11, No. 1, PP. 30-38, 2020
Plithogenic Cubic Sets
S.P. Priyadharshini1,* , F. Nirmala Irudayam 2 and F. Smarandache
1,2 Dept of Mathematics, Nirmala College for Women 3 Dept. Math and Sciences, University of New
3 ,
Dept Nirmala Women, India; priyadharshini125@gmail.com nirmalairudayam@ymail.com New Mexico, Gallup, NM, USA; smarand@unm.edu
Abstract
In this article, using the concepts of plithogenic intuitionistic fuzzy cubic set internal and external cubic sets are cubic sets were also discussed.This concept attribute decision making as this plithogenic the accuracy of the result is also so precise
* Correspondence: priyadharshini125@gmail.com , cubic set and plithogenic set, the ideas of p ntuitionistic set, Plithogenic neutrosophic cubic set are introduced and its corresponding discussed with examples. Primary properties of the P is extremely suitable for addressing problems decision neutrosophic set are described by four or more
plithogenic fuzzy cubic set, are its rimary Plithogenic neutrosophic involving multiple value of attributes and
Keywords: Fuzzy set, Intuitionistic fuzzy set, set, Plithogenic Intuitionistic fuzzy cubic set, Plithogenic neutrosophic cubic set
Neutrosophic set, Cubic set, Plithogenic set, set.
1. Introduction
Zadeh implemented Fuzzy valued fuzzy set , i.e. a fuzzy set with from the interval [ 0,1] are used to reflect, always tough for an expert to precisely enumerate their fitting to reflect this certainty degree which were frequently used in real-life
Sets[18].In [18] Zadeh gave the perception of an intervalvalued belonging function. In conventional e.g., the degree of conviction of the expert in various o their certainty; thus, rather than a particular value by an interval or even by a fuzzy set. We obtain an interval valued fuzzy sets life scenario.
Smarandache [9, 10, 11] proposed Neutrosophic sets (NSs) helpful for dealing with inadequate, functions of truth (T), indeterminacy (I) and areas of application since indeterminacy is functions are independent.
Plithogenic fuzzy cubic a fuzzy set by an intervalfuzzy concepts,numbers statements.It is value, it is more by an fuzzy set. In not the inaccuracy IFS were , and IFS, which is highly life. NSs are characterised by is very essential in several falsity
which is the generalisation of the ach to belonging and non-belonging grade with the constraint that the sum of these is y ] (NSs), a generalisation of FS uncertain, and varying data that exists in the real life s (I) falsity (F) belonging functions. This concept clearly enumerated and the truth, indeterminacy, and falsity membership
Atanassov[2] introduced intuitionistic fuzzy sets(IFS), IFS,each element is attached to both two grades is less than or equal to unity. If available knowledge is not sufficient to identify the inaccuracy of traditional fuzzy sets, IFS architecture can be viewed as an alternative / appropriate solution. Later on expanded to interval-valued IFS.
International Journal of Neutrosophic Science (IJNS)
Vol. 11, No. 1, PP. 30-38, 2020
.Wang, Smarandache, Zhang and Sunderraman[17] proposed the definition of the interval valued neutrosophic set (INS) as an extension of the neutrosophic set.The INS could reflect indeterminate, inaccurate, inadequateandunreliabledatathatoccursinthereality.
Plithogenyistheorigin,creation,productionandevolutionofnewobjectsfromthesynthesisofconflicting or non-conflicting multiple old objects. Smarandache[12] introduced the plithogenic set as a generalisation of neutrosophyin2017.
The elements of plithogenic sets are denoted by one or many number of attributes and each of it have severalvalues.Eachvaluesofattributehasitsrespective(fuzzy,intuitionisticfuzzyorneutrosophic)appurtenance degree for the element x (say) to the plithogenic set P (say) with respect to certain constraints. For the first time, Smarandache[12] introduced the contradictory (inconsistency) degree between each value of attribute and the dominant value of attribute which results in getting the better accuracy for the plithogenic aggregation operators(fuzzy,intuitionisticfuzzyorneutrosophic).
