Plithogenic Soft Set

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/341820247

Plithogenic Soft Set

Article · June 2020 DOI: 10.5281/zenodo.3783023 CITATIONS 0 READS 49 1 author:

Shawkat Alkhazaleh Jadara University 31 PUBLICATIONS 620 CITATIONS SEE PROFILE

Some of the authors of this publication are also working on these related projects:

Wrapping distribution View project Fuzzy sets View project

All content following this page was uploaded by Shawkat Alkhazaleh on 02 June 2020. The user has requested enhancement of the downloaded file.

NeutrosophicSetsandSystems,Vol.33,2020

UniversityofNewMexico

PlithogenicSoftSet

ShawkatAlkhazaleh DepartmentofMathematics,FacultyofScienceandInformationTechnology,JadaraUniversity,Irbid,Jordan shmk79@gmail.com

Abstract In1995,Smarandacheinitiatedthetheoryofneutrosophicsetasnewmathematicaltoolforhandling problemsinvolvingimprecise,indeterminacy,andinconsistentdata.Molodtsovinitiatedthetheoryofsoft setasanewmathematicaltoolfordealingwithuncertainties,whichtraditionalmathematicaltoolscannot handle.Hehasshowedseveralapplicationsofthistheoryforsolvingmanypracticalproblemsineconomics, engineering,socialscience,medicalscience,etc.In2017SmarandacheinitiatedthetheoryofPlithogenicSet andtheirproperties.Healsogeneralizedthesoftsettothehypersoftsetbytransformingthefunction F into amulti-attributefunctionandintroducedthehybridsofCrisp,Fuzzy,IntuitionisticFuzzy,Neutrosophic,and PlithogenicHypersoftSet.Inthisresearch,forthefirsttimewedefinetheconceptofPlithogenicsoftset, andgiveitsomegeneralizationsandstudysomeofitsoperations.Furthermore,Wegiveexamplesforthese conceptsandoperations.Finally,thesimilaritiesbetweentwoPlithogenicsoftsetsarealsogiven.

Keywords: Softset;Neutrosophicset;Neutrosophicsoftset;Plithogenicset;Plithogenicsoftset.

1. Introduction

Inourlifenoteverythingnonvagueorcertainwheremanythingsaroundusaresurrounded byuncertaintyandvagueness,Zadeh[13]wassuccessfullyfulfilledtheneedtorepresentuncertaindatabyintroducingtheconceptoffuzzysets.AlsoAtanassov[4]extendZadehsnotion offuzzysettoIntuitionisticfuzzysetswhichprovedtobeabettermodelofuncertainty.The wordsneutrosophyandneutrosophicwereintroducedforthefirsttimebySmarandache[9,10] asamoregeneralplatform,whichextendstheconceptsofthefuzzysetandintuitionisticfuzzy set.In1999Molodtsov[8]proposedaparameterisedfamilyofsetsnamed”softset”,todeal withuncertaintyinaparametricmanner.Smarandachedefinedaconceptofplithogenicset, whereaplithogenicset P isasetwhoseelementsarecharacterizedbyoneormoreattributes, andeachattributemayhavemanyvalues.Eachattribute’svalue v hasacorrespondingdegree ofappurtenance d(x,v)oftheelement x,totheset P ,withrespecttosomegivencriteria. Inordertoobtainbetteraccuracyfortheplithogenicaggregationoperators,acontradiction

NeutrosophicSetsandSystems,Vol.33,2020257of274 (dissimilarity)degreeisdefinedbetweeneachattributevalueandthedominant(mostimportant)attributevalue.However,therearecaseswhensuchdominantattributevaluemaynot betakingintoconsiderationormaynotexist[thereforeitisconsideredzerobydefault],or theremaybemanydominantattributevalues.Insuchcases,eitherthecontradictiondegree functionissuppressed,oranotherrelationshipfunctionbetweenattributevaluesshouldbe established.Theplithogenicaggregationoperators(intersection,union,complement,inclusionandequality)arebasedoncontradictiondegreesbetweenattributesvaluesandthefirst twoarelinearcombinationsofthefuzzyoperatorst-normandt-conorm.Plithogenicsetisa generalizationofthecrispset,fuzzyset,intuitionisticfuzzyset,andneutrosophicset,since thesefourtypesofsetsarecharacterizedbyasingleattributevalue(appurtenance):which hasonevalue(membership)forthecrispsetandfuzzyset,twovalues(membership,andnonmembership)forintuitionisticfuzzyset,orthreevalues(membership,non-membership,and indeterminacy)forneutrosophicset.Aplithogenicset,ingeneral,mayhaveelementscharacterizedbyattributeswithfourormoreattributes.In2019Abdel-Bassetetal.[1]suggestedan approachconstructedontheconnotationofplithogenictheorytechniquetocomeupwitha methodicalproceduretoassesstheinfirmaryservingunderaframeworkofplithogenictheory. Inthisresearch,theygavesomedefinitionsoftheplithogenicenvironment,whichismoregeneralandcomprehensivethanfuzzy,intuitionisticfuzzyandneutrosophicones.Abdel-Basset etal.[3]proposedmethodtoincreasetheaccuracyoftheevaluation.Thismethodisacombinationofqualityfunctiondeploymentwithplithogenicaggregationoperations.Theyapplied theaggregationoperationtoaggregatethedecisionmakersopinionsofrequirementsthatare neededtoevaluatethesupplychainsustainabilityandtheevaluationmetricsbasedonthe requirements.Alsotheyappliedtheaggregationoperationtoaggregatetheevaluationofinformationgatheringdifficulty.In2020Abdel-BassetandRehab[2]proposedamethodology asacombinationofplithogenicmulti-criteriadecision-makingapproachbasedontheTechniqueinOrderofPreferencebySimilaritytoIdealSolutionandCriteriaImportanceThrough Inter-criteriaCorrelationmethodstoestimationofsustainablesupplychainriskmanagement.

2. Preliminary

Remark2.2. Inamoregeneralway,thesummationcanbelessthan1(forincomplete neutrosophicinformation),equalto1(forcompleteneutrosophicinformation),orgreater than1(forparaconsistent/conflictingneutrosophicinformation).

Molodtsovdefinedsoftsetinthefollowingway.Let U beauniverseand E beasetof parameters.Let P (U )denotethepowersetof U and A ⊆ E.

Definition2.3. [8]Apair(F,A)iscalleda softset over U, where F isamapping

F : A → P (U ) .

Inotherwords,asoftsetover U isaparameterizedfamilyofsubsetsoftheuniverse U. For ε ∈ A,F (ε)maybeconsideredasthesetof ε-approximateelementsofthesoftset(F,A).

Definition2.4. [6]Let P (U )denotesthesetofallfuzzysetsof U .Let Ai ⊆ E .A pair(Fi,Ai)iscalledafuzzysoftsetover U ,where Fi isamappinggivenby Fi : Ai −→ P (U )

Definition2.5. [7]Let U beaninitialuniversalsetandlet E besetofparameters.Let P (U ) denotesthesetofallintuitionisticfuzzysetsof U .Apair(F,A)iscalledanintuitionistic fuzzysoftsetover U if F isamappinggivenby F : A −→ P (U ).WewriteanIntuitionistic fuzzysoftsetshortlyasIFsoftset

Definition2.6. [5]Let U beaninitialuniversesetand E beasetofparameters.Consider A ⊂ E.Let P (U )denotesthesetofallneutrosophicsetsof U .Thecollection(F,A)istermed tobetheneutrosophicsoftsetover U ,where F isamappinggivenby F : A → P (U ).

3. FormalDefinitionofSingle(Uni-Dimensional)AttributePlithogenicSet

InthissectionwerecallthedefinitionofplithogenicsetgivenbySmarandache[11,12]and somedefinitionsrelatedtothisconceptasfollows: Let U beauniverseofdiscourse,and P anon-emptysetofelements, P ⊆ U

3.1.

AttributeValueSpectrum

Definition3.1. Let A beanon-emptysetofuni-dimensionalattributes A = {α1,α2,...,αm}, m ≥ 1;and αi ∈ A beagivenattributewhosespectrumofallpossiblevalues(orstates)isthe non-emptyset S,where S canbeafinitediscreteset, S = {s1,s2,...,sl},1 ≤ l ≤∞,orinfinitely countableset S = {s1,s2,...,s∞},orinfinitelyuncountable(continuum)set S =]a,b[,a<b, where]...[isanyopen,semi-open,orclosedintervalfromthesetofrealnumbersorfromother generalset.

