More On P-Union and P-Intersection of Neutrosophic Soft Cubic Set by Ioan Degău

Neutrosophic Sets and Systems, Vol. 17, 2017

University of New Mexico

More On P-Union and P-Intersection of Neutrosophic Soft Cubic Set R. Anitha Cruz and F. Nirmala Irudayam R. Anitha Cruz, Department of Mathematics Nirmala College for Women, Coimbatore, 641018, India, anithacruz@gmail.com F.Nirmala Irudayam, Assistant Proffesor, Departmentof Mathematics, Nirmala College for Women, Coimbatore, 641018, India, nirmalairudayam@ymail.com

1 2

Abstract:The P-union ,P-intersection, P-OR and P-AND of neutrosophic soft cubic sets are introduced and their related properties are investigated. We show that the Punion and the P-intersection of two internal neutrosophic soft cubic sets are also internal neutrosophic soft cubic sets. The conditions for the P-union ( P-intersection ) of two T-external (resp. I- external, F- external) neutrosophic soft cubic sets to be T-external (resp. I- external, Fexternal) neutrosophic soft cubic sets is also dealt with.

We provide conditions for the P-union ( P-intersection ) of two T-external (resp. I- external, F- external) neutrosophic soft cubic sets to be T-internal (resp. I- internal,F- internal) neutrosophic soft cubic sets. Further the conditions for the P-union (resp. P-intersection ) of two neutrosophic soft cubic sets to be both T-external (resp. I- external, Fexternal) neutrosophic soft cubic sets and T-external (resp. I- external, F- external) neutrosophic soft cubic sets are also framed.

Keywords: Cubic set, Neutrosophic cubic set, Neutrosophic soft cubic set, T-internal (resp. I- internal,F- internal) neutrosophic soft cubic sets , T-external (resp. I- external, F- external) neutrosophic soft cubic set.

1 Introduction Florentine Smarandache[10,11] coined neutrosophic sets and neutrosophic logic which extends the concept of the classical sets, fuzzy sets and its extensions. In neutrosophic set, indeterminacy is quantified explicity and truthmembership, indeterminacy-membership and falsity – membership are independent. This assumption is very important in many applications such as information fusion in which we try to combine the data from different sensors. Pabita Kumar Majii[18] had combined the Neutrosophic set with soft sets and introduced a new mathematical model ‘ Nuetrosophic soft set’. Y. B. Jun et al[2]., introduced a new notion, called a cubic set by using a fuzzy set and an interval-valued fuzzy set, and investigated several properties. Jun et al. [19] extended the concept of cubic sets to the neutrosophic cubic sets. [1] introduced neutrosophic soft cubic set and the notion of truth-internal ( indeterminacy-internal, falsity-internal) neutrosophic soft cubic sets and truth-external ( indeterminacy-internal, falsity-internal) neutrosophic soft cubic sets As a continuation of the paper [1]We show that the Punion and the P-intersection of T-internal (resp. Iinternal,F-internal) neutrosophic soft cubic sets are also Tinternal (resp. I-internal,F-internal) neutrosophic soft cubic sets. We also provide conditions for the P-union ( Pintersection ) of two T-external (resp. I- external,Fexternal) neutrosophic soft cubic sets to be T-external (resp. I- external,F- external) neutrosophic soft cubic sets. We provide conditions for the P-union ( P-intersection ) of two T-external (resp. I- external,F- external)

neutrosophic soft cubic sets to be T-internal (resp. Iinternal,F- internal) neutrosophic soft cubic sets. We provide conditions for the P-union (resp. Pintersection ) of two NSCS to be both T-external (resp. Iexternal,F- external) neutrosophic soft cubic sets and Texternal (resp. I- external,F- external) neutrosophic soft cubic sets. 2 Preliminaries 2.1 Definition: [5] Let E be a universe. Then a fuzzy set μ over E is defined by X = { μx (x) / x: x є E }where μx is called membership function of X and defined by μx : E → [0,1]. For each x E, the value μx(x) represents the degree of x belonging to the fuzzy set X. 2.2 Definition: [2] Let X be a non-empty set. By a cubic set, we mean a structure in which A is an interval valued fuzzy set (IVF) and μ is a fuzzy set. It is denoted by . 2.3 Deﬁnition: [9]Let U be an initial universe set and E be a set of parameters. Consider A ⊂ E. Let P( U ) denotes the set of all neutrosophic sets of U. The collection ( F, A ) is termed to be the soft neutrosophic set over U, where F is a mapping given by F : A → P(U). 2.4 Definition : [4] Let X be an universe. Then a neutrosophic (NS) set λ is an object having the form λ = {< x : T(x),I(x),F(x) >: x ∈ X} where the functions T, I, F : X → ]–0, 1+[ defines respectively the degree of Truth, the degree of

R. Anitha Cruz and F. Nirmala Irudayam. More On P-Union and P-Intersection of Neutrosophic Soft Cubic Set

Neutrosophic Sets and Systems, Vol. 17, 2017

indeterminacy, and the degree of Falsehood of the element x ∈ X to the set λ with the condition. − 0 ≤ T(x) + I(x) + F(x) ≤ 3+ 2.5 Definition : [7] Let X be a non-empty set. An interval neutrosophic set (INS) A in X is characterized by the truth-membership function AT, the indeterminacymembership function AI and the falsity-membership function AF. For each point x ∈ X, AT (x),AI (x),AF (x) ⊆ [0,1]. For two INS A = {<x, [AT-(x), AT+(x)], [AI-(x), AI+(x) ], [AF-(x), AF+(x)]>: x ∈ X} and B = {<x, [BT-(x), BT+(x)], [BI-(x), BI+(x) ], [BF-(x), BF+(x)]>: x ∈ X} Then, ~ B if and only if 1. A 

AT ( x)  BT ( x) , AT ( x)  BT ( x) AI ( x)  BI ( x) , AI ( x)  BI ( x) AF ( x)  BF ( x) , AF ( x)  BF ( x)

where P is a mapping

2.7 Definition: [1] Let X be an initial universe set. A neutrosophic soft cubic set (P, A) in X is said to be • truth-internal (briefly, T-internal) if the following inequality is valid T T T (  x  X , ei  E ) ( Ae  x   e  x   Ae  x  ), (2.1) i

• indeterminacy-internal (briefly, I-internal) if the following inequality is valid I I I (  x  X , ei  E ) ( Ae ( x)  e ( x)  Ae ( x) ) , (2.2) i

• falsity-internal (briefly, F-internal) if the following inequality is valid F F F       (  x  X , ei  E ) ( Ae x  e x  Ae x ). (2.3) i

If a neutrosophic soft cubic set in X satisfies (2.1), (2.2) and (2.3) we say that ( P, A) is an internal

neutrosophic soft cubic set in 2.8 Definition: [1]

for all x∈ X.

