Intuitionistic Fuzzy Hypersoft Sets

https://communications.science.ankara.edu.tr

Commun.Fac.Sci.Univ.Ank.Ser.A1Math.Stat. Volume70,Number1,Pages443ñ455(2021)

DOI:10.31801/cfsuasmas.788329 ISSN1303ñ5991E-ISSN2618ñ6470

Receivedbytheeditors:August31,2020;Accepted:January29,2021

INTUITIONISTICFUZZYHYPERSOFTSETS

AdemYOLCU1,FlorentinSMARANDACHE2,andTahaYasinOZTURK1

1DepartmentofMathematics,KafkasUniversity,36100,Kars,TURKEY 2DepartmentofMathematics,UniversityofNewMexico, Gallup,NM87301,USA

Abstract. Inthispaper,anewenvironmentnamely,intuitionisticfuzzyhypersoftset(IFHSS)isdeÖned.Weintroducesomefundamentaloperatorsof intuitionisticfuzzyhypersoftsetssuchassubset,nullset,absoluteset,complement,union,intersection,equalsetetc.Validityandapplicationarepresented withappropriateexamples.Forgreaterprecisionandaccuracy,inthefuture, proposedoperationsindecisionmakingprocessessuchaspersonalselection, managementissuesandotherswillplayavitalrole.

1. Introduction

ThefuzzysettheoryidentiÖedin1965byZadeh[26]isoneofthemostpopular theoriesofrecenttimes.ZadehspeciÖedthatthereisaconsiderableamountof ambiguityinmostreal-lifesituationsandphysicalproblemsthattheclassicalset theoryanditsnormalmathematicaltheoriescenteredonsuchsettheorydidnot giveusthenecessaryknowledgeandinferences.Fuzzysettheoryhasbroughta greatparadigmaticchangeinmathematics,butthistheoryalsohassomestructural di¢cultiesinitsnature.FuzzysetstructureisdeÖnedwiththehelpofmembership function.Itísdi¢culttocreateamembershipfunctionforeachevent,accordingto Molodtsov,becausecreatingamembershipfunctionistooindividual.

Molodtsov[15]introducedthesoftsettheoryin1999whichhefeltwasmore practical.Thistheoryisarelativelynewapproachtosolvingproblemsinvolving decisionmakinganduncertainty.ThemajorbeneÖtofsoftsettheoryisthatin fuzzysetandothertheoriesitisfreefromdi¢culties.Softsettheoryhasbecome popularamongresearchersinashorttimeandmanyscientiÖcstudiesarecarried

2020 MathematicsSubjectClassiÖcation. 03F55,03E72,03E75.

Keywordsandphrases. Softset,uncertainty,hypersoftset,intuitionisticfuzzyhypersoftsets. yolcu.adem@gmail.com-Correspondingauthor;smarand@unm.edu;taha36100@hotmail.com 0000-0002-4317-652X;0000-0002-5560-5926;0000-0003-2402-6507.

c 2021AnkaraUniversity CommunicationsFacultyofSciencesUniversityofAnkara-SeriesA1MathematicsandStatistics 443

outonthistheoryeveryyear[8,16,17,27].Majietal.[12,14]proposedanimplementationofsoftsetsfocusedonparameterreductiontopreserveoptimumselection artifactsindecision-makingproblems.Chen[5]investigatedanewdeÖnitionand variousapplicationsofthereductionofsoftsetstoparameters.PeiandMiao[19] haveshownthatsoftsetsareaspecialclassofinformationsystems.Kongetal.[9] introducedthereductionandalgorithmofsoftsetstonormalparameters.Zouand Xiao[28]discussedthesoftdataanalysisapproach.Aktasand«aº gman[2]deÖned thealgebraicstructureofsoftsettheory.

Theintuitionisticfuzzyset(IFS)theory[3,4]wascreatedbyaddinganonmembershipfunctiontothefuzzysetstructure.Thenon-memberfunctionmakes IFSmorefunctionalindecision-makingproblems.Majietal.[10,13]combinedsoft settheorywiththetheoryoftheintuitionisticfuzzysetsandcalledintuitionistic fuzzysoftSet(IFSS).TheparameterizationandhesitancyacquiredbyIFSSfrom thismixturefacilitatesveryaccuratelythedescriptionofreal-worldsituations.IFSS isavaluablemethodforaddressingdatauncertaintyandvagueness.ManyscientiÖc paperhaveshownthesuitabilityofIFSStoissuedecisionmaking[7,11].