Y.B.Junetal[4]implementedacubicsetwhichis acombinationofafuzzysetwithanintervalvaluedfuzzy set.Internalandexternalcubicsetswerealsodescribedandsomepropertieswerestudied.
Y.B.Jun, Smarandache and Kim [4] introduced Neutrosophic cubic sets and the concept of internal and externalfortruth,falsityandindeterminacyvalues.FurthermoretheyprovidedsomanypropertiesofP(R)-union,P (R)–intersectionforinternalandexternalneutrosophiccubicsets.
In this article, using the principles of cubic sets and plithogenic set, we presented the generalisation of plithogeniccubicsetsforfuzzy,Intuitionisticfuzzyandneutrosophicsets
2. Preliminaries
Definition 2.1 [15] LetZbeauniverseofdiscourseandthefuzzyset { }Z z z z F f ∈ = ,|)( µ isdescribedbya belongingfunction fµ as, ]1,0[ : → Z fµ .
Definition 2.2 [10] Let Y be a non-empty set. Then, an interval valued fuzzy set B over Y is defined as } :/)]( ),( {[ Y yy y B y B B ∈ = + where )( x B and )( x B+ arestatedastheinferiorandsuperiordegreesofbelonging Y y∈ where + ≤ )( 0 y B 1 )( ≤ + y B correspondingly.
Definition 2.3 [2, 3] Let N beanonemptyset.Theset { }N n n A A A ∈ = φ λ ,, iscalledanintutionistic fuzzy set(inshort,IFS)of N wherethefunction ]1,0[ : → N Aλ , ]1,0[ : → N Aφ denotesthemembershipdegree(say )( n Aλ ) and non-membership degree (say )( n Aφ ) of each element N n ∈ to the set A and satisfies the constraint that )( )( 0 ≤ + ≤ n n A A φ λ
Definition 2.4 [10] Let Y beanon-emptyset.Theset } |)( ),( , { Y y y N y M y A A ∈ > < = Α iscalledaninterval valued intuitionistic fuzzy sets (IVIFS) A in Y where the functions ]1,0[ :)( → Y y M A and ]1,0[ :)( → Y y N A denotes the degree of belonging, non-belonging of A respectively. Also )]( ),( [ )( y M y M y M AU AL A = and )]( ),( [ )( y N y N y N AU AL A = , 1 )( )( 0 ≤ + ≤ y M y M AU AU foreach Y y∈
Definition 2.5 [6] Let N beanonemptyset.Theset { }N n n A A A A ∈ = γ φ λ ,,, iscalled aneutrosophicset(in short,NS)of N wherethefunction ]1,0[ : → N Aλ , ]1,0[ : → N Aφ and ]1,0[ : → N Aγ denotesthemembershipdegree (say )( n Aλ ),indeterminacydegree(say )( n Aφ ),andnon-membershipdegree(say )( n Aγ )ofeachelement N n ∈ to theset A andsatisfiestheconstraintthat )( )( )( 0 ≤ + + ≤ n n n A A A γ φ λ
International Journal of Neutrosophic Science (IJNS)
Definition 2.6 [1] Let R beanon-emptyset.Anintervalvaluedneutrosophicset(INS) A in R isdescribedby the functions of the truth-value ) ( TA , the indeterminacy)( IA and the falsity-value ) ( FA for each point R r ∈ , ]1,0[ )( ),( ),( ⊆ r A r A r A F I T
Definition 2.7 [4,5] Let E be a non-empty set. By a cubic set in E ,we construct a set which has the form } |)( {),(, E e e eBe ∈ > < = Ψ µ inwhich B isanintervalvaluedfuzzyset(IVFS)in E and µ isafuzzysetin E.
Definition 2.8 [4] Let E be a non-empty set. If )( )( )( e B e e B + ≤ ≤ µ for all E e ∈ then the cubic set > =< Ψ µ,B in E iscalledaninternalcubicset(brieflyIPS).
