Bipolartopologicalpre-closedneutrosophicsets
G.UpenderReddy1*,T.SivaNageswaraRao2,N.SrinivasaRao 3,V.VenkateswaraRao 4
Abstract
Weextendthetopologicalspacestobipolarneutrosophicstudy.Theconceptoftopologicalspacesinneutrosophictheoryrecentlydiscussedbymanyauthorsonsemigeneralizedconditionsofopenandclosedsets.In thispresentmanuscriptthetopologicalspaceisdefinedonbipolarpre-closedneutrosophicsets(BPCNS).
Keywords
Bipolarneutrosophicclosure,Bipolarneutrosophicinterior,Bipolarpre-closedneutrosophicsets(BPCNS).
AMSSubjectClassification 18B30,03E72.
1DepartmentofMathematics,NizamCollege(A),OsmaniaUniversity,Basheerbagh,Hyderabad,TS,India. 2,3,4DepartmentofMathematics,Vignan’sFoundationforScienceTechnologyandResearch(DeemedtobeUniversity),Vadlamudi,Guntur (Dt.),A.P,India.
*Correspondingauthor:yuviganga@gmail.com. ArticleHistory:Received 24 September 2020;Accepted 21 December 2020 c 2021MJM.
sets.R.Chang[3]introducedneutrosophicgeneralizedclosed setswithgeneralizedneutrosophiccontinuousmapping.In thisarticle,toevaluatethebipolarpre-closedneutrosophicsets (BPONS)inbipolarneutrosophictopologicalspaces.
Notations:
1.Introduction
NeutrosophictechniquegivesaexactinformationaboutneutralreportsandthistechniquewasderivedbyFlorentinSmarandache[5]in1998onsets,logicandprobabilityarereciprocatedwithneutrosophictechnique.Neutrosophictechniqueis consequentobtainedfromFuzzyandintuitionisticFuzzytechnique.Neutrosophictechniquemeanstodiscoverordinary featuresonnonordinaryvalues.T,I,Fareknownasneutrosophicvaluestheystandforthetruthmembership,indeterminacymembershipandfalsemembership.Thetechniqueinthis everyintentionisprojectedtocontaintheentitlementof T ,the entitlementof Iandentitlementof F with 0 ≤ T + I + F ≤ 3as theyaremanyneutrosophicrulesofinference.
J.Dezert[8]demonstratedfuzzyconceptwhileanumericalinstrumentintendedforconsideringalluncertaintiesofelementswitheveryinputtoobtain.FuzzylogicandIntuitionisticFuzzylogicwasderivedbyL.Zadeh[11]andAtanassov[9] respectively,whentheoverviewoffuzzyanywhereaswellas thecontributionwitheachcomponent.A.A.Salamaet.al[1] establishedthemodeloftopologicalspacesonneutrosophic
Thisarticlebasedonthesoftneutrosophictopology.Here westartwithsomebasicdefinitions.
Contents 1 Introduction .......................................159 2 Preliminaries ......................................160 3
..161 4 Conclusion ........................................161 References ........................................161
Bipolarpre-closedneutrosophicsets(BPCNS)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Neutrosophicset(NS)
BipolarNeutrosophicset(BNS)
Bipolarpre-openneutrosophicsets(BPONS)
Bipolarpre-closedneutrosophicsets(BPCNS).
BipolarNeutrosophictopology(BNT)
Bipolarneutrosophicclosure(BNC)
Bipolarneutrosophicinterior(BNI)
Bipolarneutrosophicopenset(BNOS)
Bipolarneutrosophicclosedset(BNCS)
Bipolarneutrosophicsemi-closed(BNSC)
Bipolarneutrosophicsemiopen(BNSO)
2.Preliminaries
Werememberseveralessentialterminologyexactingstudy ofSmarandache[18],terminologybasedonAtanassovin[10] andnotationsbasedonA.A.Salama[2]. ThenumberoftypesofNeutrosophictopologyproblemscontainedtoresidentialthroughalotofresearcherswhoassign theweightstoedgesarenotaccurateinthatanuncertaintyis thereforthatseethereferences[4,6,7,11,12,13,14,15,16, 17,19,20,21,22].Inthissection,werecallsomedefinitions andbasicresultsoffractionalcalculuswhichwillbeused throughoutthepaper.