Smarandache[21]introducedanewtechniquetodealwithuncertainty.By convertingthefunctionintomultipledecisionfunctions,hegeneralizedthesoftto hypersoftset.Manystudieshavebeendonerecentlyusinghypersoftsetstructure [1,6,18,20,22,23,24,25].

Multi-criteriadecision-making(MCDM)isconcernedwithcoordinatingandtakingcareofmattersofpreferenceandpreparation,includingmulti-criteria.Intuitionisticfuzzysoftsetenvironmentscannotbeusedtosolvecertaintypesof problemsifattributesaremorethanoneandfurtherbifurcated.Therefore,there wasaseriousneedtoidentifyanewapproachtosolvesuchproblems,soanew setting,namelytheintuitionisticfuzzyhypersoftSets(IFHSS),isestablishedfor thisreason.Inthepresentpaper,weintroduceintuitionisticfuzzyhypersoftset theory.IntuitionisticfuzzyhypersoftsettheoryisamixtureofIFStheoryandthe hypersoftsettheory.Thecomplement,subset,equalset,ìANDî,ìORî,intersection,unionnotionsaredeÖnedonintuitionisticfuzzyhypersoftsets.Thispaper alsoissupportedbymanysuitableexamples.

2. Preliminary

DeÖnition1. [3]Anintuitionisticfuzzyset H in U is H = f(u; H (u); H (u)): u 2 U g; where H : U ! [0; 1]; H : U ! [0; 1] withthecondition 0 H (u)+ H (u) 1; 8u 2 U: H ; H 2 [0; 1] denotethedegreeofmembershipandnonmembershipof u to H; respectively.Thesetofallintuitionisticfuzzysetsover U willbedenotedby IFP (U ):

DeÖnition2. [15]Let U beaninitialuniverseand E beasetofparameters.A pair (H;E) iscalledasoftsetover U; where H isamapping H : E !P(U ): In otherwords,thesoftsetisaparameterizedfamilyofsubsetsoftheset U:

DeÖnition3. [13]Let U beaninitialuniverseand E beasetofparameters: A pair (H;E) iscalledanintuitionisticfuzzysoftsetover U; where H isamapping givenby, H : E ! IFP (U ): Ingeneral,forevery e 2 E;H(e) isanintuitionisticfuzzysetof U anditis calledintuitionisticfuzzyvaluesetofparameter e: Clearly, H(e) canbewrittenas aintuitionisticfuzzysetsuchthat H(e)= f(u; H (u); H (u)): u 2 U g

DeÖnition4. [21]Let U betheuniversalsetand P (U ) bethepowersetof U Consider e1;e2;e3;:::;en for n 1,be n well-deÖnedattributes,whosecorrespondingattributevaluesareresspectivelythesets E1;E2;:::;En with Ei \ Ej = ;; for i = j and i;j 2f1; 2;:::;ng; thenthepair (H;E1 E2 ::: En) issaidtobe Hypersoftsetover U where H : E1 E2 ::: En ! P (U ):

3. IntuitionisticFuzzyHypersoftSets

DeÖnition5. Let U betheuniversalsetand IFP (U ) betheintuitionisticfuzzy powersetof U .Consider e1;e2;e3;:::;en for n 1,be n well-deÖnedattributes, whosecorrespondingattributevaluesarerespectivelythesets E1;E2;:::;En with Ei \ Ej = ;; for i = j and i;j 2f1; 2;:::;ng: Let Ai bethenonemptysubsetof Ei foreach i =1; 2;:::;n: AnintuitionisticfuzzyhypersoftsetdeÖnedasthepair (H;A1 A2 An) where; H : A1 A2 An ! IFP (U ) and H(A1 A2 An)= f< ; ( u H( )(u); H( )(u)) >: u 2 U; 2 A1 A2 An E1 E2 Eng where and arethemembershipandnon-membershipvalue,respectivelysuchthat 0 H( )(u)+ H( )(u) 1 and H( )(u); H( )(u) 2 [0; 1] Forsakeofsimplicity, wewritethesymbols for E1 E2 En; for A1 A2 An and for anelementoftheset

