Some Operations and Properties of Neutrosophic Cubic Soft Set

Abstract

In this paper we define some operations such as P-union, P-intersection, R-union, R-intersection for neutrosophic cubic soft sets (NCSSs). We prove some theorems on neutrosophic cubic soft sets. We also discuss various approaches of Internal Neutrosophic Cubic Soft Sets (INCSSs) and external neutrosophic cubic soft sets (ENCSSs). We also investigate some of their properties.

Keywords: Neutrosophic cubic soft set; Neutrosophic soft set; Cubic set; Internal neutrosophic Cubic soft set; External neutrosophic cubic soft set

Received: April 01, 2017; Accepted: April 21, 2017; Published: April 30, 2017

Introduction

Neutrosophic set [1] grounded by Smarandache in 1998, is the generalization of fuzzy set [2] introduced by Zadeh in 1965 and intuitionistic fuzzy set [3] by Atanassov in 1983. In 1999, Molodstov [4] introduced the soft set theory to overcome the inadequate of existing theory related to uncertainties. Soft set theory is free from the parameterization inadequacy syndrome of fuzzy set theory [2], rough set theory [5], probability theory for dealing with uncertainty. The concept of soft set theory penetrates in many directions such as fuzzy soft set [6-9], intuitionistic fuzzy soft set [10-13], interval valued intuitionistic fuzzy soft set [14], neutrosophic soft set [15-18], interval neutrosophic set [19,20]. In 2012, Jun et al. [21] introduced cubic set combining fuzzy set and interval valued fuzzy set. Jun et al. [21] also defined internal cubic set, external cubic set, P-union, R-union, P-intersection and R-intersection of cubic sets, and investigated several related properties. Cubic set theory is applied to CI-algebras [22], B-algebras [23], BCK/BCI-algebras [24,25], KU-Algebras ([26,27], and semi-groups [28]. Using fuzzy set and interval-valued fuzzy set Abdullah et al. [29] proposed the notion of cubic soft set [29] and defined internal cubic soft set, external cubic soft set, P-union, R-union, P-intersection and R-intersection of cubic soft sets, and investigated several related properties. Ali et al. [30] studied generalized cubic soft sets and their applications to algebraic structures. Wang et al. [31] introduced the concept of interval neutrosophic set. In 2016, Ali et al. [32] presented the concept of neutrosophic cubic set by combining the concept of neutrosophic set and interval neutrosophic set. Ali et al. [32] mentioned that neutrosophic cubic set is basically the generalization of cubic set. Ali et al. [32] also defined some new type of internal neutrosophic cubic set

ISSN 2393-8854

Pramanik S1, Dalapati S2, Alam S2 and Roy TK2

1 Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, West Bengal, India

2 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India

Corresponding author: Surapati Pramanik  sura_pati@yahoo.co.in

Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, Narayanpur, North 24 Parganas-743126, West Bengal, India.

Tel: 0332580 1826

Citation: Pramanik S, Dalapati S, Alam S, et al. Some Operations and Properties of Neutrosophic Cubic Soft Set. Glob J Res Rev. 2017, 4:2.

(INCSs) and external neutrosophic cubic set (ENCSs) namely, 1221 (),()3333 INCSorENCSINCSorENCS Ali et al. [32] also presented a numerical problem for pattern recognition. Jun et al. [33] also studied neutrosophic cubic set and proved some properties. In 2016, Chinnadurai et al. [34] introduced the neutrosophic cubic soft sets and proved some properties.

In this paper we discuss some new operations and new approach of internal and external neutrosophic cubic soft sets, and P-union, R-union, P-intersection, R-intersection. We also prove some theorems related to neutrosophic cubic soft sets.

Rest of the paper is presented as follows. Section 2 presents some basic definition of neutrosophic sets, interval-valued neutrosophic sets, soft sets, cubic set, neutrosophic cubic sets and their basic operation. Section 3 is devoted to presents some new theorems related to neutrosophic cubic soft sets. Section 4 presents conclusions and future scope of research.

Preliminaries

In this section, we recall some well-established definitions and properties which are related to the present study.

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Definition 1: Neutrosophic set [1]

Let U be the space of points with generic element in U denoted by u. A neutrosophic set λ in U is defined as λ={<u, t λ (u), i λ (u), f λ (u)>:u∈U} , where t λ (u):U →] 0, 1+ [,i λ (u):U →] 0, 1+ [, and f λ (u):U →] 0, 1+ [ and 0 ≤ t λ (u) + i λ (u) + f λ (u) ≤3+

Definition 2: Interval value neutrosophic set [31]

Let U be the space of points with generic element in U denoted by u. An interval neutrosophic set A in U is characterized by truthmembership function tA, the indeterminacy function iA and falsity membership function fA. For each u∈ U, t A (u), i A (u), fA (u) ⊆ [0, 1] and A is defined as A={<u, [ t A + (u), t A + (u)], [ i A (u), i A + (u)], [ fA (u), fA + (u)]:u∈U}.

Definition 3: Neutrosophic cubic set [32]

Let U be the space of points with generic element in U denoted by u ∈ U. A neutrosophic cubic set in U defined as N  ={< u, A (u), λ (u) >: u ∈ U} in which A (u) is the interval valued neutrosophic set and λ(u) is the neutrosophic set in U. A neutrosophic cubic set in U denoted by N = <A, λ>. We use ) U N(C  as a notation which implies that collection of all neutrosophic cubic sets in U.

