Annalsof Fuzzy Mathematicsand Informatics Volumex,No.x,(Month201y),pp.1–xx ISSN:2093–9310(printversion) ISSN:2287–6235(electronicversion) http://www.afmi.or.kr
@FMI
c ResearchInstituteforBasic Science,WonkwangUniversity http://ribs.wonkwang.ac.kr
Singlevaluedneutrosophicrelations
J.H.Kim,P.K.Lim,J.G.Lee,K.Hur
Received6March2018; Revised5April2018; Accepted24June2018
Abstract.
Weintroducetheconceptofasinglevaluedneutrosophicreflexive,symmetricandtransitiverelation.Andwestudysinglevaluedneutrosophic analoguesofmanyresultsconcerningrelationshipsbetweenordinaryreflexive,symmetricandtransitiverelations.Next,wedefinetheconceptsof asinglevaluedneutrosophicequivalenceclassandasinglevaluedneutrosophicpartition,andweprovethatthesetofallsinglevaluedneutrosophic equivalenceclassesisasinglevaluedneutrosophicpartitionandthesinglevaluedneutrosophicequivalencerelationisinducedbyasinglevalued neutrosophicpartition.Finally,wedefinean α-cutofasinglevaluedneutrosophicrelationandinvestigatesomerelationshipsbetweensinglevalued neutrosophicrelationsandtheir α-cuts.
2010AMSClassification: 04A72
Keywords: Singlevaluedneutrosophicrelation,Singlevaluedneutrosophicreflexive[respec.,symmetricandtransitive]relation,Singlevaluedneutrosophicequivalencerelation,Singlevaluedneutrosophictransitiveclosure,Singlevaluedneutrosophicvalue.
CorrespondingAuthor: J.H.Kim(junhikim@wku.ac.kr)
1. Introduction
In1965,Zadeh[28]hadintroducedtheconceptofafuzzysetasthegeneralization ofacrispset.In1971,he[27]definedthenotionsofsimilarityrelationsandfuzzy orderingsasthegeneralizationsofcrispequivalencerelationsandpartialorderings playingbasicrolesinmanyfieldsofpureandappliedscience.Afterthattime, manyresearchers[5, 6, 7, 8, 9, 10, 13, 14, 18]studiedfuzzyrelations.Inparticular, Chakrabortyetal.[5, 6, 7, 8]definedafuzzyrelationoverafuzzysetandobtained manyproperties.Furthermore,DibandYoussef[9]definedthefuzzyCartesian productoftwoordinarysets X and Y asthecollectionofall L-fuzzysetsof X × Y ,
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where L = I × I and I denotestheunitclosedinterval.In2009,Lee[14]obtained manyresultsbyusingthenotionoffuzzyrelationsintroducedbyDibandYoussef.
In1968,Atanassov[1]definedanintutionisticfuzzysetasageneralizationof afuzzyset.Afterthen,AtanassovandGargov[2, 3]introducedtheconceptof aninterval-valuedintuitionisticfuzzysetandealtwithintuitionisticfuzzylogics. Moreover,Huretal.[11]studiedthecategoryofintuitionisticH-fuzzyrelationin thesenseofatopologicaluniverse.Recently,Liuetal.[15, 16, 17]appliedthe conceptsofanintuitionisticfuzzysetandaninterval-valuedintuitionisticfuzzyset tomulti-attributegroupdecisionmakingandgroupdecisionmaking,respectively.
In1998,Smarandache[23]definedtheconceptofaneutrusophicsetasthegeneralizationofanintuitionisticfuzzyset.Alsoheintroducedneutrosophiclogics, neutrosophicsets,neutrosophicprobabilities,neutrosophicstatisticsanditsapplicationsin[21, 22].Furthermore,Salamaetal.[19, 20]introducedtheconceptof aneutrusophicrelationandstudieditssomeproperties.Recently,Bhowmikand Pal[4]introducedtheconceptofaneutrosophicrelationandstudiedsomeofits properties.Inparticular,Wangetal.[24]introducedthenotionofasinglevalued neutrosophicset.Moreover,Yangetal.[25]definedasinglevaluedneutrosophic relationandinvestigatedsomeofitsproperties.
Inthispaper,first,weintroduceasinglevaluedneutrosophicrelationfromaset X to Y andthecompositionoftwosinglevaluedneutrosophicrelations.Alsowe introducesomeoperationsbetweensinglevaluedneutrosophicrelationsandobtain someoftheirproperties.Second,weintroducetheconceptofasinglevaluedneutrosophicreflexive,symmetricandtransitiverelation.Andwestudysinglevalued neutrosophicanaloguesofmanyresultsconcerningrelationshipsbetweenordinary reflexive,symmetricandtransitiverelations.Third,wedefinetheconceptsofasinglevaluedneutrosophicequivalenceclassandasinglevaluedneutrosophicpartition, andweprovethatthesetofallsinglevaluedneutrosophicequivalenceclassesisa singlevaluedneutrosophicpartitionandthesinglevaluedneutrosophicequivalence relationisinducedbyasinglevaluedneutrosophicpartition.Finally,wedefinean α-cutofasinglevaluedneutrosophicrelationandinvestigatesomerelationships betweensinglevaluedneutrosophicrelationsandtheir α-cuts.
2. Preliminaries
Inthissection,weintroducetheconceptofsinglevaluedneutrosophicset,the complementofasinglevaluedneutrosophicset,theinclusionbetweentwosingle valuedneutrosophicsets,theunionandtheintersectionoftwosinglevaluedneutrosophicsets.
Definition2.1 ([22]). Let X beanon-emptyset.Then A iscalledaneutrosophic set(insort,NS)in X,if A hastheform A =(TA,IA,FA), where TA : X →] 0, 1+[, IA : X →] 0, 1+[, FA : X →] 0, 1+[. Sincethereisnorestrictiononthesumof TA(x), IA(x)and FA(x),foreach x ∈ X, 0 ≤ TA(x)+ IA(x)+ FA(x) ≤ 3+ .
Moreover,foreach x ∈ X, TA(x)[resp.,IA(x)and FA(x)]representthedegreeof membership[resp.,indeterminacyandnon-membership]of x to A 2
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FromExample 2.1.1 in[19],wecanseethateveryIFS(intutionisticfuzzyset) A inanon-emptyset X isanNSin X havingtheform
A =(TA, 1 (TA + FA),FA), where(1 (TA + FA))(x)=1 (TA(x)+ FA(x)).
Definition2.2 ([22]). Let A and B betwoNSsin X.Thenwecalled A iscontained in B,denotedby A ⊂ B,ifforeach x ∈ X, infTA(x) ≤ infTB (x),supTA(x) ≤ supTB (x),infIA(x) ≥ infIB (x),supIA(x) ≥ supIB (x),infFA(x) ≥ infFB (x) and supFA(x) ≥ supFB (x).
Definition2.3 ([24]). Let X beaspaceofpoints(objects)withagenericelement in X denotedby x.Then A iscalledasinglevaluedneutrosophicset(insort,SVNS) in X,if A hastheform A =(TA,IA,FA), where TA,IA,FA : X → [0, 1]. Inthiscase, TA,IA,FA arecalledtruth-membershipfunction,indeterminacymembershipfunction,falsity-membershipfunction,respectivelyandwewilldenote thesetofallSVNSsin X as SVNS(X).