DeÖnition6. i)Anintutionisticfuzzyhypersoftset (H; ) overtheuniverse U is saidtobenullintuitionisticfuzzyhypersoftsetanddenotedby 0(UIFH ; ) ifforall u 2 U and 2 ; H( )(u)=0 and H( )(u)=1: ii)Anintutionisticfuzzyhypersoftset (H; ) overtheuniverse U issaidtobe absoluteintuitionisticfuzzyhypersoftsetanddenotedby 1(UIFH ; ) ifforall u 2 U and 2 ; H( )( )=1 and H( )(u)=0

Example7. Let U bethesetofcarsgivenas U = fu1;u2;u3g alsoconsiderthe setofattributesas E1 = Fuel, E2 =Transmission, E3 = Colorandtheirrespective attributesaregivenas

E1 = Fuels = fGasoline( 1),Diesel( 2),Electric( 3)g

E2 = Transmissions = fAutomatic( 1),Manual( 2)g

E3 = Colors = fBlack( 1),Blue( 2),White( 3)g

Supposethat

A1 = f 3g;A2 = f 1; 2g;A3 = f 1; 3g

B1 = f 1; 3g;B2 = f 1g;B3 = f 1; 2g

aresubsetof Ei foreach i =1; 2; 3 Thentheintuitionisticfuzzyhypersofts (H; 1) and (G; 2) deÖnedasfollows;

(H; 1)= ( < ( 3; 1; 1); f u1 (0 6;0;3) ; u2 (0 4;0 2) g >;< ( 3; 1; 3); f u2 (0 2;0 6) ; u3 (0 3;0 1) g >; < ( 3; 2; 1); f u1 (0 7;0 3) ; u2 (0 3;0 5) ; u3 (0 1;0 6) g >;< ( 3; 2; 3); f u1 (0 8;0 2) ; u2 (0 2;0 4) g > )

(G; 2)= ( < ( 1; 1; 1); f u1 (0 3;0;3) ; u2 (0 1;0 2) ; u3 (0 6;0 4) g >;< ( 1; 1; 2); f u1 (0 7;0 2) ; u2 (0 6;0 1) g >; < ( 3; 1; 1); f u1 (0 9;0 1) ; u2 (0 2;0 7) g >;< ( 3; 1; 2); f u2 (0 5;0 3) ; u3 (0 3;0 7) g > )

Tabularformofthesesetsaregivenasfollows:

Table1. TabularformofIFHSS (H; 1)

(H; 1) u1 u2 u3 ( 3; 1; 1)(06; 03)(04; 02)(0; 1) ( 3; 1; 3)(0; 1)(02; 06)(03; 01) ( 3; 2; 1)(07; 03)(03; 05)(01; 06) ( 3; 2; 3)(08; 02)(02; 04)(0; 1)

Table2. TabularformofIFHSS (G; 2)

(G; 2) u1 u2 u3 ( 1; 1; 1)(03; 03)(01; 02)(06; 04) ( 1; 1; 2)(07; 02)(06; 01)(0; 1) ( 3; 1; 1)(09; 01)(02; 07)(0; 1) ( 3; 1; 2)(0; 1)(0:5; 0:3)(0:3; 0:7)

Corollary8. Itisclearthateachintuitionisticfuzzyhypersoftsetisalsointuitionisticfuzzysoftset.Anexampleofthissituationisprovidedbelow.

Example9. WeconsiderthatExample-7.Ifweselecttheparametersfroma singleattributesetsuchas E1 whilecreatingtheintuitionisticfuzzyhypersoftset, thentheresultingsetbecomestheintuitionisticfuzzysoftset.Therefore,itisclear thateachintuitionisticfuzzyhypersoftsetisalsointuitionisticfuzzysoftset.That is,theintuitionisticfuzzyhypersoftsetstructureisthegeneralizedversionofthe intuitionisticfuzzysoftsets.