A  B if and only if

Let

AT ( x)  BT ( x) , AT ( x)  BT ( x)

AF ( x)  BF ( x) , AF ( x)  BF ( x) for all x ∈ X. ~

AC  { x, [ A-F (x), AF (x)],[A-I (x), AI (x)],[A-T (x), AT (x)] : x  X }

be an initial universe set. A neutrosophic soft

cubic set ( P, A) in

AI ( x)  BI ( x) , AI ( x)  BI ( x)

neutrosophic soft cubic set over given by P : A  NC(U) .

is said to be

• truth-external (briefly, T -external) if the following inequality is valid T T T (  x  X , ei  E ) (e  x   ( Ae  x , Ae  x  )), (2.4) i

• indeterminacy-external (briefly, I -external) if the 4.

following inequality is (  x  X , ei  E ) (eI  x   ( Ae I  x , Ae I  x  )),

~ B { x, [min {A - (x), B (x)} , min {A  (x), B (x)}], A  T T T T [max{A-I (x), BI (x)}, max{AI (x), BI (x)}],

• falsity-external (briefly, F -external) if the following inequality is valid (  x  X , ei  E ) (eF  x   ( Ae F  x , Ae F  x )). (2.6)

[max{A-F (x), BF (x)}, max{AF (x), BF (x)}] : x  X }

5. ~ A  B { x, [max{A -T (x), BT (x)} , max {A T (x), BT (x)}] , [min{A -I (x), BI (x)}, min{A I (x), BI (x)}], [min{A -F (x), BF (x)}, min{A F (x), BF (x)}] : x  X }

2.6 Definition: [1] Let U be an initial universe set. Let NC(U) denote the set of all neutrosophic cubic sets and E be the set of A E parameters. Let then ( P, A) { P(ei )  { x, Aei ( x), ei ( x) : x U } ei  A  E }

where

Aei ( x) { x, AeT x  , AeI x , AeF x   / x U } is an i

interval neutrosophic set T I T ei ( x)  { x, (e x , e x , e x   / x U } i

valid (2.5)

, a

neutrosophic set. The pair ( P, A) is termed to be the

If a neutrosophic soft cubic set ( P, A) ) in X satisfies (2.4), (2.5) and (2.6), we say that ( P, A) is an external neutrosophic soft cubic set in X. 2.9 Definition [1] Let

(P, I) = { P(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } i

and

(Q, J) = { Q(ei ) = Bi = {< x, Be (x), e (x) > : x  X} ei  J } i

be two neutrosophic soft cubic sets in X. Let I and J be any two subsets of E (set of parameters), then we have the following 1. (P, I) = (Q, J) if and only if the following conditions are satisfied a) I = J and

R. Anitha Cruz and F. Nirmala Irudayam. More On P-Union and P-Intersection of Neutrosophic Soft Cubic Set

Neutrosophic Sets and Systems, Vol. 17, 2017

P(ei ) = Q(ei ) for all

and only if

Aei ( x)  Bei ( x)

and ei ( x)  ei ( x)

ei  I

F (ei )  P G (ei ) = F

for

all

{< x, max{ A F (x), B F (x)}, ( F   F )(x) > : x  X} e i  I  J .

x  X corresponding each ei  I .

2.11 Definition: [1] Let ( F , I ) and (G , J ) be two neutrosophic soft cubic sets (NSCS) in X where I and J are any subsets of parameter’s set E. Then we define P-intersection of neutrosophic soft cubic set as ( F , I )  p (G, J )  ( H , C) where C  I  J ,

(P, I) and (Q, J) are two neutrosophic soft cubic set then we define and denote Porder as (P, I) P (Q, J) if and only if the following conditions are satisfied c) I  J and d) P(ei )  P Q(ei ) for all ei  I

H (ei ) = F (ei )  P G(ei ) = F (ei )  P G(ei ) and H ( ei )

ei  I  J . Here

F (ei )  P G(ei ) is defined as

if and only if Aei ( x)  Bei ( x) and ei ( x)  ei ( x) for all

F (ei )  P G (ei ) =

x X

corresponding to each

{< x, min{ A e (x), B e (x) }, ( e   e )(x) > : x  X} e i  I  J

(P, I) and (Q, J) are two neutrosophic

. where A e (x), B e (x) represent interval neutrosophic sets.

ei  I . 3.

{< x, max{ A I (x), BI (x)}, ( I   I )(x) > : x  X} ei  I  J ,

soft cubic set then we define and denote P- order as (P, I) R (Q, J) if and only if the following conditions are satisfied e) I  J and

P(ei ) R Q(ei ) for all if and only

ei  I if

Aei ( x)  Bei ( x)

and ei ( x)  ei ( x)

for

all

x  X corresponding to each ei  I . 2.10 Definition: [1] Let (F , I ) and (G, J ) be two neutrosophic soft cubic sets (NSCS) in X where I and J are any two subsets of the parameteric set E. Then we define P-union of neutrosophic soft cubic set as ( F , I )  p (G, J )  ( H , C ) where C  I  J if ei  I  J   if ei  J  I  if ei  I  J 

 F (ei )  G (ei )  F (e )  G (e ) i  i P

H ( ei )

where F (ei )  P G (ei ) is defined as {< x, max{ A e (x), B e (x) }, ( e   e )(x) > : x  X} e i  I  J i

where A e (x), B e (x) represent interval neutrosophic sets. i

Hence F T (ei ) P GT (ei ) 

{< x, min{ A T (x), BT (x) }, ( T   T ) (x) > : x  X} ei  I  J , ei

F (ei )  P G (ei )  I

{< x, min{ A I (x), BI (x) }, ( I   I )(x) > : x  X} ei  I  J , ei

F F (ei )  P G F (ei )  {< x, min{ A F (x), B F (x) }, ( F   F ) (x) > : x  X} e i  I  J ei

3 More On P-union And P-intersection Of Neutrosophic Soft Cubic Set Defintion: 3.1 Let (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i i (G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i

neutrososphic soft cubic set (NSCS) in X. Then [1] P-OR is denoted by ( F , I )  p (G, J ) and deH (i , i )  F (i ) P G( i ) for all (i , i )  I  J . [2] P-AND is denoted by ( F , I )  p (G, J ) and de-

fined as ( F , I )  p (G, J )  ( H , I  J ) where

Hence F T (ei )  P GT (ei ) =

{< x, max{ A T (x), BT (x)}, ( T   T ) (x) > : x  X} ei  I  J , ei

fined as ( F , I )  p (G, J )  ( H , I  J ) where

F (ei )  P G(ei )  i

H (ei )

H (i , i )  F (i ) P G( i ) for all (i , i )  I  J .