ShawkatAlkhazaleh,PlithogenicSoftSet

3.2. AttributeValueRange

Definition3.2. Let V beanon-emptysubsetof S,where V istherangeofallattribute’s valuesneededbytheexpertsfortheirapplication.Eachelement x ∈ P ischaracterizedbyall attribute’svaluesin V = {v1,v2,...,vn},for n ≥ 1.

3.3. DominantAttributeValue

Definition3.3. Intotheattributesvalueset V ,ingeneral,thereisadominantattribute value,whichisdeterminedbytheexpertsupontheirapplication.Dominantattributevalue isdefinedasthemostimportantattributevaluethattheexpertsareinterestedin. Remark3.4. Therearecaseswhensuchdominantattributevaluemaynotbetakinginto considerationornotexist,ortheremaybemanydominant(important)attributevalueswhendifferentapproachshouldbeemployed.

3.4. AttributeValueTruth-ValueDegreeFunction

Definition3.5. Theattributevaluetruth-valuedegreefunctionis: ∀P ∈ U,d : U × V −→ ℘([0, 1]z ),so d(P,v)isasubsetof[0, 1]z ,where ℘([0, 1]z )isthepowersetofthe[0, 1]z such that z = F (forfuzzydegreeoftruth-value), z = IF (forintuitionisticfuzzydegreeoftruth-value),or z = N (forneutrosophicdegreedetruth-value).

3.5. AttributeValueContradictionDegreeFunction

Definition3.6. Theattributevaluecontradictiondegreefunctionbetweenanytwoattribute values v1 and v2,denotedby c(v1,v2)isafunctiondefineby c : V × V −→ [0, 1]suchthatthe cardinal |V |≥ 1andsatisfyingthefollowingaxioms:

(1) c(v1,v1)=0,thecontradictiondegreebetweenthesameattributevaluesiszero; (2) c(v1,v2)= c(v2,v1),commutativity.

3.6. plithogenicset

Definition3.7. Let U beauniverseofdiscourse,and P anon-emptysetofelements, P ⊆ U . a isa(multi-dimensionalingeneral)attribute, V istherangeoftheattributesvalues, d isthe degreeofappurtenanceofeachelement xsattributevaluetotheset P withrespecttosome givencriteria(x ∈ P ),and c standsforthedegreeofcontradictionbetweenattributevalues. Then(P,a,V,d,c)iscalledaplithogenicset.

ShawkatAlkhazaleh,PlithogenicSoftSet

Remark3.8. (1) d maystandfor dF , dIF or dN ,whendealingwithfuzzydegreeof appurtenance,intuitionisticfuzzydegreeofappurtenance,orneutrosophicdegreeof appurtenancerespectivelyofanelement x totheplithogenicset P ;

(2) c maystandfor cF , cIF or cN ,whendealingwithfuzzydegreeofcontradiction, intuitionisticfuzzydegreeofcontradiction,orneutrosophicdegreeofcontradiction betweenattributevaluesrespectively;

(3) Thefunctions d(.,.)and c(.,.)aredefinedinaccordancewiththeapplicationsthe expertsneedtosolve;

(4) Oneusesthenotation: x(d(x,V )),where d(x,v)= {d(x,v)∀v ∈ V }, ∀x ∈ P ;

Theplithogenicsetisanextensionofall:crispset,fuzzyset,intuitionisticfuzzyset,and neutrosophicset.InthisPaperwewillstudytheplithogenicsetasanextensionoffuzzyset, intuitionisticfuzzyset,andneutrosophicset.

In Single-ValuedFuzzySet(SVFS),theattributeis α =”appurtenance”;thesetof attributevalues V = {T },whosecardinal |V | =1;thedominantattributevalue= T ;theattributevalue appurtenancedegreefunction: d : P × V −→ [0, 1], d(x,T ) ∈ [0, 1] andtheattributevaluecontradictiondegreefunction: c : V × V −→ [0, 1], c(T,T )=0,

In Single-Valuedintuitionisticfuzzyset(SVIFS),theattributeis α =”appurtenance”; thesetofattributevalues V = {T,F },whosecardinal |V | =2; thedominantattributevalue= T ;theattributevalueappurtenancedegreefunction: d : P × V −→ [0, 1], d(x,T ) ∈ [0, 1], d(x,F ) ∈ [0, 1]with0 ≤ d(x,T )+ d(x,F ) ≤ 1;and theattributevaluecontradictiondegreefunction: c : V × V −→ [0, 1], c(T,T )= c(F,F )=0, c(T,F )=1.

In Single-ValuedNeutrosophicSet(SVNS),theattributeis α =”appurtenance”;the setofattributevalues V = {T,I,F },whosecardinal |V | =3;thedominantattributevalue= T ;theattributevalue appurtenancedegreefunction: d : P × V −→ [0, 1], d(x,T ) ∈ [0, 1], d(x,I) ∈ [0, 1], d(x,F ) ∈ [0, 1], with0 ≤ d(x,T )+ d(x,I)+ d(x,F ) ≤ 3; andtheattributevaluecontradictiondegreefunction: c : V × V −→ [0, 1], c(T,T )= c(I,I)= c(F,F )=0, c(T,F )=1, c(T,I)= c(F,I)=0 5. ShawkatAlkhazaleh,PlithogenicSoftSet

4. PlithogenicIntersection

Inthissectionwerecallthedefinitionofplithogenicintersectionoverthreecases:fuzzy, intuitionisticfuzzyandneutrosophicsetwhicharedefinedforthefirsttimebySmarandache in2017and2018[11,12].

4.1. PlithogenicFuzzyIntersection

Definition4.1. Let U = {u1,u2,...,un} A = u1 µ(u1 ) , u2 µ(u2 ) ,..., un µ(un) ,and B = u1 ν(u1 ) , u2 ν(u2 ) ,..., un ν(un) beanytwoplithogenicfuzzysetsover U .Thentheplithogenicfuzzy intersectionbetween A and B defineasfollows: A ∧FP B

4.2. PlithogenicIntuitionisticFuzzyIntersection

Definition4.2. Let U = {u1,u2,...,un}.Suppose A = u1 (µT (u1 ),µF (u1 )) , u2 (µT (u2 ),µF (u2 )) ,..., un (µT (un),µF (un)) ,and B = u1 (νT (u1 ),νF (u1 )) , u2 (νT (u2 ),νF (u2 )) ,..., un (νT (un),νF (un)) beanytwoplithogenicintuitionisticfuzzysetsover U .Thentheplithogenicintuitionistic fuzzyintersectionbetween A and B isdefinedasfollows: A ∧IP B = ui ((1 cv)(µ(ui)∧F ν(ui))+(cv)(µ(ui)∨F νF (ui)),(cv)(µ(ui)∨F ν(ui))+(1 cv)(µ(ui)∧F νF (ui))) , ∀i =1, 2,...,n Where ∧F and ∨F representfuzzyt-normandfuzzyt-conorm(s-norm)respectivelyand cv representthedegreesofcontradictions.

4.3. PlithogenicNeutrosophicIntersection

Definition4.3.

beanytwoplithogenicneutrosophicsetover U .Thentheplithogenicneutrosophicintersection between A and B isdefinedasfollows:

∀i =1, 2,...,n,where

F and

F representfuzzyt-normandfuzzyt-conorm(s-norm)respectivelyand cv representthedegreesofcontradictions.

ShawkatAlkhazaleh,PlithogenicSoftSet

5. PlithogenicUnion

Inthissectionwerecallthedefinitionofplithogenicunionoverthreecases:fuzzy,intuitionisticfuzzyandneutrosophicsetwhicharedefinedforthefirsttimebySmarandachein2017 and2018[11,12].

5.1. PlithogenicFuzzyUnion

Definition5.1. Let U , A and B asdefinedin4.1.Thentheplithogenicfuzzyunionbetween A and B isdefinedasfollows: A ∨FP B = ui (1 cv )(µ(ui)∨F ν(ui))+(cv )(µ(ui)∧F νF (ui)) , ∀i =1, 2,...,n Where ∧F and ∨F representfuzzyt-normandfuzzyt-conorm(s-norm)respectivelyand cv representthedegreesofcontradictions.

