Bipolarsoftneutrosophictopologicalregion
G.UpenderReddy1*,T.SivaNageswaraRao2,N.SrinivasaRao 3 andV.VenkateswaraRao 4
Abstract
Inthisarticledeals,differentareaswithuncertaintydatainformationinbipolarsoftneutrosophictopology.Inthe pasttime,somanyauthorsarediscussedaboutneutrosophicandbipolarneutrosophictheory.Softneutrosophic SettheorywasderivedbyMaji.Thepresentarticleextendedtobipolarsoftspatialregion.Alsoweobtained definitionsofSoftopen,softclosed,softpre-open,softpre-closedonthebipolarneutrosophic.
Keywords
Softneutrosophicset;bipolarsoftneutrosophictopology;bipolarsoftneutrosophicspatialareas.
AMSSubjectClassification
11B05.
1DepartmentofMathematics,NizamCollege(A),OsmaniaUniversity,Basheerbagh,Hyderabad,TS,India. 2,3,4DepartmentofMathematics,Vignan’sFoundationforScienceTechnologyandResearch(DeemedtobeUniversity),Vadlamudi,Guntur (Dt.),A.P,India.
*Correspondingauthor:yuviganga@gmail.com,shivathottempudi@gmail.com,srinudm@gmail.com,vunnamvenky@gmail.com.
ArticleHistory:Received 11 June 2020;Accepted 13 September 2020 c 2020MJM.
1.Introduction Everywhereintheworlduncertaintysituationsarethere ineachcase.Inparticularlymathematicstherearedifferent fieldswithuncertaintyproblems.EspeciallyFuzzytheory [14]andIntuitionistfuzzytheory[1]authorsfindoutdifferent problemsdealwithuncertainty.Byovercomethisuncertainty, Smarandache[8]derivedneutrosophictheory.Maji[3]collectivethetwotopicssoftsetsandneutrosophictheory.The authoralsohavethesomemoreresearchworkonneutrosophictheoryseethereferences[2,4,5,6,7,9,10,11,12, 13]. Notations:
Bipolarsoftneutrosophicpre-opensets(BSNPOS) 4. Bipolarsoftneutrosophicpre-closedsets(BSNPCS). 5. BipolarsoftNeutrosophictopology(BSNT) 6. Bipolarsoftneutrosophicclosure(BSNC) 7. Bipolarsoftneutrosophicinterior(BSNI) 8. Bipolarsoftneutrosophicopenset(BSNOS) 9. Bipolarsoftneutrosophicclosedset(BSNCS) 10. Bipolarsoftneutrosophicsemi-closed(BSNSC) 11. BipolarSoftneutrosophicsemiopen(BSNSO)
Thisarticlebasedonthesoftneutrosophictopology.Here westartwithsomebasicdefinitions.
2.Preliminaries
Inthissection,werecallsomedefinitionsandbasicresults offractionalcalculuswhichwillbeusedthroughoutthepaper. Definition2.1. (W, Z) isasoftsetin Ω where W : Z → ˜ P (Ψ) isamappingwhere ˜ P (Ψ) isapowersetof Ψ Weexpress (W, Z) by ˜ W ˜ W = {( f ,W ( f )) : f ∈ Z}
Contents 1 Introduction ......................................1687 2
.....................................1687 3 BipolarSoftneutrosophictopologicalspace ...1688 4 Bipolarsoftneutrosophicnearlyopensets
5 Bipolarsoftneutrosophicregion ................1689 6 Conclusion .......................................1690 References .......................................1690
Preliminaries
.....1689
2.
3.