F I (ei )  P G I (ei ) 

R. Anitha Cruz and F. Nirmala Irudayam. More On P-Union and P-Intersection of Neutrosophic Soft Cubic Set

Neutrosophic Sets and Systems, Vol. 17, 2017

Example: 3.2 Let X = {x1, x2,x3} be initial universe and E = {e1, e2} parameter’s set. Let (F, I) be a neutrosophic soft cubic set over X and defined as (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i i X F(e1) F(e2) <Ae1(x), λ e1(x) > <Ae2(x), λ e2(x) > x [0.5,0.6][0.6,0. [0.4,0. [0.3,0.6][0.2,0. [0.3,0. 7][0.5,0.6] 5,0.6] 7][0.2,0.4] 4,0.4] 1 x [0.4,0.5][0.7,0. [0.5,0. [0.3,0.5][0.6,0. [0.4,0. 8][0.2,0.3] 6,0.6] 8][0.2,0.6] 7,0.5] 2 x [0.2,0.3][0.2,0. [0.3,0. [0.4,0.7][0.2,0. [0.5,0. 3][0.3,0.5] 4,0.6] 5][0.3,0.6] 6,0.6] 3 (G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } i

G(e1) <Be1(x), μ e1(x) > x [0.7,0.9][0.3,0 [0.7,0. .5][0.3,0.4] 4,0.6] 1 x [0.5,0.6][0.3,0 [0.6,0. .7][0.1,0.2] 4,0.2] 2 x [0.3,0.4][0.1,0 [0.5,0. .2][0.2,0.4] 3,0.5] 3 ( H , I  J)  P-OR is denoted by

G(e2) < Ae2(x), μ e2(x) > [0.4,0.7][0.1,0 [0.5,0. .3][0.1,0.2] 2,0.2] [0.4,0.6][0.4,0 [0.6,0. .7][0.2,0.5] 5,0.4] [0.5,0.8][0.1,0 [0.7,0. .4][0.1,0.4] 3,0.4] ( F , I )  p (G, J )

where

I  J  {(e1 , e1 ), (e1 , e2 ), (e2 , e1 ), (e2 , e2 )}is defined

x 1

x 2

x 3

H(e1,e1) F(e1) ꓴ G(e1)

H(e1,e2) F(e1) ꓴ G(e2)

H(e2,e1) F(e2) ꓴ G(e1)

H(e2,e2) F(e2) ꓴ G(e1)

[0.7,0.9

[0.7

[0.5,0.6

[0.5

[0.7,0.9

[0.7

[0.4,00.

[0.5

][0.6,0.

,0.5

][0.6,0.

,0.5

][0.3,0.

,0.4

7][0.2,0.

,0.4

7][0.5,0

,0.6

7][0.5,0

,0.6

5][0.3,0

,0.5

7][0.2,0.

,0.4

.6]

]

.6]

]

.4]

]

[0.5,0.6

[0..

[0.4,0.6

[[0.

[0.5,0.6

[0..

[0.4,0.6]

[0.6

][0.7,0.

6,0.

][0.7,0.

6,0.

][0.6,0.

6,0.

[0.6,0.8]

,0.7

8][0.2,0

6,0.

8][0.2,0

6,0.

8][0.2,0

7,0.

[0.2,0.6]

,0.5

.3]

.5]

.6]

[0.3,0.4

[0.5

[0.5,0.8

[0.7

[0.4,0.7

[0.5

[0.5,0.8]

[0.7

][0.2,0.

,0.4

][0.2,0.

,0.4

][0.2,05

,0.6

[0.2,0.5]

,0.6

3][0.3,0

,0.6

3][0.3,0

,0.6

][0.3,06

,0.6

[0.3,0.6]

,0.6

.5]

]

.5]

]

Definition:3.3 The complement of a neutrosophic soft cubic set (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } is i

denoted by (F, I)

and deﬁned as

C c c c (F, I) = {( (F, I) = (F , I) } , where F : I  NC ( X )

and Fc (ei )  (F(ei ))c for all ei I

c (F, I) c = {( F(ei ))c = {< x, A c ei (x), ei (x) > : x  X} ei  I }.

(F, I) c 





{< x, [1  AeT ,1  AeT ],[1  Ae I ,1  Ae I ],[1  Ae F ,1  Ae F ] ,



i i T I F 1  e ,1  e ,1  e i i i

 > x  X} e  I . i

Example:3.4 Let X = {x1, x2} be initial universe and E = {e1, e2} parameter’s set. Let (F, I) be a neutrosophic soft cubic set over X and defined as (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } i i X F(e1) F(e2) < Ae1(x), < Ae2(x), λ e1(x) > λ e2(x) > x [0.3,0.5][0.1,0. [0.6,0. [0.4,0.6][0.5,0. [0.5,0. 4][0.5,0.8] 5.0.7] 7][0.6,0.9] 4,0.4] 1 x [0.6,0.8][0.4,0. [0.7,0. [0.2,0.4][0.4,0. [0.3,0. 7][0.4,0.7] 5,0.3] 7][0.3,0.6] 7,0.8] 2 Then c c c c (F, I) = {( F(ei )) = {< x, A e (x), e (x) > : x  X} e i  I } i

is defined as. X Fc (e1) < Ac e1(x), c λ e1(x) > x [0.5,0.7][0.6,0.9 [0.4,0. ][0.2,0.5] 5,0.3] 1 x [0.2,0.4][0.3,0.6 [0.3,0. ][[0.3,0.6] 5,0.7] 2

Fc (e2) Ac e1(x),

< λc e1(x) > [0.4,0.6][0.3,0. 5][0.1,0.4] [0.6,0.8][0.3,0. 6][0.4.0.7]

[0.5,0. 6,0.6] [0.7,0. 3,0.2]

Proposition :3.5 Let X be initial universe and I, J, L and S subsets of parametric set E. Then for any neutrosophic soft cubic sets , , , the following properties hold (1) if . c (2) if ⊆P then c . (3) if ⊆P and ⊆P C then ⊆P ∩P . (4) if ⊆P and ⊆P then ∪P ⊆P . (5) if ⊆P and ⊆P then ∪P ⊆P ∪P and ∩ P ⊆P ∩ P . Proof: Proof is straight forward Theorem:3.6 Let (F, I) be a neutrosophic soft cubic set over X. (1) If (F, I) is an internal neutrosophic soft cubic set, then

(F, I) c is also an internal neutrosophic soft cubic set (INSCS). (2) If (F, I) is an external neutrosophic soft cubic set, then (F, I) c is also an external Neutrosophic soft cubic set (ENSCS).

 (F(ei ))c (as ( ei )  ei )

R. Anitha Cruz and F. Nirmala Irudayam. More On P-Union and P-Intersection of Neutrosophic Soft Cubic Set

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Proof. (1) Given (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } i i

is an INSCS this implies T T T Ae x  e x   Ae x  , i i i I I I Ae ( x)  e ( x)  Ae ( x) , i i i F F F Ae x  e x  Ae x , i i i



i F 1  e i

0



i I Be ( x) i

( x) 

Be F i

x eFi x  BeiF x

for all ei  J and for all x  X. Then we have

(

I ei



)( x )  max{ Ae I  x , B e I  x }, i i

I ei

H (e i ) 

this implies 1  Te x   1  AeT x  or 1  AeT x   1  Te x  , i

for all ei  I  J and for all x  X. . Now by definition of P-union of (F, I) and (G, J) , we have (F, I) p (G, J )  ( H , C ) where I  J  C and

eI i

max{ Ae F  x , Be F  x }  (  F   F )( x )  max{ Ae F  x , Be F  x },

eF x   Ae F x  or Ae F x   eF x 





Be I ( x) i

max{ Ae I  x , B e I  x }  i i

eI x   Ae I x  or Ae I x   eI x  , i

max{ AeT  x , BeT  x }  ( T   T )( x )  max{ AeT  x , B eT  x },

 ) &

Also for (G, J) we BeT x   eT x   BeT x  ,

So we have T T T T e x   Ae x  or Ae x   e x  , i

AeT x  Te x   AeT x  , for all ei  I and for all x  X.