5.2. PlithogenicIntuitionisticFuzzyUnion

Definition5.2. Let U , A and B asdefinedin4.2.Thentheplithogenicintuitionisticfuzzy unionbetween A and B isdefinedasfollows: A ∨IP B = ui ((1 cv)(µ(ui)∨F ν(ui))+(cv)(µ(ui)∧F νF (ui)),(cv)(µ(ui)∧F ν(ui))+(1 cv)(µ(ui)∨F νF (ui))) , ∀i =1, 2,...,n Where ∧F and ∨F representfuzzyt-normandfuzzyt-conorm(s-norm)respectivelyand cv representthedegreesofcontradictions.

5.3. PlithogenicNeutrosophicUnion

Definition5.3. Let U , A and B asdefinedin4.3.Thentheplithogenicneutrosophicintersectionbetween A and B isdefinedasfollows: A ∨NP B = ui (1 cv)(µ(ui)∨F ν(ui))+(cv)(µ(ui)∧F νF (ui)), 1 2 [(µI (u1 )∧F νT (u1 ))+(µI (u1 )∨F νT (u1 ))],(cv)(µ(ui)∧F ν(ui))+(1 cv)(µ(ui)∨F νF (ui)) , ∀i =1, 2,...,n,where ∧F and ∨F representfuzzyt-normandfuzzyt-conorm(s-norm)respectivelyand cv representthedegreesofcontradictions.

5.4. Hypersoftset

Definition5.4. [11]Let U beauniverseofdiscourse, ℘(U )thepowersetof U .Let a1,a2, ,an,for n ≥ 1,be n distinctattributes,whosecorrespondingattributevaluesare respectivelythesets A1,A2, ··· ,An,with Ai ∩ Aj = ∅,for j = i,and i,j ∈{1, 2, ··· ,n.} Thenthepair (F,A1 × A2 ×···× An) ShawkatAlkhazaleh,PlithogenicSoftSet

where: F : A1 × A2 ×···× An −→ ℘(U ) iscalledaHypersoftSetover U .

6. PlithogenicSoftSet

Inthissectionwedefinetheconceptofplithogenicsoftset(InGeneral)asageneralization ofsoftset.Wealso,defineitsbasicoperationsnamely,unionandintersectionandstudytheir properties.

Definition6.1. Let U beauniverseofdiscourse,(U )z the z powersetof U suchthat z = C (Powersetof U ), z = F (Thesetofallfuzzysetof U ), z = IF (Thesetofallintuitionisticfuzzysetof U ),or z = N (Thesetofallneutrosophicsetof U ).Let a1,a2, ··· ,an,for n ≥ 1,be n distinct attributes,whosecorrespondingattributevaluesarerespectivelythesets V1,V2, ,Vn,with Vi ∩ Vj = ∅,for j = i,and i,j ∈{1, 2, ··· ,n.}.Suppose Vi = {vi1,vi2, ··· ,vimi } andlet Υ= V1 × V2 ×···× Vn.Let D =(D1,D2, Dn)thedominantattributeelementof Ai ∀i and c (Di,vij ) ,i ∈{1, 2, n} ,j ∈{1, 2, mi} theattributevaluecontradictiondegreefunction suchthat ci : Vi × Vi −→ [0, 1],Thenthepair(F z P , Υ),where: F z P :Υ −→ [0, 1]D × (U )z iscalledaPlithogenicSoftSet(P-SSInshort)over U .

Toillustratetheabovedefinitionwegivethefollowingexamplesforallcases:crisp,fuzzy, intuitionisticandneutrosophic: 6.1.

PlithogenicCrispSoftSet

Example6.2. Let U = {c1,c2,c3,c4}.Here U representsthesetofcars.Let a1 = speed,a2 = color,a3 = model,a4 = manufacturingyear.Supposetheirattributesvaluesrespectively: Speed ≡ A1 = {slow,fast,veryfast}, Color ≡ A2 = {white,yellow,red,black}, Model ≡ A3 = {model1,model2,model3,model4}, manufacturingyear ≡ A4 = {2015 andbefor, 2016, 2017, 2018, 2019}

Wherethedominantattribute’svalue D =(slow,red,model3, 2019).Let H ⊆ Υsuchthat H = { 1 =(slow,white,model1, 2015 andbefor) , 2 =(slow,yellow,model3, 2017) , 3 =(fast,red,model4, 2018)}.Defineafunction F z P : H −→ [0, 1]D × (U )C asfollows: ShawkatAlkhazaleh,PlithogenicSoftSet

F C P ( 1)= (c1 ,(0,0 4,1,1)D ) (1,1,1,1) , (c4 ,(0,0 4,1,1)D ) (1,1,1,1) ,

F C P ( 2)= (c2 ,(0,0 5,0,1)D ) (1,1,1,1) ,

F C P ( 3)= (c3 ,(0 5,0,1,1)D ) (1,1,1,1) , Thenwecanfindtheplithogeniccrispsoftset(P-CSS) F C P ,H asconsistingofthefollowing collectionofapproximations:

F C P ,H = 1, (c1 ,(0,0 4,1,1)D) (1,1,1,1) , (c4 ,(0,0 4,1,1)D) (1,1,1,1) , 2, (c2 ,(0,0 5,0,1)D) (1,1,1,1) , 3, (c3 ,(0 5,0,1,1)D) (1,1,1,1)

6.2. PlithogenicFuzzySoftSet

Example6.3. ConsiderExample6.2andsupposethatthe

F F P ( 1)= (c1 ,(0,0.4,1,1)D ) (0 6,0 7,1,1) , (c2 ,(0,0.4,1,1)D ) (0 7,0 3,0,0) , (c3 ,(0,0.4,1,1)D ) (0 1,0 3,0,0) , (c4 ,(0,0.4,1,1)D ) (0 7,0 6,1,1) ,

F F P ( 2)= (c1 ,(0,0 5,0,1)D ) (0 6,0 2,0,0) , (c2 ,(0,0 5,0,1)D ) (0 5,0 7,1,1) , (c3 ,(0,0 5,0,1)D ) (0 1,0 1,0,0) , (c4 ,(0,0 5,0,1)D ) (0 7,0 4,0,0) ,

F F P ( 3)= (c1 ,(0 5,0,1,1)D ) (0 1,0 2,0,0) , (c2 ,(0 5,0,1,1)D ) (0 1,0 3,0,0) , (c3 ,(0 5,0,1,1)D ) (0 8,0 7,1,1) , (c4 ,(0 5,0,1,1)D ) (0 1,0 1,0,0) ,

Thenwecanfindtheplithogenicfuzzysoftset(P-FSS) F F P ,H asconsistingofthefollowing collectionofapproximations:

F F P ,H = 1, (c1, (0, 0 4, 1, 1)D) (0 6, 0 7, 1, 1) , (c2, (0, 0 4, 1, 1)D) (0 7, 0 3, 0, 0) , (c3, (0, 0 4, 1, 1)D) (0 1, 0 3, 0, 0) , (c4, (0, 0 4, 1, 1)D) (0 7, 0 6, 1, 1) , 2, (c1, (0, 0 5, 0, 1)D) (0 6, 0 2, 0, 0) , (c2, (0, 0 5, 0, 1)D) (0 5, 0 7, 1, 1) , (c3, (0, 0 5, 0, 1)D) (0 1, 0 1, 0, 0) , (c4, (0, 0 5, 0, 1)D) (0 7, 0 4, 0, 0) , 3, (c1, (0 5, 0, 1, 1)D) (0 1, 0 2, 0, 0) , (c2, (0 5, 0, 1, 1)D) (0 1, 0 3, 0, 0) , (c3, (0 5, 0, 1, 1)D) (0 8, 0 7, 1, 1) , (c4, (0 5, 0, 1, 1)D) (0 1, 0 1, 0, 0) .