Definition 4: Soft set [4]

Let U be the initial universe set and E be the set of parameters. Then soft set FK over U is defined by FK ={< u, F (e)>: e ∈ K, F (e) ∈ P (U)}

Where F: K → P (U), P (U) is the power set of U and K ⊂ E.

Definition

5: Neutrosophic cubic soft set [34]

A soft set FK is said to be neutrosophic cubic soft set iff F ~ is the mapping from K to the set of all neutrosophic cubic sets in U (i.e., ) U N(C ). i.e. F : K → ) U N(C  , where K is any subset of parameter set E and U is the initial universe set. Neutrosophic cubic soft set is defined by {(,(),()/,,} iii K uAuKuU eee F λ =<>∈∈  , Where, A(ei) is the interval valued neutrosophic soft set and λ(ei) is the neutrosophic soft set.

Definition: Internal neutrosophic cubic soft set (INCSS)

A neutrosophic cubic soft set F ~ K is said to be INCSS if for all ei ∈ K E ) u )( u )( u (T TT) A()()A( e eiei i + λ ≤ ≤ , ) u )( u )( u (F FF) A()()A( e eiei i + λ ≤ ≤ , ) u )( u )( u (F FF) A()()A( e eiei i + λ ≤ ≤ , for all .uU ∈

Definition: External neutrosophic cubic soft set (ENCSS)

A neutrosophic cubic soft set FK is said to be ENCSS if for all ei ∈ K E )) u ),( u )(( u (T TT) ( ()A ()Ae eiei i + λ ∉ ,

)) u ),( u )(( u (I II) ( ()A ()Ae eiei i + λ ∉ , ()()()()((),()) iii AA uuuFFFeee λ −+ ∉ , for all u∈U.

Some theorem related to these topics

Theorem 1

Let F ~ K be a neutrosophic cubic soft set in U which is not an ENCSS. Then there exists at least one ei ∈ K ⊆ E for which there exists some u∈U such that )) u ),( u )(( u (T TT) ( ()A ()Ae eiei i + λ ∈ , )) u ),( u )(( u (I II) ( ()A ()Ae eiei i + λ ∈ , )) u ),( u )(( u (F FF) ( ()A ()Ae eiei i + λ ∈ Proof

From the definition of ENCSSs, we have )) u ),( u )(( u (T TT) ( ()A ()Ae eiei i + λ ∉ , )) u ),( u )(( u (I II) ( ()A ()Ae eiei i + λ ∉ , ()()()()((),()) iii AA uuuFFFeee λ −+ ∉ , for all u∈U, corresponding to each ei ∈ K ⊆ E.

But given that FK is not an ENCSS, so at least one ei ∈ K ⊆ E. There exists some u∈U such that )) u ),( u )(( u (T TT) ( ()A ()Ae eiei i + λ ∈ , )) u ),( u )(( u (I II) ( ()A ()Ae eiei i + λ ∈ , )) u ),( u )(( u (F FF) ( ()A ()Ae eiei i + λ ∈ Hence the proof is complete.

Theorem 2

Let F ~ K ={< u, )e(F >: e )e(F K, )e(F ∈ ) U N(C } be a NCSS in U. If F ~ K is both an INCSS and ENCSS in U for all uU ∈ , corresponding to each ei ∈ K, then, ()L ~ () U ~ ) u (T TT) ( ()A ()Ae eiei i ∪ ∈ λ , ()L ~ () U ~ ) u (I II) ( ()A ()Ae eiei i ∪ ∈ λ , ()L ~ () U ~ ) u (F FF) ( ()A ()Ae eiei i ∪ ∈ λ ,where () U ~ T) ( Aei ={ }uU ): u T( ()Aei ∈ + , ()L ~ T) ( Aei ={ }uU ): u T( ()Aei ∈ , ()UI) ( Aei = { }uU ): u I( ()Aei ∈ + , ()LI) ( Aei ={ }uU ): u I( ()Aei ∈ , () U ~ F) ( Aei = { }uU ): u F( ()Aei ∈ + , ()L ~ F) ( Aei = { }uU ): u F( ()Aei ∈ .

Proof

Suppose F ~ K be both an INCSS and ENCSS corresponding to each ei ∈ K and for all u∈U. We have )) u ),( u )(( u (T TT) ( ()A ()Ae eiei i + λ ∈ , )) u ),( u )(( u (I II) ( ()A ()Ae eiei i + λ ∈ , )) u ),( u )(( u (F FF) ( ()A ()Ae eiei i + λ ∈

Again by definition of ENCSS corresponding to each eK i ∈ and for all u∈U, we have )) u ),( u )(( u (T TT) ( ()A ()Ae eiei i + λ ∉ , )) u ),( u )(( u (I II) ( ()A ()Ae eiei i + λ ∉ , )) u ),( u )(( u (F FF) ( ()A ()Ae eiei i + λ ∉ .

Since FK is both an ENCSS and INCSS, so only possibility is that, ()()()()()()) iii AA uuoruTTTeee λ −+ = , ()()()()()()) iii AA uuoruIIIeee λ −+ = , ()()() ()()() iii AA uuoruFFFeee λ −+ = for all u∈U.

Hence proved.