Definition2.1. ANeutrosophicset(NS)istripletstructure (T, I, F ) whichisthedegreeofcontributionfunctionswiththe circumstance 0 ≤ T + I + F ≤ 3 andalsoliesamong0and 1.[5]
Definition2.2. ABipolarNeutrosophicset(BNS)isthestructurehavingsixvaluesinthatthreearepositiveandremaining threearenegative.Thestructureis T N , IN , F N , T P , IP , F P andalsothenegativevaluesliesbetween[-1,0]andpositive valuesliesbetween[0,1].
Definition2.3. Let S1 =(TS1 , IS1 , FS1 ) and S2 =(TS2 , IS2 , FS2 ) aretheBNSinthatconsidertwopossiblecasesforsubsets.
1 S1 ⊆ S2 ⇔ Ts1 ≤ Ts2 , Is1 ≤ Is2 andFs1 ≥ Fs2 2.S1 ⊆ S2 ⇔ Ts1 ≤ Ts2 , Is1 ≥ Is2 andFs1 ≥ Fs2 .
Definition2.4. Let S1 =(TS1 , IS1 , FS1 )and S2 =(TS2 , IS2 , FS2 ) aretheBNSthenfollowingmaybedefinedas
(I1)S1 ∩ S2 = TS1 ∧ TS2 , IS1 ∧ IS2 , FS1 ∨ FS2 (I1)S1 ∪ S2 = TS1 ∨ TS2 , IS1 ∧ IS2 , FS1 ∧ FS2
Theorem2.5. Let S1 =(TS1 , IS1 , FS1 ) and S2 =(TS2 , IS2 , FS2 ) aretheBNSthenthesubsequentsituationareholds.
(1)BNC(S1 ∩ S2)= BNC(S1) ∪ BNC(S2)
(2)BNC(S1 ∪ S2)= BNC(S1) ∩ BNC(S2)
Definition2.6. ABNTisanon-emptyset W isafamily BτN ofaBNSinWholdingthesubsequentsituation.
(BNT1) 0, 1 ∈ BτN
(BNT2) S1 ∩ S2 ∈ BτN foranyS1, S2 ∈ BτN
(BNT3) ∪ Si ∈ BτN forevery {Si : i ∈ I} ⊆ BτN Therefore (W, BτN )isBNT.
Example2.7. LetW = {w} and P = {< w, 0 2, 0 1, 0 5:0 4, 0 4, 0 3 >: w ∈ W } Q = {< w, 0.2, 0.6, 0.4;0.3, 0.4, 0.7 >: w ∈ W } R = {< w, 0.6, 0.3, 0.2;0.4, 0.5, 0.3 >: w ∈ W } S = {< w, 0.7, 0.2, 0.5;0.4, 0.6, 0.8 >: w ∈ W }
Thenthefamily τ = {0N , 1N , P, Q, R, S} ofbipolarneutrosophicsetinWisbipolarneutrosophictopologyW.