Furthermore,wewilldenotetheemptySVNS[resp.thewholeSVNS]in X as 0N [resp.1N ]anddefineby0N (x)=(0, 1, 1)[resp.1N =(1, 0, 0)],foreach x ∈ X
Definition2.4 ([24]). Let A ∈ SVNS(X).Thenthecomplementof A,denotedby Ac,isaSVNSin X definedasfollows:foreach x ∈ X,
TAc (x)= FA(x),IAc (x)=1 IA(x)and FAc (x)= TA(x) Definition2.5 ([26]). Let A,B ∈ SVNS(X).Then
(i) A issaidtobecontainedin B,denotedby A ⊂ B,ifforeach x ∈ X,
TA(x) ≤ TB (x),IA(x) ≥ IB (x)and FA(x) ≥ FB (x), (ii) A issaidtobeequalto B,denotedby A = B,if A ⊂ B and B ⊂ A Definition2.6 ([25]). Let A,B ∈ SVNS(X).Then (i)theintersectionof A and B,denotedby A ∩ B,isaSVNSin X definedas: A ∩ B =(TA ∧ TB ,IA ∨ IB ,FA ∨ FB ), where(TA ∧ TB )(x)= TA(x) ∨ TB (x),(FA ∨ FB )= FA(x) ∨ FB (x),foreach x ∈ X, (ii)theunionof A and B,denotedby A ∪ B,isanSVNSin X definedas: A ∪ B =(TA ∨ TB ,IA ∧ IB ,FA ∧ FB )
Result2.7 ([25],Proposition 2.1). Let A,B ∈ NS(X).Then (1) A ⊂ A ∪ B and B ⊂ A ∪ B, (2) A ∩ B ⊂ A and A ∩ B ⊂ B, (3)(Ac)c = A, (4)(A ∪ B)c = Ac ∩ Bc , (A ∩ B)c = Ac ∪ Bc
3. Singlevaluedneutrosophicrelations
Inthissection,weintroducetheconceptsofsinglevaluedneutrosophicrelation, thecompositionoftwosinglevaluedneutrosophicrelationsandtheinverseofa singlevaluedneutrosophicrelation,andstudysomepropertiesofeachconcept. Let X,Y,Z beordinarynon-emptysets.
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Definition3.1. R iscalledasinglevaluedneutrosophicrelation(inshort,SVNR) from X to Y ,ifitisaSVNSin X × Y havingtheform: R =(TR,IR,FR), where TR,IR,FR : X × Y → [0, 1]denotethetruth-membershipfunction,indeterminacymembershipfunction,falsity-membershipfunction,respectively.
Foreach(x,y) ∈ X ×Y , TR(x,y)[resp.,IR(x,y)and FR(x,y)]representthedegree ofmembership[resp.,indeterminacyandnon-membership]of(x,y)to R Inparticular,aSVNRfromfrom X to X iscalledaSVNRin X (See[25]).
TheemptySVNR[resp.thewholeSVNR]in X isdenotedby φN [resp. XN ]and definedasfollows:foreach(x,y) ∈ X × X, φN (x,y)=(0, 1, 1)[resp. XN (x,y)=(1, 0, 0)].
WewilldenotethesetofallSVNRsin X [resp.from X to Y ]as SVNR(X) [resp. SVNR(X × Y )].
Let X = {x1,x2,...,xm} andlet Y = {y1,y2,...,yn}.Then R =(TR,IR,FR) ∈ SVNR(X × Y )canbeexpressedby m × n matrix.Thiskindofmatrixexpressing aSVNRwillbecalledasinglevaluedneutrosophicmatrix.
Definition3.2 (See[25]). Let R ∈ SVNR(X × Y ).Then (i)theinverseof R,denotedby R 1,isaSVNRfrom Y to X definedasfollows: foreach(y,x) ∈ Y × X, R 1(x,y)= R(y,x),i.e., T 1 R (y,x)= TR(x,y),I 1 R (y,x)= IR(x,y),F 1 R (y,x)= FR(x,y). (ii)thecomplementof R,denotedby Rc,isaSVNRfrom X to Y definedas follows:foreach(x,y) ∈ X × Y , T c R(x,y)= FR(x,y),I c R(x,y)=1 IR(x,y),F c R(x,y)= TR(x,y)
Example3.3. Let X = {a,b,c} andlet R beaSVNRin X givenbythesingle valuedneutrosophicmatrix: R = (0 2, 0 4, 0 3)(1, 0 2, 0)(0 4, 1, 0 7) (0, 0, 0)(0 6, 0 2, 0 1)(0 3, 0 2, 0 6) (0, 0, 0)(0, 0, 0)(0 2, 0 4, 0 1)
Thentheinverseandthecomplementof R aregivenasbelow: R 1 = (0 2, 0 4, 0 3)(0, 0, 0)(0, 0, 0) (1, 0 2, 0)(0 6, 0 2, 0 1)(0, 0, 0) (0.4, 1, 0.7)(0.3, 0.2, 0.6)(0.2, 0.4, 0.1) , Rc = (0.3, 0.6, 0.2)(0, 0.8, 1)(0.7, 0, 0.4) (0, 1, 0)(0 1, 0 8, 0 6)(0 6, 0 8, 0 3) (0, 1, 0)(0, 1, 0)(0 1, 0 6, 0 2)
Remark3.4. Foreach R ∈ SVNR(X), R ∩ Rc = φN and R ∪ Rc = XN donot hold,ingeneral. 4
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ConsidertheSVNR R inExample 3.3.Then
R ∩ Rc =
R ∪ Rc =
(0 2, 0 6, 0 3)(0, 0 8, 1)(0 4, 1, 0 7) (0, 1, 0)(0 1, 0 8, 0 6)(0 3, 0 8, 0 6) (0, 1, 0)(0, 1, 0)(0.1, 0.6, 0.2) = φN ,
(0 3, 0 4, 0 2)(1, 0 2, 0)(0 7, 0, 0 4) (0, 0, 0)(0.6, 0.2, 0.1)(0.6, 0.2, 0.3) (0, 0, 0)(0, 0, 0)(0.2, 0.4, 0.1) = XN .
Definition3.5 (See[25]). Let R,S ∈ SVNR(X × Y ).Then
(i) R issaidtobecontainedin S,denotedby R ⊂ S,if TR(x,y) ≤ TS (x,y), IR(x,y) ≥ IS (x,y)and FR(x,y) ≥ FS (x,y),foreach (x,y) ∈ X × Y , (ii) R issaidtoequalto S,denotedby R = S,if R ⊂ S and S ⊂ R, (iii)theintersectionof R and S,denotedby R ∩ S,isaSVNRfrom X to Y definedas:
A ∩ B =(TA ∧ TB ,IA ∨ IB ,FA ∨ FB ), where(TA ∧ TB )(x,y)= TA(x,y) ∧ TB (x,y),(FA ∨ FB )(x,y)= FA(x,y) ∨ FB (x,y), foreach(x,y) ∈ X × Y , (iv)theunionof R and S,denotedby R ∪ S,isaSVNRin X to Y definedas:
A ∪ B =(TA ∨ TB ,IA ∧ IB ,FA ∧ FB ).
Proposition3.6 (See[25],Theorem 3.1). Let R,S,P ∈ SVNR(X × Y ).Then (1)(Rc) 1 =(R 1)c , (2)(R 1) 1 = R, (Rc)c = R, (3) R ⊂ R ∪ S and S ⊂ R ∪ S, (4) R ∩ S ⊂ R and R ∩ S ⊂ S, (5) if R ⊂ S,then R 1 ⊂ S 1 , (6) if R ⊂ P and S ⊂ P ,then R ∪ S ⊂ P , (7) if P ⊂ R and P ⊂ S,then P ⊂ R ∩ S, (8) if R ⊂ S,then R ∪ S = S and R ∩ S = R, (9)(R ∪ S) 1 = R 1 ∪ S 1 , (R ∩ S) 1 = R 1 ∩ S 1 , (10)(R ∪ S)c = Rc ∩ Sc , (R ∩ S)c = Rc ∪ Sc
Proof. TheproofsaresimilartoTheorem 3.1 in[25].
FromDefinitions 3.2 and 3.5,wecaneasilyobtainthefollowingresults.
Proposition3.7. Let R,S,P ∈ SVNR(X × Y ).Then (1)(Idempotentlaws): R ∪ R = R, R ∩ R = R, (2)(Commutativelaws): R ∪ S = S ∪ R, R ∩ S = S ∩ R, (3)(Associativelaws): R ∪ (S ∪ P )=(R ∪ S) ∪ P , R ∩ (S ∩ P )=(R ∩ S) ∩ P , (4)(Distributivelaws): R ∪ (S ∩ P )=(R ∪ S) ∩ (R ∪ P ), R ∩ (S ∪ P )=(R ∩ S) ∪ (R ∩ P ), (5)(Absorptionlaws): R ∪ (R ∩ S)= R, R ∩ (R ∪ S)= R.