1. BipolarSoftNeutrosophic(BSN)
BipolarsoftNeutrosophicset(BSNS)
Definition2.2. AbipolarneutrosophicsetBon Ψ isdefinedas: B = {< z, εBN (z) , φBN (z) , ϕBN (z) , εBP (z) , φBP (z) , ϕBP (z) >: z ∈ Ψ} where εBP, φBP, ϕBP : Ψ →] 0, 1+[ and εBN , φBN , ϕBN : Ψ →] 1, 0 [ and 3 ≤ εBN (z)+ φBN (z)+ ϕBN (z)+ εBP (z)+ φBP (z) +ϕBP (z) ≤ 3+
Definition2.3. Let Ψ bethesetandZbeparameterset. Let P (Ψ) representedthesetofallBSNSof Ψ . Then (W, Z) isknownasBSNSover Ψ where W : Z → P (Ψ) isamapping. WeexpresstheBSNS (W, Z) by ˜ WNu Thatis, ˜ WNu = {( f , {< z, εNWNu (z), φNWNu (z), ϕNWNu (z), εPWNu (z), φPWNu (z), ϕPWNu (z) >: z ∈ Ψ}) f ∈ Z}
Definition2.4. ThecomplementoftheBSNS ˜ WNu isrepresentedby ( ˜ WNu)C andisdefinedby ( ˜ WNu)C = {( f , {< z, ϕNWNu (z), φNWNu (z), εNWNu (z), ϕPWNu (z), φPWNu (z), εPWNu (z) >: z ∈ Ψ}) f ∈ Z}
Definition2.5. ForanytwoBSNS WNu and ˜ SNu over Ψ , WNu isaBSNsubsetof ˜ SNu if εNWNu (z) ≤ εNSNu (z) ; εPWNu (z) ≤ εPSNu (z) φNWNu (z) ≤ φNSNu (z);φPWNu (z) ≤ φPSNu (z) ϕNWNu (z) ≥ ϕNSNu (z) ; ϕPWNu (z) ≥ ϕPSNu (z) forallf ∈ Zandz ∈ Ψ.
Definition2.6. ABSNS ˜ WNu over Ψ issaidtobenullBSNS if ϕNWNu (z)= 0 ; ϕPWNu (z)= 0 φNWNu (z)= 0;φPWNu (z)= 0;εNWNu (z)= 1 ; εPWNu (z)= 1 forallf ∈ Zandz ∈ Ψ .Itisdenotedby Φ⊕ Nu
Definition2.7. ABSNS ˜ WNu over Ψ issaidtobeabsolute BSNSif
ϕNWNu (z)= 1 ; ϕPWNu (z)= 1 φNWNu (z)= 1; φPWNu (z)= 1; εNWNu (z)= 0 ; εPWNu (z)= 0 forallf ∈ Zandz ∈ Ψ Itisrepresentedby ΨNu Definition2.8. ThedisjunctionoftwoBSNS WNu and SNu is representedby WNu ∪ SNu andisdefinedby UNu = WNu ∪ SNu asfollows
ϕNWNu (z) iff ∈ B C ϕNSNu (z) iff ∈ C B min ϕNWNu (z) , ϕNSNu (z) iff ∈ B ∩ C
ϕPUNu (z)=
ϕPWNu (z) iff ∈ B C ϕPSNu (z) iff ∈ C B min ϕPWNu (z) , ϕPSNu (z) iff ∈ B ∩ C
Definition2.9. TheconjunctionoftwoBSNS ˜ WNu and SNu is representedby ˜ WNu ∩ SNu andisdefinedby ˜ UNu = ˜ WNu ∩ SNu ,asfollows εNUNu (z)= min{εNWNu (z), εNSNu (z)}; εPUNu (z)= min{εPWNu (z), εPSNu (z)}
φNUNu (z)= φNWNu (z)+φNSNu (z) 2 ; φPUNu (z)= φPWNu (z)+φPSNu (z) 2 ϕNUNu (z)= max{ϕNWNu (z), ϕNSNu (z) }; ϕPUNu (z)= max{ϕPWNu (z), ϕPSNu (z) }
3.BipolarSoftneutrosophictopological space
Definition3.1. LetBSNS(Ψ, Z) bethefamilyofallBNSS over Z and NBSτ ∗ ⊂ BSNS (Ψ, Z) Then NBSτ ∗ isknownas bipolarsoftneutrosophictopology(BSNT)on (Ψ, Z) ifthe subsequentcircumstancesaresatisfied
(i) ΦBNu, ˜ ΨBNu ∈ NBSτ ∗ (ii) NBSτ ∗ isclosedunderarbitrarydisjunction. (iii) NBSτ ∗ isclosedunderinfiniteconjunction. Thenthetriplet Ψ, ˜ NBSτ ∗, Z isknownasBSNTspace. Theelementsof ˜ NBSτ ∗ areknownBSNOSin Ψ, ˜ NSτ ∗, Z ABSNS ˜ WBNu inBSNS(Ψ, Z) issoftclosedin Ψ, ˜ NBSτ ∗, Z ifitscomplement ˜ WBNu C isBSNOSin Ψ, ˜ NBSτ ∗, Z TheBSNclosureof ˜ WBNu istheBSNS, BNu ≈ SCL(WBNu)= ∩{ ˜ SBNu : ˜ SBNu isbipolarneutrosophic softclosedand WBNu ⊆ ˜ SBNu } TheBSNinteriorof WNu istheBNSS, BNu ≈ SINT (WBNu)= ∪{SBNu : SBNu isbipolarneutrosophic softclosedand WBNu ⊆ SBNu }. Itiseasytoseethat WBNu isBSNopeniff WBNu = BNu ≈ SINT WBNu andBSNclosedifandonlyifBNu ≈ SCL ˜ WBNu .