x  ( Ae F x , Ae F x i i F F Ae x  Ae x  1 i i



soft cubic sets. So for (F, I) we have i

0  Ae I  x   Ae I  x   1 ,

eF i

Since (F, I) and (G, J) are internal neutrosophic

F F F Ae I ( x)  eI ( x)  Ae I ( x) , Ae x   e x   Ae x  i i i i i i

Te x   ( AeT x , AeT x  )

(2) (F, I)  p (G , J ) is an INSCS

for all ei  I and for all x  X.

x  )

 )

x   1 , eI x   ( Ae I x , Ae I x  i i i

x ,1 

i Ae F i

be internal nuetrosophic cubic soft sets. Then, (1) (F, I)  p (G , J ) is an INSCS

i i i I I I e  x   ( Ae  x , Ae  x  ) i i i F F e x  ( Ae x , Ae F x i i i

x  

x   (1 

i F Ae i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J }

an ENSCS this implies Te  x   ( AeT  x , AeT  x ) ,

0

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } is i i

AeT i

Proof: (1)

i AeT i

(F, I) = { F(ei ) = {< x, A e (x), e (x) > : x  X} e i  I } an

Given

Since

Theorem: 3.7 Let



Hence (F, I) is an ENSCS .

(F, I) c is an INSCS .



for all ei  I and for all x  X.

(2)

1  eI x   (1  Ae I x ,1  Ae I x  ) ,

thisimplies 1  AeT  x   1  Te  x   1  AeT  x  , i i i I I  1  Ae ( x)  1  e ( x)  1  Ae I ( x) , i i i F F F   1  Ae x   1  e x   1  Ae x 

Hence

Thus 1  Te x   (1  AeT x ,1  AeT x ) ,



1  eF x   1  Ae F x  or 1  Ae F x   1  eF x  , for all ei  I and for all x  X.

for all ei  I and for all x  X.

1  eI x   1  Ae I x  or 1  Ae I x   1  eI x  ,

 F (e i )  G (e i )  F (e )  G (e ) i p i 

if e  I  J   if e  J  I  if e  I  J 

if ei  I  J , then F (ei )  p G(ei ) is defined as F (ei )  p G(ei ) =

H (e i ) =

 x, max{Ae ( x), Be ( x)},(e i

 ei )(x), x  X , ei I  J

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(G, J)* = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } res

where

F (e i )  p G (e i ) T

 x, max{A





( x), B Te ( x)}, (ei   ei ) T ( x), x  X , ei  I  J ,

T ei

F (ei )  p G (ei ) I



 x, max{A (x), B (x)},( I

F F (ei )  p G F (ei )

 x, max{A (x), B F



Theorem 3.9 For two ENSCSs

 ei ) I ( x), x  X , ei I  J ,



(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i



ei  I  J .

If ei  I  J or ei J  I then the result is trivial. Hence (F, I) p (G, J ) is an INSCS in all cases.

(F, I)* and(G, J )* are

F (ei )  p G(ei ) is defined as

 ei ) ( x)  x  X , e  I  J



. Also

ENSCS. Then for

x 

(F, I) we

 ( Ae I i

x 

, Ae I i

have

Te x   ( AeT x , AeT x  ) ,

x )

i F e i

for all ei  I and for all x  X

far

have

AeT x   Te x   AeT x , Ae I ( x)  eI ( x)  Ae I ( x) ,

i Ae F i

x  

i eF i

x  

i Ae F i

x 

for all ei  I and for all x  X. (G, J)

And for

BeT i

we have

x  

eT i

for all ei  J and for all x  X. min{ AeT i

x 

min{ Ae I i

, BeT i

x 

x }  (

, Be I i

T ei

x }  (



I ei

T ei



)( x ) 

I ei

min{ AeT i

)( x ) 



x  ,



, BeT i

min{ Ae I i

x },

x , Bei I x } i

for all ei  J and for all x  X.

Also

given

that

(F, I)* = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

this

implies

Ae I ( x)  eI ( x)  Ae I ( x) , i

i AeT i Ae F x i

x eTi x  AeiT x ,   eFi x  Aei F x

for all ei  I and for all x  X. And BeT  x  Te  x   BeT  x  i I Be ( x) i

sets

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } ,

we interchange λ and μ , Then the new neutrosophic soft cubic set (NSCS) are denoted and defined as * (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

INSCS

(NSCS) (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

cubic

x )

(G, J) we have

and



eI i

( x)  Be I ( x) , Be F x   eF x   Be F x  . i

for all ei  J and for all x  X.

soft

x 

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J} are

for all ei  I  J and for all x  X. Hence (F, I) p (G, J ) is an INSCS . Definition: 3.8 Given two neutrosophic

x 

i F , Ae i

min{ Ae F  x , Be F x }  ( F   F )( x)  min{ Ae F x , Be F x } i

eF x   ( Be F x , Be F x )

x  BeT i  Be F x i

x 

i F  ( Ae i

eT x   ( BeT x , BeT x ) , eI x   ( Be I x , Be I x ) ,

Be I ( x)  eI ( x)  Be I ( x) , Be F x   eF x  i

then

given that (F, I) and (G, J) are INSCS.

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and

eI i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } are

H ( ei )  F ( ei )  p G ( ei )  i

INSCS

Proof: Since

and H (ei )  F (ei )  p G(ei ) . If

 x, min {Ae ( x), Be ( x)},(e

(F, I) P (G, J ) is an INSCS in X.

Since (F, I) p (G, J )  ( H , C ) where I  J  C

ei I  J then

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } in

( x)}, (ei  ei ) F ( x), x  X , ei I  J .

Thus (F, I) p (G, J ) is an INSCS if

(2)

pectively.

Since (F, I) and (G, J)

are ENSCS and (F, I) and (G , J ) are INSCS. Thus by definition of ENSCS and INSCS all the possibilities are under 1) (a1) Te x  AeT x   eT x  AeT x  (a2) (a3)

(b1)

i I e i F e i T e i

x   x    x 

i I Ae i F Ae i T Be i

i I  e i F  e i T  e i

x  x  x 

i I x  Ae x i x  Ae F x i T x  Be x i

  

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(b2) eI x  Be I x   eI x  Be I x  (b3) 2) (a1) (a2) (a3) (b1) (b2) (b3) 3) (a1) (a2) (a3) (b1) (b2) (b3) 4) (a2) (a2) (a2) (b1) (b2) (b2)

F I (e i )  P G I (e i ) 

i i i i eF x  Be F x  eF x  Be F x i i i i AeT x  eT x  AeT x  Te x i i i i I I I I Ae x  e x  Ae x  e x i i i i F F F Ae x  e x  Ae x  eF x i i i i T T T T Be x  e x  Be x  e x i i i i I I I I Be x  e x  Be x  e x i i i i Be F x  eF x  Be F x  eF x i i i i Te x  AeT x  eT x  AeT x i i i i I I I I e x  Ae x  e x  Ae x i i i i F F F e x  Ae x  e x  Ae F x i i i i T T T T Be x  e x  Be x  e x i i i i I I I I Be x  e x  Be x  e x i i i i Be F x  eF x  Be F x  eF x i i i i AeT x  eT x  AeT x  Te x i i i i I I I I Ae x  e x  Ae x  e x i i i i F F F Ae x  e x  Ae x  eF x i i i i T T T T e x  Be x  e x  Be x i i i i I I I I e x  Be x  e x  Be x i i i i eF x  Be F x  eF x  Be F x i i i i

