HewaydaElGhawalby1 andA.A.Salama2
1 FacultyofEngineering,Port-SaidUniversity,Egypt. hewayda2011@eng.psu.edu.eg 2 FacultyofScience,Port-SaidUniversity,Egypt. drsalama44@gmail.com
Abstract. Inthispaperwepresentanewneutrosophiccrispfamilygeneratedfromthethreecomponents’neutrosophiccrispsetspresentedby Salama[4].TheideabehindSalam’sneutrosophiccrispsetwastoclassify theelementsofauniverseofdiscoursewithrespecttoanevent”A”into threeclasses:oneclasscontainsthoseelementsthatarefullysupportive toA,anotherclasscontainsthoseelementsthattotallyagainstA,anda thirdclassforthoseelementsthatstandinadistancefrombeingwith oragainstA.Ouraimhereistostudytheelementsoftheuniverseof discoursewhichtheirexistenceisbeyondthethreeclassesoftheneutrosophiccrispsetgivenbySalama.Byaddingmorecomponentswewillget afourcomponents’neutrosophiccrispsetscalledtheUltraNeutrosophoc CrispSets.Fourtypesofset’soperationsisdefinedandthepropertiesof thenewultraneutrosophiccrispsetsarestudied.Moreover,adefinition oftherelationbetweentwoultraneutrosophiccrispsetsisgiven.
Keywords: CrispSetsOperations,,CrispSetsRelations,FuzzySets,NeutrosophicCrispSets
1Introduction
EstablishedbyFlorentinSmarandache,neutrosophywaspresentedasthestudy oforigin,nature,andscopeofneutralities,aswellastheirinteractionswithdifferentideationalspectra.Themainideawastoconsideranentity,”A”inrelation toitsopposite”Non-A”,andtothatwhichisneither”A”nor”Non-A”,denoted by”Neut-A”.Andfromthenon,neutrosophybecamethebasisofneutrosophic logic,neutrosophicprobability,neutrosophicset,andneutrosophicstatistics. Accordingtothistheoryeveryidea”A”tendstobeneutralizedandbalanced by”neutA”and”nonA”ideas-asastateofequilibrium.Inaclassicalway ”A”,”neutA”,and”antiA”aredisjointtwobytwo.Nevertheless,sinceinmany casesthebordersbetweennotionsarevagueandimprecise,itispossiblethat ”A”,”neutA”,and”antiA”havecommonpartstwobytwo,orevenallthreeof themaswell.
In[10],[11],[12],Smarandacheintroducedthefundamentalconceptsof neutrosophicsets,thathadledSalamaetal.toprovideamathematicaltreatment
fortheneutrosophicphenomenawhichalreadyexistedinourrealworld(see forinstance[1],[2],[3],[4],[5],[6],[8],[9],andthereferencestherein).Moreover, theworkofSalamaetal.formedastartingpointtoconstructnewbranchesof neutrosophicmathematics.Hence,neutrosophicsettheoryturnedouttobea generalizationofboththeclassicalandfuzzycounterparts.
In[4],Salamaintroducedtheconceptofneutrosophiccrispsetsasatriple structureoftheform, AN = ⟨A1, A2, A3⟩.Thethreecomponents- A1, A2, and A3referstothreeclassesoftheelementsoftheuniverse X withrespecttoanevent A.Where A1 istheclasscontainingthoseelementsthatarefullysupportiveto A, A3 forthoseelementsthattotallyagainst A,and A2 forthoseelementsthat standinadistancefrombeingwithoragainst A.Thethreeclassesaresubsets of X.Furthermore,in[7]theauthorssuggestedthreedifferenttypesofsuch neutrosophiccrispsetsdependingonwhitherthereisanoverlapbetweenthe threeclassesornot,andwhithertheirunioncoverstheuniverseornot.
Thepurposeofthispaper,istoinvestigatetheelementsoftheuniverse whichhavenotbeensubjectedtotheclassification;thoseelementsbelongingto thecomplementoftheunionofthethreeclasses.Hence,afourthcomponentis tobeaddedtothealreadyexistedthreeclassesinSalama’ssense.