Theorem3.2. Let Ψ, ˜ NBSτ ∗, Z beaBSNTSover (Ψ, Z) and WBNu and SBNu ∈ (Ψ, Z) then (i) BNu ≈ SINT WBNu ⊂ WBNu and BNu ≈ SINT WBNu is thelargestopenset. (ii) WBNu ⊂ WBNu impliesBNu ≈ SINT WBNu ⊂ BNu ≈ SINT ˜ WBNu (iii) BNu ≈ SINT ˜ WBNu isanBSNOS. ThatisBNu ≈ SINT ˜ WBNu ∈ ˜ NBSτ ∗
Bipolarsoftneutrosophictopologicalregion—1688/1690
εNUNu
z
εNWNu (z) iff ∈ B C εNSNu
z) iff ∈ C B max εNWNu (z) , εNSNu (z) iff ∈ B ∩ C εPUNu (z)= εPWNu (z) iff ∈ B C εPSNu (z) iff ∈ C B max εPWNu (z) , εPSNu (z) iff ∈ B ∩ C φNUNu (z)= φNWNu (z) iff ∈ B C φNSNu (z) iff ∈ C B max φNWNu (z) , φNSNu (z) iff ∈ B ∩ C φPUNu (z)= φPWNu (z) iff ∈
φPSNu
z
max φPWNu
z
φPSNu
z
ϕNUNu
(
)=
(
B C
(
) iff ∈ C B
(
) ,
(
) iff ∈ B ∩ C
(z)=
1688
(iv) WBNu isBSNOBNu ≈ SINT WBNu = WBNu (v) BNu ≈ SINT BNu ≈ SINT ˜ WBNu = BNu ≈ SINT ˜ WBNu (vi) BNu ≈ SINT ΦBNu = ΦBNu, BNu ≈ SINT ˜ ΨBNu = ΨBNu (vii) BNu ≈ SINT ˜ WBNu ∩ SBNu = BNu ≈ SINT WBNu ∩ BNu ≈ SINT ˜ SBNu (viii) BNu ≈ SINT WBNu ∪ BNu ≈ SINT SBNu ⊂ BNu ≈ SINT ˜ WBNu ∪ SBNu Theorem3.3. Let Ψ, NBSτ ∗, Z beaBSNTS (Ψ, Z) and WBNu and SBNu ∈ (Ψ, Z) then (i) ˜ WBNu ⊂ BNu ≈ SCL ˜ WBNu andBNu ≈ SCL ˜ WBNu arethesmallestclosedsets
(ii) WBNu ⊂ WBNu impliesBNu ≈ SCL WBNu ⊂ BNu ≈ SCL WBNu (iii) BNu ≈ SCL WBNu isBSNCS. ThatisBNu ≈ SCL WBNu ∈ NBSτ ∗ C (iv) WBNu isbipolarneutrosophicsoftclosed BNu ≈ SCL WBNu = WBNu (v) BNu ≈ SCL BNu ≈ SCL WBNu = BNu ≈ SCL ˜ WBNu (vi) BNu ≈ SCL ΦBNu = ΦBNu, BNu ≈ SCL ˜ ΨBNu = ˜ ΨBNu (vii)BNu ≈ SCL WBNu ∩ BNu ≈ SCL SBNu ⊂ BNu ≈ SCL WBNu ∩ ˜ SBNu (viii) BNu ≈ SCL WBNu ∪ BNu ≈ SCL SBNu = BNu ≈ SCL ˜ WBNu ∪ SBNu
4.Bipolarsoftneutrosophicnearlyopen sets
Definition4.1. Let Ψ, NSτ ∗, Z beaBSNTSand WBNu beaBSNOSin (Ψ, Z),then WBNuisknownas (i)Bipolarsoftneutrosophic α -open ⇔ WBNu ⊆ BNu ≈ SINT BNu ≈ SCL BNu ≈ SINT WBNu (ii)Bipolarsoftneutrosophicpre-open ⇔ ˜ WBNu ⊆ BNu ≈ SINT BNu ≈ SCL ˜ WBNu (iii)Bipolarsoftneutrosophicsemi-open ⇔ WBNu ⊆ BNu ≈ SCL BNu ≈ SINT WBNu (iv)Bipolarsoftneutrosophic β -open ⇔ WBNu ⊆ BNu ≈ SCL BNu ≈ SINT BNu ≈ SCL WBNu (v)Bipolarsoftneutrosophicregular-open ⇔ WBNu ⊆ BNu ≈ SINT BNu ≈ SCL WBNu