F IF P ( 1)= (c1 ,(0,0 4,1,1)D) ((0 6,0 2),(0 7,0 1),(1,0),(1,0)) , (c2 ,(0,0 4,1,1)D) ((0 7,0 1),(0 3,0 5),(0,1),(0,1)) , (c3 ,(0,0 4,1,1)D) ((0 1,0 8),(0 3,0 5),(0,1),(0,1)) , (c4 ,(0,0 4,1,1)D) ((0 7,0 2),(0 6,0 3),(1,0),(1,0)) ,

F IF P ( 2)= (c1 ,(0,0 5,0,1)D) ((0 6,0 3),(0 2,0 6),(0,1),(0,1)) ,

(c2 ,(0,0 5,0,1)D) ((0 5,0 4),(0 7,0 2),(1,0),(1,0)) ,

F IF P ( 3)= (c1 ,(0 5,0,1,1)D) ((0 1,0 7),(0 2,0 7),(0,1),(0,1)) , (c2 ,(0 5,0,1,1)D) ((0 1,0 8),(0 3,0 5),(0,1),(0,1)) ,

ShawkatAlkhazaleh,PlithogenicSoftSet

(c3 ,(0,0 5,0,1)D) ((0 1,0 7),(0 1,0 8),(0,1),(0,1)) ,

(c4 ,(0,0 5,0,1)D) ((0 7,0 2),(0 4,0 4),(0,1),(0,1)) ,

(c3 ,(0 5,0,1,1)D) ((0 8,0 1),(0 7,0 2),(1,0),(1,0)) , (c4 ,(0 5,0,1,1)D) ((0 1,0 8),(0 1,0 7),(0,1),(0,1)) ,

Thenwecanfindtheplithogenicintuitionisticfuzzysoftset(P-IFSS) F IF P ,H asconsisting ofthefollowingcollectionofapproximations:

F IF P ,H = 1, (c1,(0,0 4,1,1)D) ((0 6,0 2),(0 7,0 1),(1,0),(1,0)),

2, (c1,(0,0 5,0,1)D) ((0 6,0 3),(0 2,0 6),(0,1),(0,1)),

3, (c1,(0 5,0,1,1)D) ((0 1,0 7),(0 2,0 7),(0,1),(0,1)),

6.4.

(c2,(0,0 4,1,1)D) ((0 7,0 1),(0 3,0 5),(0,1),(0,1)), (c3,(0,0 4,1,1)D) ((0 1,0 8),(0 3,0 5),(0,1),(0,1)), (c4,(0,0 4,1,1)D) ((0 7,0 2),(0 6,0 3),(1,0),(1,0)) ,

(c2,(0,0 5,0,1)D) ((0 5,0 4),(0 7,0 2),(1,0),(1,0)),

(c2,(0 5,0,1,1)D) ((0 1,0 8),(0 3,0 5),(0,1),(0,1)),

PlithogenicNeutrosophicSoftSet

(c3,(0,0 5,0,1)D) ((0 1,0 7),(0 1,0 8),(0,1),(0,1)), (c4,(0,0 5,0,1)D) ((0 7,0 2),(0 4,0 4),(0,1),(0,1)) ,

(c3,(0 5,0,1,1)D) ((0 8,0 1),(0 7,0 2),(1,0),(1,0)), (c4,(0 5,0,1,1)D) ((0 1,0 8),(0 1,0 7),(0,1),(0,1))

Example6.5. ConsiderExample6.2and U = {c1,c2,c3

F N P ( 1)= (c1 ,(0,0 4,1,1)D) ((0 6,0 3,0 1),(0 7,0 2,0 1),(1,0,0),(1,0,0)) ,

F N P ( 2)= (c1 ,(0,0 5,0,1)D) ((0 6,0 3,0 1),(0 2,0 6,0 2),(0,0,1),(0,0,1)) ,

},supposethatthe

(c2 ,(0,0 4,1,1)D) ((0 7,0 1,0 2),(0 3,0 5,0 2),(0,0,1),(0,0,1)) , (c3 ,(0,0 4,1,1)D) ((0 1,0 1,0 8),(0 3,0 6,0 1),(0,0,1),(0,0,1)) ,

(c2 ,(0,0 5,0,1)D) ((0 5,0 2,0 3),(0 7,0 2,0 1),(1,0,0),(1,0,0)) ,

(c3 ,(0,0 5,0,1)D) ((0 1,0 2,0 7),(0 1,0 1,0 8),(0,0,1),(0,0,1)) ,

F N P ( 3)= (c1 ,(0 5,0,1,1)D) ((0 1,0 2,0 7),(0 2,0 1,0 7),(0,0,1),(0,0,1)) , (c2 ,(0 5,0,1,1)D) ((0 1,0 1,0 8),(0 3,0 2,0 5),(0,0,1),(0,0,1)) , (c3 ,(0 5,0,1,1)D) ((0 8,0 1,0 1),(0 7,0 1,0 2),(1,0,0),(1,0,0)) ,

Thenwecanfindtheplithogenicneutrosophicsoftset(P-NSS) F N P ,H asconsistingofthe followingcollectionofapproximations:

F N P ,H = 1, (c1,(0,0 4,1,1)D) ((0 6,0 3,0 1),(0 7,0 2,0 1),(1,0,0),(1,0,0)), (c2,(0,0 4,1,1)D) ((0 7,0 1,0 2),(0 3,0 5,0 2),(0,0,1),(0,0,1)), (c3,(0,0 4,1,1)D) ((0 1,0 1,0 8),(0 3,0 6,0 1),(0,0,1),(0,0,1)) ,

2, (c1,(0,0 5,0,1)D) ((0 6,0 3,0 1),(0 2,0 6,0 2),(0,0,1),(0,0,1)),

(c2,(0,0 5,0,1)D) ((0 5,0 2,0 3),(0 7,0 2,0 1),(1,0,0),(1,0,0)), (c3,(0,0 5,0,1)D) ((0 1,0 2,0 7),(0 1,0 1,0 8),(0,0,1),(0,0,1)) ,

3, (c1,(0 5,0,1,1)D) ((0 1,0 2,0 7),(0 2,0 1,0 7),(0,0,1),(0,0,1)), (c2,(0 5,0,1,1)D) ((0 1,0 1,0 8),(0 3,0 2,0 5),(0,0,1),(0,0,1)), (c3,(0 5,0,1,1)D) ((0 8,0 1,0 1),(0 7,0 1,0 2),(1,0,0),(1,0,0))

7. UnionandIntersection

Inthissection,weintroducethedefinitionsofunionandintersectionofplithogenicsoftsets, derivetheirproperties,andgivesomeexamples.

Definition7.1. The union oftwoplithogenicsoftsets(F z P ,A)and(Gz P ,B)over U ,denoted by (F z P ,A) z P (Gz P ,B),istheplithogenicsoftset(H z P , Ω)whereΩ=

where ∨z P isaz-plithogenicunion.

Definition7.2. The intersection oftwoplithogenicsoftsets(F z P ,A)and(Gz P ,B)over U , denotedby ShawkatAlkhazaleh,PlithogenicSoftSet

 

F z P (ε) , if ε ∈ A B

(F z P ,A) z P (Gz P ,B),istheplithogenicsoftset(H z P , Ω)whereΩ= A ∪ B, and ∀ ε ∈ Ω, H z P (ε)=

Gz P (ε) , if ε ∈ B A

F z P (ε) ∧z P Gz P (ε) , if ε ∈ A ∩ B

where ∧z P isaz-plithogenicintersection.

7.1. PlithogenicFuzzySoftUnion

Example7.3. ConsiderExample6.2Let

A = a1 =(slow,white,model1, 2015 andbefor) ,a2 =(slow,yellow,model3, 2017) , a3 =(fast,red,model4, 2018) and B = b1 =(slow,white,model1, 2015 andbefor) ,b2 =(slow,yellow,model3, 2017) , b3 =(fast,red,model4, 2018) ,b4 =(slow,red,model1, 2019)

Suppose F F P ,A and GF P ,A aretwoplithogenicfuzzysoftsetsover U suchthat F F P ,A = a1, (c1, (0, 0.4, 1, 1)D) (0 6, 0 7, 1, 1) , (c2, (0, 0.4, 1, 1)D) (0 7, 0 3, 0, 0) , (c3, (0, 0.4, 1, 1)D) (0 1, 0 3, 0, 0) , (c4, (0, 0.4, 1, 1)D) (0 7, 0 6, 1, 1) , a2, (c1, (0, 0.5, 0, 1)D) (0 6, 0 2, 0, 0) , (c2, (0, 0.5, 0, 1)D) (0 5, 0 7, 1, 1) , (c3, (0, 0.5, 0, 1)D) (0 1, 0 1, 0, 0) , (c4, (0, 0.5, 0, 1)D) (0 7, 0 4, 0, 0) , a3, (c1, (0.5, 0, 1, 1)D) (0.1, 0.2, 0, 0) , (c2, (0.5, 0, 1, 1)D) (0.1, 0.3, 0, 0) , (c3, (0.5, 0, 1, 1)D) (0.8, 0.7, 1, 1) , (c4, (0.5, 0, 1, 1)D) (0.1, 0.1, 0, 0)