Definition3.8. Let(Rj )j∈J ⊂ SVNR(X × Y ).Then 5
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(i)thetheintersectionof(Rj )j∈J ,denotedby j∈J Rj (simply, Rj ),isaSVNR from X to Y definedas: Rj =( TRj , IRj , FRj ), (ii)thetheunionof(Rj )j∈J ,denotedby j∈J Rj (simply, Rj ),isaSVNR from X to Y definedas: Rj =( TRj , IRj , FRj )
ThefollowingsaretheimmediateresultofDefinitions 3.2, 3.5 and 3.8 Proposition3.9. Let R ∈ SVNR(X ×Y ) andlet (Rj )j∈J ⊂ SVNR(X ×Y ).Then (1)( Rj )c = Rc j , ( Rj )c = Rc j , (2) R ∩ ( Rj )= (R ∩ Rj ), R ∪ ( Rj )= (R ∪ Rj )
Definition3.10. Let R ∈ SVNR(X × Y )andlet S ∈ SVNR(Y × Z).Thenthe compositionof R and S,denotedby S ◦ R,isaSVNRfrom X to Z definedas: S ◦ R =(TS◦R,IS◦R,FS◦R),
whereforeach(x,z) ∈ X × Z, TS◦R(x,z)= y∈Y (TR(x,y) ∧ TS (y,z)), IS◦R(x,z)= y∈Y (IR(x,y) ∨ IS (y,z)), FS◦R(x,z)= y∈Y (FR(x,y) ∨ FS (y,x)).
Proposition3.11. (1) P ◦ (S ◦ R)=(P ◦ S) ◦ R),where R ∈ SVNR(X × Y ), S ∈ SVNR(Y × Z) and P ∈ SVNR(Z × W ) (2) P ◦ (R ∪ S)=(P ◦ R) ∪ (P ◦ S),where R,S ∈ SVNR(X × Y ) and P ∈ SVNR(Y × Z). (3) If R ⊂ S,then P ◦ R ⊂ P ◦ S,where R,S ∈ SVNR(X × Y ) and P ∈ SVNR(Y × Z). (4)(S ◦ R) 1 = R 1 ◦ S 1,where R ∈ SVNR(X × Y ) and S ∈ SVNR(Y × Z)
Proof. (1)Let R ∈ SVNR(X × Y ),S ∈ SVNR(Y × Z)and P ∈ SVNR(Z × W ) andlet(x,w) ∈ (X × Z).Then TP ◦(S◦R)(x,w)= z∈Z (TS◦R(x,z) ∧ TP (z,w)) = z∈Z ([ y∈Y (TR(x,y) ∧ TS (y,z)] ∧ TP (z,w)) = y∈Y (TR(x,y) ∧ [ z∈Z (TS (y,z) ∧ TP (z,w))]) = y∈Y (TR(x,y) ∧ TP ◦S (y,w)) = T(P ◦S)◦R)(x,w). Similarly,wecanprovethat IP ◦(S◦R)(x,w)= I(P ◦S)◦R)(x,w)and FP ◦(S◦R)(x,w)= F(P ◦S)◦R)(x,w).Thustheresultholds.
(2)Let R,S ∈ SVNR(X × Y )and P ∈ SVNR(Y × Z)andlet(x,z) ∈ X × Z Then
TP ◦(R∪S)(x,z)= y∈Y (TR∪S (x,y) ∧ TP (y,z))
= y∈Y ([TR(x,y) ∨ TS (x,y)] ∧ TP (y,z))
=[ y∈Y (TR(x,y) ∧ TP (y,z)] ∨ [ y∈Y (TS (x,y) ∧ TP (y,z)]
= TP ◦R(x,z) ∨ TP ◦S (x,z) 6
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= T(P ◦R)∪(P ◦S)(x,z).
Similarly,wecanseethat IP ◦(R∪S)(x,z)= I(P ◦R)∪(P ◦S)(x,z)and FP ◦(R∪S)(x,z)= F(P ◦R)∪(P ◦S)(x,z).Thustheresultholds.
(3)Let R,S ∈ SVNR(X × Y )and P ∈ SVNR(Y × Z).Suppose R ⊂ S andlet (x,z) ∈ X × Z.Then TP ◦R(x,z)= y∈Y (TR(x,y) ∧ TP (y,z)) ≤ y∈Y (TS (x,y) ∧ TP (y,z)) [Since R ⊂ S, TR(x,y) ≤ TS (x,y)] = TP ◦S (x,z).
Similarly,wecanprovethat IP ◦R(x,z) ≥ IP ◦S (x,z)and FP ◦R(x,z) ≥ FP ◦S (x,z). Thustheresultholds.
(4)Let R ∈ SVNR(X × Y )and S ∈ SVNR(Y × Z)andlet(x,z) ∈ X × Z.Then T(S◦R) 1 (z,x)= T(S◦R)(x,z) = y∈Y (TR(x,y) ∧ TS (y,z)) = y∈Y (TS 1 (z,y) ∧ TR 1 (y,x)) = TR 1 ◦S 1 (z,x).
Similarlywecanseethat I(S◦R) 1 (z,x)= IR 1 ◦S 1 (z,x)and F(S◦R) 1 (z,x)= FR 1 ◦S 1 (z,x).Thustheresultholds.
Remark3.12. (1)ForanySVNRs R and S, S ◦ R = R ◦ S,ingeneral. (2)Forany R,S ∈ SVNR(X × Y )and P ∈ SVNR(Y × Z), P ◦ (R ∩ S) = (P ◦ R) ∩ (P ◦ S),ingeneral.
Example3.13. Let X = Y = {a,b}, Z = {x,y}.ConsidertwoSVNRs R and S in X,andanSVNR P from X to Z givenbyfollowingsinglevaluedneutrosophic matrices:
R = (0.6, 0.3, 0.4)(0.7, 0.2, 0.1) (0 4, 0 6, 0 3)(0 6, 0 4, 0 2) , S = (0.7, 0.4, 0.2)(0.4, 0.6, 0.4) (0.5, 0.2, 0.6)(0.3, 0.6, 0.5) and P = (0 7, 0 2, 0 3)(0 4, 0 6, 0 4) (0.4, 0.6, 0.2)(0.8, 0.2, 0.3) Then TP ◦(R∩S)(a,x)=0.6 =0.4= T(P ◦R)∩(P ◦S)(a,x).Thus P ◦ (R ∩ S) =(P ◦ R) ∩ (P ◦ S).
4. Singlevaluedneutrosophicreflexve,smmetricandansitive relations
Inthissection,weintroducesinglevaluedneutrosophicreflexve,smmetricand ansitiverelationsandobtainsomepropertiesrelatedtothem. Definition4.1 ([25]). Thesinglevaluedneutrosophicidentityrelationin X,denotedby IX (simply, I),isaSVNRin X definedas:foreach(x,y) ∈ X × X, TIX (x,y)= 1if x = y 0if x = y, IIX (x,y)= 0if x = y 1if x = y, FIX (x,y)= 0if x = y 1if x = y.
Itisclearthat I = I 1 and I c =(I c) 1 7
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Definition4.2 ([25]). R ∈ SVNR(X)issaidtobe:
(i)reflexive,ifforeach x ∈ X,TR(x,x)1,IR(x,x)= FR(x,x)=0, (ii)anti-reflexive,ifforeach x ∈ X,TR(x,x)=0,IR(x,x)= FR(x,x)=1.
FromDefinitions 4.1 and 4.2,itisobviousthat R isneutrosophicreflexiveifand onlyif I ⊂ R.
Thefollowingsaretheimmediateresultsoftheabovedefinition.
Proposition4.3 (See[19],Theorem 2.5.2). Let R ∈ SVNR(X).
(1) R isreflexiveifandonlyif R 1 isreflexive.
(2) If R isreflexive,then R ∪ S isreflexive,foreach S ∈ SVNR(X) (3) If R isreflexive,then R ∩ S isreflexiveifandonlyif S ∈ SVNR(X) is reflexive.
ThefollowingsaretheimmediateresultofDefinitions 3.2, 3.5 and 4.2
Proposition4.4. Let R ∈ SVNR(X)
(1) R isanti-reflexiveifandonly R 1 isanti-reflexive.
(2) If R isanti-reflexive,then R ∪ S isanti-reflexiveifandonlyif S ∈ SVNR(X) isanti-reflexive.
