Some Operations and Properties of Neutrosophic Cubic Soft Set

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.


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][7][8][9], intuitionistic fuzzy soft set [10][11][12][13], interval valued intuitionistic fuzzy soft set [14], neutrosophic soft set [15][16][17][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 (INCSs) and external neutrosophic cubic set (ENCSs) namely,  [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. 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 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., 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  Some theorem related to these topics

Theorem 1
Let FK be a neutrosophic cubic soft set in U which is not an ENCSS. Then there exists at least one e i ∈ K ⊆ E for which there exists some u∈U such that , for all u∈U, corresponding to each e i ∈ K ⊆ E.
But given that FK is not an ENCSS, so at least one e i ∈ K ⊆ E.
There exists some u∈U such that Hence the proof is complete.

Theorem 2
} be a NCSS in U. If FK is both an INCSS and ENCSS in U for all U u∈ , corresponding to each e i ∈ K, then,

Proof
Suppose FK be both an INCSS and ENCSS corresponding to each e i ∈ K and for all u∈U. We have Again by definition of ENCSS corresponding to each K ei ∈ and for all u∈U, we have Since FK is both an ENCSS and INCSS, so only possibility is that, Hence proved.

Definition
Let F K 1 and G K 2 be two neutrosophic cubic soft sets in U and K 1 , K 2 be any two subsets of K. Then, we define the following: If F K1 and G K 2 are two NCSSs then we define P-order as ⊆ P F K 1 G K 2 iff the following conditions are satisfied: 3. If F K1 and G K 2 are two neutrosophic cubic soft sets, then we define the R-order as ⊆ R F K1 G K 2 iff the following conditions are satisfied:

Definition
Let G K2 and G K 2 be two NCSSs in U and K 1 , K 2 be any two subsets of parameter set K. Then we define P-union as ∪P

Definition
Let F K1 and G K 2 be two NCSSs in U and K 1 , K 2 be any two subsets of parameter set K. The P-intersection of F K1 and G K 2 is denoted

Definition: Compliment
The compliment of F K1 denoted by c K1 F is defined by

Some properties of P-union and P-intersection
Hence the proof.

Proof 2:
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 Here

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 Hence the proof.
ii.  T  T  I  I  T  T  I  I e  e  e   T  T  I  I e  e  e   T  T  I  I Hence the proof.

Theorem 4
Let F I be a NCSS over U,

Proof
Since F I and G J are INCSSs, so for F I we have , where K ∈ I ∩ J.
ii. Now, and by definition, Since G J and G J are INCSSs then we have for F I , Hence H K is an INCSS.

Theorem 6
Let F I and G J be any two INCSSs over U having the conditions: Then R-union of F I and G J is also INCSS.

Proof
Since F I and G J are INCSSs in U.
So for F I , we have From F I and G J we get , ∀ e i ∈ I ∩ J,∀ u∈U. If e i ∈ I -J or e i ∈ J -I then, the result is trivial.

Theorem 7
Let F I and G J be any two INCSSs over U satisfying the condition:

Proof
Since F I and G J are INCSSs in U, we have, Also for G J we have, Given condition that

Theorem 8
Let F I and G J be any two ENCSSs then is an ENCSS.

Proof
Since F I and are ENCSSs, we have   Now by definition of P-union, For e i ∈ I -J or e i ∈ J -I these results are trivial.
Hence the proof.
ii. Since F I and G J are ENCSSs, then  , ∀ e i ∈ I ∩ J ∀ u ∈ U Now, by definition of P-intersection is an ENCSS.

Theorem 9
Let F I and G J be any two INCSSs in U such that

Proof:
Since F I and G J are INCSSs in U.
we have, Since, Given condition is

Let
are two NCSSs in U. We defined new NCSSs by interchanging the neutrosophic part of the two NCSSs. We denoted its by ii(a).

Case 1
If H C = ) ( F ei , if e i ∈ I -J, then from i(a)., and ii(a). We have

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.