GF P ,B = b1, (c1, (0, 0 4, 1, 1)D) (0 5, 0 6, 1, 1) , (c2, (0, 0 4, 1, 1)D) (0 8, 0 4, 0, 0) , (c3, (0, 0 4, 1, 1)D) (0 2, 0 2, 0, 0) , (c4, (0, 0 4, 1, 1)D) (0 7, 0 5, 1, 1) , b2, (c1, (0, 0 5, 0, 1)D) (0.5, 0.3, 0, 0) , (c2, (0, 0 5, 0, 1)D) (0.7, 0.7, 1, 1) , (c3, (0, 0 5, 0, 1)D) (0.3, 0.2, 0, 0) , (c4, (0, 0 5, 0, 1)D) (0.8, 0.5, 0, 0) , b3, (c1, (0 5, 0, 1, 1)D) (0.5, 0.6, 0, 0) , (c2, (0 5, 0, 1, 1)D) (0.8, 0.3, 0, 0) , (c3, (0 5, 0, 1, 1)D) (0.2, 0.7, 1, 1) , (c4, (0 5, 0, 1, 1)D) (0.6, 0.3, 0, 0) , b4, (c1, (0, 0, 1, 1)D) (0.2, 0.3, 1, 0) , (c2, (0, 0, 1, 1)D) (0.1, 0.3, 0, 0) , (c3, (0, 0, 1, 1)D) (0.8, 0.7, 1, 1) , (c4, (0, 0, 1, 1)D) (0.1, 0.1, 1, 0) .

Byusingbasicfuzzyunion(maximum)andbasicfuzzyintersection(minimum)wehave: ShawkatAlkhazaleh,PlithogenicSoftSet

F F P ,A F P GF P ,B = H F P ,K where

H F P ,K = k1, (c1, (0, 0 4, 1, 1)D) (0 6, 0 66, 1, 1) , (c2, (0, 0 4, 1, 1)D) (0 8, 0 36, 0, 0) , (c3, (0, 0 4, 1, 1)D) (0 2, 0 26, 0, 0) , (c4, (0, 0 4, 1, 1)D) (0 7, 0 56, 1, 1) ,

k2, (c1, (0, 0 5, 0, 1)D) (0.6, 0.25, 0, 0) , (c2, (0, 0 5, 0, 1)D) (0.7, 0.7, 1, 1) , (c3, (0, 0 5, 0, 1)D) (0.3, 0.15, 0, 0) , (c4, (0, 0 5, 0, 1)D) (0.8, 0.45, 0, 0) , k3, (c1, (0.5, 0, 1, 1)D) (0 3, 0 6, 0, 0) , (c2, (0.5, 0, 1, 1)D) (0 45, 0 3, 0, 0) , (c3, (0.5, 0, 1, 1)D) (0 5, 0 7, 1, 1) , (c4, (0.6, 0, 1, 1)D) (0 35, 0 3, 0, 0) ,

k4, (c1, (0, 0, 1, 1)D) (0 2, 0 3, 1, 0) , (c2, (0, 0, 1, 1)D) (0 1, 0 3, 0, 0) , (c3, (0, 0, 1, 1)D) (0 8, 0 7, 1, 1) , (c4, (0, 0, 1, 1)D) (0 1, 0 1, 1, 0) .

TodescribetheresultinExampleabovelet’scompute (c1 ,(0,0 4,1,1)D ) (0 6,0 7,1,1) ∨F P (c1 ,(0,0 4,1,1)D ) (0 5,0 6,1,1) as follows: (1 0)∗max(0.6, 0.5)+0∗min(0.6, 0.5)=0.6,(1 0.4)∗max(0.7, 0.6)+0.4∗min(0.7, 0.6)=0.66, (1 1) ∗ max(1, 1)+1 ∗ min(1, 1)=1and(1 1) ∗ max(1, 1)+1 ∗ min(1, 1)=1

7.2. PlithogenicFuzzySoftIntersection

Example7.4. ConsiderExample7.3Byusingbasicfuzzyunion(maximum)andbasicfuzzy intersection(minimum)wehave: F F P ,A F P GF P ,B = H F P ,K where H F P ,K = k1, (c1, (0, 0 4, 1, 1)D) (0 5, 0 64, 1, 1) , (c2, (0, 0 4, 1, 1)D) (0 7, 0 34, 0, 0) , (c3, (0, 0 4, 1, 1)D) (0 1, 0 18, 0, 0) , (c4, (0, 0 4, 1, 1)D) (0 7, 0 52, 1, 1) , k2, (c1, (0, 0.5, 0, 1)D) (0.5, 0.25, 0, 0) , (c2, (0, 0.5, 0, 1)D) (0.5, 0.7, 1, 1) , (c3, (0, 0.5, 0, 1)D) (0.1, 0.15, 0, 0) , (c4, (0, 0.5, 0, 1)D) (0.7, 0.45, 0, 0) , k3, (c1, (0.5, 0, 1, 1)D) (0 3, 0 2, 0, 0) , (c2, (0.5, 0, 1, 1)D) (0 45, 0 3, 0, 0) , (c3, (0.5, 0, 1, 1)D) (0 5, 0 7, 1, 1) , (c4, (0.5, 0, 1, 1)D) (0 35, 0 1, 0, 0) , k4, (c1, (0, 0, 1, 1)D) (0 2, 0 3, 1, 0) , (c2, (0, 0, 1, 1)D) (0 1, 0 3, 0, 0) , (c3, (0, 0, 1, 1)D) (0 8, 0 7, 1, 1) , (c4, (0, 0, 1, 1)D) (0 1, 0 1, 1, 0) .

TodescribetheresultinExampleabovelet’scompute (c1 ,(0,0 4,1,1)D ) (0 6,0 7,1,1) ∧F P (c1 ,(0,0 4,1,1)D ) (0 5,0 6,1,1) as follows: (1 0)∗min(0.6, 0.5)+0∗max(0.6, 0.5)=0.5,(1 0.4)∗min(0.7, 0.6)+0.4∗max(0.7, 0.6)=0.64, (1 1) ∗ min(1, 1)+1 ∗ max(1, 1)=1and(1 1) ∗ min(1, 1)+1 ∗ max(1, 1)=1 7.3.

1),

,1))

,0

,1,1)D) ((0 1,0 8),(0 3,0 5),(0,1),(0,1))

,1,1)D) ((0 7,0 2),(0 6,0 2),(1,0),(1,0)) , a

((0 6,0 3),(0 2,0 7),(0,1),(0,1)) , (c2,(0,0 5,0,1)D) ((0 5,0 4),(0 7,0 3),(1,0),(1,0)) , (c3,(0,0 5,0,1)D) ((0 1,0 8),(0 1,0 9),(0,1),(0,1)) , (c4,(0,0 5,0,1)D) ((0 7,0 3),(0 4,0 5),(0,1),(0,1)) ,

a3, (c1,(0 5,0,1,1)D) ((0 1,0 8),(0 2,0 8),(0,1),(0,1)) , (c2,(0 5,0,1,1)D) ((0 1,0 9),(0 3,0 7),(0,1),(0,1)) , (c3,(0 5,0,1,1)D) ((0 8,0 1),(0 7,0 2),(1,0),(1,0)) , (c4,(0 5,0,1,1)D) ((0 1,0 8),(0 1,0 9),(0,1),(0,1))

GIF P ,B = b1, (c1,(0,0 4,1,1)D) ((0 5,0 4),(0 6,0 3),(1,0),(1,0)) , (c2,(0,0 4,1,1)D) ((0 8,0 2),(0 4,0 6),(0,1),(0,1)) ,

b2, (c1,(0,0 5,0,1)D) ((0 5,0 5),(0 3,0 7),(0,1),(0,1)) ,

b3, (c1,(0 5,0,1,1)D) ((0 5,0 4),(0 6,0 4),(0,1),(0,1)) ,

b4, (c1,(0,0,1,1)D) ((0 2,0 8),(0 3,0 6),(0,1),(0,1)) ,

(c2,(0,0 5,0,1)D) ((0 7,0 2),(0 7,0 2),(1,0),(1,0)) ,

(c2,(0 5,0,1,1)D) ((0 8,0 1),(0 3,0 7),(0,1),(0,1)) ,

(c2,(0,0,1,1)D) ((0 1,0 9),(0 3,0 7),(0,1),(0,1)) ,

(c3,(0,0 4,1,1)D) ((0 2,0 7),(0 2,0 8),(0,1),(0,1)) , (c4,(0,0 4,1,1)D) ((0 7,0 2),(0 5,0 4),(1,0),(1,0)) ,