Definition

Let FK1 and G K 2 be two neutrosophic cubic soft sets in U and K1, K2 be any two subsets of K. Then, we define the following:

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1. FK1 = G ~ K 2 if K1 = K2 and ∀ = () G ~ ()F ~ eeiiK ei ∈ ()and()()uU()B Ae eeei ii1i2 ∈ ∀ = = ⇒ λ λ

2. If F ~ K1 and G K 2 are two NCSSs then we define P- order as ⊆ FP K1 G ~ K 2 iff the following conditions are satisfied: i. K1 ⊆ K2, and ii. () G ~ ()F ~ ee iPi ⊆ for all eKi1 ∈ iff ()and()()uU()B Ae eeei ii1i2 ∈ ∀ ⊆ ⊆ λ λ corresponding to each ei ∈ K1, where, ()()B Ae ei i ⊆ ⇒ ) u )( u T(T) A()B( e ei i ≤ ) u )( u T(T) A()B( e ei i + + ≤ , ) u )( u I(I) A()B( e ei i ≥ , ) u )( u I(I) A()B( e ei i + + ≥ , ) u )( u F(F) A()B( e ei i ≥ , ) u )( u F(F) A()B( e ei i + + ≥ for all u∈U, and ) u )( u T(T) e (i e)2(i1 λ λ ≤ , ) u )( u I(I) e (i e)2(i1 λ λ ≥ , ) u )( u F(F) e (i e)2(i1 λ λ ≥

3. If F ~ K1 and G K 2 are two neutrosophic cubic soft sets, then we define the R-order as ⊆ FR ~ K1G K 2 iff the following conditions are satisfied: i. K1 ⊆ K2 and ii. () G ~ ()F ~ ee iRi ⊆ for all ei ∈ K1 iff ()and()()uU()B Ae eee2 Ri1i ∈ ∀ ≥ λ λ ⊆ corresponding to each ei ∈ K1 Definition

Let G ~ K 2 and G ~ K 2 be two NCSSs in U and K1, K2 be any two subsets of parameter set K. Then we define P-union as ∪ FP ~ K1 G K 2 = H K 3 , where K3 ∈ K1 ∪ K2, ()F ~ () H ~ ee ii = , if ei ∈ K1 – K2 = () G ~ ei , if ei ∈ K2 – K1 = () G ~ ()F ~ ee iPi ∨ , if ei ∈ K1 ∩ K2 Here }(),():uU ,A (){uF ~ eee ii1i ∈ ∀ > < = λ , }(),():uU,B (){u G ~ eee ii2 ∈ ∀ > < = λ and }[,],[,],[,],uU , (){u AF eTTIIF) ( ()A ()A ()A ()A ()A Ae eieieieiei i ∈ ∀ > < = + + + , }[,],[,],[,],uU , (){u BF eTTIIF) ( ()B ()B ()B ()B ()B iBe eieieieiei i ∈ ∀ > < = + + + , }),,uU u ),( u ),( u ,( (){uK eTIFe1 1ii e)(1e)1(1e)1(11 ∈ ∀ ∈ ∀ > < = λ λ λ λ , e)},foralluUandeKK}e),(e)},rmax{((e),B(A,rmax{ e){u( e)G( F2 iPiii1i2i1 ∩ ∈ ∈ λ λ < = ∨ .

Definition

Let F ~ K1 and G ~ K 2 be two NCSSs in U and K1, K2 be any two subsets of parameter set K. The P-intersection of FK1 and G K 2 is denoted by ∩ FP K1G K 2 = N K 3 where 312KKK =∩ and N K 3 defined as N ~ K 3 = ()()12 P FGifiii eee KK∈∩ ∧ . Here, = ∧ () ()GF ~ eeP} uUand forall ()},rmin{(),()},(),BA,rmin{ {uK eeeeeK2 121 ∩ ∈ ∈ < λ λ

Definition: Compliment

Where, })],uU u),1( u1([ )], u),1( u1([ )], u),( u,[( ){u u A(F 1T1TIIF) AAAAA)A c eeiiei ∈ ∀ > < = + , }),uU u ),1( u ),1( u ,( ){u u (F1TI) ()()( c e e1ie1ie1i i ∈ ∀ > < = λ λ λ λ

Some properties of P-union and P-intersection

FG K K2 1 P∪ = F ~ GK K1 2 P∪ G ~ F ~ K K2 1 P∩ = F ~ G ~ K K1 2 P∩

Proof 1:

FG K K2 1 P∪ = H K 3 where K3 = K1 ∪ K2 ()(),12HFifiii eee KK=∈−  = (),21Gifii ee KK∈−  = e),ifeKK( G ~ e)F( ~ iPii12 ∩ ∈ ∨ Here, ,})},uU u),( u)},rmax{( u),B( uA(,rmax{ (){u ()G FK eeeK2 ()i1 ()2 iPi1e eeiei i ∩ ∈ ∀ ∈ ∀ > < = λ λ ∨ GFK K1 2 P∪ = RK 3 , R K 3 = () G ~ ei , if ei ∈ K2 – K1 = F() ~ ei , if ei ∈ K1 – K2 = ()(),12 P GFifiii eee KK∈∩ ∨ ,})},uU u),( u)},rmax{( u),A( uB(,rmax{ (){u()F GK eeeK2 iP2)1)1 e eeiei i ∩ ∈ ∀ ∈ ∀ > < = λ λ ∨ ,})},uU u ),( u )},rmax{( u ),B( u A(,rmax{ {uK eK2()1 ()2 1e eieiei i ∩ ∈ ∀ ∈ ∀ > < = λ λ = ()F()Ge ei iP ∨

Hence the proof.

Proof 2:

S ~ K 3 = G ~ F ~ K K2 1P ∩ = ,})},uU u ),( u )},rmin{( u ),B( u A(,rmin{ {uK eK2()1 ()2 1e eieiei i ∩ ∈ ∀ ∈ ∀ > < λ λ GFK K1 2 P∩ = Q K 3 = ,})},uU u ),( u )},rmin{( u ),A( u B(,rmin{ {uK eK2()1()1 2e e eiei ∩ ∈ ∀ ∈ ∀ > < λ λ = ,}))},uU u ,( u },rmin{( ) u ),B( u A(,rmin{ {uK eK2()21 1e eieiei i ∩ ∈ ∀ ∈ ∀ > < λ λ = FG ~ K K2 1P ∩

Hence the proof.

