Fuzzy Neutrosophic Equivalence Relations

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1 wwwijirdm Jur 6 Vl 5 ssue SS 78 Olie u eutrsphi Equivlee eltis J Mrti Je eserh Shlr Deprtmet f Mthemtis irml Cllege fr Wme Cimtre mil du di rkiri riipl irml Cllege fr Wme Cimtre mil du di strt: his pper itrdues the ept f fu eutrsphi equivlee reltis d disuss sme f their prperties ls we defie fu eutrsphi trsitive lsure d ivestigte their prperties Kewrds: u eutrsphi equivlee relti fu eutrsphi equivlee lss fu eutrsphi trsitive lsure MSC : 3993E99 trduti eltis re suitle tl fr desriig rrespdees etwee jets Crisp reltis like hve served well i develpig mthemtil theries he use f fu reltis rigited frm the servti tht rel life jets e relted t eh ther t erti degree u reltis re le t mdel vgueess ut the t mdel uertit tuitiisti fu sets s defied tssv [45] give us w t irprte uertit i dditil degree995 lretie Smrdhe [ 3]eteded the ept f ituitiisti fu sets t tri mpet lgi set with-stdrd itervl mel eutrsphi set Mtivted this ept rkiriet l [] defied the fu eutrsphi set i whih the -stdrd itervl is tke s stdrd itervl 996 ustie d urill [7] itrdued the ept f ituitiisti fu reltis d studied sme f its prperties 3 Deshrijver d Kerre [9] ivestigted sme prperties f the mpsiti f ituitiisti fu reltis this pper we itrdue d stud sme prperties f fu eutrsphi equivlee reltis d fu eutrsphi trsitive lsures relimiries Defiiti : [] u eutrsphi set the uiverse f disurse is defied s where : [] d Defiiti :[] u eutrsphi set is suset f u eutrsphi set ie fr ll if Defiiti 3: [] Let e -empt set d he m m mi mi mi m Defiiti 4: [] he differee etwee tw u eutrsphi sets d is defied s \ mi mi m e tw u eutrsphi sets EOL JOUL O OVVE ESECH & DEVELOME ge 9

2 wwwijirdm Jur 6 Vl 5 ssue Defiiti 5: [] u eutrsphi set ver the uiverse is sid t e ull r empt u eutrsphi set if fr ll t is deted Defiiti 6: [] u eutrsphi set ver the uiverse is sid t e slute uiverse u eutrsphi set if fr ll t is deted Defiiti 7: [] he mplemet f u eutrsphi set is deted d is defied s where he mplemet f u eutrsphi set ls e defied s Defiiti 8: [3] fu eutrsphi set relti is defied s fu eutrsphi suset f hvig the frm { : } where : Stisf the diti We will dete with the set f ll fu eutrsphi susets i Defiiti 9: [3] Give ir fu eutrsphi relti etwee d we defie etwee d mes f t whih we ll iverse relti f Defiiti : [3] Let d e tw fu eutrsphi reltis etwee d fr ever We defie 3 { 4 { 5 { : } Defiiti : [3] Let e t-rms r t-rms t eessril dul tw tw d We will ll mpsed relti t the e defied / Where { } Wheever { } + + { } 3 he hie f the t-rms d t-rms i the previus defiiti is evidetl ditied the fulfilmet f EOL JOUL O OVVE ESECH & DEVELOME ge