Definition 2.9 [4] Let E be a non-empty set. If ))( ),( ( )( e B e B e + ∉ µ for all E e ∈ then the cubic set > =< Ψ µ,B in E iscalledanexternalcubicset(brieflyECS)
Definition 2.10 [10] Plithogenic Fuzzy set for an interval valued (IPFS) is defined as )]1,0[( :, C W C dC c → × ∈ ∀ , W w∈ ∀ and),(wcd is an open, semi-open, closed interval included in [0, 1] and C([0,1])isthepowersetoftheunitinterval[0,1](i.e)allsubsetsof[0,1].
Definition 2.11 [10] A set which has the form )]1,0[( :, 2 C W C dC c → × ∈ ∀ and W w∈ ∀ is called interval valued plithogenic intuitionistic fuzzy set (IPIFS) where),(wcd is an open, semi-open, closed for the interval includedin]1,0[
Definition 2.12 [10] A set which has the form )]1,0[( :, 3 C W C dC c → × ∈ ∀ and W w∈ ∀ is called interval valuedplithogenicneutrosophicset(IPNS)where),(wcd isanopen,semi-open,closedforthe intervalincludedin ]1,0[
3. Plithogenic Cubic sets
3.1 Plithogenic fuzzy Cubic sets
Definition 3.1.1 For a non empty set Y, the Plithogenic fuzzy cubic set (PFCS) is defined as } |)(),(, { Y y y yBy ∈ > < = Λ λ inwhich B isanintervalvaluedplithogenicfuzzysetinYand λ isafuzzysetinY.
Example 3.1.2 The following values of attribute represents the criteria “Musical instruments” : Piano (the dominantone),guitar,saxophone,violin.
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Piano Guitar Saxophone Violin AppurtenanceDegree )(yB [0.5,0.6] [0.2,0.8] [0.4,0.7] [0.1,0.3] )( y λ 0.3 0.4 0.6 0.2
Definition 3.1.3 For a non-empty set Y, the PFCS > =< Λ λ ,B in Y is called an internal plithogenic fuzzy cubicset(brieflyIPFCS)if )( )( )( y B y y B i i d i d + ≤ ≤ λ forall Y y ∈ and di representsthecontradictorydegreeand theirrespectivevalueofattributes.
Example 3.1.4 Thefollowingvaluesofattributerepresentsthecriteria“Color”:Yellow(thedominantone), green,orangeandred.
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Yellow Green Orange Red
Vol. 11, No. 1, PP. 30-38, 2020 DOI: 10.5281/zenodo.4030378
International Journal of Neutrosophic Science (IJNS)
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AppurtenanceDegree )(yB [0.2,0.3] [0.6,0.8] [0.3,0.6] [0.4,0.9] )( y λ 0.2 0.7 0.5 0.8
Definition 3.1.5 For a non-empty set Y, the PFCS > =< Λ λ,B in Y is called an external plithogenic fuzzy cubicset(brieflyEPFCS)if ))( ),( ( )( y B y B y i i d d i + ∉ λ forall Y y ∈ andanddi representsthecontradictorydegree andtheirrespectivevalueofattributes.
Example 3.1.6 Thefollowingvaluesofattributerepresentsthecriteria“Subjects”:Mathematics(thedominant one),Physics,ChemistryandBiology
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Mathematics Physics Chemistry Biology AppurtenanceDegree )(yB [0.4,0.6] [0.5,0.8] [0.2,0.3] [0.5,0.9] )( y λ 0.7 0.1 0.6 0.3
Remark 3.1.7 Ifanyoneoftheattributevalue ( ) y y i = λ forall Y y ∈ and i representsthe valuesofattribute then Λ isneitheranIPFCSnoranEPFCS.
3.2 Plithogenic Intuitionistic fuzzy cubic set
Definition 3.2.1 For a non empty set Y, the Plithogenic Intuitionistic fuzzy cubic set (PIFCS) is defined as } |)(),(, { Y y y yBy ∈ > < = Λ λ in which B is an interval valued Plithogenic Intuitionistic fuzzy set in Y and λ is a intuitionisticfuzzysetinY.
