfy 𝐵𝐴 = 𝐴 The 𝐴𝑋 = 𝐵 becomes [⟨1, 1, 0⟩⟨1, 1, 0⟩ ⟨0, 0, 1⟩⟨1, 1, 0⟩][⟨𝑥𝑇 11,𝑥𝐼 11,𝑥𝐹 11⟩⟨𝑥𝑇 12,𝑥𝐼 12,𝑥𝐹 12⟩ ⟨𝑥𝑇 21,𝑥𝐼 21,𝑥𝐹 21⟩⟨𝑥𝑇 22,𝑥𝐼 22,𝑥𝐹 22⟩] = [⟨1, 1, 0⟩⟨1, 1, 0⟩ ⟨0, 0, 1⟩⟨1, 1, 0⟩] ⟨𝑥𝑇 11,𝑥𝐼 11,𝑥𝐹 11⟩ = ⟨min 𝑘 (𝜎𝑘1,𝑏𝑘1), min 𝑘 (𝜎 ′ 𝑘1,𝑏′ 𝑘1), max 𝑘 (𝜎 ′′ 𝑘1,𝑏′′ 𝑘1)⟩ = ⟨0, 0, 1⟩ ⟨𝑥𝑇 12,𝑥𝐼 12,𝑥𝐹 12⟩ = ⟨min 𝑘 (𝜎𝑘1,𝑏𝑘2), min 𝑘 (𝜎 ′ 𝑘1,𝑏′ 𝑘2), max 𝑘 (𝜎 ′′ 𝑘1,𝑏′′ 𝑘2)⟩ = ⟨1, 1, 0⟩ ⟨𝑥𝑇 21,𝑥𝐼 21,𝑥𝐹 21⟩ = ⟨min 𝑘 (𝜎𝑘2,𝑏𝑘1), min 𝑘 (𝜎 ′ 𝑘2,𝑏′ 𝑘1), max 𝑘 (𝜎 ′′ 𝑘2,𝑏′′ 𝑘1)⟩ = ⟨1, 1, 0⟩ ⟨𝑥𝑇 22,𝑥𝐼 22,𝑥𝐹 22⟩ = ⟨min 𝑘 (𝜎𝑘2,𝑏𝑘2), min 𝑘 (𝜎 ′ 𝑘2,𝑏′ 𝑘2), max 𝑘 (𝜎 ′′ 𝑘2,𝑏′′ 𝑘2)⟩ = ⟨1, 1, 0⟩ Therefore ˆ 𝑋 = [⟨0, 0, 1⟩⟨1, 1, 0⟩ ⟨1, 1, 0⟩⟨1, 1, 0⟩] . Clearly 𝐴 ˆ 𝑋𝐴 = 𝐴 Hence ˆ 𝑋 isthemaximumg-inverseof 𝐴 ˆ 𝑋𝐴 = 𝐴. Togettheminimalsolution:Letusfindtheminimum ˆ 𝐵 from ˆ 𝐵𝐴 = 𝐴 andusing theminimum ˆ 𝐵 in 𝐴𝑋 = 𝐵 wecanfindtheminimum ˆ 𝑋 Consider [⟨1, 1, 0⟩⟨1, 1, 0⟩ ⟨0, 0, 1⟩⟨1, 1, 0⟩][⟨1, 1, 0⟩⟨1, 1, 0⟩ ⟨1, 1, 0⟩⟨0, 0, 1⟩] = [⟨1, 1, 0⟩⟨1, 1, 0⟩ ⟨1, 1, 0⟩⟨0, 0, 1⟩] Step1.Determinetheset 𝐽𝑖𝑗 ( ˆ 𝐵) 11

𝐽11( ˆ 𝐵)= {𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨1, 1, 0⟩},𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}} = ⟨1, 1, 0⟩ = {⟨1, 1, 0⟩, ⟨1, 1, 0⟩} = {1, 2} 𝐽12( ˆ 𝐵)= {𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨1, 1, 0⟩},𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨0, 0, 1⟩}} = ⟨1, 1, 0⟩ = {⟨1, 1, 0⟩, ⟨0, 0, 1⟩} = {1} 𝐽21( ˆ 𝐵)= {𝑚𝑖𝑛{⟨0, 0, 1⟩, ⟨1, 1, 0⟩},𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}} = ⟨1, 1, 0⟩ = {⟨0, 0, 1⟩, ⟨1, 1, 0⟩} = {2} 𝐽22( ˆ 𝐵)= {𝑚𝑖𝑛{⟨0, 0, 1⟩, ⟨1, 1, 0⟩},𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨0, 0, 1⟩}} = ⟨0, 0, 1⟩ = {⟨0, 0, 1⟩, ⟨0, 0, 1⟩} = {1, 2}