Definition:

R-union and R-intersection

Let F ~ K1 and G ~ K 2 be two NCSSs over U. Then R-union is denoted as G ~ F ~ K K2 1 R∪ = N K 3 , where K3 = K1 ∪ K2 and N K 3 Then R-union is defined as N K 3 = ()Nei = F() e , if ei ∈ K1 – K2 = ()Gei , if ei ∈ K2 – K1 = ()(),12 R FGifiii eee KK∈∩ ∨ Here () G ~ F() ~ ee iRi ∨ defined as 1()2()12 ()(){,max{(),()},min{(),()},,} ii ii R ii i FGurAuBuruuuU ee ee ee e KK λλ =< >∀∈∀∈∩ ∨

cccc ii i FFuAuuuU K ee ee e K λ ==< >∀∈∈

The compliment of FK1 denoted by c K1 F ~ is defined by ,}(),():uU ,A (){uF ~ ()K Feeeee1 Kiii1ii 1 ∈ ∈ ∀ > < = = λ , 1 1 ()(){,(),():,} ii

R-intersection is denoted as FG K K2 1R ∩ = JK 3 where K3 ∈ K1 ∩ K2.Then R-intersection is defined as:

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Proof

Theorem 3

Let U be the initial universe and I, J, L, S any four subsets of E, then for four corresponding neutrosophic cubic soft sets FGHT IJLS ,,, the following properties hold i.IfH ~ F ~ H,then ~ GFGand~ ~ JPIPLIPJL ⊆ ⊆ ⊆ Proof: ,IJ()I ()G FeFGee IPJi ⊆ ∈ ∀ ≤ ⇒ ⊆ ()H(),J G ~ JLandeGHee JPLi ∈ ∀ ≤ ⊆ ⇒ ⊆ eIeJ ii ∈ ⇒ ∈ , since I ⊆ J e)H( ~ e)( G ~ ii ≤ ∀ ei ∈ I ()H() ()G Fe eei ii ≤ ≤ ⇒ e)e)H(( Fi i ≤ , ∀ ei∈ I ⊆ J ⊆ L H ~ F ~ IPL ⊆ ⇒ Hence the proof. ii.c PI c J) F G)(FG( IPJthen ⊆ ⊆ ifI=J. Proof: If I = J ⇒ I ⊆ J and J ⊆ I FG IPJ ⊆ ⇒ IJ ⊆ and () ()G Fe ei ≤ (by definition) () ()G Fe ei ≤ ⇒ ,()and()()foralluUA()Be eee2 ii1i ∈ ⊆ ⊆ λ λ

()(),()(),()(),()(), ()(),()()

) u )( u T(T) ()( e e2i1i λ λ ≤ , ), u )( u I(I) ()( e e2i1i λ λ ≥ ) u )( u F(F) ()( e e2i1i λ λ ≥ c I F) ( = },I):uU u ),( u ,A( {uei cc e)(ie1 i ∈ ∈ ∀ > λ < where ()()()()() () (){,[(),()],[1(),1()],[1(),1()],}, 11 i iiii ii

c AAAA AA AuuuuuuuuuU e eeee ee TT IIFF −+ −+−+ =< −−−−>∀∈ }.),uU u ),1( u ),1( u ,( ){u u (F1TI) ()()( c e e1ie1ie)1i(i1 ∈ ∀ > < = λ λ λ λ And c J G)( = },I):uU u ),( u ,B( {uei cc e)(i e2 i ∈ ∈ ∀ > λ < , where ()()()()() () (){,[(),()],[1(),1()],[1(),1()],}, 11 i iiii ii

c BBBB BB BuuuuuuuuuU e eeee ee TT IIFF −+ −+−+ =< −−−−>∀∈ }.),uU u ),1( u ),1( u ,( ){u u (F1TI) ()( (2 2 c e ee2ie)i(i 2 ∈ ∀ > < = λ λ Now, ) u )( u )( u )( u (T TTT) ( ()B ()A ()B Ae eieiei i ≤ ≥ ⇒ Adding 1 both sides we get, ) u )( u (T 1T1) ( ()B Ae ei i ≥ , Similarly, ) u )( u (T 1T1) ( ()B Ae ei i + + ≥ , ) u )( u (I 1I1) ( ()B Ae ei i + + ≤ , ) u )( u (I 1I1) ( ()B Ae ei i + + ≤ , ) u )( u (F 1F1) ( ()B Ae ei i ≤ , ) u )( u (F 1F1) ( ()B Ae ei i + + ≤ (1) And, ) u )( u (T 1T1) ()( e e2i1i λ λ ≥ , ) u )( u (I 1I1) ()( e e2i1i λ λ ≤ , ) u )( u (F 1F1) ()( e e2i1i λ λ ≤ (2) & (2) ⇒ c PI c J) F ~ G)( ~ ( ⊆ since J ⊆ I. Hence the proof. Theorem 4

Let F ~ I be a NCSS over U, If F ~ I is an INCSS then F ~ c I is also an INCSS. If FI is an ENCSS then F ~ c I is also an ENCSS.

i.} ,I):uU u ),( u ,( {ueFA ~ Ii e)(ie1 i ∈ ∈ ∀ > < = λ isanINCSS, ) u )( u )( u (T TT) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (F FF) A()()A( e eie1i i + λ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ I Now, ) u )1( u )( u (T 1T1T) A()()A( e eie1i i + λ ≥ ≥ , ) u )1( u )1( u (I 1II) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )1( u 1(F FF) A()()A( e eie1i i + λ ≤ ≤ ∀ u ∈ U, ∀ ei ∈ I.