(3) If R isanti-reflexive,then R ∩ S isanti-reflexive,foreach S ∈ SVNR(X)
Proposition4.5. Let R,S ∈ SVNR(X).If R and S arereflexive,then S ◦ R is reflexive.
Proof. Let x ∈ X.Since R and S arereflexive, TR(x,x)=1,IR(x,x)= FR(x,x)=0 and TS (x,x)=1,IR(x,x)= FS (x,x)=0 Thus TS◦R = y∈X (TR(x,y) ∧ TS (y,x)) =[ x=y∈X (TR(x,y) ∧ TS (y,x))] ∨ (TR(x,x) ∧ TS (x,x)) =[ x=y∈X (TR(x,y) ∧ TS (y,x))] ∨ (1 ∧ 1) =1.
Ontheotherhand,
IS◦R = y∈X (IR(x,y) ∨ IS (y,x)) =[ x=y∈X (IR(x,y) ∨ IS (y,x))] ∧ (IR(x,x) ∨ IS (x,x)) =[ x=y∈X (IR(x,y) ∨ IS (y,x))] ∧ (0 ∨ 0) =0.
Similarly, FS◦R =0.So S ◦ R isreflexive.
Definition4.6. Let R =(TR,IR,FR) ∈ SVNR(X).Then (i)[19, 25] R issaidtobesymmetric,ifforeach x,y ∈ X, TR(x,y)= TR(y,x),IR(x,y)= IR(y,x),FR(x,y)= FR(y,x), (ii)[19] R issaidtobeanti-symmetric,ifforeach(x,y) ∈ X × X with x = y, TR(x,y) = TR(y,x),IR(x,y) = IR(y,x),FR(x,y) = FR(y,x), 8
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FromDefinitions 4.2 and 4.6,itisobviousthat φN isasymmetricandantireflexiveSVNR, XN and I aresymmetricandreflexiveSVNRsand I c isanantireflexiveSVNR.
ThefollowingistheimmediateresultofDefinitions 3.5 and 4.6
Result4.7 ([25],Theorem 3.1). Let R ∈ SVNR(X).Then R issymmetriciff R = R 1
Proposition4.8. Let R ∈ SVNR(X).If R issymmetric,then R 1 issymmetric. Proposition4.9. Let R,S ∈ SVNR(X).If R and S aresymmetric,then R ∪ S and R ∩ S aresymmetric.
Proof. Let(x,y) ∈ X × X.Since R and S andaresymmetric, TR(x,y)= TR(y,x),IR(x,y)= IR(y,x),FR(x,y)= FR(y,x) and TS (x,y)= TS (y,x),IS (x,y)= IS (y,x),FS (x,y)= FS (y,x)
Thus TR∪S (x,y)= TR(x,y) ∨ SR(x,y)= TR(y,x) ∨ SR(y,x)= TR∪S (y,x). Similarly,wecanseethat IR∪S (x,y)= IR∪S (y,x)and FR∪S (x,y)= FR∪S (y,x). So R ∪ S issymmetric. Similarly,wecanprovethat R ∩ S issymmetric.
Remark4.10. R and S arensymmetric,but S ◦ R isnotsymmetric,ingeneral. Example4.11. Let X = {a,b,c} andconsidertwoNRs R and S in X givenbythe followingsinglevaluedneutrosophicmatrices: R = (0.2, 0.4, 0.3)(1, 0.2, 0)(0.4, 1, 0.7) (1, 0.2, 0)(0.6, 0.2, 0.1)(0.3, 0.2, 0.6) (0.4, 1, 0.7)(0.3, 0.2, 0.6)(0.2, 0.4, 0.1) and S = (0 2, 0 4, 0 3)(0, 0 2, 0 6)(0 2, 0 6, 0 3) (0, 0.2, 0.6)(0.6, 0.2, 0.1)(0.3, 0.2, 0.6) (0.2, 0.6, 0.3)(0.3, 0.2, 0.6)(0.2, 0.4, 0.1) .
Thenclearly, R and S aresymmetric.But TS◦R(a,b)=0 6 =0 2= TS◦R(b,a) Thus S ◦ R isnotnsymmetric.
Thefollowinggivestheconditionforitsbeingsymmetric.
Proposition4.12. Let R,S ∈ SVNR(X).Let R and S besymmetric.Then S ◦ R issymmetricifandonlyif S ◦ R = R ◦ S
Proof. Suppose S ◦ R issymmetric.Since R and S andaresymmetric,byResult 4.7, R = R 1 and S = S 1.Thus S ◦ R =(S ◦ R) 1 [BythehypothesisandResult 4.7] = R 1 ◦ S 1 [ByProposition 3.11] = R ◦ S.
Conversely,suppose S ◦ R = R ◦ S.Then (S ◦ R) 1 = R 1 ◦ S 1 [ByProposition 3.11] 9
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= R ◦ S
[Since R and S andaresymmetric, R = R 1 and S = S 1] = S ◦ R.[Bythehypothesis] Thiscompletestheproof.
ThefollowingistheimmediateresultofProposition 4.12 Corollary4.13. If R issymmetric,then Rn issymmetric,forallpositiveinteger n,where Rn = R ◦ R ◦ ...n times. Definition4.14. (See[25]) R ∈ SVNR(X)issaidtobetransitive,if R ◦ R ⊂ R, i.e., R2 ⊂ R.
Proposition4.15. Let R ∈ SVNR(X).If R istransitive,then R 1 isso.
Proof. Let(x,y) ∈ X × X.Then TR 1 (x,y)= TR(y,x) ≥ TR◦R(y,x) = z∈X (TR(y,z) ∧ TR(z,x)) = z∈X (TR 1 (z,y) ∧ TR 1 (x,z)) = z∈X (TR 1 (x,z) ∧ TR 1 (z,y)) = TR 1 ◦R 1 (x,y). Similarly,wecanprovethat IR 1 (x,y) ≤ IR 1 ◦R 1 (x,y)and FR 1 (x,y) ≤ FR 1 ◦R 1 (x,y). Thustheresultholds.
Proposition4.16. Let R ∈ SVNR(X).If R istransitive,thensois R2
Proof. Let(x,y) ∈ X × X.Then TR2 (x,y)= z∈X (TR(x,z) ∧ TR(z,y)) ≥ z∈X (TR2 (x,z) ∧ TR2 (z,y)) = TR2 ◦R2 (x,y). Similarly,wecanseethat IR2 (x,y) ≤ IR2 ◦R2 (x,y)and FR2 (x,y) ≤ FR2 ◦R2 (x,y). Thustheresultholds.
Proposition4.17. Let R,S ∈ SVNR(X).If R and S aretransitive,then R ∩ S istransitive.
Proof. Let(x,y) ∈ X × X.Then T(R∩S)◦(R∩S)(x,y)= z∈X (TR∩S (y,z) ∧ TR∩S (z,x)) = z∈X ([TR(x,z) ∧ TS (x,z)] ∧ [TR(z,y) ∧ TS (z,y)]) = z∈X ([TR(x,z) ∧ TR(z,y)] ∧ [TS (x,z) ∧ TS (z,y)]) =( z∈X [TR(x,z) ∧ TR(z,y)]) ∧ ( z∈X [TS (x,z) ∧ TS (z,y)]) = TR◦R(x,y) ∧ TS◦S (x,y) ≤ TR(x,y) ∧ TS (x,y)[Since R and S aretransitive] = TR∩S (x,y). Similarly,wecanprovethat
I(R∩S)◦(R∩S)(x,y) ≥ IR∩S (x,y)and F(R∩S)◦(R∩S)(x,y) ≥ FR∩S (x,y). Thustheresultholds.
Remark4.18. Fortwosinglevaluedneutrosophictransitiverelation R and S in X, R ∪ S isnottransitive,ingeneral. 10
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Example4.19. Let X = {a,b} andconsidertwoSVNRs R and S in X givenby followingsinglevaledneutrosophicmatrices:
R = (0 8, 0 5, 0 4)(0 6, 0 4, 0 5) (0 7, 0 6, 0 2)(0 7, 0 6, 0 3) and S = (0 7, 0 4, 0 2)(0 4, 0 6, 0 4) (0 5, 0 4, 0 3)(0 5, 0 4, 0 4) . Thenwecaneasilyseethat R and S aretransitive.Ontheotherhand, R ∪ S = (0.8, 0.4, 0.2)(0.6, 0.4, 0.4) (0 7, 0 4, 0 2)(0 7, 0 4, 0 3)
Then T(R∪S)◦(R∪S)(a,b)=0 7 ≥ 0 6= TR∪S (a,b).Thus R ∪ S isnottransitive.