3 wwwijirdm Jur 6 Vl 5 ssue Defiiti: [3] he relti is lled the relti f idetit if he mplemetr relti is defied 3u eutrsphi equivlee reltis Defiiti 3: Let e set d let he the mpsiti f d ls edefied s fllws : fr [ ] [ ] [ d ] EOL JOUL O OVVE ESECH & DEVELOME ge Defiiti 3: Let e set d let 3 he 3 3 f d the prtiulr if the f the 6 d 8 rpsiti 3: Let d e fu eutrsphi reltis set f the rf: rf fllws frm defiiti 3 Defiiti 33: fu eutrsphi relti set is lled fu eutrsphi equivlee relti i shrt E if it stisfies the fllwig ditis: fr eh i t is fu eutrsphi refleive ie ii t is fu eutrsphi smmetri ie iii t is fu eutrsphi trsitive ie We will dete the set f ll Es s E he fllwig prpsiti is the immedite result f defiiti 3 rpsiti 3: Let e set d let i f is fu eutrsphi refleive respetivel smmetri trsitive the is fu eutrsphi refleive respetivel smmetri trsitive ii f is fu eutrsphi refleive respetivel smmetri trsitive the is fu eutrsphi refleive respetivel smmetri trsitive iii f is fu eutrsphi refleive the iv f is fu eutrsphi smmetri the d re smmetri d v f d re fu eutrsphi refleive respetivel smmetri trsitive he refleive respetivel smmetri trsitive is fu eutrsphi

4 wwwijirdm Jur 6 Vl 5 ssue vi f d re fu eutrsphismmetri the is fu eutrsphi smmetri rf: t is the immedite result f defiiti 33 he fllwig tw results re esil see rpsiti 34: Let e set f E the rpsiti 35: Let { α e -empt fmil f Es set he α E Hwever i geerl U α eed } αγ t e E Emple 3: Let { } αγ Let d e the s represeted mtries re give elw E d is the fu eutrsphi relti represeted the fllwig mtri he lerl > 6 O the ther hd 4 > 3 d < is t fu eutrsphi trsitive Hee E hus S rpsiti 36: Let d e fu eutrsphi refleive reltis set he rf: Let he [ t t ] t Sie d re fu eutrsphi refleive Similrl t [ t t ] αγ is ls fu eutrsphi refleive relti EOL JOUL O OVVE ESECH & DEVELOME ge

5 wwwijirdm Jur 6 Vl 5 ssue hus fr eh Hee is fu eutrsphi relti rpsiti 37: Let e set d let E f the E rf: Let Sie d re fu eutrsphi refleive [ ] Similrl d [ ] hus S is fu eutrsphi refleive Let [ ] [ ] Sie d re fu eutrsphi smmetri [ ] Similrl he [ ] [ ] S is fueutrsphi smmetri O the ther hd prpsiti3 Sie d re fu eutrsphi trsitive E Hee Defiiti 34: Let e fu eutrsphi equivlee relti set d let We defie mple mppig : s fllws : fr eh he lerl S he fu eutrsphi set i is lled fu eutrsphi equivlee lss f tiig he set { : } is lled the fu eutrsphi qutiet set f d deted / herem 3: Let e set d let E he the fllwig hld: if d l if fr fr i ii if d l if iii U iv here eists surjeti p / rf: : lled the turl mppig defied p fr eh i Suppse Hee Sie is fu eutrsphi equivlee relti EOL JOUL O OVVE ESECH & DEVELOME ge 3

6 wwwijirdm Jur 6 Vl 5 ssue Cversel suppse he [ ] Similrl Let he Sie is fu eutrsphi trsitive [ ] the similr rgumets we hve Hee he prfs f ii iii d iv re es his mpletes the prf Defiiti 35: Let e set let d let { α } αγ lled the E geerted d deted t is esil see tht hus e the fmil f ll the Es tiig he α is e e is the smllest fu eutrsphi equivlee relti tiig Defiiti 36: Let e set d let he the fu eutrsphi trsitive lsure f deted fllws: U where rpsiti 37: Let e set d let he i iif there eists Emple 3: Let { } he 3 Mrever suh tht i whih urs times + the αγ is defied s is the smllest fu eutrsphi trsitive relti tiig d let e the defied s fllws: S hus EOL JOUL O OVVE ESECH & DEVELOME ge 4