Forthepurposeofthispaper,wewillcategorizetheneutrosophiccrispsets ofsomeuniverse X,intotwocategoriesaccordingtowhithertheunionofthe threeclassescoverstheuniverseornot.Thefirstcategorywillcontainallthe neutrosophiccrispsetswhosecomponentsdoesnotcovertheuniverse,whether theyaremutuallyexclusiveortheyhavesomecommonpartsin-between;two bytwo,orevenallthethreeofthem.whilethesecondcategorywillcontain theremainingneutrosophiccrispsetswhosecomponentscoversthewhole universe.
Theremainingofthispaperisorganizedasfollows:in(sec.2)weintroduce somebasicdefinitionnecessaryforthiswork.Theconceptoftheultraneutrosophiccrispsetsisintroducedin(sec.3).Furthermore,fourtypesofultra neutrosophiccrispsets’operationsanditspropertiesarepresentedin(sec.4) and(sec.5).Hence,theproductandrelationbetweenultraneutrosophiccrisp setsaredefinedin(sec.6),(sec.7)and(sec.8).Finally,conclusionsaredrawn andfuturedirectionsofresearcharesuggestedin(sec.9).
2Preliminaries
theform AN = ⟨A1,ϕ, A3⟩,where A3 ⊆ Ac 1.Moreover,intheintuitionisticfuzzy sense AN = ⟨A1, (A1 ∪ A3)c , A3⟩
2.1.2Definition[4] Thecomplementofaneutrosophiccrispsetisdefinedas: coAN = ⟨coA1, coA2, coA3⟩
2.1.3Definition[7] Aneutrosophiccrispset AN = ⟨A1, A2, A3⟩ iscalled: – Aneutrosophiccrispsetoftype1,ifsatisfyingthat: Ai ∩ Aj = ϕ, where i ≠ j and ⋃3 i=1 Ai ⊂ X, ∀i, j = 1, 2, 3
– Aneutrosophiccrispsetoftype2,ifsatisfyingthat: Ai ∩ Aj = ϕ, where i ≠ j and ⋃3 i=1 Ai = X, ∀i, j = 1, 2, 3
– Aneutrosophiccrispsetoftype3,ifsatisfyingthat: ⋂3 i=1 Ai = ϕ, and ⋃3 i=1 Ai = X, ∀i, j = 1, 2, 3
2.2NeutrosophicCrispSetsOperationsofType1[7]
Foranytwoneutrosophiccrispsets AN and BN,wehavethat:
AN ⊆ BN if A1 ⊆ B1, A2 ⊆ B2, and A3 ⊇ B3, AN = BN ifandonlyif Ai = Bi for i = 1, 2, 3 hence,wecandefinethefollowing:
AN ∪ BN = ⟨A1 ∪ B1, A2 ∪ B2, A3 ∩ B3⟩
AN ∩ BN = ⟨A1 ∩ B1, A2 ∩ B2, A3 ∪ B3⟩
2.3NeutrosophicCrispSetsOperationsofType2[7]
Foranytwoneutrosophiccrispsets AN and BN,wehavethat:
AN ⊆ BN if A1 ⊆ B1, A2 ⊇ B2, and A3 ⊇ B3, AN = BN ifandonlyif Ai = Bi for i = 1, 2, 3 hence,wecandefinethefollowing:
AN ∪ BN = ⟨A1 ∪ B1, A2 ∩ B2, A3 ∩ B3⟩
AN ∩ BN = ⟨A1 ∩ B1, A2 ∪ B2, A3 ∪ B3⟩
3UltraNeutrosophicCrispSets
Inthissectionweconsiderelementsin X whichdonotbelongtoanyofthe threeclassesoftheneutrosophiccrispsetdefinedin(2.1.1).