(c3,(0,0 5,0,1)D) ((0 3,0 6),(0 2,0 8),(0,1),(0,1)) , (c4,(0,0 5,0,1)D) ((0 8,0 1),(0 5,0 5),(0,1),(0,1)) ,

(c3,(0 5,0,1,1)D) ((0 2,0 7),(0 7,0 2),(1,0),(1,0)) , (c4,(0 5,0,1,1)D) ((0 6,0 3),(0 3,0 7),(0,1),(0,1))

(c3,(0,0,1,1)D) ((0 8,0 2),(0 7,0 3),(1,0),(1,0)) , (c4,(0,0,1,1)D) ((0 1,0 9),(0 1,0 8),(0,1),(0,1))

Byusingbasicfuzzyunion(maximum)andbasicfuzzyintersection(minimum)wehave:

F IF P ,A IF P GIF P ,B = H IF P ,K where

HIF P ,K = k1, (c1,(0,0 4,1,1)D) ((0 6,0 3),(0 66,0 3),(1,0),(1,0)) , (c2,(0,0 4,1,1)D) ((0 8,0 2),(0 4,0 54),(0,1),(0,1)) , (c3,(0,0 4,1,1)D) ((0 2,0 7),(0 26,0 63),(0,1),(0,1)) , (c4,(0,0 4,1,1)D) ((0 7,0 2),(0 56,0 26),(1,0),(1,0)) ,

k2, (c1,(0,0 5,0,1)D) ((0 6,0 3),(0 25,0 7),(0,1),(0,1)) , (c2,(0,0 5,0,1)D) ((0 7,0 2),(0 7,0 25),(1,0),(1,0)) , (c3,(0,0 5,0,1)D) ((0 3,0 6),(0 15,0 85),(0,1),(0,1)) , (c4,(0,0 5,0,1)D) ((0 8,0 1),(0 45,0 5),(0,1),(0,1)) ,

k3, (c1,(0 5,0,1,1)D) ((0 3,0 6),(0 6,0 4),(0,1),(0,1)) , (c2,(0 5,0,1,1)D) ((0 45,0 5),(0 3,0 7),(0,1),(0,1)) , (c3,(0 5,0,1,1)D) ((0 5,0 4),(0 7,0 2),(1,0),(1,0)) , (c4,(0 5,0,1,1)D) ((0 35,0 55),(0 3,0 7),(0,1),(0,1))

k4, (c1,(0,0,1,1)D) ((0 2,0 8),(0 3,0 6),(0,1),(0,1)) , (c2,(0,0,1,1)D) ((0 1,0 9),(0 3,0 7),(0,1),(0,1)) , (c3,(0,0,1,1)D) ((0 8,0 2),(0 7,0 3),(1,0),(1,0)) , (c4,(0,0,1,1)D) ((0 1,0 9),(0 1,0 8),(0,1),(0,1))

7.4. PlithogenicIntuitionisticFuzzySoftIntersection

Example7.6. ConsiderExample7.5Byusingbasicfuzzyunion(maximum)andbasicfuzzy intersection(minimum)wehave:

F IF P ,A IF P GIF P ,B = H IF P ,K where

HIF P ,K = k1, (c1,(0,0 4,1,1)D) ((0 6,0 4),(0 64,0 3),(1,0),(1,0)) ,

(c2,(0,0 4,1,1)D) ((0 7,0 2),(0 34,0 56),(0,1),(0,1)) , (c3,(0,0 4,1,1)D) ((0 1,0 8),(0 24,0 68),(0,1),(0,1)) , (c4,(0,0 4,1,1)D) ((0 7,0 2),(0 54,0 32),(1,0),(1,0)) ,

k2, (c1,(0,0 5,0,1)D) ((0 5,0 4),(0 25,0 7),(0,1),(0,1)) , (c2,(0,0 5,0,1)D) ((0 5,0 4),(0 7,0 25),(1,0),(1,0)) ,

(c3,(0,0 5,0,1)D) ((0 1,0 8),(0 15,0 85),(0,1),(0,1)) , (c4,(0,0 5,0,1)D) ((0 7,0 3),(0 45,0 5),(0,1),(0,1)) ,

k3, (c1,(0 5,0,1,1)D) ((0 3,0 6),(0 2,0 8),(0,1),(0,1)) , (c2,(0 5,0,1,1)D) ((0 45,0 5),(0 3,0 7),(0,1),(0,1)) , (c3,(0 5,0,1,1)D) ((0 5,0 4),(0 7,0 2),(1,0),(1,0)) , (c4,(0 5,0,1,1)D) ((0 35,0 55),(0 1,0 8),(0,1),(0,1))

k4, (c1,(0,0,1,1)D) ((0 2,0 8),(0 3,0 6),(0,1),(0,1)) , (c2,(0,0,1,1)D) ((0 1,0 9),(0 3,0 7),(0,1),(0,1)) , (c3,(0,0,1,1)D) ((0 8,0 2),(0 7,0 3),(1,0),(1,0)) , (c4,(0,0,1,1)D) ((0 1,0 9),(0 1,0 8),(0,1),(0,1)) 7.5. PlithogenicNeutrosophicSoftUnion

A = a1 =(slow,white,model1, 2015 andbefor) ,a2 =(slow,yellow,model3, 2017) , a3 =(fast,red,model4, 2018) and B = b1 =(slow,white,model1, 2015 andbefor) ,b2 =(slow,yellow,model3, 2017) , b3 =(fast,red,model4, 2018)

ShawkatAlkhazaleh,PlithogenicSoftSet

Suppose F N P ,A and GN P ,B aretwoplithogenicneutrosophicsoftsetsover U suchthat

F N P ,A = a1, (c1 ,(0,0 4,1,1)D) ((0 6,0 3,0 1),(0 7,0 2,0 1),(1,0,0),(1,0,0)) ,

a2, (c1 ,(0,0 5,0,1)D)

((0 6,0 3,0 1),(0 2,0 6,0 2),(0,0,1),(0,0,1)) ,

a3, (c1 ,(0 5,0,1,1)D) ((0 1,0 2,0 7),(0 2,0 1,0 7),(0,0,1),(0,0,1)) ,

GN P ,B = b1, (c1 ,(0,0 4,1,1)D) ((0 5,0 2,0 3),(0 6,0 3,0 1),(1,0,0),(1,0,0)) ,

(c2 ,(0,0 4,1,1)D) ((0 7,0 1,0 2),(0 3,0 5,0 2),(0,0,1),(0,0,1)) ,

(c2 ,(0,0 5,0,1)D) ((0 5,0 2,0 3),(0 7,0 2,0 1),(1,0,0),(1,0,0)) ,

(c2 ,(0 5,0,1,1)D) ((0 1,0 1,0 8),(0 3,0 2,0 5),(0,0,1),(0,0,1)) ,

(c3 ,(0,0 4,1,1)D) ((0 1,0 1,0 8),(0 3,0 6,0 1),(0,0,1),(0,0,1)) ,

(c3 ,(0,0 5,0,1)D) ((0 1,0 2,0 7),(0 1,0 1,0 8),(0,0,1),(0,0,1)) ,

(c3 ,(0 5,0,1,1)D) ((0 8,0 1,0 1),(0 7,0 1,0 2),(1,0,0),(1,0,0))

(c2 ,(0,0 4,1,1)D) ((0 8,0 1,0 1),(0 2,0 3,0 5),(0,0,1),(0,0,1)) ,

b3, (c1 ,(0 5,0,1,1)D) ((0 3,0 2,0 5),(0 3,0 2,0 5),(0,0,1),(0,0,1)) ,

(c3 ,(0,0 5,0,1)D) ((0 1,0 1,0 8),(0 2,0 2,0 6),(0,0,1),(0,0,1)) ,

(c3 ,(0,0 4,1,1)D) ((0 3,0 1,0 6),(0 3,0 4,0 3),(0,0,1),(0,0,1)) , b2, (c1 ,(0,0 5,0,1)D) ((0 5,0 3,0 2),(0 4,0 4,0 2),(0,0,1),(0,0,1)) , (c2 ,(0,0 5,0,1)D) ((0 4,0 3,0 3),(0 8,0 1,0 1),(1,0,0),(1,0,0)) ,