7 wwwijirdm Jur 6 Vl 5 ssue Hee is fu eutrsphi trsitive rpsiti 39: f is fu eutrsphi smmetrithe s is rf: r d [ ] Similrl [ ] is fu eutrsphi smmetri fr ther prf: Hee t is ler tht is fu eutrsphi smmetri Suppse is fu eutrsphi smmetrilet he k + [ ] [ ] is fu eutrsphi smmetri [ k ] [ k ] k [ k ] k Similrl k + k is fu eutrsphi smmetri fr k > We shw tht k + k + d k + k [ k ] [ ] k [ k ] k k + S f is fu eutrsphi smmetri fr rpsiti 3: Let e set d let d E rf: t is ler tht Hee Hee is fu eutrsphi smmetri he f the the Suppse defiiti 3 Suppse d E k fr k k k fr > he it is ler tht k he defiiti 3 Suppse he k + k S herem 3: e f is fu eutrsphi relti set the [ ] hus + + k k k+ fr Hee EOL JOUL O OVVE ESECH & DEVELOME ge 5

8 wwwijirdm Jur 6 Vl 5 ssue EOL JOUL O OVVE ESECH & DEVELOME ge 6 rf: Let [ ] he lerl prpsiti 38 is fu eutrsphi trsitivelet Sie d hus S is fu eutrsphi refleive t is ler tht [ ] is fu eutrsphi smmetri prpsiti 38 is fu eutrsphi smmetri Hee E w E K suh tht K he K d K K hus K defiiti34[ ] K K fr S K Hee [ ] e his mpletes the prf rpsiti 3: Let e set d let E We defie s fllws: ie he E rf: prpsiti 38 is fu eutrsphi trsitive Let Sie d re fu eutrsphi refleive [ ] [ ] [ ] hus is fu eutrsphi refleive w let Sie d re fu eutrsphi smmetri [ ] [ ] [ ] [ ] [ ] [ ] hus is fu eutrsphi smmetri Hee E he fllwig result gives ther desripti fr f tw Es d herem 33: Let e set d let E f E the where detes the lest upper ud fr { } with respet t the ilusi rf: Let he he [ ] Sie is fu eutrsphi refleive Similrl [ ] hus Similrl we hve S is upper ud fr { } with respet t w let e fu eutrsphi equivlee relti suh tht d Let he [ ] [ ] sie is fu eutrsphi trsitive Similrl d [ ] [ ]

9 wwwijirdm Jur 6 Vl 5 ssue hus S is the lest upper ud fr { } with respet t Hee rpsiti 3: Let e set f E rf: Suppse E the [ ] e he therem 3 E Sie d hus prpsiti 39 E d hus Hee he fllwig is the immedite result f rpsiti 3 d rpsiti 3 Crllr 3: Let e set f E Sie defiiti 3 d O the ther hd sie prpsiti 39 suh tht the S 4 eferees i rkiri Sumthi d J Mrti Je u eutrsphi Sft plgil Spes tertil jurl f mthemtil rhives ii rkiri J Mrti Je Mre u eutrsphi sets d u eutrsphi plgil spes tertil jurl f ivtive reserh d studies M 4 vl 3 ssue iii rkiri J Mrti Jeu eutrsphi reltiscmmuited iv K tssv tuitiisti fu sets u sets d sstems 986:87-96 v K tssv Mre ituitiisti fu sets u sets d sstems98933:37-46 vi uhesu Sme servtis tuitiisti fu reltis timert Semir util equtis-8 vii ustie H urillstrutures ituitiisti fu reltis u sets d sstems Vl viii ChkrrthMK M Ds Studies i fu relti ver fu susets u sets d sstems Vl i DeshrijverG Ee Kerre O the mpsiti f ituitiisti fureltis u sets d sstems Vl DDuis d H rde lss f u mesures sed trigulr rmsstj Geerl sstems i Mukerjee Sme servtis fu reltis ver fu susets u sets d sstems Vl ii MurliV u equivlee reltis u sets d sstems iii Smrdhe eutrsphi set geerliti f the ituitiisti fu sets terj ure pplmth EOL JOUL O OVVE ESECH & DEVELOME ge 7

Introduction to Matrix Algebra

Introduction to Matrix Algebra Itrodutio to Mtri Alger George H Olso, Ph D Dotorl Progrm i Edutiol Ledership Applhi Stte Uiversit Septemer Wht is mtri? Dimesios d order of mtri A p q dimesioed mtri is p (rows) q (olums) rr of umers,