Example 3.2.2 The following values of attribute represents the criteria “Mobile phone brands” : Apple (the dominantone),Samsung,Nokia,Lenova.
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Apple Samsung Nokia Lenova AppurtenanceDegree )(yB ([0.2,0.5], [0.3,0.6]) ([0.5,0.8], [0.1,0.7]) ([0.4,0.7], [0.2,0.6]) ([0.1,0.3], [0.4,0.7]) )( y λ ([0.1,0.7]) ([0.2,0.4]) ([0.1,0.8]) ([0.2,0.5])
Definition 3.2.3 For a non-empty set Y, the PIFCS > =< Λ λ,B in Y is called an internal plithogenic intuitionistic fuzzy cubic set (briefly IPIFCS) if )( )( )( y B y y B i i d i d + ≤ ≤ λ for all Y y ∈ and di represents the contradictorydegreeandtheirrespectivevalueofattributes.
Example 3.2. Thefollowingvaluesofattributerepresentsthecriteria“Proficiency”:Excellent(thedominant one),good,average,poor
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Excellent Good Average Poor AppurtenanceDegree )(yB ([0.3,0.5], [0.1,0.6]) ([0.4,0.6], [0.5,0.8]) ([0.1,0.8], [0.7,0.8]) ([0.2,0.3], [0.1,0.6]) )( y λ ([0.4,0.5]) ([0.6,0.7]) ([0.2,0.5]) ([0.3,0.5])
DOI: 10.5281/zenodo.4030378
International Journal of Neutrosophic Science (IJNS)
Definition 3.2.5 For a non-empty set Y, the PIFCS > =< Λ λ,B in Y is called an external plithogenic intuitionistic fuzzy cubic set (briefly EPIFCS) if ))( ),( ()( y B y B y i i d d i + ∉ λ for all Y y ∈ and di represents the contradictorydegreeandtheirrespectivevalueofattributes.
Example 3.2.6 The following values of attribute represents the criteria “ Mode of Transport” : Bus (the dominantone),Car,Lorry,Train
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Bus Car Lorry Train AppurtenanceDegree )(yB ([0.4,0.6], [0.3,0.5]) ([0.5,0.7], [0.2,0.4]) ([0.2,0.6], [0.3,0.7]) ([0.3,0.6], [0.1,0.4]) )( y λ ([0.1,0.6]) ([0.4,0.5]) ([0.7,0.2]) ([0.7,0.6])
3.3 Plithogenic Neutrosophic cubic set
Definition 3.3.1 Let Ω be an universal set and Y be a non empty set. The structure } |)(),(, { Y y y yBy ∈ > < = Λ λ is said to be Plithogenic Neutrosophic cubic set (PNCS) in Y, where ( ) { } )( ),( ),( y B y B y B B F d I d T d i i i = is an interval valued Plithogenic Neutrosophic set in Y and )}( ),( ),( {( y y y F i I i T i λ λ λ λ= isaneutrosophicsetinY.
The pair > =< Λ λ,B is called plithogenic neutrosophic cubic set over Ω where Λ is a mapping given by )( : Ω → Λ NC B .Thesetofallplithogenicneutrosophiccubicsetsover Ω willbedenotedby Ω NP
Example 3.3.2 consider the attribute “Size” which has the following values: Small (the dominant one), medium,big,verybig.
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Small Medium Big Verybig AppurtenanceDegree )(yB ([0.3,0.5], [0.1,0.8], [0.1,0.9])
([0.1,0.3], [0.4,0.8], [0.4,0.9])
([0.1,0.5], [0.2,0.6], [0.6,0.9])
([0.2,0.6], [0.2,0.5], [0.7,0.9]) )( y λ ([0.4,0.5,0.8]) ([0.3,0.7,0.4]) ([0.2,0.5,0.8]) ([0.5,0.2,0.8])
Definition 3.3.3 Foranon-emptysetY,theplithogenicneutrosophiccubicset > =< Λ λ,B inYiscalled
(i) truthinternalif )( )( )( y B y y B T d T i T d i i + ≤ ≤ λ forall Y y ∈ anddi representsthecontradictorydegreeand theirrespectivevalueofattributes. (3.1)
(ii) indeterminacy internal if )( )( )( y B y y B I d I i I d i i + ≤ ≤ λ for all Y y ∈ and di represents the contradictory degreeandtheirrespectivevalueofattributes.(3.2)
(iii) falsity internal if )( )( )( y B y y B F d F i F d i i + ≤ ≤ λ for all Y y ∈ and di represents the contradictory degree andtheirrespectivevalueofattributes. (3.3).