Let 𝛽1 = 𝐽11( ˆ 𝐵) × 𝐽12( ˆ 𝐵)= {1, 2}×{1} = {(1, 2), (2, 1)} 𝛽2 = 𝐽21( ˆ 𝐵) × 𝐽22( ˆ 𝐵)= {2}×{1, 2} = {(2, 1), (2, 2)}

Step2. Determinetheset 𝐾(𝛽𝑘,𝑗) for 𝑘 =1, 2 and 𝑗 =1, 2

For 𝛽1 =(1, 1)𝐾(𝛽1, 1)= {1, 2} 𝐾(𝛽1, 2)= {𝜙}

For 𝛽1 =(2, 1)𝐾(𝛽1, 1)= {2} 𝐾(𝛽1, 2)= {1}

For 𝛽2 =(2, 1)𝐾(𝛽2, 1)= {2} 𝐾(𝛽2, 2)= {1}

For 𝛽2 =(2, 2)𝐾(𝛽2, 1)= {𝜙} 𝐾(𝛽2, 2)= {1, 2}

Writingthevaluesintabularformweget

(𝛽1,𝑗) 1 2 (1, 1) {1, 2} 𝜙 (2, 1) {2} 1 (𝛽2,𝑗) 1 2 (2, 1) {2} {1} (2, 2) {𝜙} {1, 2}

Step3. Foreach 𝛽𝑘 letusgeneratethe 𝑔(𝛽𝑘) tuples

For 𝛽1 =(1, 1) 𝑔1(𝛽1)=(1, 1) 𝑔1(𝛽1)=max 𝑘∈𝐾(𝛽,1){⟨1, 1, 0⟩, ⟨1, 1, 0⟩} = ⟨1, 1, 0⟩ 𝑔2(𝛽1)= ⟨0, 0, 1⟩

For (𝛽1)=(2, 1)

𝑔1(𝛽1)= ⟨1, 1, 0⟩

𝑔2(𝛽1)= ⟨1, 1, 0⟩

For 𝑔1(𝛽1)=(2, 1) 𝑔1(𝛽1)= ⟨1, 1, 0⟩ 𝑔2(𝛽1)= ⟨1, 1, 0⟩

For 𝑔1(𝛽1)=(2, 2) 𝑔1(𝛽2)= ⟨0, 0, 1⟩ 𝑔2(𝛽2)= 𝑚𝑎𝑥{⟨1, 1, 0⟩, ⟨0, 0, 1⟩} = ⟨1, 1, 0⟩

Thecorrespondingtabularformsaregivenby (𝛽1,𝑗) 𝑔(𝛽1) (1, 1) (⟨1, 1, 0⟩, ⟨0, 0, 1⟩) (2, 1) (⟨1, 1, 0⟩, ⟨1, 1, 0⟩)

(𝛽2,𝑗) 𝑔(𝛽2) (2, 1) (⟨0, 0, 1⟩, ⟨1, 1, 0⟩) (2, 1) (⟨0, 0, 1⟩, ⟨1, 1, 0⟩)

Bypairwisecomparisonwecanfindouttheminimumineachoftheabove table,weget ˆ 𝐵 = [⟨1, 1, 0⟩⟨0, 0, 1⟩ ⟨0, 0, 1⟩⟨1, 1, 0⟩]

Usingtheminimum ˆ 𝐵 in 𝐴𝑋 = 𝐵 wecanfindtheminimum ˆ 𝑋 Now 𝐴𝑋 = ˆ 𝐵 is [⟨1, 1, 0⟩⟨1, 1, 0⟩ ⟨1, 1, 0⟩⟨0, 0, 1⟩][⟨0, 0, 1⟩⟨1, 1, 0⟩ ⟨1, 1, 0⟩⟨1, 1, 0⟩] = [⟨1, 1, 0⟩⟨0, 0, 1⟩ ⟨0, 0, 1⟩⟨1, 1, 0⟩]