5. Singlevalurdneutrosophictransitiveclosure
Inthissection,wedefinetheconceptofthesinglevaluedneutrosophictransitive closureofanSVNRandstudysomeofitsproperties.
Definition5.1. Let R ∈ SVNR(X).Thenthesinglevaluedneutrosophictransitive closureof R,denotedby ˆ R,isdefinedas: ˆ R = R ∪ R2 ∪
ThefollowingistheimmediateresultofDefinition 5.1
Proposition5.2. Let R ∈ SVNR(X).Then (1) ˆ R istransitive. (2) R istransitiveiff R = ˆ R
Proposition5.3. Let R,S ∈ SVNR(X).If R ⊂ S,then ˆ R ⊂ ˆ S. Proof. ByDefinition 5.1, ˆ R = R ∪ R2 ∪ and ˆ S = S ∪ S2 ∪ Since R ⊂ S,by Proposition 3.11, R ◦ R ⊂ S ◦ R ⊂ S ◦ S.Then R2 ⊂ S2.Thus R3 ⊂ S3 andsoon. So ˆ R ⊂ ˆ S
Proposition5.4. Let R,S ∈ SVNR(X).If R issymmetric,then ˆ R issymmetric. Proof. ByCorollary 4.13, R2 , R3,...,aresymmetric.ThenbyProposition 4.9, ˆ R is symmetric.
Proposition5.5. Let R ∈ SVNR(X).Then ( ˆ R) 1 = ˆ R 1 Proof. (Rn) 1 =(R ◦ R ◦ ... ◦ R) 1 ntimes = R 1 ◦ R 1 ◦ ◦ R 1 =(R 1)n =(R 1)n Then ( ˆ R) 1 =(R ∪ R2 ∪ ) 1 = R 1 ∪ (R2) 1 ∪ = R 1 ∪ (R 1)2 ∪ = ˆ R 1
Proposition5.6. Forany R ∈ SVNR(X),Then ˆ R istheintersectionofallsingle valuedneutrosophictransitiverelationscontaining R 11
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Proof. Let R ∈ SVNR(X)andlet
R∗ = {RT : RT isatransitiverelationcontaining R}. Thenclearly, R∗ isthesmallesttransitiverelationcontaining R.Since ˆ R isatransitiverelationcontaining R, R∗ ⊂ ˆ R.
Conversely,let RT beanytransitiverelationcontaining R.ThenbyProposition 5.3, R ⊂ ˆ RT .Since RT istransitive,byProposition 5.2, ˆ RT = RT .Thus R ⊂ ˆ RT , foreach RT .So ˆ R ⊂ R∗.Thiscompletestheproof.
6. Singlevaluedneutrosophicequivalencerelation
Inthissection,wedefinetheconceptofasinglevaluedneutrosophicequivalence classandasinglevaluedneutrosophicpartition,andweprovethatthesetofall singlevaluedneutrosophicequivalenceclassesisaneutrosophicpartitionandinduce thesinglevaluedneutrosophicequivalencerelationfromasinglevaluedneutrosophic partition.
Definition6.1. R ∈ SVNR(X × X)iscalleda: (i)tolerancerelationon X,ifitisreflexiveandsymmetric, (ii)similarity(orequivalence)relationon X,ifitisreflexive,symmetricand transitive.
(iii)orderrelationon X,ifitisreflexive,anti-symmetricandtransitive. Wewilldenotethesetofalltolerance[resp.,equivalenceandorder]relationson X as SVNT (X)[resp., SVNE(X)and SVNO(X)].
ThefollowingistheimmediateresultofPropositions 4.3, 4.9 and 4.17.
Proposition6.2. Let (Rj )j∈J ⊂ SVNT (X) [resp., SVNE(X) and SVNO(X)]. Then Rj ∈ SVNT (X) [resp., SVNE(X) and SVNO(X)].
Proposition6.3. Let R ∈ SVNE(X).Then R = R ◦ R
Proof. FromDefinition 4.14,itisclearthat R ◦ R ⊂ R. Let(x,y) ∈ X × X.Then TR◦R(x,y)= z∈X (TR(x,z) ∧ TR(z,y)) ≥ TR(x,x) ∧ TR(x,y) =1 ∧ TR(x,y)[Since R isreflexive] = TR(x,y) and IR◦R(x,y)= z∈X (IR(x,z) ∨ IR(z,y)) ≤ IR(x,x) ∨ IR(x,y) =0 ∨ IR(x,y)[Since R isreflexive] = IR(x,y). Similarly, FR◦R(x,y) ≤ FR(x,y).Thus R ◦ R ⊃ R.So R ◦ R = R.
Definition6.4. Let A ∈ SVNS(X).Then A issaidtobenormal,if x∈X TA(x)= 1, x∈X IA(x)= x∈X FA(x)=0.
Definition6.5. Let R ∈ SVNE(X)andlet x ∈ X.Thenthesinglevalued neutrosophicequivalenceclassof x by R,denotedby Rx,isaSVNSin X defined as: Rx =(TRx ,IRx ,FRx ), 12
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where TRx ,IRx ,FRx : X → [0, 1]aremappings and TRx (y)= TR(x,y),IRx (y)= IR(x,y),FRx (y)= FR(x,y),foreach y ∈ X. Wewilldenotethesetofallsinglevaluedneutrosophicequivalenceclassby R as X/R anditwillbecalledthesinglevaluedneutrosophicquotientsetof X by R
Proposition6.6. Let R ∈ SVNE(X) andlet x,y ∈ X.Then (1) Rx isnormal,infact, Rx =0N , (2) Rx ∩ Ry =0N iff R(x,y)=(0, 1, 1), (3) Rx = Ry iff R(x,y)=(1, 0, 0)
Proof. (1)Since R isreflexive, TR(x,x)= TRx (x)=1, IR(x,x)= IRx (x)=0and FR(x,x)= FRx (x)=0. Thus y∈X TRx (y)=1, y∈X IRx (y)=0and y∈X FRx (y)=0.So Rx isnormal. Moreover, Rx =(1, 0, 0) =(0, 1, 1)=0N (x) Hence Rx =0N (2)Suppose Rx ∩ Ry =0N andlet z ∈ X.Then 0= TRx∩Ry (z)
= TRx (z) ∧ TRy (z)
= TR(x,z) ∧ TR(y,z)[ByDefinition 6.5] = TR(x,z) ∧ TR(z,y)[Since R issymmetric] and 1= IRx∪Ry (z) = IRx (z) ∨ IRy (z) = IR(x,z) ∨ IR(y,z)[ByDefinition 6.5] = IR(x,z) ∨ FIR(z,y).[Since R issymmetric] Thus 0= z∈X (TR(x,z) ∧ TR(z,y)) = TR◦R(x,y)
= TR(x,y)[ByProposition 6.3] and 1= z∈X (IR(x,z) ∨ IR(z,y)) = IR◦R(x,y) = IR(x,y)[ByProposition 6.3].
Similarly, FR(x,y)=1.So R(x,y)=(0, 1, 1). Thesufficientconditioniseasilyproved.