IfaplithogenicneutrosophiccubicsetinYsatisfiestheaboveequationsweconcludethat Λ isaninternal plithogenicneutrosophiccubicset(IPNCS)inY.
Example 3.3.4 The following values of attribute represents the criteria “Sports”: Volley ball(the dominant one),Basketball,Cricket,Bat-minton
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Volleyball Basketball Cricket Bat-minton
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AppurtenanceDegree )(yB ([0.2,0.5], [0.6,0.8], [0.1,0.5])
([0.1,0.3], [0.4,0.5], [0.6,0.9])
([0.5,0.8], [0.1,0.4], [0.6,0.9])
([0.3,0.6], [0.1,0.5], [0.7,0.9]) )( y λ ([0.3,0.6,0.4]) ([0.2,0.5,0.8]) ([0.7,0.2,0.8]) ([0.4,0.3,0.9])
Definition 3.3.5 Foranon-emptysetY, theplithogenicneutrosophiccubicset > =< Λ λ,B inYiscalled
(i) truth externalif ))( ),( ( )( y B y B y T d T d T i i i + ∉ λ for all Y y ∈ and di represents the contradictory degree andtheirrespectivevalueofattributes. (3.4)
(ii) indeterminacyexternalif ))( ),( ()( y B y B y I d I d I i i i + ∉ λ forall Y y ∈ anddi representsthecontradictory degreeandtheirrespectivevalueofattributes. (3.5)
(ii) falsity external if ))( ),( ()( y B y B y F d F d F i i i + ∉ λ for all Y y ∈ and di represents the contradictory degreeandtheirrespectivevalueofattributes. (3.6)
Ifaplithogenic neutrosophiccubicsetinY satisfiesthe aboveequations, weconclude that Λ isanexternal plithogenicneutrosophiccubicset(EPNCS)inY.
Example 3.3.6 Thefollowingvaluesofattributerepresentsthecriteria“Seasons”:Spring(thedominantone), Winter,Summer,Autumn
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Spring Winter Summer Autumn AppurtenanceDegree )(yB ([0.3,0.6], [0.7,0.5], [0.4,0.8])
([0.1,0.3], [0.2,0.3], [0.6,0.8])
([0.1,0.4], [0.2,0.6], [0.8,0.9])
([0.3,0.5], [0.2,0.7], [0.6,0.8]) )( y λ ([0.1,0.2,0.9]) ([0.4,0.5,0.9]) ([0.7,0.9,0.6]) ([0.1,0.8,0.3])
Theorem 3.3.7 LetYbeanonemptysetand > =< Λ λ,B beaPNCSinY whichisnotexternal.Thenthere exists Y y ∈ such that ))( ),( ( )( y B y B y T d T d T i i i + ∈ λ , ))( ),( ( )( y B y B y I d I d I i i i + ∈ λ or ))( ),( ( )( y B y B y F d F d F i i i + ∈ λ where di represents the contradictory degree and their respective value of attributes..
Proof. Directproof
Theorem 3.3.8 Let Y be a non empty set and > =< Λ λ,B be a PNCS in Y. If Λ is both T-internal and Texternal,then ( ) { } { } ( ) Y y y B Y y y B y Y y T d T d T i i i ∈ ∪ ∈ ∈ ∈ ∀ +)( )( )(λ where di representsthecontradictorydegree andtheirrespectivevalueofattributes.