                  

  I I I I  x, max{ Aei ( x), B ei ( x)}, (  e   e ( x), x  X , ei  I  J  i i   ,

F F (ei )  P G F (ei )    F F F F  x, max{ A ei ( x), B ei ( x)}, (  e   e ( x), x  X , ei  I  J  i i   for all ei  I  J and for all x  X. . Case: 1 If H (ei )  F (ei ) that is if ei I  J then from (1)(a1) and (2)(a1) , we have

Te  x   AeT  x and Te  x   AeT  x  i

where

 T T T  x, max{ Aei ( x), B ei ( x)}, (  e i 

for all ei  I and for all x  X. Thus

I  J  C and

 ei )(x), x  X , ei I  J

F T (ei )  P GT (ei ) 

H (e i ) =

where

for all ei  I  J and for all x  X. Case: 2 If H (ei )  G(ei ) that is if ei  J  I then from (1)(b1) and (2)(b1) , we have T T e x  Be x and eT x  BeT x 

if e  I  J   H (e i )  if e  J  I  if e  I  J  if ei  I  J , then F (ei )  R G(ei ) is defined as

 x, max{Ae ( x), Be ( x)},(e

Ae F x  eF x   Ae F x  ,

 F (e i )  G (ei )  F (e )  G (e ) i P i 

F (ei )  P G(ei ) =

for all ei  I  J and for all x  X. Similarly we can prove for (1)(a2) , (2)(a2) and (1)(a3) , (2)(a3). Ae I x  eI x   Ae I x  and Thus

Since P-union of (F, I) and (G, J) is denoted and defined as (F, I) P (G, J )  ( H , C )

for all ei  I and for all x  X. Thus AeT x  Te x   AeT x  ,



BeT x  eT x   BeT x  , i

for all ei  J - I and for all x  X. Similarly we can prove for (1)(b2) and (2)(b2) and (1)(b3) and (2)(b3). Thus Be I x  eI x   Be I x  and i F Be i

x 

i eF i

x  

i Be F i

x  ,

for all ei  J  I and for all x  X. Case: 3 If H (ei )  F (ei )  P G(ei ) that is if ei  I  J , then from (1)(a1) and (1)(b1) , we have T T T Ae x  e x   Ae x  for all ei  I and for all x  X. i

and    T )( x), x  X , ei  I  J  B T x   T x   B T x  for all e  J and for all x  X. ei ei ei i  ei

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eT x   ( BeT x , BeT x ) , eI x   ( Be I x , Be I x ) ,

Hence (i) ei  I  J then 



max{ AeT  x , BeT  x }   T   T ( x)  max{ AeT  x , BeT  x } . i

 ei

ei 

Similarly we can prove (1)(a2) , (1)(b2) and (1)(a3), (1)(b3) . Thus max{Ae I i

x 

I , Be i

   x }   I   I ( x)  max{Ae I  x , Be I  x } i i e e i   i 



, max{ Ae F  x , Be F  x }   F   F ( x)  max{ Ae F  x , Be F  x } i

 ei

ei 

. Thus in all the three cases (F, I) P (G, J ) is an INSCS in

eF x   ( Be F x , Be F x ) for all ei  J and for all x  X.

eT x   ( AeT x , AeT x ) , eI x   ( Ae I x , Ae I x ) ,

i F e i

x 

i F  ( Ae i

i F , Ae i

x 

x )

for all ei  I and for all x  X.

Te x   ( BeT x , BeT x ) , eI x   ( Be I x , Be I x  ) ,

i F e i

x 

i F  ( Be i

i F , Be i

x 

x ) for all

ei  J and for all x  X

respectively. Thus we have

 T    T  e ei  i

Theorem: 3.10 For two ENSCSs

 I      I ( x)  { max{ Ae I  x , Be I  x }, max{ Ae I  x , Be I  x }},  e  i i i i e i   i

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } in i

 ( x) { max{ AeT  x , BeT  x }, max{ Ae T  x , Be T  x }} ,  i i i i 

 F      F ( x )  { max{ Ae F  x , Be F  x } , max{ Ae F  x , Be F  x }}  e i i i i ei   i

for all ei  I  J and for all x  X. Thus

have

X if * (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and

   e  e ( x)  max{ Ae x , Be x } i i i  i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } are

for all ei  I  J and for all x  X. Also since (F, I) P (G, J )  ( H , C ) where I  J  C and

INSCS in X then (F, I) P (G, J ) is an INSCS in X. Proof: By similar way to Theorem 3.9 we can obtain the result. Theorem: 3.11 Let

H (e i )



 F (e i )  G (ei )  F (e )  G (e ) i p i 

if e  I  J   if e  J  I  if e  I  J 

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and

if e  I  J , then F (ei )  p G(ei ) is defined as

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be

F (ei )  p G(ei ) =

ENSCSs in X such that * (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J)* = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i

ENSCS in X. Then P-union of (F, I) and (G, J) is an ENSCS in X. Proof: Since (F, I) , (G, J) , (F, I)* and(G, J )* are ENSCS so by definition of an external soft cubic set for (F, I) ,

(G, J) , (F, I)* and(G, J )* we have T T T I I I e x   ( Ae x , Ae x  ) , e x   ( Ae x , Ae x ) , i F e i

x 

i F  ( Ae i

x 

i F , Ae i

x  ) ,

 x, max{Ae ( x), Be ( x)},(e i

H (e i ) =



 ei )(x), x  X , ei I  J .

where

F T (ei )  p GT (ei )    T T T T  x, max{ Aei ( x), B ei ( x)},     ei  ei 

   ( x), x  X , ei  I  J     

F I (e i )  p G I (e i )     I   I I I   x, max{ A ei ( x), B ei ( x)},      ( x), x  X , ei  I  J  ei   ei  

for all ei  I and for all x  X.