DeÖnition10. Let U beaninitialuniversesetand (H; 1); (G; 2) betwointuitionisticfuzzyhypersoftsetsovertheuniverse U .Wesaythat (H; 1) isan intuitionisticfuzzyhypersoftsubsetof (G; 2) anddenote (H; 1) (G; 2) if i) 1 2 ii)Forany 2 1;H() G(); Thatisforall u 2 U and 2 1; H( )(u) G( )(u) and H( )(u) G( )(u):

Example11. WeconsiderthatattributesinExample-7andLet (H; 1), (G; 2) betwointuitionisticfuzzyhypersoftsetoverthesameuniverse U = fu1;u2;u3g: Tabularformsof (H; 1) and (G; 2) arefollowing:

Table3. TabularformofIFHSS (H; 1)

(H; 1) u1 u2 u3 ( 3; 1; 1)(05; 04)(01; 08)(03; 04) ( 3; 1; 3)(0:2; 0:7)(0:3; 0:6)(0:1; 0:6) ( 3; 2; 1)(0; 1)(0; 1)(0; 1) ( 3; 2; 3)(0; 1)(0; 1)(0; 1)

Table4. TabularformofIFHSS (G; 2)

(G; 2) u1 u2 u3 ( 3; 1; 1)(0:6; 0:1)(0:2; 0:5)(0:7; 0:3) ( 3; 1; 3)(0:6; 0:4)(0:4; 0:6)(0:7; 0:4) ( 3; 2; 1)(0; 1)(0; 1)(0; 1) ( 3; 2; 3)(04; 03)(08; 02)(06; 04)

Itisclearthat (H; 1) (G; 2) wecanalsowrittenas

(H; 1) (G; 2)=

8 > > > < > > > : < ( 3; 1; 1); f u1 (05;0;4) ; u2 (01;08) ; u3 (03;04) g >; < ( 3; 1; 3); f u1 (02;0;7) ; u2 (03;06) ; u3 (01;06) g >; < ( 3; 2; 1); f u1 (0;1) ; u2 (0;1) ; u3 (0;1) g >; < ( 3; 2; 3); f u1 (0;1) ; u2 (0;1) ; u3 (0;1) g >

9 > > > = > > > ;

9 > > > = > > > ; e 8 > > > < > > > : < ( 3; 1; 1); f u1 (06;0;1) ; u2 (02;05) ; u3 (07;03) g >; < ( 3; 1; 3); f u1 (06;0;4) ; u2 (04;06) ; u3 (07;04) g >; < ( 3; 2; 1); f u1 (0;1) ; u2 (0;1) ; u3 (0;1) g >; < ( 3; 2; 3); f u1 (04;03) ; u2 (08;02) ; u3 (06;04) g >

DeÖnition12. Let U beaninitialuniversesetand (H; 1); (G; 2) betwointuitionisticfuzzyhypersoftsetsovertheuniverse U: Wesaythat (H; 1) isan intuitionisticfuzzyhypersoftequal (G; 2) anddenote (H; 1)=(G; 2) ifforall u 2 U and 2 ; H( )(u)= G( )(u) and H( )(u)= G( )(u)

Theorem13. Let U beaninitialuniverseset, 1; 2; 3 and (H; 1); (G; 2), (K; 3) beintuitionisticfuzzyhypersoftsetsovertheuniverse U: Then, i) (H; 1) 1(UIFH ; ); ii) 0(UIFH ; ) (H; 1); iii) (H; 1) e (G; 2) and (G; 2) e (K; 3) ) (H; 1) e (K; 3)

Proof. i) (H; 1) 1(UIFH ; ); since H( )(u) H( )(u)=1 and H( )(u) H( )(u)=0 forall 2 , u 2 U; ii)0(UIFH ; ) (H; 1); since 0= ;(u) H(")(u) and 1= ;(u) H( )(u) for all 2 , u 2 U; iii) (H; 1) (G; 2) ) H( )(u) G( )(u) and H( )(u) H( )(u) forall 2 , u 2 U: Also (G; 2) e (K; 3) ) G( )(u) K( )(u) and G( )(u) K( )(u) forall 2 , u 2 U: Therefore H( )(u) K( )(u) and H( )(u) K( )(u) Thus, weobtain (H; 1) (K; 3) DeÖnition14. Thecomplementofintutionisticfuzzyhypersoftset (H; ) over theuniverse U isdenotedby (H; )c anddeÖnedas (H; )c =(H c ; ); where H c :(E1 E2 En)= ! IFP (U ) and H c( )=(H( ))c forall Thusif (H; )= f< ; ( u H( )(u); H( )(u) ) >: u 2 U; 2 )g; then (H; )c = f< ; ( u H( )(u); H( )(u) ) >: u 2 U; 2 )g: Example15. AccordingtoExample-7,considertheintuitionisticfuzzyhypersoft set (H; ) overtheuniverse U = f 1; 2; 3; 4g

