THE THEORY OF NEUTROSOPHIC CUBIC SETS AND THEIR APPLICATIONS IN PATTERN RECOGNITION

In this study, we presented the concept of neutrosophic cubic set by extending the concept of cubic set to neutrosophic set. We also defined internal neutrosophic cubic set (INCS) and external neutrosophic cubic set (ENCS). Then, we proposed some new type of INCS and ENCS is called 1 3 -INCS (or 2 3 -ENCS ), 2 3 -INCS(or 1 3 -ENCS). Then we study some of their relevant properties. Finally, we introduce an adjustable approach to NCS based decision making by similarity measure and an illustrative example is employed to show that they can be successfully applied to problems that contain uncertainties.


Introduction
Fuzzy set is firstly introduced by Zadeh [23] to represent the degree of certainty of expert's in different statements.Zadeh also proposed the concept of a linguistic variable with application in [24].Then Peng et al. [12] presented an application in multi-criteria decision-making problems.After Zadeh, Türks ¸en [18] extend fuzzy set to an interval valued fuzzy set.Interval values fuzzy sets have many applications in real life such as Sambuc [16], Kohout [9], Mukherjee and Sarkar [25] also gave its applications in Medical, Türks ¸en [19,20] in interval valued logic.
Jun et al. [7] introduced cubic set which is basically the combination of fuzzy sets with interval valued fuzzy sets.They also defined internal (external) cubic sets and investigate some of their properties.The notions of cubic subalgebras/ideals in BCK/BCI algebras are also introduced in [5].Jun et al. [6] gave cubic matics, Quaid-e-Azam University Islamabad, Pakistan.Tel./Fax: +92 3340078958; E-mail: mumtazali7288@gmail.com.q-ideals in BCI-algebras and relationship between a cubic ideal and a cubic q-ideal.Also, they established conditions for a cubic ideal to be cubic q-ideal, characterizations of a cubic q-ideal and cubic extension property for a cubic q-ideal.The concept of cubic sub LA--semihypergroup is proposed by Aslam et al. [1].They also introduced results on cubichyperideals and cubic bi--hyperideals in left almost -semihypergroups.Smarandache [17] introduced the theory of neutrosophic logic and sets in 1995.Neutrosophy provides a foundation for a whole family of new mathematical theories with the generalization of both classical and fuzzy counterparts.In a neutrosophic set, an element has three associated defining functions such as truth membership function T, indeterminate membership function I and false membership function F defined on a universe of discourse X.These three functions are independent completely.Neutrosophic set is basically studies the origin, nature and scope of neutralities and their interactions with ideational spectra.Neutrosophic set generalizes the concept of classical fuzzy set [23], and so on.The neutrosophic set has vast applications in various fields [2-4, 8, 10, 13-15, 22].After Smarandache [17], Wang et al. [21] introduced the concept of interval neutrosophic sets which are the generalization of fuzzy, interval valued fuzzy set and neutrosophic sets.Also the interval neutrosophic set studied in [26].
In this paper, we introduce neutrosophic cubic set and define some new notions such as internal (external) neutrosophic cubic sets.The notion of neutrosophic cubic set generalizes the concept of cubic set.We also investigate some of the core properties of neutrosophic cubic set.By using these new notions we then construct a decision making method called neutrosophic cubic method.We finally present an application which shows that the methods can be successfully applied to many problems containing uncertainties.