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For each e  I  J , Take

F F (ei )  p G F (ei ) 

   F F F F    T  x, max{Aei ( x), B ei ( x)},     ( x), x  X , ei  I  J   ei  min max{AeT x , BeT x }, max{AeT x , BeT x } and i i i i   e e   i  i  eTi  max { min{ AT x , B T x }, min{ AT x , B T x } ei ei ei ei By definition of an external soft cubic set   (F, I) P (G, J ) is an ENSCS in X. . T Then  ei is one of AeT x , B T x  , AeT x , BeT x  . ei Example: 3.12 i i i Let (P, I ) and (Q, J ) be neutrosophic soft cubic sets in X T Now we consider  ei = AT x  or AT x  only, as the where ei ei (P, I) = P(e1 ) remaining cases are similar to this one. = {< x, [0.3,0.5],[0.2,0.5],[0.5,0.7], 0.8,0.3,0.4 > e1  I } eTi = If then AeT x  , i (Q, J) = Q(e1 ) T T T  eTi = Be  x   Be  x   AT x   Ae  x  and so = {< x, [0.7,0.9][0.6,0.8][0.4,0.7], 0.4,0.7,0.3 > e1  J } i i i ei for all x  X T Then (P, I ) and (Q, J ) are T-external neutrosophic cubic Bei  x  sets in X and (P, I) P (Q, J ) = T T  T T  thus B T x   (Aei  Bei ) ( x)  (Aei  Bei ) ( x) = ( P, I )  (Q, J )  P  Q(e1 ) ei

 {< x, [0.3,0.5][0.2,0.5],[0.4,0.7], 0.4,0.3,0.3 > e1  I  J }

for all x  X . (P, I) P (Q, J ) is not an T-external neutrosophic cubic set since  T   T x  = 0.4 (0.3,0.5) =  e1  ∈ e    AT  BT 1  x  ,  AT  BT   x    e1 

e1 

 e1

e1 

 

From the above example it is clear that P-intersection of Texternal neutrosophic soft cubic sets may not be an Texternal neutrosophic soft cubic set. We provide a condition for the P-intersection of T-external (resp. Iexternal and F-external) neutrosophic soft cubic sets to be T-external (resp. I-external and F-external) neutrosophic soft cubic set. Theorem: 3.13 Let

(F, I) = { F(ei ) = {< x, A e (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i

T- ENSCSs in X such that     max { min{A  T  x , B T  x } , min{A T  x , B  T  x }, ei ei ei ei  T     T     ( x )   ei     ei  T  T  T  T  min  max{Ae  x , Be  x } , max{Ae  x , Be  x } i i i i   

     

(3.7)

for all ei  I and for all ei J and for all x  X. Then (F, I) P (G, J ) is also an T- ENSCS.

BeT x  i





  Te  eT (x) . i  i

T =  ei



 T T  e  e (x)  i  i

AeT x   BeT x  . i i

from

this

 T T  e  e (x) i i 

BeT x   i

can



AeT x  i

write





  Te  eT (x) = i  i

AeT x  i

AeT x  i

 or

 B T x  . e i

 T T BeT x   AT x       (x)  e e e i i i  i  T  T Ae x   Be x  it is contradiction to the fact that

For this case i

(F, I) and (G, J) are T-ENSCS.

where H (ei ) = F (ei )  p G(ei ) is defined as

For

=  x, min{ Aei ( x), Bei ( x)}, (ei  ei )(x), x  X , ei I  J .

BeT x   i

AeT x   B T x  ei i

Proof Consider (F, I) P (G, J )  ( H , C ) where I  J  C F (ei )  p G(ei ) = H (ei )



 T T T T  T T  Hence  e  e (x)  (Aei  Bei ) ( x) , ( Aei  Bei ) ( x) i  i . T If  ei = AT x , then BeT x   AeT x   BeT x  ei i i i T and so  ei = max{AeT x , BeT x } . i i T Assume that  ei = AT x  then BeT x   AT x   ei ei i

the

case

AeT x   BeT x  i i

T Be  x   AT x  i ei

have



 T T  e  e (x) = i i   T T  e  e (x)  i  i

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Theorem : 3.15

(A

T ei







 BeTi ) ( x) , ATei  BeTi ) ( x) because  Te  eT (x) = i  i

AeT x  i

(ATei

 BeTi

) ( x) . Again assume that

 eTi

BeT x  then   AeT x  BeT x  i i i  T T  e  e (x)  AT x   B T x  . From this we can ei ei i  i write   AeT x  BeT x  i

 T T  e  e (x)  i i  

AeT x   BeT x  i i

or AT x   B T x  e e i



Let

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i

F- ENSCSs in X such that     min  max{ A F x , B  F x } , max{ A F x , B  F x },  e e e e i i i i    F  F   e  e ( x)   i  i    max { min{ A F x , B  F x } , min{ A F x , B  F x }  e e e e i i i i    …………..(3.9) for all ei  I and for all ei J and for all x  X. Then (F, I) P (G, J ) is also an F- ENSCS. Proof : By similar way to Theorem 3.13, we can obtain the

  Te  eT (x) = AeT x   B T  x  . For this case result i i e  i

i  T T AeT x   BeT x    e  e (x )  AeT x   BeT x  i i i i i  i it is contradiction to the fact that (F, I) and (G, J) are T-

ENSCS. And if we take the case AT x   B T x   e e i

 T T  e  e (x) = AT x   ei i  i

BeT x  , i

(A

 T T  e  e (x)  i  i

T ei

get

have

 BeTi ) ( x) , ATei  BeTi ) ( x)



 T    T (x) = AT x  = (ATe  BeT ) ( x) . i i ei  because  ei ei Hence in all the cases (F, I) P (G, J ) is an T-ENSCS in

Corallary:3.16 Let

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

ENSCSs in X. Then P-intersection (F, I) P (G, J ) is also an ENSCS in X when the conditions (3.7), (3.8)and (3.9) are valid. Theorem: 3.17 If neutrosophic

soft

cubic

set

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } in i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i i

satisfy

the following condition

  min  max{AT  x , B T  x } , max{AT  x , B T  x } ei ei ei ei  

Theorem: 3.14

 

 (x ) i

 

  Te  eT

Let

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i

I- ENSCSs in X such that     min  max{ A I x , B  I x }, max{ A I x , B  I x },  ei ei ei ei    I  I   e  e ( x)   i  i    I  I  I  I  max { min{ A x , B x }, min{ A x , B x }  e e e e i i i i   

(3.8)

for all ei  I and for all ei J and for all x  X. Then (F, I) P (G, J ) is also an I – ENSCS Proof: By similar way to Theorem 3.13, we can obtain the result.

 

 max{ min{AeT x , BeT x }, min{AeT x , BeT x } ….(11.1) i

then the (F, I) P (G, J ) is both an T-Internal Neutrosophic Soft Cubic Set and T-External Soft Neutrosophic Cubic Set in X. Proof: Consider where (F, I) P (G, J )  ( H , C )

I J  C

where

F (ei )  p G(ei )

H (e i ) =

defined as F (ei )  P G(ei ) =

H (ei )

{< x, min{ Ae (x), Be (x)}, ( e  e )(x) > : x  X} ei  I  J } where

R. Anitha Cruz and F. Nirmala Irudayam. More On P-Union and P-Intersection of Neutrosophic Soft Cubic Set

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(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } in

F T (ei )  P GT (ei ) =

x, min{ A T ei

.For T

e

(x)}, ( T ei

(x), BT ei

 T ei

ei  I  J Take

each



  T  e  max{ min{AT  x , B T  x }, min{AT  x , B T  x } . i ei ei ei ei   T

T T T T Ae  x , Be  x  , Ae  x , Be  x  . i i i i T Ae  x  i

T consider  ei = AT x , or e

cases are similar to this one. If

BeT x   BeT x   i i



 

Then

Now we

= AT  x  then ei

BeT x  i

T  Ae x , and so  ei = i  T I T this implies A T  x  =  ei =  e  e (x) = i ei  i

 eTi =

BeT x  . i

T Ae  x , i

 T      T (x) = AT  x   A T  x  , which implies e e ei ei i  i that

T T  BeT x  = ( A ei  Bei ) ( x) . i



 T T T T  T T  Hence  e  e (x)  (Aei  Bei ) ( x) , ( Aei  Bei ) ( x) i  i and

(ATei

 BeTi )  ( x )

 T     T (x )   ei ei 



( A Tei



 BeTi )  ( x) .