⇒ F ~ c I is an INCSS. ii. For F ~ K , an ENCSS, then )) u ),( u )(( u (T TT) ( ()A ()Ae eiei1i + λ ∉ , )) u ),( u )(( u (I II) ( ()A ()Ae eiei1i + λ ∉ , )) u ),( u )(( u (F FF) ( ()A ()Ae eiei1i + λ ∉ , ∀ u ∈ U, ∀ ei ∈ I, and )1 u )( u 0(TT) ( ()A Ae ei ≤ ≤ ≤ + , )1 u )( u 0(II) ( ()A Ae ei i ≤ ≤ ≤ + , )1 u )( u 0(FF) ( ()A Ae ei i ≤ ≤ ≤ + . ⇒ 11 ()()()() ()()()() iiii AA uuoruu eeee TTTT λλ≤≥−+ , ⇒ 11 ()()()() ()()()() iiii AA uuoruu eeee IIII λλ≤≥−+ , ⇒ 11 ()()()() ()()()() iiii AA uuoruu eeee FFFF λλ≤≥−+ ∀ u ∈ U, ∀ ei ∈ I ⇒ 11 ()()() ()1()() 11()1() iiii AA u uor u u eeee TTTT λλ −+ ≥− ≤− , ⇒ 11 ()()() ()1()() 11()1() iiii AA u uor uu eeee IIII λλ −+ ≥− ≤− ⇒ 11 ()()() ()1()() 11()1() iiii AA u uor u u eeee FFFF λλ −+ ≥− ≤−

Thus, )) u 1(),( u 1())( u (T1TT () ( ()A ()Ae eei1i + ∉ λ , )) u 1(),( u 1())( u (I1II () ( ()A ()Ae eiei1i + ∉ λ and )) u 1(),( u 1())( u (F1FF () ( ()A ()Ae eiei1i + ∉ λ ∀ u ∈ U, ∀ ei ∈ I ∀ u ∈ U, ∀ ei ∈ I. Hence F ~ c I is an ENCSS.

Theorem 5

Let F ~ I and G ~ J be any two INCSSs then G ~ FJ IP ∪ is an INCSS. G ~ F ~ IPJ ∩ is an INCSS. Proof

Since F ~ I and G ~ J are INCSSs, so for F ~ I we have ) u )( u )( u (T TT) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ ) u )( u )( u (F FF) A()()A( e eie1i i + λ ≤ ≤ , for all u ∈ U, ei ∈ I.

Also for G J we have ) u )( u )( u (T TT) ( ()()B Be eie2i i + λ ≤ ≤ , ) u )( u )( u (I II) B()()B( e eie2i i + λ ≤ ≤ , ) u )( u )( u (F FF) B()()B( e eie2i i + λ ≤ ≤ , ∀ u ∈ U and ∀ ei ∈ J. Now, ) uT({ max) ( Aei , ()12()()()() ()}max{(),()}max{(),()} ii ii B AB uuuuu eeeee TTTTT λλ ++ ≤≤ , ()()21()()()() max{(),()}max{(),()}max{(),()} ii ii ii AB AB uuuuuu eeeeee IIIIII λλ ++ ≤≤ , ()()()()()() max{(),()}max{(),()}max{(),()} AB AB uuuuuu eeeeee FFFFFF λλ ≤≤ ∀ u ∈ U, ∀ ei ∈ I ∪ J

Now by the definition of P-union G ~ FJ IP ∪ is an INCSS. H ~ G ~ F ~ IPJK = ∩ , where K ∈ I ∩ J. ii. Now, () ()G Fe Hei KP ∧ = and by definition, () G ~ () Fe eP i ∧ },K}:uU ) u ),( u )},rmin{( u ),( u ,rmin{( {ue ABi () ()2 ()()1e eeiei ∈ ∈ > < = λ λ , Since G ~ J and G ~ J are INCSSs then we have for FI , ) u )( u )( u (T TT) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ ) u )( u )( u (F FF) A()()A( e eie1i i + λ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ I And for G ~ J , ) u )( u )( u (T TT) ( ()()B Be eie2i i + λ ≤ ≤ , ) u )( u )( u (I II) B()()B( e eie2i i + λ ≤ ≤ , ) u )( u )( u (F FF) B()()B( e eie2i i + λ ≤ ≤ , ∀ u ∪ U, ∀ ei ∈ J ⇒ ()()1 ()2()()() min{(),()}min{(),()}min{(),()} ii ii ii AB AB uuuuuu eeeeee TTTTTT λλ ++ ≤≤ , ()2 ()1()()()() min{(),()}min{(),()}min{(),()} ii ii B AB A uuuuuu eeeee e I IIIII λλ ++ ≤≤ ()12 ()()()()() min{(),()}min{(),()}min{(),()} i ii ii B AB A uuuuuu eeeee e F FFFFF λλ ++ ≤≤ ∀ ei∈K. Hence H K is an INCSS.