(3)Suppose Rx = Ry andlet z ∈ X.Then R(x,z)= R(y,z).Inparticular, R(x,y)= R(y,y).Since R isreflexive, R(x,y)=(1, 0, 0). Conversely,suppose R(x,y)=(1, 0, 0)andlet z ∈ X.Since R istransitive, R ◦ R ⊂ R.Then
TR(x,y) ∧ TR(y,z) ≤ TR(x,z), IR(x,y) ∨ IR(y,z) ≥ IR(x,z), FR(x,y) ∨ FR(y,z) ≥ FR(x,z)
Since R(x,y)=(1, 0, 0), TR(x,y)=1and IR(x,y)= FR(x,y)=0.Thus TR(y,z) ≤ TR(x,z),IR(y,z) ≥ IR(x,z),FR(y,z) ≥ FR(x,z)
So TRy (z) ≤ TRx (z),IRy (z) ≥ IRx (z),FRy (z) ≥ FRx (z).Hence Ry ⊂ Rx Similarly,wecanseethat Rx ⊂ Ry .Therefore Rx = Ry 13
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Definition6.7. LetΣ=(Aj )j∈J ⊂ SVNS(X).ThenΣiscalledasinglevalued neutrosophicpartitionof X,ifitsatisfiesthefollowings: (i) Aj isnormal,foreach j ∈ J , (ii)either Aj = Ak or Aj = Ak,forany j,k ∈ J , (iii) j∈J Aj =1N
ThefollowingistheimmediateresultofProposition 6.6 andDefinition 6.7. Corollary6.8. Let R ∈ SVNE(X).Then X/R isasinglevaluedneutrosophic partitionof X Proposition6.9. Let Σ beasinglevaluedneutrosophicpartitionof X.Wedefine R(Σ)=(TR(Σ),IR(Σ),FR(Σ)) as:foreach (x,y) ∈ X × X, TR(Σ)(x,y)= A∈Σ [TA(x) ∧ TA(y)],
IR(Σ)(x,y)= A∈Σ [IA(x) ∨ IA(y)], FR(Σ)(x,y)= A∈Σ [FA(x) ∨ FA(y)], where TR(Σ),IR(Σ),FR(Σ) : X × X → [0, 1] aremappings. Then R(Σ) ∈ SVNE(X). Proof. Let x ∈ X.ThenbyDefinition 6.7 (iii), TR(Σ)(x,x)= A∈Σ (TA(x) ∧ TA(x)= A∈Σ (TA(x)=1 and IR(Σ)(x,y)= A∈Σ (IA(x) ∨ IA(y)= A∈Σ (IA(x)=0= FR(Σ)(x,y)
Thus R(Σ)isreflexive. Fromthedefinitionof R(Σ),itisclearthat R(Σ)issymmetric. Let(x,y) ∈ X × X.Then TR(Σ)◦R(Σ)(x,y) = z∈X [TR(Σ)(x,z) ∧ TR(Σ)(z,y)] = z∈X [ A∈Σ(TA(x) ∧ TA(z)) ∧ B∈Σ(TB (z) ∧ TB (y))] = z∈X [( A∈Σ TA(z) ∧ B∈Σ TB (z)) ∧ (TA(x) ∧ TB (y))] = z∈X [(1 ∧ 1) ∧ (TA(x) ∧ TB (y))][Since A and B arenormal] = z∈X [TA(x) ∧ TB (y)] = TR(Σ)(x,y).
Similarly,wecanprovethat IR(Σ)◦R(Σ)(x,y)= IR(Σ)(x,y)and FR(Σ)◦R(Σ)(x,y)= FR(Σ)(x,y).Thus R(Σ)istransitive.So R(Σ) ∈ SVNE(X).
Proposition6.10. Let R,S ∈ SVNE(X).Then R ⊂ S iff Rx ⊂ Sx,foreach x ∈ X
Proof. Suppose R ⊂ S andlet x ∈ X.Let y ∈ X.Thenbythehypothesis, TRx (y)= TR(x,y) ≤ TS (x,y)= TSx (y), 14
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IRx (y)= IR(x,y) ≥ IS (x,y)= ISx (y),
FRx (y)= FR(x,y) ≥ FS (x,y)= FSx (y)
Thus Rx ⊂ Sx Theconversecanbeeasilyproved.
Proposition6.11. Let R,S ∈ SVNE(X).Then S ◦ R ∈ NE(X) iff S ◦ R = R ◦ S
Proof. Suppose S ◦ R = R ◦ S.Since R and S arereflexive,byProposition 4.5, S ◦ R isreflexive.Since R and S aresymmetric,bythehypothesisandProposition 4.12, S ◦ R issymmetric.Thenitissufficienttoshowthat S ◦ R istransitive. (S ◦ R) ◦ (S ◦ R)= S ◦ (R ◦ S) ◦ R [ByProposition 3.11 (2)] = S ◦ (S ◦ R)◦) =(S ◦ S) ◦ (R ◦ R) ⊂ S ◦ R Thus S ◦ R istransitive.So S ◦ R ∈ SVNE(X). Theconverseisimmediate.
Proposition6.12. Let R,S ∈ SVNE(X).If R ∪ S = S ◦ R, then R ∪ S ∈ SVNE(X)
Proof. Suppose R ∪ S = S ◦ R. Since R and S arereflexive,byResult 4.3 (2), R ∪ S isneutrosophicreflexive.Since R and S aresymmetric,bythehypothesisand Proposition 4.8, R ∪ S issymmetric.Thenbythehypothesis, S ◦ R issymmetric. ThusbyProposition 4.12, S ◦R = R◦S.SobyProposition 6.11, S ◦R ∈ SVNE(X). Hence R ∪ S ∈ SVNE(X).
7. Relationshipsbetweenaneutrosophicrelationandits α-cut
For Tα,Iα,Fα ∈ [0, 1], α =(Tα,Iα,Fα)willbecalledasinglevaluedneutrosophic value.Fortwosinglevaluedneutrosophicvalues α and β,
(i) α ≤ β iff Tα ≤ Tβ ,Iα ≥ Iβ and Fα ≥ Fβ (ii) α<β iff Tα <Tβ ,Iα >Iβ and Fα >Fβ Inparticular,theform α =(α, 1 α, 1 α)iscalledasinglevaluedneutrosophic constantanddenotedby α∗ . Wewilldenotethatsetofallsinglevaluedneutrosophicvalues[resp.constant] as SVNV [resp. SVNC].
Definition7.1. Let R ∈ SVNR(X × Y )andlet α ∈ SVNV
(i)Thestrong α-levelsubsetorstrong α-cutof R,denotedby[R]α,isanordinary relationfrom X to Y definedas:
[R]α = {(x,y) ∈ X × Y : TR(x,y) >Tα,IR(x,y) <Iα,FR(x,y) <Fα}.
(ii)The α-levelsubsetor α-cutof R,denotedby[R]α,isanordinaryrelation from X to Y definedas:
[R]α = {(x,y) ∈ X × Y : TR(x,y) ≥ Tα,IR(x,y) ≤ Iα,FR(x,y) ≤ Fα}.
Definition7.2. Let R ∈ SVNR(X × Y )andlet α∗ ∈ SVNC 15
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(i)Thestrong α∗-levelsubsetorstrong α∗-cutof R,denotedby[R]α∗ ,isan ordinaryrelationfrom X to Y definedas:
[R]α∗ = {(x,y) ∈ X × Y : TR(x,y) >α,IR(x,y) < 1 α,FR(x,y) < 1 α}
(ii)The α∗-levelsubsetor α∗-cutof R,denotedby[R]α∗ ,isanordinaryrelation from X to Y definedas:
[R]α∗ = {(x,y) ∈ X × Y : TR(x,y) ≥ α,IR(x,y) ≤ 1 α,FR(x,y) ≤ 1 α}
Example7.3. InExample 3.3,
[R](0 2,0 3,0 1) = {(x,y) ∈ X × X : TR(x,y) ≥ 0 2,IR(x,y) ≥ 0 3,FR(x,y) ≤ 0 1} = φ, [R](0 2,0 3,0 1) = {(c,c)}, and[R](0 2,0 3,0 1) = φ, [R](0 2,0 3,0 8) = {(c,c)}, [R](0 2,0 3,0 8) = {(a,a), (a,c), (c,c)}
[R]0 2∗ =[R](0 2,0 2,0 9) = {(a,a), (c,c)} =[R]0 2∗
Proposition7.4. Let R,S ∈ SVNR(X × Y ) andlet α,β ∈ SVNV.
(1) If R ⊂ S,then [R]α ⊂ [S]α and [R]α ⊂ [S]α. (2) If α ≤ β,then [R]β ⊂ [R]α and [R]β ⊂ [R]α.