Proof. The conditions (3.1) and (3.4) implies that )( )( )( y B y y B T d i T d i i + ≤ ≤ λ and ))( ),( ()( y B y B y T d T d T i i i + ∉ λ forall Y y ∈ andi=1,2,3,…n.
Thenitindicatesthat )( )( y B y T d T i i = λ or )( )( y B y T d T i i + = λ ,andhence { } { } Y y y B Y y y B y T d T d T i i i ∈ ∪ ∈ ∈ +)( )( )(λ where di represents the contradictory degree and their respectivevalueofattributes.
Hencetheproof.
Vol. 11, No. 1, PP. 30-38, 2020 DOI: 10.5281/zenodo.4030378
International Journal of Neutrosophic Science (IJNS)
Correspondinglythesubsequenttheoremsholdfortheindeterminateandfalsityvalues.
Theorem 3.3.9 Let Y be a non empty set and > =< Λ λ,B be a PNCS in Y. If Λ is both I-internal and Iexternal,then ( ) { } { } ( ) Y y y B Y y y B y Y y I d I d I i i i ∈ ∪ ∈ ∈ ∈ ∀ +)( )( )(λ where di representsthecontradictorydegree andtheirrespectivevalueofattributes.
Theorem 3.3.10 Let Y be a non empty set and > =< Λ λ,B be a PNCS in Y. If Λ is both F-internal and F external,then ( ) { } { } ( ) Y y y B Y y y B y Y y F d F d F i i i ∈ ∪ ∈ ∈ ∈ ∀ +)( )( )(λ where di representsthecontradictorydegree andtheirrespectivevalueofattributes.
Definition 3.3.11 Let Y be a non empty set, The complement of > =< Λ λ,B is said to be the PNCS > =< Λ cc c B λ , where ( ) { } )( ),( ),( y B y B y B B F d c I d c T d c c i i i = is an interval valued PNCS in Y and )}( ),( ),( {( y y y F i c I i c T i c λ λ λ λ= isaneutrosophicsetinY.
Theorem 3.3.12 Let Y be a non empty set and > =< Λ λ,B be a PNCS in Y. If Λ is both T-internal and T external,thenthecomplement > =< Λ cc c B λ, of > =< Λ λ,B isan T-Internaland T-externalPNCSinY.
Proof. Let Y be a non empty set .If > =< Λ λ,A is an T-internal and T-external PNCS in Y, then )( )( )( y B y y B T d i T d i i + ≤ ≤ λ and ))( ),( ()( y B y B y T d T d T i i i + ∉ λ for all Y y ∈ and i = 1, 2, 3,…,n. It follows that )( 1 )( 1)( 1 y B y y B T d i T d i i + ≤ ≤ λ and. )).( 1,)( 11()( y B y B y T d T d T i i i + ∉ λ Therefore > =< Λ cc c B λ, is an T Internaland T-externalPNCSinY.
Correspondinglythesubsequenttheoremsholdfortheindeterminateandfalsityvalues.
Theorem 3.3.13 Let Y be a non empty set and > =< Λ λ,B be a PNCS in Y. If Λ is both I-internal and I external,thenthecomplement > =< Λ cc c B λ , of > =< Λ λ,B isan I-internaland I-externalPNCSinY.
Theorem 3.3.14 Let Y be a non empty setand > =< Λ λ,B be a PNCS in Y. If Λ is both F-internal and F external,thenthecomplement > =< Λ cc c B λ , of > =< Λ λ,B isan F-internaland F-externalPNCS inY.
Definition 3.3.15 Let N P B Ω >∈ =<Λ λ , .
If )( )( )( y B y y B T d T i T d i i + ≤ ≤ λ , )( )( )( y B y y B I d I i I d i i + ≤ ≤ λ , ))( ),( ()( y B y B y F d F d F i i i + ∉ λ or )( )( )( y B y y B T d T i T d i i + ≤ ≤ λ , )( )( )( y B y y B F d F i F d i i + ≤ ≤ λ ))( ),( ()( , y B y B y I d I d I i i i + ∉ λ or )( )( )( y B y y B F d F i F d i i + ≤ ≤ λ , )( )( )( y B y y B I d I i I d i i + ≤ ≤ λ ))( ),( ()( , y B y B y T d T d T i i i + ∉ λ for all Y y ∈ .Then Λ iscalled 3 2 IPNCSor 3 1 EPNCS.