Definition4.2. Let Ψ, ˜ NBSτ ∗, Z beaBSNTSand e ˜ WBNu ∈ (Ψ, Z),then ˜ WNu isknownas (i)Bipolarsoftneutrosophic α -closed ⇔ BNu ≈ SCL BNu ≈ SINT BNu ≈ SCL WBNu ⊆ WBNu
(ii)Bipolarsoftneutrosophicpre-closed ⇔ BNu ≈ SCL BNu ≈ SINT WBNu ⊆ WBNu
(iii)Bipolarsoftneutrosophicsemi-closed ⇔
BNu ≈ SINT BNu ≈ SCL WBNu ⊆ WBNu
(iv)Bipolarsoftneutrosophic β -closed ⇔ BNu ≈ SINT BNu ≈ SCL BNu ≈ SINT ˜ WBNu ⊆ ˜ WBNu
(v)Bipolarsoftneutrosophicregular-closed ⇔ ˜ WBNu = BNu ≈ SCL BNu ≈ SINT ˜ WBNu
5.Bipolarsoftneutrosophicregion
Topologydealswithsurfaceareastudyinthatanalysis ofGeographicalinformationsystems(GIS)andGeospatial databases.Thereisalotofproblemsontheuncertaintyon theregions.Further,gothroughthesomedefinitionsand proposalsforaBSNTregion,whichsupplyahypothetical structureforthemodelingofBSNTrelationssurroundedby uncertainregions.
Definition5.1. Let Ψ, ˜ NBSτ ∗, Z beaBNSTSover (Ψ, Z) and ˜ WBNu ∈ BSNS (Ψ, Z).ThenBSNboundaryof ˜ WBNu is definedby ℑWBNu = BNu ≈ SCL WBNu ∩ BNu ≈ SCL WBNu C
Definition5.2. Let Ψ, ˜ NBSτ ∗, Z beaBSNTSover (Ψ, Z) ThentheBSNexteriorof ˜ WBNu ∈ BSNS (Ψ, Z) isrepresented by WBNu ext andisdefinedby WBNu ext = BNu ≈ SINT WBNu C Theorem5.3. Let WBNu and SBNu betwoBSNSover (Ψ, Z). Then (i) WBNu ext = BNu ≈ SINT WBNu C (ii) WBNu ∪ ˜ SBNu ext = WBNu ext ∩ ˜ SBNu ext (iii) ˜ WBNu ext ∪ SBNu ext ⊂ ˜ WBNu ∩ SBNu ext Theorem5.4. Let Ψ, NBSτ ∗, Z beaBSNTSover (Ψ, Z) and ˜ WBNu, SBNu ∈ BSNS (Ψ, Z) Then (i) ℑWBNu C = BNu ≈ SINT ˜ WBNu ∪ BNu ≈ SINT ˜ WBNu C (ii)BNu ≈ SCL ˜ WBNu = BNu ≈ SINT ˜ WBNu ∪ ℑ ˜ WBNu (iii) ℑ ˜ WBNu = BNu ≈ SCL ˜ WBNu ∩ BNu ≈ SCL ˜ WBNu C (iv) ℑWBNu ∩ BNu ≈ SINT WBNu = ˜ ΦBNu (v) ℑ ℑ ℑ ℑWBNu = ℑ ℑWBNu Definition5.5. Let Ψ, ˜ NBSτ ∗, Z beaBSNTSover(Ψ, Z) Thenacoupleofnon-emptyBSNOSare WBNu, ˜ SBNu isknown asaBSNseparationof Ψ, NBSτ ∗, Z if ΨBNu = ˜ WBNu ∪ SBNu and ˜ WBNu ∩ SBNu = ΦBNu Definition5.6. ABSNTS Ψ, NBSτ ∗, Z isknownasBSN connectediftheredoesnotpresentaBSNseparationof Ψ, ˜ NBSτ ∗, Z . Otherwise Ψ, ˜ NBSτ ∗, Z isknownasBSNdisconnected. Next,wegothroughamodelforspatialBSNregionbasedon BSNconnectedness.