(c2 ,(0 5,0,1,1)D) ((0 1,0 1,0 8),(0 5,0 1,0 4),(0,0,1),(0,0,1)) , (c3 ,(0 5,0,1,1)D) ((0 6,0 2,0 2),(0 6,0 2,0 2),(1,0,0),(1,0,0))

Byusingbasicfuzzyunion(maximum)andbasicfuzzyintersection(minimum)wehave:

F N P ,A N P GN P ,B = H N P ,K where

HN P ,K = k1, (c1,(0,0 4,1,1)D) ((0 6,0 25,0 1),(0 66,0 25,0 1),(1,0,0),(1,0,0)) , (c2,(0,0 4,1,1)D) ((0 8,0 1,0 1),(0 26,0 4,0 32),(0,0,1),(0,0,1)) , (c3,(0,0 4,1,1)D) ((0 3,0 1,0 6),(0 3,0 5,0 18),(0,0,1),(0,0,1)) , k2, (c1,(0,0 5,0,1)D) ((0 6,0 3,0 1),(0 3,0 5,0 2),(0,0,1),(0,0,1)) , (c2,(0,0 5,0,1)D) ((0 5,0 25,0 3),(0 75,0 15,0 1),(1,0,0),(1,0,0)) , (c3,(0,0 5,0,1)D) ((0 1,0 15,0 7),(0 15,0 15,0 7),(0,0,1),(0,0,1)) , k3, (c1,(0 5,0,1,1)D) ((0 2,0 2,0 6),(0 3,0 15,0 5),(0,0,1),(0,0,1)) , (c2,(0 5,0,1,1)D) ((0 1,0 1,0 8),(0 5,0 15,0 4),(0,0,1),(0,0,1)) , (c3,(0 5,0,1,1)D) ((0 7,0 15,0 15),(0 7,0 15,0 2),(1,0,0),(1,0,0))

7.6. PlithogenicNeutrosophicSoftIntersection

Example7.8. ConsiderExample7.7Byusingbasicfuzzyunion(maximum)andbasicfuzzy intersection(minimum)wehave:

F N P ,A N P GN P ,B = H N P ,K where

HN P ,K = k1, (c1,(0,0 4,1,1)D) ((0 5,0 25,0 3),(0 64,0 25,0 1),(1,0,0),(1,0,0)) ,

(c3,(0,0 4,1,1)D) ((0 1,0 1,0 8),(0 3,0 5,0 22),(0,0,1),(0,0,1)) , k2, (c1,(0,0 5,0,1)D) ((0 5,0 3,0 2),(0 3,0 5,0 2),(0,0,1),(0,0,1)) , (c2,(0,0 5,0,1)D) ((0 4,0 25,0 3),(0 75,0 15,0 1),(1,0,0),(1,0,0)) , (c3,(0,0 5,0,1)D) ((0 1,0 15,0 8),(0 15,0 15,0 7),(0,0,1),(0,0,1)) , k3, (c1,(0 5,0,1,1)D) ((0 2,0 2,0 6),(0 2,0 15,0 7),(0,0,1),(0,0,1)) , (c2,(0 5,0,1,1)D) ((0 1,0 1,0 8),(0 3,0 15,0 5),(0,0,1),(0,0,1)) , (c3,(0 5,0,1,1)D) ((0 7,0 15,0 15),(0 6,0 15,0 2),(1,0,0),(1,0,0))

(c2,(0,0 4,1,1)D) ((0 7,0 1,0 2),(0 24,0 4,0 38),(0,0,1),(0,0,1)) ,

Proposition7.9. Let F z P , Gz P and H z P beanythreePSSsover U .Thenthefollowingresults hold:

proof Fromunionandintersectiondefinitionsandthefactthatfuzzyset,intuitionistic fuzzyandneutrosophicsetarecommutativeandassociative,wecangettheproof.

Proposition7.10. Let (F z P ,E), (Gz P ,E) and (H z P ,E) beanythreePSSsover U .Thenthe followingresultshold: ShawkatAlkhazaleh,PlithogenicSoftSet

(1) F z P z P (Gz P z P H z P )=(F z P z P Gz P ) z P (F z P z P H z P ) , (2) F z P z P (Gz P z P H z P )=(F z P z P Gz P ) z P (F z P z P H z P ) proof. Let z ≡ Fuzzy, ∀x ∈ E,andwithoutlossofgeneralitysuppose c =0 (1) λF F P (x) F P (GF P (x) F P H F P (x)) (x)= s λF F P (x) (x) ,λ(GF P (x) F P H F P (x)) (x) = s λF F P (x) (x) ,t λGF P (x) (x) ,λH F P (x) (x) = t{s λF F P (x) (x) ,λGF P (x) (x) ,s λF F P (x) (x) ,λH F P (x) (x) } = t λ(F F P (x) F P GF P (x)) (x) ,λ(F F P (x) F P H F P (x)) (x) = λ(F F P (x) F P GF P (x)) F P (F F P (x) F P H F P (x)) (x) For z ≡ IF and z ≡ N usethesamemethod.

(2) Wecanusethesamemethodin(a).

8. PlithogenicSoftSimilarity

InthissectionweintroduceameasureofsimilaritybetweentwoP-FSSs(z ≡ F )andwe leavetheothersimilaritywhen(z ≡ IFandN )forfutureresearch.Thesettheoreticapproach hasbeentakeninthisregardbecauseitispopularandveryeasyforcalculation.

Definition8.1. Let F F P ,E and GF P ,E betwoP-FSSsover(U,E)asinDefinition6.1. Similarity between F F P ,E and GF P ,E ,denotedby S F F P ,GF P ,isdefinedasfollows: S F F P ,GF P = 1 |E| |E| k=1 Mk where Mk =1

|e| i=1 |Fj (eik) Gj (eik)| |U | j=1

|U | j=1

|e| i=1 |Fj (eik)+ Gj (eik)| Where e ∈ E.

Definition8.2. Let F F P ,E , GF P ,E and H F P ,E betwoP-FSSsover(U,E).Wesaythat F F P ,E and GF P ,E are significantlysimilar if S F F P ,GF P 1 2 .

Proposition8.3. Let F F P ,E and GF P ,E beanytwoP-FSSsover (U,E) suchthat F F P or GF P anon-zeroP-FSS.Thenthefollowingholds:

(1) S F F P ,GF P = S GF P ,F F P , (2) 0 S F F P ,GF P 1, (3) F F P = GF P ⇒ S F F P ,GF P =1, (4) F F P ⊆ GF P ⊆ H F P ⇒ S F F P ,H F P S GF P ,H F P , ShawkatAlkhazaleh,PlithogenicSoftSet

(5) If F F P ,A F P GF P ,B = ∅⇒ S F F P ,GF P =0

proof. Proofs(a)-(d)followsfromDefinition8.1,Wewillgivetheproofof(e). Forthelefthandsidewehave F F P ,A F P GF P ,B = ∅,then

t F F P (eik),GF P (eik) =0, ∀i,k.Now, byusingDefinition8.1wehave Mk =1

|e| i=1 |Fj (eik) Gj (eik)| |U | j=1

|U | j=1

|e| i=1 |Fj (eik)+ Gj (eik)| .

Since t F F P (eik),GF P (eik) =0,then Fj (eik) Gj (eik)= Fj (eik)+ Gj (eik)andthisgives |U | j=1

|e| i=1 |Fj (eik) Gj (eik)| |U | j=1

|e| i=1 |Fj (eik) Gj (eik)| =1.Then

Mk =1 1=0.