Theorem 6 Let F ~ I and G ~ J be any two INCSSs over U having the conditions: () max{() i A u e T , ()12()() ()}min{(),()} ii B uuuTTTeee λλ ≤ ()()21()() max{(),()}min{(),()} ii ii AB uuuu eeee IIIIλλ ≤ and ()()12()() max{(),()}min{(),()} ii ii AB uuuu eeee FFFFλλ ≤ ∀ ei ∈ I ∩ J, ∀ u ∈ U. Then R-union of F ~ I and G J is also INCSS. Proof

Since FI and G ~ J are INCSSs in U. So for FI , we have ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (F FF) A()()A( e eie1i i + λ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ I. Also for G J we have ) u )( u )( u (T TT) B()()B( e eie2i i + λ ≤ ≤ , ) u )( u )( u (I II) B()()B( e eie2i i + λ ≤ ≤ , ) u )( u )( u (F FF) B()()B( e eie2i i + λ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ J

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= ()Ge , if ei ∈ I-J. = ()(), R iii FGifIJ eee ∈∩ ∨  Her () F()G ~ ee R ∨ defined as 1()2() ()(){,max{(),()},max{(),()},,} ii ii R ii i FGurAuBuruuuUIJ ee ee ee e λλ =< ∀∈∀∈∩ ∨ ,

Since FI and G ~ J are INCSSs so from the given condition and definition of INCSS we can write, 1()2() max{(),()}min{(),()}max{(),()} ii ii ii uuuuuu ee ee ee AB AB λλ ++ ≤≤ , ∀ ei ∈ I ∩ J, ∀ u∈U. If ei ∈ I - J or ei ∈ J - I then, the result is trivial. Thus G ~ F ~ IRJ ∪ is an INCSS.

Theorem 7

Let F ~ I and G ~ J be any two INCSSs over U satisfying the condition: 1()2() min{(),()}max{(),()} ii ii ruuruu ee ee AB λλ ++ ≥ , ∀ ei ∈ I ∩ J, ∀ u ∈ U.

Then G ~ F ~ IRJ ∩ is an INCSS.

Proof

Since F ~ I and G ~ J are INCSSs in U, we have, ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ and ) u )( u )( u (F FF) A()()A( e eie1i i + λ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ I.

Also for G ~ J we have, ) u )( u )( u (T TT) B()()B( e eie2i i + λ ≤ ≤ , ) u )( u )( u (I II) B()()B( e eie2i i + λ ≤ ≤ and ) u )( u )( u (F FF) B()()B( e eie2i i + λ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ J. ⇒ 1()2() min{(),()}max{(),()} ii ii uuuu ee ee AB λλ ≤ , also R K I J F G H = ∩  , where K = I ∩ J and G ~ FJ IR ∩ () F()G ~ ee iRi ∧ = , if ei ∈ I ∩ J, then () G ~ F()e ei iR ∧ defined as ()F()Ge ei iR ∧ = ,})},uU u ),( u )},rmax{( u ),B( u A(,rmin{ {uK e ()i ()2 1e eieiei i ∈ ∀ ∈ ∀ < λ λ .

Given condition that 1()2() min{(),()}max{(),()} ii ii ruuruu ee ee AB λλ ++ ≥ , ∀ ei ∈ I ∩ J, ∀ u ∈ U. Thus from given condition and definition of INCSSs

1()2() min{(),()}max{(),()}min{(),()} ii ii ii ruuruuruu ee ee ee AB AB λλ ++ ≤≤ . Hence G ~ F ~ IRJ ∩ is an INCSS. Theorem 8

Let FI and G J be any two ENCSSs then FG IPJ ∪ is an ENCSS.

and H ~ K = F() ei , if ei ∈ I-J.

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) u )( u )( u (F FF) B()()B( e eie2i i + λ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ J.

by definition of P-union, FGH,whereKIJ IPJK

K) ( Fei

, if ∀ ei

I - J.

() G ~ e , if ∀ ei ∈ J - I.

() G ~ ()F ~ ee iPi

, if ∀ ei ∈ I

J. Here () F()G ~ ee iP

defined as

),( u)},rmax{( u),B( uA(,rmax{ (){u ()G Fe eei iP1)2) eeee i

uUandIJ forall )},

Thus G ~ F ~ IPJ ∪ is a ENCSSs if ∀ ei∈ I ∩ J. For ei ∈ I - J or ei ∈ J - I these results are trivial. Hence the proof. ii. Since F ~ I and G ~ J are ENCSSs, then 1 ()()()()((),()) iii AA uuuTTTeee λ −+ ∉ , 1 ()()()()((),()) iii AA uuuIIIeee λ −+ ∉ , 1 ()()()()((),()) iii AA uuuFFFeee λ −+ ∉ , ∀ u ∈ U, ∀ ei ∈ J and )1 u)( u 0(TT) ( ()A Ae ei ≤ ≤ ≤ + , )1 u)( u 0(II( (A Ae ei i ≤ ≤ ≤ + and )1 u )( u 0(FF) ( ()A Ae ei i ≤ ≤ ≤ + . ⇒ 22 ()()()() ()()()() iiii BB uuoruu eeee TTTT λλ≤≥−+ , ⇒ 2()()2()() ()()()() iiii BB uuoruu eeee IIII ≤≥−+ , ⇒ 22 ()()()() ()()()() iiii BB uuoruu eeee FFFF λλ≤≥−+ , ∀ u ∈ U, ∀ ei ∈ I For G ~ J , we have )), u ),( u )(( u (T TT) ( ()B ()Be eiei2i + λ ∉ )) u ),( u )(( u (I II) ( ()B ()B 2e eiei i + λ ∉ , )), u ),( u )(( u (F FF) ( ()B ()Be eiei2i + λ ∉ ∀ u ∈ U, ∀ ei ∈ I and )1 u )( u 0(TT) ( ()B Be ei i ≤ ≤ ≤ + , )1 u )( u 0(II) ( ()B Be ei i ≤ ≤ ≤ + , )1 u )( u 0(FF) ()B Be ei i ≤ ≤ ≤ + ⇒ 22 ()()()() ()()()() BB uuoruu eeee TTTT λλ≤≥−+ , ⇒ 2()()2()() ()()()() iiii BB uuoruu eeee IIII ≤≥−+ , ⇒ 22 ()()()() ()()()() iiii BB uuoruu eeee FFFF λλ≤≥−+ , ∀ u ∈ U, ∀ ei ∈ I ⇒ 1()2() min{(),()}{min{(),()},min{(),()} ii ii ii uuuuuu eeee ee ABAB λλ −−++ ∉