3.1Definition
Let X beanygivenuniverse,theultraneutrosophiccrispsetisdefinedas: ˘ A = ⟨A1, A2, A3, MA⟩, where MA = co( 3 ⋃ i=1 Ai)
Thefamilyofallultraneutrosophiccrispsetsin X willbedenotedby ˘ U(X)
3.2Definition
Thecomplementofanyultraneutrosophiccrispset ˘ A,isdefinedas: co ˘ A = ⟨coA1, coA2, coA3, coMA⟩
4UltraNeutrosophicCrispSetsOperations
4.1UltraOperationsofTypeI
Foranytwoultraneutrosophiccrispsets ˘ A and ˘ B,wehavethat: ˘
A ⊆I ˘ B if A1 ⊆ B1, A2 ⊆ B2, A3 ⊇ B3, and MA ⊇ MB, ˘ A = ˘ B ifandonlyif Ai = Bifori = 1, 2, 3andMA = MB
hence,wecandefinethefollowing: ˘ A ⊎I ˘ B = ⟨A1 ∪ B1, A2 ∪ B2, A3 ∩ B3, MA ∩ MB⟩ ˘ A I ˘ B = ⟨A1 ∩ B1, A2 ∩ B2, A3 ∪ B3, MA ∪ MB⟩
4.2UltraOperationsofTypeII
Foranytwoultraneutrosophiccrispsets ˘ A and B,wehavethat: A ⊆II ˘ B if A1 ⊆ B1, A2 ⊇ B2, A3 ⊇ B3, and MA ⊇ MB, ˘ A = ˘ B ifandonlyif Ai = Bifori = 1, 2, 3andMA = MB hence,wecandefinethefollowing: ˘ A ⊎II ˘ B = ⟨A1 ∪ B1, A2 ∩ B2, A3 ∩ B3, MA ∩ MB⟩ ˘ A II ˘ B = ⟨A1 ∩ B1, A2 ∪ B2, A3 ∪ B3, MA ∪ MB⟩
4.3UltraOperationsofTypeIII
Foranytwoultraneutrosophiccrispsets ˘ A and ˘ B,wehavethat:
˘ A ⊆III ˘ B if A1 ⊆ B1, A2 ⊆ B2, A3 ⊇ B3 and MA ⊆ MB, ˘ A = ˘ B ifandonlyif Ai = Bifori = 1, 2, 3andMA = MB
hence,wecandefinethefollowing: ˘ A ⊎III ˘ B = ⟨A1 ∪ B1, A2 ∪ B2, A3 ∩ B3, MA ∪ MB⟩ ˘ A III ˘ B = ⟨A1 ∩ B1, A2 ∩ B2, A3 ∪ B3, MA ∩ MB⟩
4.4UltraOperationsofTypeIV
Foranytwoultraneutrosophiccrispsets ˘ A and ˘ B,wehavethat: ˘ A ⊆IV ˘ B if A1 ⊆ B1, A2 ⊇ B2, A3 ⊇ B3 and MA ⊆ MB, A = ˘ B ifandonlyif Ai = Bifori = 1, 2, 3andMA = MB hence,wecandefinethefollowing: ˘ A ⊎IV ˘ B = ⟨A1 ∪ B1, A2 ∩ B2, A3 ∩ B3, MA ∪ MB⟩ ˘ A IV ˘ B = ⟨A1 ∩ B1, A2 ∪ B2, A3 ∪ B3, MA ∩ MB⟩
4.5Definition