Proof. (1)Let(x,y) ∈ [R]α.Then TR(x,y) ≥ Tα, IR(x,y) ≤ Iα and FR(x,y) ≤ Fα. Since R ⊂ S, TR(x,y) ≤ TS (x,y), IR(x,y) ≥ IS (x,y)and FR(x,y) ≥ FS (x,y).Thus SR(x,y) ≥ Tα, IS (x,y) ≤ Iα and FS (x,y) ≤ Fα.Hence[R]α ⊂ [S]α. Theproofofthesecondpartissimilar.
(2)Let(x,y) ∈ [R]β .Then TR(x,y) ≥ Tβ , IR(x,y) ≤ Iβ and FR(x,y) ≤ Fβ Since α ≤ β, Tα ≤ Tβ , Iα ≥ Iβ and Fα ≥ Fβ .Thus TR(x,y) ≥ Tα, IR(x,y) ≤ Iα and FR(x,y) ≤ Fα.So(x,y) ∈ [R]α.Hence[R]β ⊂ [R]α Theproofofthesecondpartissimilar.
ThefollowingistheparticularcaseoftheaboveProposition.
Corollary7.5. Let R,S ∈ SVNR(X × Y ) andlet α∗,β∗ ∈ SVNC
(1) If R ⊂ S,then [R]α∗ ⊂ [S]α∗ and [R]α∗ ⊂ [S]α∗ (2) If α∗ ≤ β∗,then [R]β∗ ⊂ [R]α∗ and [R]β∗ ⊂ [R]α∗
Proposition7.6. Let R ∈ SVNR(X × Y )
(1)[R]r isanordinaryrelationfrom X to Y ,foreach r ∈ SVNV (2)[R]r isanordinaryrelationfrom X to Y ,foreach r ∈ SVNV,where Tr ∈ [0.1) and Ir ,Fr ∈ (0, 1]. (3)[R]r = s<r [R]s,foreach r ∈ SVNV,where Tr ∈ (0, 1] and Ir ,Fr ∈ [0, 1). (4)[R]r = s>r [R]s,foreach r ∈ SVNV,where Tr ∈ [0, 1) and Ir ,Fr ∈ (0, 1].
Proof. Theproofsof(1)and(2)areclearfromDefinition 7.1 (3)FromProposition 7.4,itisobviousthat {[R]r : r ∈ SVNV} isadescending familyofordinaryrelationsfrom X to Y .Let r ∈ SVNV suchthat Tr ∈ (0, 1]and Ir ,Fr ∈ [0, 1).Thenclearly,[R]r ⊂ s<r [R]s.Assumethat(x,y) / ∈ [R]r .Then TR(x,y) <Tr or IR(x,y) >Ir or FR(x,y) >Fr .
Suppose TR(x,y) <Tr .Thenthereexists Ts ∈ (0, 1]suchthat TR(x,y) <Ts < Tr .Thus(x,y) / ∈ [R]s,i.e.,(x,y) / ∈ s<r [R]s.So s<r [R]s ⊂ [R]r .Hence [R]r = s<r [R]s
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Suppose IR(x,y) >Ir or FR(x,y) >Fr .Theneachcasecanbesimilarlyproved. (4)AlsofromProposition 7.4,itisobviousthat {[R]r : r ∈ SVNV]} isadescendingfamilyofordinaryrelationsfrom X to Y .Let r ∈ SVNV suchthat Tr ,Ir ∈ [0, 1) and Fr ∈ (0, 1].Thenclearly,[R]r ⊃ s>r Rs.Assumethat(x,y) / ∈ [R]r .Then TR(x,y) ≤ Tr or IR(x,y) ≤ Ir or FR(x,y) ≥ Fr
Suppose TR(x,y) ≤ Tr .Thenthereexists Ts ∈ [0, 1)suchthat TR(x,y) ≤ Tr <Ts Thus(x,y) / ∈ [R]s,i.e.,(x,y) / ∈ s>r [R]s.So s>r [R]s ⊂ [R]r .Hence [R]r = s>r [R]s
Suppose IR(x,y) ≤ Ir or FR(x,y) ≥ Fr .Theneachcasecanbesimilarlyproved.
ThefollowingistheparticularcaseoftheaboveProposition.
Corollary7.7. Let R ∈ SVNR(X × Y )
(1)[R]r∗ isanordinaryrelationfrom X to Y ,foreach r∗ ∈ SVNC (2)[R]r∗ isanordinaryrelationfrom X to Y ,foreach r∗ ∈ SVNC,where r ∈ [0, 1). (3)[R]r∗ = s∗<r∗ [R]s∗ ,foreach r∗ ∈ SVNV,where r ∈ (0, 1]. (4)[R]r∗ = s∗>r∗ [R]s∗ ,foreach r∗ ∈ SVNC,where r ∈ [0, 1).
Proposition7.8. Let X,Y benon-emptysetsandlet {Rr : r ∈ [0, 1]} beanonemptydescendingfamilyofordinaryrelationsfrom X to Y suchthat R0 = X × Y (1) Wedefine TR,IR,FR : X × Y → [0, 1] asfollows:foreach (x,y) ∈ X × Y , TR(x,y)= {r ∈ [0, 1]:(x,y) ∈ Rr }, IR(x,y)= FR(x,y) = {r ∈ [0, 1]:(x,y) / ∈ Rr } = {(1 r) ∈ [0, 1]:(x,y) ∈ Rr } =1 {r ∈ [0, 1]:(x,y) ∈ Rr }. Then R ∈ SVNR(X × Y ).
(2) Foreach r ∈ (0, 1],if Rr = s<r Rs,then [R]r∗ = Rr .
(3) Foreach r ∈ [0, 1),if Rr = s>r Rs,then [R]r∗ = Rr
Intheaboveproposition, R iscalledthesinglevaluedneutrosophicrelationfrom X to Y inducedby {Rr : r ∈ [0, 1]}
Proof. (1)Itisobviousfromthedefinitionof R.
(2)Suppose Rr = s<r Rs,foreach r ∈ (0, 1]andlet(x,y) ∈ Rr .Then TR(x,y)= {r ∈ [0, 1]:(x,y) ∈ Rr }≥ r and IR(x,y)= FR(x,y)=1 {r ∈ [0, 1]:(x,y) / ∈ Rr }≤ 1 r. Thus(x,y) ∈ Rr .So Rr ⊂ [R]r∗ ,foreach r ∈ (0, 1]. Nowlet(x,y) ∈ [R]r∗ .Then TR(x,y) ≥ r,IR(x,y) ≤ 1 r,FR(x,y) ≤ 1 r,say TR(x,y) ≥ r.Thusbythedefinitionof R, TR(x,y)= {k ∈ [0, 1]:(x,y) ∈ Rk} = s ≥ r.
Let > 0.Thenthereexists k ∈ (0, 1]suchthat s <k and(x,y) ∈ Rk.Thus r <s <k and(x,y) ∈ Rk.So(x,y) ∈ Rr .Since > 0isarbitrary, bythehypothesis,(x,y) ∈ Rr .Hence[R]r∗ ⊂ Rr .Therefore[R]r∗ = Rr ,foreach r ∈ (0, 1].
(3)Bythesimilarargumentoftheproofof(2),itisproved. 17
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ThefollowingistheimmediateresultofCorollary 7.7 andProposition 7.8 Corollary7.9. Let X,Y benon-emptysets,let R ∈ SVNR(X × Y ) andlet {[R]r∗ : r ∈ [0, 1]} beafamilyofallordinaryrelationsfrom X to Y .Wedefinemappings TS ,IS ,FS : X × Y →] 0, 1+[ asfollows:foreach (x,y) ∈ X × Y , TS (x,y)= {r ∈ [0, 1]:(x,y) ∈ [R]r∗ }, IS (x,y)= FS (x,y)=1 {r ∈ [0, 1]:(x,y) ∈ [R]r∗ } Then S ∈ SVNR(X × Y ) and R = S
Fromtheabovecorollary,wehavethefollowing.
Corollary7.10. Let X,Y benon-emptysetsandlet ,R,S ∈ SVNR(X × Y ).Then R = S iff [R]r∗ =[S]r∗ ,foreach r ∈ [0, 1],oralternatively,iff [R]r∗ =[S]r∗ ,for each r ∈ [0, 1]
Definition7.11. Let X,Y benon-emptysets,let R beanordinaryrelationfrom X to Y andlet RN ∈ SVNR(X × Y ).Then RN issaidtobecompatiblewith R,if R = S(RN ),where S(RN )= {(x,y): TRN (x,y) > 0,IRN (x,y) < 1,FRN (x,y) < 1}.