Since, FGH,whereKIJ IRJK ∪ = = ∪ HK = F() ~ ei , if ∀ ei ∈ I - J. = ()Gei , if ∀ ei ∈J - I. = ()(),. R iii FGifIJ eee∀∈∩ ∨

Here () G ~ F()e ei iR ∨ defined as }. ,IJ)},uU u),( u)},rmin{( u),B( uA(,rmax{ (){u G ~ ()F ~ eee ()2)iiRi1e eeei i ∩ ∈ ∀ ∈ ∀ > < = λ λ ∨

Given condition is 1()2() min{(),()}max{(),(),, ii ii uu uuIJanduU ee ee ee AB λλ ≤ ∀∈∀∈∀∈ ⇒ 1()2() min{(),()}(max{(),()},max{(),()}) ii ii ii uuuuuu eeee ee ABAB λλ −−++ ∉ Hence G ~ F ~ IRJ ∪ is an ENCSS.

Definition

Let = FI} ,I):uU u ),( u ,( (){u Fe eAi ()i()1e ei i ∈ ∀ ∈ ∀ > < = λ and = G J} ,J):uU u ),( u ,( (){u Ge eBi ()i()2e ei i ∈ ∀ ∈ ∀ > < = λ are two NCSSs in U. We defined new NCSSs by interchanging the neutrosophic part of the two NCSSs. We denoted its by FI , G J and defined by } ,I(),():uU ,A {ue Feei Ii2i ∈ ∈ ∀ > < = λ  , },J(),():uU,B {ue Geei Ji1i ∈ ∈ ∀ > < = λ respectively.

Theorem 10

Theorem 9 Let F ~ I and G ~ J be any two INCSSs in U such that 1()2() min{(),()}max{(),(),,, ii ii R ii I J uu uuIJanduUthenisanENCSS ee ee ee AB FG λλ ≤ ∀∈∀∈∀∈ ∪ Proof: Since F ~ I and G J are INCSSs in U. we have, ) u )( u )( u (T TT) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (F FF) A()()A( e eie1i i + λ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ I and ) u )( u )( u (T TT) ( ()()B Be eie2i i + λ ≤ ≤ , ) u )( u )( u (I II) B()()B( e eie2i i + λ ≤ ≤ ,

 and G J  are INCSSs in U. Then FG ~ IPJ ∪ is an INCSS in U. Proof

F ~ I and G J are ENCSSs and FI

 },I):uU u ),( u ,( {ue Ai ()()2e ei i ∈ ∀ ∈ ∀ > < λ and },J):uU u ),( u ,( {ue GBi ()J()1e ei i ∈ ∀ ∈ ∀ > < = λ are INCSSs, then ) u )( u )( u (T TT) A()()A( e eie2i i + λ ≤ ≤ , ) u )( u )( u (I II) A()()A( e eie2i i + λ ≤ ≤ , ) u )( u )( u (F FF) A()()A( e eie2i i + λ ≤ ≤ , ∀ u ∈ U, ei ∈ I. and ) u )( u )( u (T TT) B()()B( e eie1i i + λ ≤ ≤ , ) u )( u )( u (I II) B()()B( e eie1i i + λ ≤ ≤ , ) u )( u )( u (F FF) B()()B( e eie1i i + λ ≤ ≤ , ∀ u ∈ U, ∀