– Thedifferencebetweenanytwoultraneutrosophiccrispsets ˘ A and ˘ B,is definedas ˘ A \ ˘ B = ˘ A co ˘ B
– Thesymmetricdifferencebetweenanytwoultraneutrosophiccrispsets ˘ A and B,isdefinedas ˘ A ⨁ B = ˘ A \ B ⊎ B \ ˘ A
neutrosophiccrispsetsoperationsverifythefollowingproperties:
1.Associativelaws: ˘ A ⊎ (B ⊎ ˘ C) = ( ˘ A ⊎ B) ⊎ ˘ C ˘ A (B ˘ C) = ( ˘ A B) ˘ C
2.Commutativelaws: ˘ A ⊎ ˘ B = ˘ B ⊎ ˘ A ˘ A ˘ B = ˘ B ˘ A
3.Distributivelaws: ˘ A ⊎ ( ˘ B ˘ C) = ( ˘ A ⊎ ˘ B) ( ˘ A ⊎ ˘ C) ˘ A ( ˘ B ⊎ ˘ C) = ( ˘ A ˘ B) ⊎ ( ˘ A ˘ C)
4.Idempotentlaws: ˘ A ⊎ ˘ A = ˘ A ˘ A ˘ A = ˘ A
5.Absorptionlaws: ˘ A ⊎ ( ˘ A ˘ B) = ˘ A ˘ A ( ˘ A ⊎ ˘ B) = ˘ A
6.Involutionlaw: co(co ˘ A) = ˘ A
7.DeMorgan’slaws: co(A ⊎ ˘ B) = coA co ˘ B co(A ˘ B) = coA ⊎ co ˘ B
Proof
Forexplanation,wewillshowtheproofofthefirstassociativelawfortypeI, theproofofthefirstdistributivelawfortypeII,theproofofthefirstabsorption lawfortypeIII,andtheproofofthefirstDeMorgan’slawfortypeIV.usingthe definitions
i) ˘ A ⊎I ( ˘ B ⊎I ˘ C) = ⟨A1 ∪ (B1 ∪ C1), A2 ∪ (B2 ∪ C2), A3 ∩ (B3 ∩ C3), MA ∩ (MB ∩ MC)⟩ = ⟨(A1 ∪ B1) ∪ C1, (A2 ∪ B2) ∪ C2, (A3 ∩ B3) ∩ C3, (MA ∩ MB) ∩ MC)⟩ = ( ˘ A ⊎I ˘ B) ⊎I ˘ C
ii) A ⊎II ( ˘ B II C) = ⟨A1 ∪ (B1 ∩ C1), A2 ∩ (B2 ∪ C2), A3 ∩ (B3 ∪ C3, MA ∩ (MB ∪ MC)⟩ = ⟨(A1 ∪ B1) ∩ (A1 ∪ C1), (A2 ∩ B2) ∪ (A2 ∩ C2), (A3 ∩ B3) ∪ (A3 ∩ C3), (MA ∩ MB) ∪ (MA ∩ MC)⟩ = ( ˘ A ⊎II ˘ B) II ( ˘ A ⊎II ˘ C) iii) ˘ A ⊎III ( ˘ A III ˘ B) = ⟨A1 ∪ (A1 ∩ B1), A2 ∪ (A2 ∩ B2), A3 ∩ (A3 ∪ B3), MA ∪ (MA ∩ MB)⟩ = ⟨A1, A2, A3, MA⟩ = A iv) co( ˘ A ⊎IV ˘ B) = ⟨co(A1 ∪ B1), co(A2 ∩ B2), co(A3 ∩ B3), co(MA ∪ MB)⟩ = ⟨coA1 ∩ coB1, coA2 ∪ coB2, coA3 ∪ coB3, coMA ∩ coMB)⟩ = co ˘ A IV co ˘ B
Notethat: thesameprocedurecanbeappliedtoproveanyofthelawsgivenin (5)foralltypes:I,II,II,andIV.