Example 3.3.16 Thefollowingvaluesofattributerepresentsthecriteria“Typesofhumans”:Funloving(the dominantone),Sensitive,Determined,Serious ContradictoryDegree 0 0.50 0.75 1
Vol. 11, No. 1, PP. 30-38, 2020 DOI: 10.5281/zenodo.4030378
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Valueofattributes Funloving Sensitive Determined Serious AppurtenanceDegree )(yB ([0.1,0.6], [0.3,0.5], [0.4,0.8])
([0.1,0.3], [0.2,0.3], [0.6,0.8])
([0.1,0.4], [0.2,0.6], [0.8,0.9])
([0.3,0.5], [0.2,0.7], [0.6,0.8]) )( y λ ([0.1,0.4,0.9]) ([0.2,0.5,0.7]) ([0.7,0.5,0.8]) ([0.3,0.4,0.3])
Definition 3.3.16 Let N P B Ω >∈ =<Λ λ , If )( )( )( y B y y B T d T i T d i i + ≤ ≤ λ , ))( ),( ()( y B y B y I d I d I i i i + ∉ λ , ))( ),( ()( y B y B y F d F d F i i i + ∉ λ or )( )( )( y B y y B F d F i F d i i + ≤ ≤ λ , ))( ),( ( )( y B y B y T d T d T i i i + ∉ λ , ))( ),( ()( y B y B y I d I d I i i i + ∉ λ or )( )( )( y B y y B I d I i I d i i + ≤ ≤ λ , ))( ),( ()( y B y B y F d F d F i i i + ∉ λ , ))( ),( ( )( y B y B y T d T d T i i i + ∉ λ forall Y y ∈ .
Then Λ iscalled 3 1 IPNCSor 3 2 EPNCS.
Example 3.3.17 The following values of attribute represents the criteria “ Social Networks ” : Whatsapp, Facebook(the dominantone), Instagram, Linkedinn
ContradictoryDegree 0 0.50 0.75 1 Valueofattributes Whatsapp Facebook Instagram Linkedin AppurtenanceDegree )(yB ([0.2,0.6], [0.7,0.5], [0.4,0.8])
([0.1,0.3], [0.2,0.4], [0.6,0.8])
Theorem 3.3.14 Let N P B Ω >∈ =<Λ λ , . Then
(i) AllIPNCSisageneralisationof ICS. (ii) AllEPNCSisa generalisationof ECS. (iii) AllPNCSisa generalisationofCS.
Proof. Directproof.
4. Conclusions and future work
([0.1,0.4] , [0.2,0. 6], [0.8,0.9])
([0.3,0.5], [0.2,0.7], [0.6,0.8]) )( y λ ([0.3,0.2,0.9]) ([0.4,0.3, 0.9]) ([0.7,0.2, 0.6]) ([0.2,0.5, 0.4])
In this article,We have introduced the plithogenic fuzzy cubic set, plithogenic intuitionistic fuzzy cubic set, plithogenic neutrosophic cubic sets and their corresponding internal and external cubic sets are defined with examples. Furthermore some of the properties of plithogenic neutrosophic cubic sets are investigated. In the consecutive research, we will study the P-Union, P-Intersection, R-Union, R-Intersection of plithogenic neutrosophiccubic sets.
Funding: “Thisresearchreceivednoexternalfunding”
Conflicts of Interest: “The authorsdeclare noconflictofinterest.”
Vol. 11, No. 1, PP. 30-38, 2020 DOI: 10.5281/zenodo.4030378
International Journal of Neutrosophic Science (IJNS)
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Vol. 11, No. 1, PP. 30-38, 2020 DOI: 10.5281/zenodo.4030378