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

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. Iexternal, Fexternal) neutrosophic soft cubic sets to be T-external (resp. Iexternal, Fexternal) neutrosophic soft cubic sets is also dealt with. We provide conditions for the P-union ( P-intersection ) of two T-external (resp. Iexternal, Fexternal) neutrosophic soft cubic sets to be T-internal (resp. Iinternal,Finternal) 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. Iexternal, Fexternal) neutrosophic soft cubic sets and T-external (resp. Iexternal, Fexternal) neutrosophic soft cubic sets are also framed.


INTRODUCTION
Every real situation does not have a crisp or an exact solution hence there is some degree of uncertainty. To deal with uncertainty many Mathematician developed many theories. In 1965 Zadeh [19] introduced the concept of Fuzzy set were we consider the degree of belongingness to a set as a membership function. Following him in 1986 Atanassov [3] introduced the degree of non membership and defined intuitionistic fuzzy set. Further researches were done in these fields but these two sets were not enough to meet all the uncertainties in real physical problems. Hence In 1995 Smarandache [5,6] coined neutrosophic logic and neutrosophic sets to deal with truth , indeterminate and falsehood. On other hand in 1999 Molodtsov [4] introduced soft set which helps the view an environment in a parameterized manner. Pabita Kumar Majii [5][6][7] had combined the Neutrosophic set with soft sets and introduced 'Neutrosophic soft set'. Y. B. Jun et al. [16][17][18] coined cubic set by using a fuzzy set and an interval-valued fuzzy set, and also 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 consider R-unions and R-intersections of T-external (I-external, F-external) neutrosophic soft cubic sets. We provide examples to show that the R-intersection and the R-union of T-external (resp. I-external and F-external) neutrosophic soft cubic sets may not be a T-external (resp. I-external and F-external) neutrosophic soft cubic set. We also discuss conditions for the R-union of T-external (resp. I-external and F-external) neutrosophic soft cubic sets to be a T-external (resp. I-external and F-external) neutrosophic soft cubic set. Further the condition for the R-intersection of T-external (resp. I-external and F-external) neutrosophic soft cubic sets to be a T-external (resp. I-external and F-external) neutrosophic soft cubic set. [19] Let E be a universe. Then a fuzzy set μ over E is defined by X = { μ x (x) / x: x є E }where μ x is called membership function of X and defined by μ x : E → [0,1]. For each x E, the value μ x (x) represents the degree of x belonging to the fuzzy set X. 2.2 Definition: [16] Let X be a non-empty set. By a cubic set, we mean a structure Ξ = ∈ { }

Definition
x A x x x X , ( ), ( ) | µ in which A is an interval valued fuzzy set (IVF) and μ is a fuzzy set. It is denoted by A,µ . 2.3 Definition: [5] 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: [9] 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 indeterminacy, and the degree of falsehood of the element x ∈ X to the set λ with the condition.  ( ) ( ) ( ) > ∈ is a neutrosophic set. The pair (P, A) is termed to be the neutrosophic soft cubic set over X where P is a mapping given by p: A NC(X) → .

Definition
: [1] Let X be an initial universe set. A neutrosophic soft cubic set (P,M) in X is said to be • truth-internal (briefly, T-internal) if the following inequality is valid • indeterminacy-internal (briefly, I-internal) if the following inequality is valid • falsity-internal (briefly, F-internal) if the following inequality is valid If a neutrosophic soft cubic set in X satisfies (2.1), (2.2) and (2.3) we say that (P,M) is an internal neutrosophic soft cubic in X.

Definition: [1]
Let X an initial universe set. A neutrosophic soft cubic set (P,M) in X is said to be • truth-external (briefly, T-external) if the following inequality is valid • indeterminacy-external (briefly, I-external) if the following inequality is valid be two neutrosophic soft cubic sets in X. Let M and N be any two subsets of E (set of parameters), then we have the following for all x X ∈ corresponding to each e M i ∈ . 2.10 Definition: [1] Let (P,M) and (Q,N) be two neutrosophic soft cubic sets (NSCS) in X where I and J are any two subsets of the parametric set E. Then we define R-union of neutrosophic soft cubic set as ( , ) ( , ) ( , ) 2.12 Definition: [2] The complement of a neutrosophic soft cubic set ∈ } } is denoted by (F, I) C and defined as   R-OR is denoted by ( ,

MORE ON R-UNION AND R-INTERSECTION OF NEUTROSOPHIC SOFT CUBIC SET
Proposition: 3.3 Let X be initial universe and I,J,L and S subsets of E. Then for any neutrosophic soft cubic sets A B C D , , , , , the following properties hold Proof: Straight forward.
for all e M and for all x X. i ∈ ∈ Also given that max{ , Since (P,M) and (Q,N) are INSCS so from above given condition and definition of an INSCS we can write, max{ , for all e M and for all x X. i ∈ ∩ ∈ N Thus from given condition and ) )} for all e M and for all x X. From the above example it is clear that R-union of T-ENSCS may not be T-ENSCS. We provide a condition for the R-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.
for all e M for all e and for all x X.
it is contradiction to the fact that and are T-ENSCS.

From this we can write
is contradiction to the fact that and are T-ENSCS. And if we take the case Similarly we have the following theorems for alle Mand for alle Nand for all x X.
is also an I-ENSCS.
for alle Mand for alle N and for all x X.
is also F-ENSCS.
From the above example it is clear that R-intersection of T-ENSCS may not be an T-ENSCS. We provide a condition for the R-intersection 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.
contradiction to the fact that and (P,M) are (Q.N) T-ENSCS.
contradiction to the fact that and are T-ENSCS. And if we take the case Similarly we have the following theorems.
for all e Mand for alle Nand for all x X.