Step.4 Determinetheset 𝐽𝑖𝑗 ( ˆ 𝐵) 𝐽11( ˆ 𝑋)= {𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨0, 0, 1⟩},𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}} = ⟨1, 1, 0⟩ = {⟨0, 0, 1⟩⟨1, 1, 0⟩ = {2} 𝐽12( ˆ 𝑋)= {𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨1, 1, 0⟩},𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}} = ⟨0, 0, 1⟩ = {⟨1, 1, 0⟩⟨1, 1, 0⟩ = {𝜙}

21( ˆ 𝑋)= {𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨0, 0, 1⟩},𝑚𝑖𝑛⟨{⟨0, 0, 1⟩, ⟨1, 1, 0⟩}} = ⟨0, 0, 1⟩ = {⟨0, 0, 1⟩⟨0, 0, 1⟩ = {1, 2} 𝐽22( ˆ 𝑋)= {𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨1, 1, 0⟩},𝑚𝑖𝑛⟨{⟨0, 0, 1⟩, ⟨1, 1, 0⟩}} = ⟨1, 1, 0⟩

𝛽2 = 𝐽21 ˆ 𝐵 × 𝐽22 ˆ 𝐵 = {1, 2}×{1} = {(1, 1)(2, 1)}

Step5. Determinetheset 𝐾(𝛽𝑘,𝑗) fork=1,2andj=1,2

For 𝛽1 = {2} 𝐾(𝛽1, 1)= 𝜙 𝐾(𝛽1, 2)= {1}

For 𝛽2 = {2, 1}𝐾(𝛽2, 1)= {2} 𝐾(𝛽2, 2)= {1} 𝑘(𝛽1,𝑗) 1 2 {2}× 𝜙 𝜙 {1} 𝑘(𝛽2,𝑗) {1, 2} 𝜙 (1, 1) {1, 2} 𝜙 (2, 1) {2} {1}

Step6:Foreach 𝛽𝑘 Letasgeneratethe 𝑔(𝛽𝑘) tuples.

For 𝛽1 = {2}× 𝜙 𝑔1(𝛽1)= ⟨0, 0, 1⟩ 𝑔2(𝛽2)= ⟨1, 1, 0⟩ 𝐹𝑜𝑟𝛽2 =(1, 1) 𝑔1(𝛽2)= ⟨1, 1, 0⟩ 𝑔2(𝛽2)= ⟨0, 0, 1⟩ 𝐹𝑜𝑟𝛽2 =(2, 1) 𝑔1(𝛽2)= ⟨1, 1, 0⟩ 𝑔2(𝛽2)= ⟨0, 0, 1⟩

Thecorrespondingtabularformsaregivenby 𝛽1 𝑔(𝛽1) {2}× 𝜙 ⟨0, 0, 1⟩, ⟨1, 1, 0⟩ 𝛽2 𝑔(𝛽2) (1, 1) ⟨1, 1, 0⟩, ⟨0, 0, 1⟩ (2, 1) ⟨1, 1, 0⟩, ⟨0, 0, 1⟩

Togetthe 𝑋 selectaminimumrowfromeachtable,thatis ˇ 𝑋 = [⟨0, 0, 1⟩⟨1, 1, 0⟩ ⟨1, 1, 0⟩⟨0, 0, 1⟩] Clearlythis ˇ 𝑋 willsatisfy 𝐴𝑋𝐴 = 𝐴 andweobservethat [𝑋, ˆ 𝑋]= {[⟨0, 0, 1⟩⟨1, 1, 0⟩ ⟨1, 1, 0⟩⟨𝛼,𝛼 ′ ,𝛼 ′′ ⟩] ∣0 ≤ 𝛼 ≤ 1, 0 ≤ 𝛼 ′ ≤ 1 and 0 ≤ 𝛼 ′′ ≤ 1 with 𝛼 + 𝛼 ′ + 𝛼 ′′ ≤ 3} isthesetofallg-inversein [ ˇ 𝑋, ˆ 𝑋]

Conclusion: Themaximumandminimumsolutionoftherelationalequation 𝑥𝐴 = 𝑏 and 𝐴𝑥 = 𝑏 hasbeenobtained.Usingthisrelationalequationmaximumand minimumg-inverseofafuzzyneutrosophicsoftmatrixarealsofound.

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