Definition2.8. Let (W, BτN ) beBNTspace S1 = TS1 , IS1 , FS1 beaBNSin W thenBNCandBNIof S1 aredefinedasfollows
BNcl(S1)= ∩ {R1, r1 isaBNcsinWandS1 ⊆ R1}
BNint(S1)= ∪ {R2, r2 isaBNosinWandR2 ⊆ S1}
Itpreservesaswellshowthat BNcl(S1) isBNCsetand BNint(S1) isBNOsetinW. (i)S1 isBNOS⇔ S1 = BNint(S1) (ii)S1 isBNCS⇔ S1 = BNcl(S1)
Theorem2.9. Let(W, BτN ) beBNTspaceand S1 ∈ (W, BτN ) wehave
(a) BNcl (C (S1))= C (BNint (S1))
(b) BNint (C (S1))= C (BNcl (S1))
Theorem2.10. Let (W, BτN ) beBNTspaceand S1, S2 are twoBNSinWthenthesubsequentrulesarehold.
a) BNint (S1) ⊆ S1 b) S1 ⊆ BNcl (S1) c) S1 ⊆ S2 ⇒ BNint (S1) ⊆ BNint (S2) d) S1 ⊆ S2 ⇒ BNcl (S1) ⊆ BNcl (S2)
e) BNint (BNint (S1))= BNint (S1) f ) BNcl (BNcl (S1))= BNcl (S1) g) BNint (S1 ∩ S2)= BNint (S1) ∩ BNint (S2) h) BNcl (S1 ∪ S2)= BNcl (S1) ∪ BNcl (S2) i) BNint (0N )= BNcl (0N )= 0N j) BNint (1N )= BNcl (1N )= 1N k) S1 ⊆ S2 ⇒ BNC (S1) ⊆ BNC (S2)
l) BNCL (S1 ∩ S2) BNcl (S1) ∩ BNcl (S2)
m) BNint (S1 ∪ S2) ⊇ BNint (S1) ∪ BNint (S2)
Definition2.11. Let (W, BτN ) beBNTspaceand S1 =(TS1 , IS1 , FS1 ) beaBNSinWthenS1 isaBNSO ifS1 ⊆ BNcl (BNint (S1)) andalso BNSCifBNint (BNcl (S1)) ⊆ S1 ThecomplementofBNSOsetisaBNSCset.
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3.Bipolarpre-closedneutrosophicsets (BPCNS)
Definition3.1. Let S1 beBNSofaBNTspaces W .Then S1 isknownasBPCNSof W ifthereexistsa BNC setsuchthat BNcl (BNC (S1)) ⊆ S1 ⊆ BNC (S1)
Theorem3.2. Anysubset S1 ina BNTSW is BNCS setiff BNcl (BNint (S1)) ⊆ S1 Proof: consider BNcl (BNint (S1)) ⊆ S1
ThenBNC = BNcl(S1) clearlyBNcl (BNint (S1)) ⊆ S1 ⊆ BNC(S1) ThereforeS1isBNPCset
Ontheotherhand,assumethatLet S1 be BNPC containedin W
Then BNC (BNint (S1)) ⊆ S1 ⊆ BNC (S1) forsome BNS closed setBNC. butBNcl (S1) ⊆ BNC (S1). Hencethetheoremproved.
Theorem3.3. Let (W, BτN ) be BNTS and S1beaBNSof W thenS1 isaBPCNSiffBNC (S1) isBPONSinW.
Proof: LetS1 beaBPCNSsubsetofW. ClearlyBNcl (BNint (S1)) ⊆ S1
Byapplyingcomplementontogethersides
BNC (S1) ⊆ BNC (BNcl (BNint (S1)))
BNC (S1) ⊆ BNint (BNcl (BNC (S1)))
HenceBNC (S1)isBNPOset
Onthecontrary,assumethatBNC (S1) isBNPOset i.e.BNC (S1) ⊆ BNint (BNcl (BNC (S1))) Byapplyingcomplementontogethersidesobtained
BNcl (BNint (S1)) ⊆ S1 isBPCNS. Hencethetheoremproved.
Theorem3.4. Consider (W, BτN ) beaBNTspaces.Then conjunctionoftwoBPCNSisalsoaBPCNS.