5.1Proposition
Let ˘ Ai, i ∈ J,beanarbitraryfamilyofultraneutrosophiccrispsetson X;then wehavethefollowing:
1.TypeI: i∈JI ˘ Ai = ⟨⋂i∈J Ai1, ⋂i∈J Ai2, ⋃i∈J Ai3, ⋃i∈J MAi⟩ i∈JAi = ⟨⋃i∈J Ai1, ⋃i∈J Ai2, ⋂i∈J Ai3, ⋂i∈J MAi⟩
2.TypeII: i∈J ˘ Ai = ⟨⋂i∈J Ai1, ⋃i∈J Ai2, ⋃i∈J Ai3, ⋃i∈J MAi⟩ i∈J ˘ Ai = ⟨⋃i∈J Ai1, ⋂i∈J Ai2, ⋂i∈J Ai3, ⋂i∈J MAi⟩
3.TypeIII: i∈J ˘ Ai = ⟨⋂i∈J Ai1, ⋂i∈J Ai2, ⋃i∈J Ai3, ⋂i∈J MAi⟩ i∈J ˘ Ai = ⟨⋃i∈J Ai1, ⋃i∈J Ai2, ⋂i∈J Ai3, ⋃i∈J MAi⟩
4.TypeIV: i∈J ˘ Ai = ⟨⋂i∈J Ai1, ⋃i∈J Ai2, ⋃i∈J Ai3, ⋂i∈J MAi⟩ i∈J ˘ Ai = ⟨⋃i∈J Ai1, ⋂i∈J Ai2, ⋂i∈J Ai3, ⋃i∈J MAi⟩
6TheUltraCartesianProductofUltraNeutrosophicCrisp Sets
Consideranytwoultraneutrosophiccrispsets, A on X and B on Y;where A = ⟨A1, A2, A3, MA⟩ and ˘ B = ⟨B1, B2, B3, MB⟩ Theultracartesianproductof ˘ A and ˘ B isdefinedasthequadruplestructure: ˘ A × ˘ B = ⟨A1 × B1, A2 × B2, A3 × B3, MA × MB⟩
whereeachcomponentisasubsetofthecartesianproduct X × Y; Ai × Bi = {(ai, bi) ∶ ai ∈ Ai and bi ∈ Bi}, ∀i = 1, 2, 3and MA × MB = {(ma, mb) ∶ ma ∈ MA and mb ∈ MB} 6.1Corollary Ingeneralif A ≠ ˘ B,then A × ˘ B ≠ ˘ B × A 7UltraNeutrosophicCrispRelations
Anultraneutrosophiccrisprelation ˘ R fromanultraneutrosophiccrispset ˘ A to ˘ B,namely ˘ R ∶ ˘ A → ˘ B,isdefinedasaquadruplestructureoftheform ˘ R = ⟨R1, R2, R3, RM⟩,where Ri ⊆ Ai × Bi, ∀i = 1, 2, 3and RM ⊆ MA × MB, thatis Ri = {(ai, bi) ∶ ai ∈ Ai and bi ∈ Bi} RM = {(ma, mb) ∶ ma ∈ MA and mb ∈ MB}
7.1DomainandRangeofUltraNeutrosophicCrispRelations
Foranyultraneutrosophiccrisprelation ˘ R ∶ ˘ A → ˘ B,wedefinethefollowing:
– Theultradomainof R,isdefinedas: uDom( ˘ R) = ⟨dom(R1), dom(R2), dom(R3), dom(RM)⟩
– Theultrarangeof ˘ R,isdefinedas: uRng( ˘ R) = ⟨rng(R1), rng(R2), rng(R3), rng(RM)⟩
– TheDomainof ˘ R,isdefinedas: Dom( ˘ R) = dom(R1) ∪ dom(R2) ∪ dom(R3) ∪ dom(RM)
– TheRangeof ˘ R,isdefinedas: Rng(R) = rng(R1) ∪ rng(R2) ∪ rng(R3) ∪ rng(RM)
7.2Corollary
Fromthedefinitionsgivenin7.1,onemaynoticethatforanyultraneutrosophic crisprelation ˘ R ∶ ˘ A → ˘ B,wehave:
– Thedomainof R isacrispsubsetof X,namely, Dom(R) ⊆ X
– Therangeof ˘ R isacrispsubsetof Y,namely, Rng( ˘ R) ⊆ Y.