(H; 1)= ( < ( 3; 1; 1); f u1 (0 6;0;3) ; u2 (0 4;0 2) g >;< ( 3; 1; 3); f u2 (0 2;0 6) ; u3 (0 3;0 1) g >; < ( 3; 2; 1); f u1 (0 7;0 3) ; u2 (0 3;0 5) ; u3 (0 1;0 6) g >;< ( 3; 2; 3); f u1 (0 8;0 2) ; u2 (0 2;0 4) g > )

Thenthecomplementof (H; ) iswrittenas;

(H; 1)c = ( < ( 3; 1; 1); f u1 (0 3;0;6) ; u2 (0 2;0 4) ; u3 (1;0) g >;< ( 3; 1; 3); f u1 (1;0) ; u2 (0 6;0 2) ; u3 (0 1;0 3) g >; < ( 3; 2; 1); f u1 (0 3;0 7) ; u2 (0 5;0 3) ; u3 (0 6;0 1) g >;< ( 3; 2; 3); f u1 (0 2;0 8) ; u2 (0 4;0 2) g > )

Theorem16. Let (H; ) beanyintuitionisticfuzzyhypersoftsetovertheuniverse U: Then, i) ((H; )c)c =(H; ) ii) 0c (UIFH ; ) =1(UIFH ; ) iii) 1c (UIFH ; ) =0(UIFH ; )

Proof. Proofsaretrivial.

DeÖnition17. Let U beaninitialuniverseset, 1; 2 and (H; 1); (G; 2) betwointuitionisticfuzzyhypersoftsetsovertheuniverse U: Theunionof (H; 1) and (G; 2) isdenotedby (H; 1)[(G; 2)=(K; 3) where 3 = 1 [ 2 and K( )(u)= 8 < : H() if 2 1 2 G() if 2 2 1 max(H(); (G()) if 2 1 \ 2 K( )(u)= 8 < : H() if 2 1 2 G() if 2 2 1 min(H(); (G()) if 2 1 \ 2

Theorem18. Let U beaninitialuniverseset, 1; 2; 3 and (H; 1); (G; 2), (S; 3) beintuitionisticfuzzyhypersoftsetsovertheuniverse U: Then; i) (H; 1)[(H; 1)=(H; 1) ii) 0(UIFH ; ) [(H; 1)=(H; 1) iii) (H; 1)[ 1(UIFH ; ) =1(UIFH ; ) iv) (H; 1)e [(G; 2)=(G; 2)[(H; 1) v) ((H; 1)[(G; 2)) [(S; 3)=(H; 1)[ ((G; 2)[(S; 3))

Proof. Proofsaretrivial.

DeÖnition19. Let U beaninitialuniverseset, 1; 2 and (H; 1); (G; 2) betwointuitionisticfuzzyhypersoftsetsovertheuniverse U: Theintersectionof (H; 1) and (G; 2) isdenotedby (H; 1)\(G; 2)=(K; 3) where 3 = 1 \ 2, (K; 3)= f< ; ( u (minf H( )(u); G( )(u)g; maxf H( )(u); G( )(u)g)) >: u 2 U; 2 )g

Theorem20. Let U beaninitialuniverseset, 1; 2; 3 and (H; 1); (G; 2), (S; 3) beintuitionisticfuzzyhypersoftsetsovertheuniverse U: Then; i) (H; 1)\(H; 1)=(H; 1) ii) 0(UIFH ; ) \(H; 1)=0(UIFH ; ) iii) (H; 1)\1(UIFH ; ) =(H; 1) iv) (H; 1)\(G; 2)=(G; 2)\(H; 1) v) ((H; 1)\(G; 2)) \(S; 3)=(H; 1)\ ((G; 2)\(S; 3))