Proof: Consider S2 and S2 arebipolarneutrosophicpreclosedsetson (W, BτN ). ThenBNcl (BNint (S1)) ⊆ S1,BNcl (BNint (S2)) ⊆ S2 ConsiderS1 ∩ S2 ⊇ BNcl (BNint (S1)) ∩ BNcl (BNint (S2))
S1 ∩ S2 ⊇ BNcl (BNint (S1)) ∩ (BNint (S2))
S1 ∩ S2 ⊇ BNcl (BNint (S1 ∪ S2)) BNcl (BNint (S1 ∩ S2)) ⊆ S1 ∩ S2
HenceS1 ∩ S2 isBPCNS.
Remark3.5. TheunionofanytwoBPCNSneednotbea BPCNSon (W, BτN )
Theorem3.6. Let {S1}α ∈∆ beacollectionofBPCNSon (W, BτN ) then ∩ α ∈∆ S1 α ∈∆ BPCNSon (W, BτN ) Proof: AneutrosophicsetBNCα suchthat
BNCα (BNint (S1)) ⊆ S1 α ⊆ BNCα (S1) forall α ∈ ∆ Then ∩ α ∈∆ BNCα (BNint (S1)) ⊆∩ α ∈∆ S1 α ⊆∩ α ∈∆ BNCα (S1) ∩ α ∈∆
BNclα (BNint (S1)) ⊆∩ α ∈∆ S1 α
Hence ∩ α ∈∆ S1 α isBPCNSon(W, BτN )
Theorem3.7. EveryBNCSintheBNTspaces (W, BτN ) is BPCNSin (W, BτN ) Proof: LetS1 beBNCSmeansS1 = BNcl (S1) andalso BNint (S1) ⊆ S1 Fromthat BNcl (BNint (S1)) ⊆ BNcl (S1) , BNcl (BNint (S1)) ⊆ S1, sinceS1 = BNcl (S1) HenceS1 isaBPCNS.
Theorem3.8. Let S1 beaBNCSinBNTspaces (W, BτN ) and suppose BNint (S1) ⊆ S2 ⊆ S1then S2 isBPCNSon (W, BτN ) Proof: LetS1 beaBNSinBNTspaces (W, BτN ) SupposeBNint (S1) ⊆ S2 ⊆ S1 ThereexistaBNCS,suchthat
BNC (BNint (S1)) ⊆ S2 ⊆ S1 ⊆ BNC. ThenS2 ⊆ BNCandalsoBNint(S2) ⊆ S2 ⊆ BNC. Thus,BNcl(BNint(S2)) ⊆ S2 HenceS2isbipolarneutrosophicpre-closedseton (W, BτN ).
Theorem3.9. Let W1 and W2 areBNTspaceswiththeconditions W1 isBNmultiplicationassociatedto W2 thentheBN multiplicationassociatedto S1 × S2 isaBPCNSoftheBN multiplicationassociatedtotopologicalspace W1 ×W2.Where BPCNSS1 ofW1 andaBPCNSS2 ofW2 Proof:LetS1 andS2 areBPCNS. BNC1 (BNint (S1)) ⊆ S1 ⊆ BNC1 and BNC2 (BNint (S2)) ⊆ S2 ⊆ BNC2 Formtheabove, BNC1 (BNint (S1)) × BNC2 (BNint (S2)) ⊆ S1 × S1 ⊆ BNC1 × BNC2 (BNC1 × BNC2)(BNint(S1 × S2)) ⊆ S1 × S2 ⊆ BNC1 × BNC2 HenceS1 × S2 isBPCNSinBNTspaceW1 × W2
4.Conclusion
Inthisarticle,BipolartopologicalPre-closedneutrosophic setsexplainedonpreclosedbipolarneutrosophictheory.we discussedaboutsomebasicdefinitionsaboutneutrosophic topologicalspace,bipolarpre-closedneutrosophicsetetc.,. Furtherweobtainedtheresultsbasedonpre-closedsetswith similarresults
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ISSN(P):2319 3786 MalayaJournalofMatematik ISSN(O):2321 5666
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