Example7.12. (1)Let X,Y benon-emptysets,let φX×Y betheordinaryempty relationfrom X to Y andlet0N,X×Y bethesinglevaluedneutrosophicempty relationfrom X to Y definedby0N,X×Y =(0, 1, 1),foreach x ∈ X.Thenclearly, S(0N,X×Y )= φX×Y .Thus0N,X×Y iscompatiblewith φX×Y (2)Let X,Y benon-emptysets,let X × Y bethewholeordinaryrelationfrom X to Y andlet1N,X×Y bethesinglevaluedneutrosophicwholerelationfrom X to Y definedby0N,X×Y =(1, 0, 0),foreach x ∈ X.Thenclearly, S(1N,X×Y )= X × Y . Thus1N,X×Y iscompatiblewith X × Y . (3)Let X,Y benon-emptysets,let r ∈ (0, 1)befixed.Wedefinethemappings TR,IR,FR : X × Y → [0, 1]asfollows:foreach(x,y) ∈ X × Y , TR(x,y)= r,IR(x,y)= FR(x,y)=1 r.
Thenclearly, R ∈ SVNR(X × Y )and S(R)= r∗∈SVNC [R]r∗ .Thus R iscompatiblewith r∗∈SVNC [R]r∗ .
Fromthefollowingresult,everyordinaryrelationcanbeconsiderasasingle valuedneutrosophicrelation.
Proposition7.13. Let X,Y benon-emptysets,let R beanordinaryrelationfrom X to Y andlet r ∈ (0, 1].Thenthereexists Rr∗ ∈ SVNR(X × Y ) suchthat Rr∗ is compatiblewith R and [Rr∗ ]r∗ = R.
Inthiscase, Rr∗ willbecalledan r∗-thsinglevaluedneutrosophicrelationfrom X to Y
Proof. Wedefinethemappings TR,IR,FR : X × Y → [0, 1]asfollows:foreach (x,y) ∈ X × Y ,
TRr∗ (x,y)= r if(x,y) ∈ R 0if(x,y) / ∈ R, IRr∗ (x,y)= FRr∗ (x,y)= 1 r if(x,y) ∈ R 1if(x,y) / ∈ R.
Thenclearly, Rr∗ ∈ SVNR(X × Y )and[Rr∗ ]r∗ = R.Moreover,bythedefinition of Rr∗ , S(Rr∗ )= R.Thus Rr∗ iscompatiblewith R 18
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ThefollowingistheimmediateresultofDefinitions 3.5 and 7.1.
Proposition7.14. Let R,S ∈ SVNR(X × Y ) andlet α ∈ SVNV.Then (1)[R ∪ S]α =[R]α ∪ [S]α, [R ∪ S]α =[R]α ∪ [S]α, (2)[R ∩ S]α =[R]α ∩ [S]α, [R ∩ S]α =[R]α ∩ [S]α
ThefollowingistheimmediateresultofDefinition 7.2 andProposition 7.14
Corollary7.15. Let R,S ∈ SVNR(X × Y ) andlet α∗ ∈ SVNC.Then (1)[R ∪ S]α∗ =[R]α∗ ∪ [S]α∗ , [R ∪ S]α∗ =[R]α∗ ∪ [S]α∗ , (2)[R ∩ S]α∗ =[R]α∗ ∩ [S]α∗ , [R ∩ S]α∗ =[R]α∗ ∩ [S]α∗ .
FromDefinitions 4.2, 4.6 and 7.1 itisclearthat R ∈ SVNR(X)isreflexive[resp. symmetric],then[R]α and[R]α areordinaryreflexive[resp.symmetric]on X,for each α ∈ SVNV
Proposition7.16. Let R ∈ SVNR(X × Y ) andlet α ∈ SVNV.If R istransitive, then [R]α and [R]α areordinarytransitiveon X
Proof. Suppose R istransitive.Then R ◦ R ⊂ R,i.e., TR◦R ⊂ TR, IR◦R ⊃ IR and FR◦R ⊃ FR. Let(x,z) ∈ [R]α ◦ [R]α.Thenthereexists y ∈ X suchthat (x,z), (z,y) ∈ [R]α.Thus
TR(x,z) ≥ Tα,IR(x,z) ≤ Iα,FR(x,z) ≤ Fα and TR(z,y) ≥ Tα,IR(z,y) ≤ Iα,FR(z,y) ≤ Fα So TR(x,z) ∧ TR(z,y) ≥ Tα,IR(x,z) ∨ IR(z,y) ≤ Iα,FR(x,z) ∨ FR(z,y) ≤ Fα. Since R ◦ R ⊂ R, TR(x,y) ≥ TR(x,z) ∧ TR(z,y),IR(x,y) ≤ IR(x,z) ∨ IR(z,y), FR(x,y) ≤ FR(x,z) ∧ FR(z,y).Hence TR(x,y) ≥ Tα,IR(x,y) ≤ Iα,FR(x,y) ≤ Fα, i.e.,(x,y) ∈ [R]α.Therefore[R]α isordinarytransitive. Theprofofthesecondpartissimilar.
FromDefinitions 4.2, 4.6 and 7.2 itisclearthat R ∈ NR(X)isreflexive[resp. symmetric],then[R]α∗ and[R]α∗ areordinaryreflexive[resp.symmetric]on X,for each α∗ ∈ NCV.Moreover,weobtainthefollowingfromProposition 7.16
Corollary7.17. Let R ∈ NR(X × Y ) andlet α∗ ∈ NCV.If R istransitive,then [R]α∗ and [R]α∗ areordinarytransitiveon X
Thefollowingsaretheimmediateresultsof 4.2, 4.6,Proposition 7.16 andCorollary 7.17.
Corollary7.18. Let R ∈ SVNE(X) andlet α ∈ SVNV.Then [R]α and [R]α are ordinaryequivalencerelationon X
Corollary7.19. Let R ∈ SVNE(X) andlet α∗ ∈ SVNC.Then [R]α∗ and [R]α∗ areordinaryequivalencerelationon X 19
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8. Conclusions
Fromnowon,wedealtwithpropertiesofsinglevaluedneutrosophicreflexive, symmetric,transitiverelationsandsinglevaluedneutrosophicequivalencerelations. Inparticular,wedefinedasinglevaluedneutrosophicequivalenceclassofapoint inaset X moduloasinglevaluedneutrosophicequivalencerelation R andasingle valuedneutrosophicpartitionofaset X.Andweprovedthatthesetofallsingle valuedneutrosophicequivalenceclassesisasinglevaluedneutrosophicpartition andinducedthesinglevaluedneutrosophicequivalencerelationbyasinglevalued neutrosophicpartition.However,wedidnotdealwiththequotientof S by R,for anySVNRs R and S suchthat R ⊂ S anddecompositionofamapping f : X → Y by Vneutrosophicrelations.Furthermore,wedefined α-cutofaSVNRandinvestigated somerelationshipsbetweenSVNRsandtheir α-cuts.
Inthefuture,wewillsolvebytheabovetwoproblemsanddealwithsinglevalued neutrosophicrelationsinafixedSVNS A
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J.Kim (junhikim@wku.ac.kr)
DepartmentofMathematicsEducation,WonkwangUniversity,460,Iksan-daero, Iksan-Si,Jeonbuk54538,Korea
P.K.Lim (pklim@wku.ac.kr)
DivisionofMathematicsandInformationalStatistics,InstituteofBasicNaturalScience,WonkwangUniversity,460,Iksan-daero,Iksan-Si,Jeonbuk54538,Korea
J.G.Lee (jukolee@wku.ac.kr)
DivisionofMathematicsandInformationalStatistics,InstituteofBasicNaturalScience,WonkwangUniversity,460,Iksan-daero,Iksan-Si,Jeonbuk54538,Korea
K.Hur (kulhur@wku.ac.kr)
DivisionofMathematicsandInformationalStatistics,InstituteofBasicNatural Science,WonkwangUniversity,460,Iksan-daero,Iksan-Si,Jeonbuk54538,Korea
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