Proof. Proofsaretrivial. DeÖnition21. Let U beaninitialuniverseset, 1; 2 and (H; 1); (G; 2) betwointuitionisticfuzzyhypersoftsetsovertheuniverse U: Thedi§erenceof (H; 1) and (G; 2) isdenotedby (H; 1) e n(G; 2)=(K; 3) where (H; 1)\(G; 2)c =(K; 3)

Example22. WeconsiderthatattributesinExample-7andLet (H; 1), (G; 2) beanytwointuitionisticfuzzyhypersoftsetoverthesameuniverse U = fu1;u2;u3g: TheIFHSS (H; 1) and (G; 2) deÖnedasfollows;

(H; 1)= ( < ( 3; 1; 1); f u1 (0 6;0;3) ; u2 (0 4;0 2) g >;< ( 3; 1; 3); f u2 (0 2;0 6) ; u3 (0 3;0 1) g >; < ( 3; 2; 1); f u1 (0 7;0 3) ; u2 (0 3;0 5) ; u3 (0 1;0 6) g >;< ( 3; 2; 3); f u1 (0 8;0 2) ; u2 (0 2;0 4) g > )

(G; 2)= ( < ( 1; 1; 1); f u1 (0 3;0;3) ; u2 (0 1;0 2) ; u3 (0 6;0 4) g >;< ( 1; 1; 2); f u1 (0 7;0 2) ; u2 (0 6;0 1) g >; < ( 3; 1; 1); f u1 (0 9;0 1) ; u2 (0 2;0 7) g >;< ( 3; 1; 2); f u2 (0 5;0 3) ; u3 (0 3;0 7) g > )

Theunion,intersectionanddi§erenceoperatorofaboveIFHSSiswrittenas; (H; 1)[(G; 2)= 8 > > > < > > > : < ( 3; 1; 1); f u1 (09;0;1) ; u2 (04;02) g >;< ( 3; 1; 3); f u2 (02;06) ; u3 (03;01) g >; < ( 3; 2; 1); f u1 (07;03) ; u2 (03;05) ; u3 (01;06) g >;< ( 3; 2; 3); f u1 (08;02) ; u2 (02;04) g >; < ( 1; 1; 1); f u1 (03;0;3) ; u2 (01;02) ; u3 (06;04) g >;< ( 1; 1; 2); f u1 (07;02) ; u2 (06;01) g > < ( 3; 1; 2); f u2 (05;03) ; u3 (03;07) g >

(H; 1)\(G; 2)= < ( 3; 1; 1); f u1 (0:6; 0; 3) ; u2 (0:2; 0:7) g > and

(H; 1) e n(G; 2)= < ( 3; 1; 1); f u1 (01; 0; 9) ; u2 (04; 02) g >

Theorem23. Let U beaninitialuniverseset, 1; 2 and (H; 1); (G; 2) be twointuitionisticfuzzyhypersoftsetsovertheuniverse U: ThenDe-MorganLaws arehold.

i) ((H; 1)~[(G; 2))c =(H; 1)c ~ \(G; 2)c ii)((H; 1)\(G; 2))c =(H; 1)c[(G; 2)c Proof. Weonlyprove ((H; 1)[(G; 2))c =(H; 1)c\(G; 2)c .Theotherpropertiescanbesimilarlyproved.Supposethat ((H; 1)[(G; 2))c =(K; 1 [ 2) and (H; 1)c\(G; 2)c =(I; 1 [ 2) Forany 2 1 [ 2; weconsiderthefollowing cases.

Case1: 2 1 2 Then K()= H c()= I() Case2: 2 2 1 Then K()= Gc()= I() Case3: 2 1 \ 2 Then K()=( H( )(u) \ G( )(u); H( )(u) [ G( )(u))= H c() \ Gc()= I(): Therefore, K and I aresameoperators,andso ((H; 1)[(G; 2))c =(H; 1)c\(G; 2)c

DeÖnition24. Let U beaninitialuniverseset, 1; 2 and (H; 1); (G; 2) betwointuitionisticfuzzyhypersoftsetsovertheuniverse U: The"AND"operation onthemisdenotedby (H; 1) ^ (G; 2)=(K; 1 2) isgivenas;

(K; 1 2)= < ( 1; 2); ( u K( 1; 2)(u); K( 1; 2)(u)) >: u 2 U; ( 1; 2) 2 1 2 where K( 1; 2)(u)=min H( 1)(u); G( 2)(u) K( 1; 2)(u)=max H( 1)(u); G( 2)(u)