Definition5.7. Let Ψ, NBSτ ∗, Z beaBSNTS.AspatialBSN regionin (Ψ, Z) isanonemptyBSNsubset ˜ WBNu suchthat (i)BNu ≈ SINT ˜ WBNu isBSNconnected. (ii) ˜ WBNu = BNu ≈ SCL BNu ≈ SINT ˜ WBNu
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6.Conclusion
Inthisarticle,Bipolarsoftneutrosophictopologicalregion explainedonsoftopen,softclosed,softpre-openandsoft pre-closedonthebipolarneutrosophictheory.Wediscussed aboutsomebasicdefinitionsaboutneutrosophictopological space,bipolarsoftneutrosophicsetetc.,.Furtherweobtained theresultsbasedonsoftopenandsoftclosedwithsimilar resultssoftpre-openandsoftpre-closedsetsontopological region.
References
[1] K.Atanassov,Intuitionsticfuzzysets, FuzzySetSyst., 20(1986),87–96.
[2] Ch.ShashiKumar,T.SivaNageswaraRao,Y.Srinivasa Rao,V.VenkateswaraRao,InteriorandBoundaryvertices ofBSVNeutrosophicGraphs, Jour.ofAdv.Researchin Dynamical&ControlSystems, 12(6)(2020),1510-1515.
[3] P.K.Maji,Neutrosophicsoftset, AnnalsofFuzzyMathematicsandInformatics, 5(1)(2013),157–168.
[4] S.Broumi,A.Bakali,M.Talea,F.Smarandacheand V.VenkateswaraRao,IntervalComplexNeutrosophic GraphofType1, NeutrosophicOperationalResearch, (2018),88–107.
[5] S.Broumi,A.Bakali,M.Talea,F.Smarandacheand V.VenkateswaraRao,BipolarComplexNeutrosophic GraphsofType1, NewTrendsinNeutrosophicTheory andApplications, 2(2018),189-208.
[6] S.Broumi,M.Talea,A.Bakali,F.Smarandache,Prem KumarSingh,M.Murugappan,andV.Venkateswara Rao,NeutrosophicTechniqueBasedEfficientRouting ProtocolForMANETBasedOnItsEnergyAndDistance, NeutrosophicSetsandSystems, 24(2019),61–69.
[7] S.Broumi,P.K.Singh,M.Talea,A.Bakali,F.SmarandacheandV.VenkateswaraRao,Single-valuedneutrosophictechniquesforanalysisofWIFIconnection, AdvancesinIntelligentSystemsandComputing, 915(2013), 405–512.
[8]
F.Smarandache,Neutrosophicset-ageneralizationof theintuitionsticfuzzyset, InternationalJournalofPure andAppliedMathematics, 24(3)(2005),287–294.
[12] V.VenkateswaraRao,Y.SrinivasaRao,NeutrosophicPreopenSetsandPre-closedSetsinNeutrosophicTopology, InternationalJournalofChemTechResearch, 10(10)(2017),449-458.
[13] Y.SrinivasaRao,Ch.ShashiKumar,T.SivaNageswara Rao,V.VenkateswaraRao,SingleValuedNeutrosophic detourdistance, JournalofCriticalReviews,7(8)(2020), 810–812.
[14] L.A.Zadeh,Fuzzysets, InformationandControl, 8(3)(1965),338–353.
ISSN(P):2319 3786 MalayaJournalofMatematik ISSN(O):2321 5666
[9]
F.Smarandache,S.Broumi,P.K.Singh,C.Liu,V. VenkateswaraRao,H.-L.YangandA.Elhassouny,Introductiontoneutrosophyandneutrosophicenvironment, InNeutrosophicSetinMedicalImageAnalysis,(2019), 3–29.
[10] T.SivaNageswaraRao,Ch.ShashiKumar,,Y.Srinivasa Rao,V.VenkateswaraRao,DetourInteriorandBoundary verticesofBSVNeutrosophicGraphs, InternationalJournalofAdvancedScienceandTechnology, 29(8)(2020), 2382-2394.
[11] T.SivaNageswaraRao,G.UpenderReddy,V.Venkates waraRao,Y.SrinivasaRao,BipolarNeutrosophicWeakly BG∗ -ClosedSets, HighTechnologyLetters,26(8)(2020), 878–887.
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