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definition of ENCSSs and INCSSs all the possibility are as under: i(a). 12 ()()()() ()()()(), iiii AA uuuu eeee TTTT λλ ≤≤≤−+ ) u )( u )( u )( u (I III) ()A()()A( e eie2iei1i + λ λ ≤ ≤ ≤ , ) u )( u )( u )( u (F FFF) ()A()()A( e eie2iei1i + λ λ ≤ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ I. i(b). ) u )( u )( u )( u (T TTT) ()B()()B( e eie1iei2i + λ λ ≤ ≤ ≤ , ) u )( u )( u )( u (I III) ()B()()B( e eie1iei2i + λ λ ≤ ≤ ≤ ) u )( u )( u )( u (F FFF) ()B()()B( e eie1iei2i + λ λ ≤ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ I ii(a). ) u )( u )( u )( u (T TTT) A()()A()( e e1ieie2i i λ λ ≤ ≤ ≤ + ) u )( u )( u )( u (I III) A()()A()( e e1ieie2i i λ λ ≤ ≤ ≤ + ,uU),I u )( u )( u )( u (e FFFFi A()()A()() e e1ieie2i i ∈ ∀ ∈ ∀ ≤ ≤ ≤ λ λ + ii(b). ) u )( u )( u )( u (T TTT) ()( B()()Be e2ieie1i i λ + λ ≤ ≤ ≤ ) u )( u )( u )( u (I III) ()( B()()Be e2ieie1i i λ + λ ≤ ≤ ≤ , ) u )( u )( u )( u (F FFF) ()( B()()Be e2ieie1i i λ + λ ≤ ≤ ≤ , ∀ ei ∈ I, ∀ u ∈ U iii(a). ), u )( u )( u )( u (T TTT) ()A()()A( e eie2iei1i + λ λ ≤ ≤ ≤ ) u )( u )( u )( u (I III) ()A()()A( e eie2iei1i + λ λ ≤ ≤ ≤ , ) u )( u )( u )( u (F FFF) ()A()()A( e eie2iei1i + λ λ ≤ ≤ ≤ , ∀ ei ∈ I and ∀ u ∈ U iii(b). ) u )( u )( u )( u (T TTT) ()( B()()Be e2ieie1i i λ + λ ≤ ≤ ≤ ) u )( u )( u )( u (I III) ()( B()()Be e2ieie1i i λ + λ ≤ ≤ ≤ ) u )( u )( u )( u (F FFF) ()( B()()Be e2ieie1i i λ + λ ≤ ≤ ≤ , ∀ ei ∈ I ∀ u ∈ U, iv(a). ) u )( u )( u )( u (T TTT) A()()A()( e e1ieie2i i λ λ ≤ ≤ ≤ + , ) u )( u )( u )( u (I III) A()()A()( e e1ieie2i i λ λ ≤ ≤ ≤ + , ,uU),I u )( u )( u )( u (e FFFFi A()()A()() e e1ieie2i i ∈ ∀ ∈ ∀ ≤ ≤ ≤ λ λ + iv(b). ) u )( u )( u )( u (T TTT) ()B()()B( e eie1iei2i + λ λ ≤ ≤ ≤ , ) u )( u )( u )( u (I III) ()B()()B( e eie1iei2i + λ λ ≤ ≤ ≤ , ) u )( u )( u )( u (F FFF) ()B()()B( e eie1iei2i + λ λ ≤ ≤ ≤ , ∀ u ∈ U, ∀ ei ∈ I Since p-union of .GH,whereCIJ ~ FC IPJ ∪ = = ∪ FG IPJ ∪ = F(), ~ ei if ei ∈ I - J = (), G ~ e if ei ∈ J – I. = ()(),. P iii FGifIJ eee ∈∩ ∨ Here ()F()Ge ei P ∨ defined as } )},uUandIJ u),( u)},rmax{( u),B( uA(,rmax{ (){u ()G Fe ee( ()2 Pi1e eeiei ∩ ∈ ∀ ∈ ∀ < = λ λ ∨ Case 1 If H ~ C = F() ei , if ei ∈ I – J, then from i(a)., and ii(a). We have

) u )( u T(T) ()A( e ei1iλ = ) u )( u I(I) ()A( e ei1iλ = , ) u )( u F(F) ()A( e ei1iλ = and ) u )( u T(T) ()A( e ei1i + λ = , ) u )( u I(I) ()A( e ei1i + λ = , ) u )( u F(F) ()A( e ei1i + λ = , ∀ ei ∈ I and ∀ u ∈ U. Thus ) u )( u )( u (T TT) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (F FF) A()()A( e eie1i i + λ ≤ ≤ ∀ ei ∈ I ∀ u ∈ U.

Case 2

H C = eJI i ∈ , if eJI i ∈ then from i(b) and ii(b). , we have ) u )( u T(T) ()B( e ei2i λ = , ) u )( u I(I) ()B( e ei2i λ = , ) u )( u F(F) ()B( e ei2i λ = and ) u )( u T(T) ()B( e ei2i + λ = , ) u )( u I(I) ()B( e ei2i + λ = , ) u )( u F(F) ()B( e ei2i + λ = , ∀ ei ∈ J ∀ u ∈ U. Thus ) u )( u )( u (T TT) B()()B( e eie2i i + λ ≤ ≤ , ) u )( u )( u (I II) B()()B( e eie2i i + λ ≤ ≤ , ) u )( u )( u (F FF) B()()B( e eie2i i + λ ≤ ≤ , ∀ ei ∈ J - I ∀ u ∈ U. Case 3

H C = ()() P iii FGifIJ eee ∈∩ ∨ , then from i(a) and i(b)., we have ) u )( u )( u (T TT) ()A( A()1e eiei i + λ ≤ ≤ , ) u )( u )( u (I II) A()()A( e eie1i i + λ ≤ ≤ , ) u )( u )( u (F FF) ()A( A()1e eiei i + λ ≤ ≤ ∀ ei ∈ J, ∀ u ∈ U and ) u )( u )( u (T TT) ()B( ()2 Be eiei i + λ ≤ ≤ , ) u )( u )( u (I II) ()B( ()2 Be eiei i + λ ≤ ≤ , ) u )( u )( u (F FF) ()B( ()2 Be eiei i + λ ≤ ≤ ∀ ei ∈ J, ∀ u ∈ U Hence, if ei ∈ I ∩ J, then } ) u ),( u {( ))max u )( u }(( ) u ),( u {( maxB ABAe eieieieiei i) ( ()21 + + ≤ ∨ ≤ λ λ in all the three cases.

G ~ FJ IP ∪ is an INCSS in U.

Conclusion

In this paper we have defined some operations such as P-union, P-intersection, R-union, R-intersection for neutrosophic cubic soft sets. We have also defined some operation of INCSSs and ENCSSs. We have proved some theorems on INCSSs and ENCSSs. We have discussed various approaches INCSSs and ENCSSs. We hope that proposed theorems and operations will be helpful to multi attribute group decision making problems in neutrosophic cubic soft set environment.

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