DeÖnition25. Let U beaninitialuniverseset, 1; 2 and (H; 1); (G; 2) betwointuitionisticfuzzyhypersoftsetsovertheuniverse U: The"OR"operation onthemisdenotedby (H; 1) _ (G; 2)=(K; 1 2) isgivenas; (K; 1 2)= < ( 1; 2); ( u K( 1; 2)(u); K( 1; 2)(u)) >: u 2 U; ( 1; 2) 2 1 2 where K( 1; 2)(u)=max H( 1)(u); G( 2)(u) K( 1; 2)(u)=min H( 1)(u); G( 2)(u)

Theorem26. Let U beaninitialuniverseset, 1; 2; 3 and (H; 1); (G; 2), (S; 3) beintuitionisticfuzzyhypersoftsetsovertheuniverse U: Then; i) (H; 1) _ [(G; 2) _ (S; 3)]=[(H; 1) _ (G; 2)] _ (S; 3) ii) (H; 1) ^ [(G; 2) ^ (S; 3)]=[(H; 1) ^ (G; 2)] ^ (S; 3)

Proof. Straightforward.

Theorem27. Let U beaninitialuniverseset, 1; 2 and (H; 1); (G; 2) betwointuitionisticfuzzyhypersoftsetsovertheuniverse U: Then; i) [(H; 1) _ (G; 2)]c =(H; 1)c ^ (G; 2)c ii) [(H; 1) ^ (G; 2)]c =(H; 1)c _ (G; 2)c :

Proof. Weonlyprove(i).Theotherpropertiescanbesimilarlyproved. Forall ( 1; 2) 2 1 2 and u 2 U;

(H; 1) _ (G; 2)= <u; max H( 1)(u); G( 2)(u) ; min H( 1)(u); G( 2)(u) > ; [(H; 1) _ (G; 2)]c = <u; min H( 1)(u); G( 2)(u) ; max H( 1)(u); G( 2)(u) > Ontheotherhand, (H; 1)c = <u; H( 1)(u); H( 1)(u) >: 1 2 1 (G; 2)c = <u; G( 2)(u); G( 2)(u) >: 2 2 2

Then, (H; 1)c ^ (G; 2)c = <u; min H( 1)(u); G( 2)(u) ; max H( 1)(u); G( 2)(u) > =[(H; 1) _ (G; 2)]c Hence, [(H; 1) _ (G; 2)]c =(H; 1)c ^ (G; 2)c isobtained.

Example28. WeconsiderthatattributesinExample-7.Thenthefuzzyhypersoft sets (H; 1) and (G; 2) deÖnedasfollows;

(H; 1)= ( < ( 3; 1; 1); f u1 (0 6;0;3) ; u2 (0 4;0 2) g >;< ( 3; 1; 3); f u2 (0 2;0 6) ; u3 (0 3;0 1) g >; < ( 3; 2; 1); f u1 (0 7;0 3) ; u2 (0 3;0 5) ; u3 (0 1;0 6) g >;< ( 3; 2; 3); f u1 (0 8;0 2) ; u2 (0 2;0 4) g > )

(G; 2)= ( < ( 1; 1; 1); f u1 (0 3;0;3) ; u2 (0 1;0 2) ; u3 (0 6;0 4) g >;< ( 1; 1; 2); f u1 (0 7;0 2) ; u2 (0 6;0 1) g >; < ( 3; 1; 1); f u1 (0 9;0 1) ; u2 (0 2;0 7) g >;< ( 3; 1; 2); f u2 (0 5;0 3) ; u3 (0 3;0 7) g > )

Letísassume ( 3; 1; 1)= m1, ( 3; 1; 3)= m2, ( 3; 2; 1)= m3, ( 3; 2; 3)= m4 in (H; 1) and ( 1; 1; 1)= n1, ( 1; 1; 2)= n2, ( 3; 1; 1)= n3; ( 3; 1; 2)= n4 in (G; 2) foreasieroperation.Thetabularformsofthesesetsareasfollows.

Thenthe"AND"and"OR"operationsofthesesetsaregivenasbelow.