Example8.4. Let U = {c1,c2,c3,c4} and a1 =speed, a2 =color, a3 =model, a4 =manufacturingyear.Supposetheirattributesvaluesrespectively:

Speed ≡ A1 = {slow,fast,veryfast}, Color ≡ A2 = {white,yellow,red,black},

Model ≡ A3 = {model1,model2,model3,model4}, manufacturingyear ≡ A4 = {2015 andbefor, 2016, 2017, 2018, 2019}.Suppose

E = e1 =(slow,white,model1, 2015 andbefor) ,e2 =(slow,yellow,model3, 2017) , e3 =(fast,red,model4, 2018)

Suppose F F P ,E and GF P ,E aretwoplithogenicfuzzysoftsetsover U suchthat F F P ,E = e1, (c1, (0, 0 4, 1, 1)D) (0 6, 0 7, 1, 1) , (c2, (0, 0 4, 1, 1)D) (0 7, 0 3, 0, 0) , (c3, (0, 0 4, 1, 1)D) (0 1, 0 3, 0, 0) , (c4, (0, 0 4, 1, 1)D) (0 7, 0 6, 1, 1) , e2, (c1, (0, 0 5, 0, 1)D) (0 6, 0 2, 0, 0) , (c2, (0, 0 5, 0, 1)D) (0 5, 0 7, 1, 1) , (c3, (0, 0 5, 0, 1)D) (0 1, 0 1, 0, 0) , (c4, (0, 0 5, 0, 1)D) (0 7, 0 4, 0, 0) , e3, (c1, (0 5, 0, 1, 1)D) (0 1, 0 2, 0, 0) , (c2, (0 5, 0, 1, 1)D) (0 1, 0 3, 0, 0) , (c3, (0 5, 0, 1, 1)D) (0 8, 0 7, 1, 1) , (c4, (0 5, 0, 1, 1)D) (0 1, 0 1, 0, 0)

ShawkatAlkhazaleh,PlithogenicSoftSet

GF P ,E = e1, (c1, (0, 0.4, 1, 1)D) (0.5, 0.6, 1, 1) , (c2, (0, 0.4, 1, 1)D) (0.8, 0.4, 0, 0) , (c3, (0, 0.4, 1, 1)D) (0.2, 0.2, 0, 0) , (c4, (0, 0.4, 1, 1)D) (0.7, 0.5, 1, 1) , e2, (c1, (0, 0.5, 0, 1)D) (0.5, 0.3, 0, 0) , (c2, (0, 0.5, 0, 1)D) (0.7, 0.7, 1, 1) , (c3, (0, 0.5, 0, 1)D) (0.3, 0.2, 0, 0) , (c4, (0, 0.5, 0, 1)D) (0.8, 0.5, 0, 0) , e3, (c1, (0.5, 0, 1, 1)D) (0.5, 0.6, 0, 0) , (c2, (0.5, 0, 1, 1)D) (0.8, 0.3, 0, 0) , (c3, (0.5, 0, 1, 1)D) (0.2, 0.7, 1, 1) , (c4, (0.5, 0, 1, 1)D) (0.6, 0.3, 0, 0)

Thenwecanfindthesimilaritybetween F F P and GF P asfollows: M1 =1

4 i=1 |Fj (ei1) Gj (ei1)| 4 j=1

4 j=1

4 i=1 |Fj (ei1)+ Gj (ei1)| Now,for j =1wehave 4 i=1 |F1(ei1) G1(ei1)| = |(0 6 0 5)| + |(0 7 0 6)| + |(1 1)| + |(1 1)| =0 2 and 4 i=1 |F1(ei1)+ G1(ei1)| = |(0.6+0.5)| + |(0.7+0.6)| + |(1+1)| + |(1+1)| =6.4 For j =2wehave 4 i=1 |F2(ei1) G2(ei1)| = |(0.7 0.8)| + |(0.3 0.4)| + |(0 0)| + |(0 0)| =0.2 and 4 i=1 |F2(ei1)+ G2(ei1)| = |(0.7+0.8)| + |(0.3+0.4)| + |(0+0)| + |(0+0)| =2.2. For j =3wehave 4 i=1 |F3(ei1) G3(ei1)| = |(0 1 0 2)| + |(0 3 0 2)| + |(0 0)| + |(0 0)| =0 2 and 4 i=1 |F3(ei1)+ G3(ei1)| = |(0 1+0 2)| + |(0 3+0 2)| + |(0+0)| + |(0+0)| =0 8 For j =4wehave ShawkatAlkhazaleh,PlithogenicSoftSet

4 i=1 |F4(ei1) G4(ei1)| = |(0.7 0.7)| + |(0.6 0.5)| + |(1 1)| + |(1 1)| =0.1 and 4 i=1 |F4(ei1)+ G4(ei1)| = |(0.7+0.7)| + |(0.6+0.5)| + |(1+1)| + |(1+1)| =6.5. Then M1 =1

4 i=1 |Fj (ei1) Gj (ei1)| 4 j=1

4 j=1

4 i=1 |Fj (ei1)+ Gj (ei1)| =1 [0 2+0 2+0 2+0 1] [6.4+2.2+0.8+6.5] =0 96 Similarlyweget M2 =0 92and M3 =0 78.ThenthesimilaritybetweenthetwoP-FSSs F F P and GF P isgivenby S F F P ,F F P = 1 3

3 k=1 Mk = 0 96+0 92+0 78 3 ∼ = 0 89

9. Conclusionsandfutureresearch

InthispaperwehavedefinedtheconceptofPlithogenicsoftset,andgavesomegeneralizationsofthisconcept.Wealsogaveexamplesfortheseconcepts.Westudiedthebasic propertiesoftheseoperations.Animportantresultofsuchapaperisthatnewquestionscan beusedasanideaforfurtherresearch,assucharesearchalwaysunearthsfurtherquestions. Theworkpresentedinthispaperposesinterestingnewquestionstoresearchersandprovides thetheoreticalframeworkforfurtherstudyonplithogenicsoftsets.Basedontheprevious resultswemaysuggestproblemsinrelationtoourresearchthatweanticipatetoventure elsewhereinthecasesofoperationsinplithogenicsoftsets,similarityinplithogenicsoftsets andapplicationsonplithogenicsoftsets.Wecaninvestigatethefollowingtopicsforfurther works:

(1) Todefineandstudythe”AND”and”OR”operationsofplithogenicsoftset.

(2) TodefineandstudytheapplicationofsimilaritymeasureofplithogenicsoftsetonDM andMD.

(3) TodefineandstudytheapplicationofplithogenicsoftsetoperationsonDMandMD.

(4) Togeneralizedplithogenicsoftsetstoplithogenicsoftmultisets,expertset,possibility setandetc.

Acknowledgements

ThisresearchwassupportedbytheDeanshipofScientificResearch,JadaraUniversity. ShawkatAlkhazaleh,PlithogenicSoftSet

References

1. MohamedAbdel-Basset,MohamedEl-hoseny,AbduallahGamal,andFlorentinSmarandache.Anovel modelforevaluationhospitalmedicalcaresystemsbasedonplithogenicsets. ArtificialIntelligencein Medicine,100:101710,2019.

2. MohamedAbdel-BassetandRehabMohamed.Anovelplithogenictopsis-criticmodelforsustainablesupply chainriskmanagement. JournalofCleanerProduction,247:119586,2020.

3. MohamedAbdel-Basset,RehabMohamed,AbdEl-NasserH.Zaied,andFlorentinSmarandache.Ahybrid plithogenicdecision-makingapproachwithqualityfunctiondeploymentforselectingsupplychainsustainabilitymetrics. Symmetry,11(7):903,2019.

4. KrassimirT.Atanassov.Intuitionisticfuzzysets. FuzzySetsandSystems,20(1):87–96,1986.

5. P.K.Maji.Neutrosophicsoftset. AnnalsofFuzzyMathematicsandInformatics,5(1):157–168,2013.

6. P.K.Maji,R.Biswas,andA.R.Roy.Fuzzysoftset. JournalofFuzzyMathematics,9(3):589–602,2001.

7. P.K.Maji,R.Biswas,andA.R.Roy.Intutionisticfuzzysoftsets. JournalofFuzzyMathematics,9(3):677–693,2001.

8. D.Molodtsov.Softsettheory-firstresults. ComputersandMathematicswithApplications,37(2):19–31, 1999.

9. F.Smarandache. AUnifyingFieldinLogics.Neutrosophy:NeutrosophicProbability,SetandLogic.AmericanResearchPress,Rehoboth,USA,1998.

10. F.Smarandache.Neutrosophicset,ageneralisationoftheintuitionisticfuzzysets. InternationalJournalof PureandAppliedMathematics,24(3):287–297,2005.

11. F.Smarandache. Plithogeny,PlithogenicSet,Logic,Probability,andStatistics.PonsEditions,Brussels, Belgium,2017.

12. F.Smarandache.Plithogenicset,anextensionofcrisp,fuzzy,intuitionisticfuzzy,andneutrosophicsets revisited. NeutrosophicSetsandSystems,21:153–166,2018.

13. L.A.Zadeh.Fuzzysets.8:338–353,1965.

Received:6,December,2019/Accepted:3,March,2020