– Theultredomainof ˘ Risaquadruplestructurewhosecomponentsarecrisp subsetsof X;furthermore, dom(Ri) ⊆ Ai, i = 1, 2, 3and dom(RM) ⊆ MA
– Theultrerangeof ˘ R isaquadruplestructurewhosecomponentsarecrisp subsetsof Y;furthermore, rng(Ri) ⊆ Bi, i = 1, 2, 3and rng(RM) ⊆ MB
7.3Definition
Anultraneutrosophiccrispinverserelation R 1 isanultraneutrosophiccrisp relationfromanultraneutrosophiccrispset ˘ B to ˘ A, ˘ R 1 ∶ ˘ B → ˘ A,andtobe definedasaquadruplestructureoftheform: ˘ R 1 = ⟨R 1 1 , R 1 2 , R 1 3 , R 1 M ⟩,where R 1 i ⊆ Bi × Ai, ∀i = 1, 2, 3and RM ⊆ MB × MA, thatis: R 1 i = {(bi, ai) ∶ (ai, bi) ∈ Ri} R 1 M = {(mb, ma) ∶ (ma, mb) ∈ RM}
7.4Corollary
oranyultraneutrosophiccrisprelation ˘ R ∶ ˘ A → B,wehavethat: Dom( ˘ R 1) = Rng( ˘ R) Rng( ˘ R 1) = Dom( ˘ R) uDom( ˘ R 1) = uRng( ˘ R) uRrng( ˘ R 1) = uDom( ˘ R)
8CompositionofUltraNeutrosophicCrispRelations
Considerthethreeultraneutrosophiccrispsets: ˘ A of X, ˘ B of Y and ˘ C of Z;and, thetwoultraneutrosophiccrisprelations: ˘ R ∶ ˘ A → ˘ B and ˘ S ∶ ˘ B → ˘ C;where ˘ R = ⟨R1, R2, R3, RM⟩,and ˘ S = ⟨S1, S2, S3, SM⟩.Thecompositionof ˘ R and ˘ S,is denotedanddefinedas: ˘ R ˘ S = ⟨R1 ◦ S1, R2 ◦ S2, R3 ◦ S3, RM ◦ SM⟩ suchthat, Ri ◦ Si ∶ Ai → Ci,where, Ri ◦ Si = {(ai, ci) ∶∃bi ∈ Bi, (ai, bi) ∈ Ri and (bi, ci) ∈ Si}; RM ◦ SM ∶ MA → MC,where, RM ◦ SM = {(ma, mc) ∶∃mb ∈ MB, (ma, mb) ∈ RM and (mb, mc) ∈ SM}
8.1Corollary
Foranytwoultraneutrosophiccrisprelations: ˘ R ∶ ˘ A → B and ˘ S ∶ B → ˘ C; uDom( ˘ R⊙ ˘ S) ⊆ uDom( ˘ R) uRng( ˘ R⊙ ˘ S) ⊆ uRng(˘ S)
8.2Corollary
Considerthethreeultraneutrosophiccrisprelations: ˘ R ∶ ˘ A → B, ˘ S ∶ B → ˘ C,and K ∶ C → ˘ D;wehavethat: ˘ R⊙ (˘ S⊙ ˘ K) = ( ˘ R⊙ ˘ S) ⊙ ˘ K
9ConclusionandFutureWork
Inthispaperwehavepresentedanewconceptofneutrosophiccrispsets, called”TheUltraNeutrosophicCrispSets”,asaquadrablestructure.Thefirst threecomponentsrepresentaclassificationoftheuniverseofdiscoursewith respecttosomeevent;whilethefourthcomponentdealswiththeelements whichhavenotbeensubjectedtothatclassification.Whiletheelementsofthe firstandthethirdareconsideredtobewelldefined,thereisablurryaboutthe behavioroftheelementsinbothsecondandfourthcomponents.Consequently, fourtypesofset’soperationswereestablishedandthepropertiesofthenew ultraneutrosophiccrispsetswerestudiedaccordingtodifferentexpectations abouttheperformanceofthesecondandthefourthcomponents.Moreover,the definitionoftherelationbetweentwoultraneutrosophiccrispsetsweregiven. Finally,theconceptsofproductandcompositionofultraneutrosophiccrispsets wereintroduced.
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