Table5. TabularformofFHSS (H; 1)

(H; 1) u1 u2 u3 m1 (06; 0; 3)(04; 02)(0; 1) m2 (0; 1)(02; 06)(03; 01) m3 (07; 03)(03; 05)(01; 06) m4 (08; 02)(02; 04)(0; 1)

Table6. TabularformofFHSS (G; 2)

(G; 2) u1 u2 u3 n1 (03; 0; 3)(01; 02)(06; 04) n2 (07; 02)(06; 01)(0; 1) n3 (09; 01)(02; 07)(0; 1) n4 (0; 1)(0:5; 0:3)(0:3; 0:7)

Table7. TabularformofFHSS (H; 1) ^ (G; 2)

(H; 1) ^ (G; 2) u1 u2 u3 m1 n1 (03; 03)(01; 02)(0; 1) m1 n2 (0:6; 0:3)(0:4; 0:2)(0; 1) m1 n3 (0:6; 0:3)(0:2; 0:7)(0; 1) m1 n4 (0; 1)(0:4; 0:3)(0; 1) m2 n1 (0; 1)(01; 06)(03; 04) m2 n2 (0; 1)(02; 06)(0; 1) m2 n3 (0; 1)(02; 07)(0; 1) m2 n4 (0; 1)(02; 06)(03; 07) m3 n1 (03; 03)(01; 05)(01; 06) m3 n2 (07; 03)(03; 05)(0; 1) m3 n3 (07; 03)(02; 07)(0; 1) m3 n4 (0; 1)(0:3; 0:5)(0:1; 0:7) m4 n1 (0:3; 0:3)(0:1; 0:4)(0; 1) m4 n2 (0:7; 0:2)(0:2; 0:4)(0; 1) m4 n3 (0:8; 0:2)(0:2; 0:7)(0; 1) m4 n4 (0; 1)(02; 04)(0; 1)

4. Conclusion

TheaimofthispaperistoovercometheuncertaintytroubleinmoreparticularwaybywayofcombingIntuitionisticfuzzysetwithHypersoftset.Some operationsofIntuitionisticFuzzyHypersoftsetsuchassubset,equalset,union, intersection,complement,AND,ORoperationsarepresented.BydeÖningthese notions,thefoundationoftheintuitionisticfuzzyhypersoftsetstructurewasbuilt.

Table8. TabularformofFHSS (H; 1) _ (G; 2)

(H; 1) _ (G; 2) u1 u2 u3 m1 n1 (06; 03)(04; 02)(06; 04) m1 n2 (07; 02)(06; 01)(0; 1) m1 n3 (09; 01)(04; 02)(0; 1)

m1 n4 (06; 03)(05; 02)(03; 07) m2 n1 (0:3; 0:3)(0:2; 0:2)(0:6; 0:1) m2 n2 (0:7; 0:2)(0:6; 0:1)(0:3; 0:1)

m2 n3 (0:9; 0:1)(0:2; 0:6)(0:3; 0:1) m2 n4 (0; 1)(0:5; 0:3)(0:3; 0:1)

m3 n1 (07; 03)(03; 02)(06; 04) m3 n2 (07; 02)(06; 01)(01; 06) m3 n3 (09; 01)(03; 05)(01; 06) m3 n4 (07; 03)(05; 03)(03; 06) m4 n1 (08; 02)(02; 02)(06; 04) m4 n2 (08; 02)(06; 01)(0; 1) m4 n3 (09; 01)(02; 04)(0; 1) m4 n4 (0:8; 0:2)(0:5; 0:3)(0:3; 0:7)

ThevalidityandimplementationoftheproposedoperationsanddeÖnitionsare validatedthroughpresentingsuitableinstance.Matrices,similaritymeasure,singleandmulti-valued,intervalvalued,functions,distancemeasures,algorithms: scorefunction,VIKOR,TOPSIS,AHPofIntutionisticFuzzyHypersoftsetswill befuturework.Wehopethat,thisstudywillplayacriticalfunctioninfuture decision-makingresearch.

AuthorsContributionStatement

Theauthorscontributedequallytothiswork. Allauthorsofthesubmittedresearchpaperhavedirectlyparticipatedintheplanning,execution,oranalysisofstudy.

DeclarationofCompetingInterest

Theauthorsdeclarethatthereisnoconáict ofinterestregardingthepublicationofthisarticle.

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