Fundamental concepts
In this section, we present basic definitions of fuzzy sets [23], interval valued fuzzy sets [18], neutrosophic sets [17], interval valued neutrosophic sets [21] and cubic sets [7].For more detail of these sets, we refer to the earlier studies [1, 5-7, 17, 18, 21, 23].Definition 1. [23] Let E be a universe.Then a fuzzy set over E is defined by X = { x (x)/x : x ∈ E} where μ x is called membership function of X and defined by x : E → [0.1].For each x E, the value μ x (x) represents the degree of x belonging to the fuzzy set X. Definition 2. [18] Let E be a universe.Then, an interval valued fuzzy set A over E is defined by where A − (x) and A + (x) are referred to as the lower and upper degrees of membership x ∈ E where 0 ≤ A − (x) + A + (x) ≤ 1, respectively.Definition 3. [7] Let X be a non-empty set.By a cubic set, we mean a structure = { x, A(x), μ(x) |x ∈ X} in which A is an interval valued fuzzy set (IVF) and μ is a fuzzy set.It is denoted by A, μ .Definition 4. [7] Let X be a non-empty set.A cubic set = A, μ in X is called an internal cubic set (ICS) if A − (x) ≤ μ(x) ≤ A + (x) for all x ∈ X, where A − and A + are lower fuzzy set and upper fuzzy set in X respectively.Definition 5. [7] Let X be a non-empty set.A cubic set = A, μ in X is called an external cubic set (ECS) if μ(x) / ∈ (A − (x), A + (x)) for all x ∈ X. Definition 6. [7] Let 1 = A 1 , μ 1 and 2 = A 2 , μ 2 be cubic sets in X.Then we define Definition 7. [7] For any where j ∈ , we define Definition 8. [17] Let X be an universe.Then a neutrosophic (NS) set λ is an object having the form where the functions T, I, F : X−→ ] -0, 1 + [ define 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.
For two NS, the operations are defined as follows: 1.
Let X be a non-empty set.An interval neutrosophic set (INS) A in X is characterized by the truth-membership function A T , the indeterminacy-membership function A L and the falsity-membership function A F .For each point x ∈ X, A T (x)

Neutrosophic cubic set
In this section, we extend the notion of cubic set to neutrosophic set and introduced neutrosophic cubic set.We also defined internal neutrosophic cubic set (INCS) and external neutrosophic cubic set (ENCS).Some of it is quoted from [1,[5][6][7]25].Definition 10.Let X be an universe.Then a neutrosophic cubic set (NCS) set is an object having the form where A is an interval neutrosophic set in X and λ is a neutrosophic set in X.We simply denoted a neutrosophic cubic set as = A, λ .
Note that the sets of all neutrosophic cubic sets over X will be denoted by C X N .
Example 1. Suppose that X = {x 1 , x 2 , x 3 } be a universe set.Then an interval neutrosophic set A of X defined by Then = A, λ is a neutrosophic cubic set in X.
) for all x ∈ X, then is called an external neutrosophic cubic set (ENCS).

Example 3. Let
N which is not external neutrosophic cubic set.Then, there exist x ∈ X such that If is both INCS and ENCS.Then Proof.Suppose that = A, λ is both INCS and ENCS.Then by Definitions (3.1.3)and (3.1.4),we have and I(x) / ∈ (A − I (x), A + I (x)) or and I(x) / ∈ (A − I (x), A + I (x)) or and F(x) / ∈ (A − F (x), A + F (x)), for all x ∈ X, then is called 1  3 -INCS or 2 3 -ENCS.

Every INCS is a generalization of the ICS. ii. Every ENCS is a generalization of the ECS. iii. Every NCS is the generalization of cubic set.
Proof.The proofs are directly followed from above definitions.
Proof.It is easy.
given by Definition 18 is metrics, where R + is the set of all non-negative real numbers.
Proof.The proof is straightforward.
Proof.The proof is straightforward.

An application in pattern recognition problem
In this section we proposed an decisin making method based on similarity measures of two NCS in pattern recognition problems which is adapted from [25].In this method, we assume that if similarity between the ideal pattern and sample pattern is greater than or equal to 0.5, then the sample pattern belongs to the family of ideal pattern in consideration.The algorithm of this method is as follows: Algorithm: Step 1. Construct an ideal NCS = A, λ on X.
Step 4. If d( , j ) ≤ 0.5 then the pattern j is to be recognized to belong to the ideal Pattern and if d( , j ) > 0.5 then the pattern j is to be recognized not to belong to the ideal Pattern .Now we give an example which is adapted from [25].Example 6.Here a fictitious numerical example is given to illustrate the application of similarity measures between two NCSs in pattern recognition problem.In this example we take three sample patterns which are to be recognized.Let X = {x 1 , x 2 , x 3 } be the universe.Also let be NCS set of the ideal pattern and j = A j , λ j , j = 1, 2, 3 be the NCSs of three sample patterns.

Conclusion
Neutrosophic cubic set (NCS) is a combination of a neutrosophic set with interval neutrosophic set.It is basically the generalization of cubic set.In this paper, we introduced some new type of notions with their basic properties.In the future, we will apply the sets to algebraic structures such as; sub-algebras, ideals, BCK/BCI algebras, q-ideals, and so on.