= A T  x  then BeT x   A T  x   BeT x  , i i ei ei  T T T  T and so  e  e (x) = A T  x  = ( A ei  Bei ) ( x ) . i i ei   If

 eT

Hence



(ATei

 BeTi ) ( x) , ( ATei

following

condition

 (x ) i

 

 

 max{ min{Ae I x , Be I x } , min{Ae I x , Be I x } …….(11.2) i

then the (F, I) P (G, J ) is both an I-internal neutrosophic soft cubic set and an I-external soft neutrosophic cubic set in X. Theorem :3.19 If neutrosophic

soft

cubic

set

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } in i i B T  x   B T  x  = X satisfy the following condition ei ei

Thus

 T T  e  e (x) = i  i

the

  eI  eI

only, as the remaining T ei

satisfy

  min  max{A I  x , B  I  x } , max{A I  x , B  I  x } ei ei ei ei  

  min  max{AT  x , B T  x }, max{AT  x , B T  x } and ei ei ei ei  

 e is one of i

)(x) > : x  X} ei  I X J}

 BeTi ) ( x)



 T T  e  e (x)  i  i and

  min  max{ A F x , B  F x }, max{ A F x , B  F x } ei ei ei ei  

    F   F (x)  ei ei     max { min{ A F x , B  F x } , min{ A F x , B  F x } e e e e i i i i   ……….(11.3)then the

(F, I) P (G, J ) is both

an F-internal neutrosophic soft cubic set and an F-external soft neutrosophic cubic set in X. Corollary:3.20 Let (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  I } i

NSCSs in X. Then P-intersection (F, I) P (G, J ) is also an ENSCS and an INSCS in X when the conditions (11.1), (11.2)and (11.3) are valid.

(ATei  BeTi )  ( x)   Te  eT (x)  ( ATei  BeTi )  ( x) . i  i The following example shows that the P-union of TConsequently we note that (F, I) P (G, J ) is both T-internal neutrosophic soft cubic set and T-external soft neutrosophic cubic set in X. Similarly we have the following theorems Theorem 3.18 If neutrosophic soft cubic set (F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

external neutrosophic soft cubic sets may not be an Texternal neutrosophic soft cubic set. Example 3.21. Let (P, I ) and (Q, J ) be neutrosophic soft cubic sets in X where (P, I) = P(e1 ) = {< x, [0.3,0.5],[0.2,0.5],[0.5,0.7], 0.8,0.3,0.4 > e1  I }

(Q, J) = Q(e1) = {< x, [0.7,0.9][0.6,0.8][0.4,0.7], 0.4,0.7,0.3 > e1  J}

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Then (P, I ) and (Q, J ) are T-external neutrosophic cubic sets in X and ( P, I )  (Q, J )  P  Q (e1 )  {< x, [0.7,0.9][0.6,0.8],[0.5,0.7], 0.8,0.7,0.4 > } ( P, I )  p (Q, J ) is not an T-external neutrosophic cubic set in X since  T   T x  = 0.8 ∈ (0.7,0.9) =  e1 e2          AT  B T   x  ,  AT  B T   x   e   . e1 e1  e1  1     We consider a condition for the P-union of T-external (resp. I-external and F-external) neutrosophic soft cubic sets to be T-external (resp. I-external and F-external) neutrosophic soft cubic set.

= AT x  or AeT x , only as the remaining ei i cases are similar to this one. consider

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i

T- ENSCSs in X such that     max { min{A  T  x , B T  x } , min{A T  x , B  T  x },  ei ei ei ei  T     T     ( x )    ei     ei  T  T  T  T  min  max{Ae  x , Be  x } , max{Ae  x , Be  x }  i i i i    

…………..(12.1) for all ei  I and for all ei J and for all x  X. Then (F, I) P (G, J ) is also an T- ENSCS.

 eT

T ei



 BeTi ) ( x) , ATei  BeTi ) ( x) . If  eTi =



BeT x  . i

from

 F (e i ) if e  I  J    H (e i )  G (e i ) if e  J  I   F (e )  G (e ) if e  I  J  i p i  where H (ei ) = F (ei )  p G(ei ) is defined as

 T T  e  e (x) i i 



 ei )(x), x  X , ei I  J ,

where

F T (ei )  p G T (ei ) 

 x, max{A (x), B (x)},( If e  I  J , i

T ei



 eT )( x), x  X , ei I  J , i

For

this

then

 AeT x  i

the

can

write

 T T  e  e (x)  i  i BeT x   AT x  = ei i



AeT x  i

A T  x   BeT x  i ei

 x, max{Ae ( x), Be ( x)},(e

AeT x , i

 eTi = BeT x   AeT x   BeT x  and so i i i T  T  T max{Ae x , Be x } . Assume that  ei = AT x  then ei i i  T T BeT x   AT x    e  e (x)  AT  x  ei i i ei  i BeT x   i

H (e i ) =

then

T BeT x   BeT x   A T  x   AeT x , and so  ei = i i i ei T (ATei  BeTi ) ( x) = BeT x  .Thus AT  x  =  ei i ei    T T   T   T (x ) . Hence  e  e (x)  e e i i  i  i

Proof: Consider (F, I) P (G, J )  ( H , C ) where I  J  C and

F (ei )  p G(ei ) =

AeT x  i

(A

Theorem 3.22 Let

 eT

BeT x  . i

BeT x   i

case

AeT x  i



 T T  e  e (x)  AT x   B T x  it is contradiction to ei ei i  i the fact that (F, I) and (G, J) are T-ENSCS. For the case BeT x   i AeT x  i





AeT x  = i

B T  x  ei

have



 T T  e  e (x) i  i



 T T  e  e (x) i  i



T T  T T  because   (Aei  Bei ) ( x) , Aei  Bei ) ( x) T T T     T  min  max{ A x , B x } , max{ A x , B x } e e e e i i i i   T   (A ei  BeTi )  ( x) = A T  x  =  T   T (x) . and ei  ei ei   eTi  max { min{ AT x , B T x }, min{ AT x , B T x } Again assume that  T = BT x  then T ei ei ei ei Ae x   ei   ei i  eTi is Then one of   T T BeT x    e  e (x )  AeT x   BeT x  , so from i i i i  i T  T  T  T Now we AeT x , Be x  ,  ei Ae x , Be x  . i i i i

 eTi

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this

can

AeT x  i

write

 T T  e  e (x)  i  i



AeT x  i

 T T =  e  e (x)  i  i

BeT x  i

AeT x  i



or AT x  e i

BeT x  . i



BeT x  i



BeT x  i

For this case

     Te  eT (x)  AeT x   BeT x  i i i  i it is contradiction to the fact that (F, I) and (G, J) are TENSCS. And if we take the case AT x   B T x  = AeT x  i

BeT x  i

 T T  e  e (x)  i  i  T T  e  e (x) i  i

(A

T ei

AeT x   BeT x  , i i

get

have





 BeTi ) ( x) , ATei  BeTi ) ( x) because (ATei  BeTi ) ( x)

 T T = B T x  =  e  e (x) . If ei  I  J or ei  J  I ,then ei i  i we have trivial result. Hence (F, I) P (G, J ) is an TENSCS in X. Similarly we have the following theorems Theorem:3.23 Let

(F, I) = { F(ei ) = {< x, A e (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i

T- ENSCSs in X such that     max { min{A  T  x , B T  x } , min{A T  x , B  T  x },  ei ei ei ei  T     T     ( x )    ei   ei  min  max{A  T  x , B T  x } , max{A T  x , B  T  x }  ei ei ei ei    

…………..(12.2) for all ei  I and for all ei J and for all x  X. Then (F, I) P (G, J ) is also an T- ENSCS. Theorem :3.24 Let

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i

…………..(12.3) for all ei  I and for all ei J and for all x  X. Then (F, I) P (G, J ) is also an T- ENSCS.

Corollary:3.25 Let

(F, I) = { F(ei ) = {< x, Ae (x), e (x) > : x  X} ei  I } and i

(G, J) = { G(ei ) = {< x, Be (x), e (x) > : x  X} ei  J } be i i ENSCSs in X. Then (F, I) P (G, J ) is also an ENSCS in X when the conditions (12.1), (12.2)and (12.3) are valid. References [1] R. Anitha Cruz and F. Nirmala Irudayam, Neutrosophic Soft Cubic Set, International Journal of Mathematics Trends and Technology, 46 (2) (2017), 88-94. [2] Y.B. Jun, C.S. Kim and K.O. Yang, Cubic sets, Annals of Fuzzy Mathematics and Informatics 4(3) (2012), 83–98. [3] Turksen,“Interval valued fuzzy sets based on normal forms”. Fuzzy Sets and Systems, 20, (1968), pp. 191–210. [4] F. Smarandache, A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic, Rehoboth: American Research Press, 1999. [5] L. A. Zadeh, Fuzzy Sets, Inform. Control 8 (1965), 338–353. [6] I. B. Turksen, Interval-valued strict preference with Zadeh triples, Fuzzy Sets and Systems 78 (1996) 183– 195. [7] H. Wang, F. Smarandache, Y. Q. Zhangand, R. Sunderraman, Interval Neutrosophic Sets and logic: Theory and Applictionsin Computing, Hexis; Neutrosophic book series, No. 5, 2005. [8] D. Molodtsov , Soft Set Theory–First Results, Computers and Mathematics with Application, 37 (1999) 19–31. [9] Pabitra Kumar Maji, “Neutrosophic soft set” Annals of Fuzzy Mathematics and Informatics 5 (2013), 157–168. [10] F. Smarandache, Neutrosophic set, a generalization of the intuitionistic fuzzy sets, Inter. J. Pu Bre Appl. Math. 24 (2005) 287–297. [11] F. Smarandache, Neutrosophy and Neutrosophic Logic, First International Conference on Neutrosophy, Neutrosophy Logic, Set, Probability and Statistics, University of New Mexico, Gallup, NM 87301, USA (2002). [12] L. J. Kohout, W. Bandler, Fuzzy interval inference utilizing the checklist paradigm and BKrelational products, in: R.B. Kearfort et al. (Eds.), Applications of Interval Computations, Kluwer, Dordrecht, 1996, pp. 291–335. [13] R. Sambuc, Functions Φ-Flous, Application a l’aide au Diagnostic en Pathologie Thyroidienne, These de Doctorat en Medecine, Marseille, 1975.

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[14] I. B. Turksen, Interval-valued fuzzy sets and compensatory AND, Fuzzy Sets and Systems 51 (1992) 295–307. [15] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inform. Sci. 8 (1975) 199–249. [16] P. K. Maji , R. Biswas ans A. R. Roy, “Fuzzy soft sets”, Journal of Fuzzy Mathematics, Vol. 9, no. 3, 2001, pp. 589-602, [17] P.K.Maji, R. Biswas ans A.R.Roy, “Intuitionistic Fuzzy soft sets”, The journal of fuzzy Mathematics, Vol 9, (3)(2001), 677 – 692. [18] Pabitra Kumar Maji, Neutrosophic soft set, Annals of Fuzzy Mathematics and Informatics, Volume 5, No.1,(2013).,157-168. [19] Jun YB, Smarandache F, KimCS (2017) Neutrosophic cubic sets. New Mathematics and Natural Computation 13:41-54. [20] Abdel-Basset, M., Mohamed, M., & Sangaiah, A. K. (2017). Neutrosophic AHP-Delphi Group decision making model based on trapezoidal neutrosophic numbers. Journal of Ambient Intelligence and Humanized Computing. [21] Abdel-Basset, M., Mohamed, M., Hussien, A. N., & Sangaiah, A. K. (2017). A novel group decisionmaking model based on triangular neutrosophic numbers. Soft Computing, 1-15. http://doi.org/ 10.1007/s00500-017-2758-5 [22] F. Smarandache, M. Ali, Neutrosophic Triplet as extension of Matter Plasma, Unmatter Plasma, and Antimatter Plasma, 69th Annual Gaseous Electronics Conference, Bochum, Germany, Veranstaltungszentrum & Audimax, Ruhr-Universitat, October 1014, 2016.

[23] F. Smarandache, Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications. Pons Editions, Bruxelles, 325 p., 2017. [24] Topal, S. and Öner, T. Contrastings on Textual Entailmentness and Algorithms of Syllogistic Logics, Journal of Mathematical Sciences, Volume 2, Number 4, April 2015, pp 185-189. [25] Topal S. An Object- Oriented Approach to Counter-Model Constructions in A Fragment of Natural Language, BEU Journal of Science, 4(2), 103111, 2015. [26] Topal, S. A Syllogistic Fragment of English with Ditransitive Verbs in Formal Semantics, Journal of Logic, Mathematics and Linguistics in Applied Sciences, Vol 1, No 1, 2016. [27] Topal, S. and Smaradache, F. A Lattice-Theoretic Look: A Negated Approach to Adjectival (Intersective, Neutrosophic and Private) Phrases. The 2017 IEEE International Conference on INnovations in Intelligent SysTems and Applications (INISTA 2017); (accepted for publication). [28] Taş, F. and Topal, S. Bezier Curve Modeling for Neutrosophic Data Problem. Neutrosophic Sets and Systems, Vol 16, pp. 3-5, 2017. [29] Topal, S. Equivalential Structures for Binary and Ternary Syllogistics, Journal of Logic, Language and Information, Springer, 2017, DOI: 10.1007/s10849017-9260-4

Received: August 1, 2017. Accepted: August 21, 2017.

R. Anitha Cruz and F. Nirmala Irudayam. More On P-Union and P-Intersection of Neutrosophic Soft Cubic Set