The theory of neutrosophic cubic sets and their applications in pattern recognition

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Journal of Intelligent & Fuzzy Systems 30 (2016) 1957–1963DOI:10.3233/IFS-151906IOS Press

1957

The theory of neutrosophic cubic sets andtheir applications in pattern recognition

Mumtaz Alia,∗, Irfan Delib and Florentin Smarandachec

aDepartment of Mathematics, Quaid-e-Azam University Islamabad, PakistanbMuallim Rıfat Faculty of Education, Kilis 7 Aralık University, Kilis, TurkeycUniversity of New Mexico, Gallup, New Mexico, USA

Abstract. In this study, we presented concept of neutrosophic cubic set by extending the concept of cubic set to neutrosophicset. We also defined internal neutrosophic cubic set (INCS) and external neutrosophic cubic set (ENCS). Then, we proposedsome new type of INCS and ENCS is called 1

3 -INCS (or 23 -ENCS), 2

3 -INCS (or 13 -ENCS). Then we study some of their

relevant properties. Finally, we introduce an adjustable approach to NCS based decision making by similarity measure andan illustrative example is employed to show that they can be successfully applied to problems that contain uncertainties.

Keywords: Cubic set, neutrosophic cubic set, internal (external) neutrosophic cubic set, decision making

1. Introduction

Fuzzy set is firstly introduced by Zadeh [23] to rep-resent the degree of certainty of expert’s in differentstatements. Zadeh also proposed the concept of a lin-guistic variable with application in [24]. Then Penget al. [12] presented an application in multi-criteriadecision–making problems. After Zadeh, Turksen[18] extend fuzzy set to an interval valued fuzzy set.Interval values fuzzy sets have many applications inreal life such as Sambuc [16], Kohout [9], Mukherjeeand Sarkar [25] also gave its applications in Medical,Turksen [19, 20] in interval valued logic.

Jun et al. [7] introduced cubic set which is basicallythe combination of fuzzy sets with interval valuedfuzzy sets. They also defined internal (external) cubicsets and investigate some of their properties. Thenotions of cubic subalgebras/ideals in BCK/BCI alge-bras are also introduced in [5]. Jun et al. [6] gave cubic

∗Corresponding author. Mumtaz Ali, Department of Mathe-matics, Quaid-e-Azam University Islamabad, Pakistan. Tel./Fax:+92 3340078958; E-mail: [email protected].

q-ideals in BCI-algebras and relationship between acubic ideal and a cubic q-ideal. Also, they establishedconditions for a cubic ideal to be cubic q-ideal, char-acterizations of a cubic q-ideal and cubic extensionproperty for a cubic q-ideal. The concept of cubicsub LA-�-semihypergroup is proposed by Aslam etal. [1]. They also introduced results on cubic �-hyperideals and cubic bi-�-hyperideals in left almost�-semihypergroups.

Smarandache [17] introduced the theory of neutro-sophic logic and sets in 1995. Neutrosophy providesa foundation for a whole family of new mathemati-cal theories with the generalization of both classicaland fuzzy counterparts. In a neutrosophic set, an ele-ment has three associated defining functions such astruth membership function T, indeterminate mem-bership function I and false membership function Fdefined on a universe of discourse X. These threefunctions are independent completely. Neutrosophicset is basically studies the origin, nature and scopeof neutralities and their interactions with ideationalspectra. Neutrosophic set generalizes the concept ofclassical fuzzy set [23], and so on. The neutrosophic

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1958 M. Ali et al. / The theory of neutrosophic cubic sets and their applications in pattern recognition

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 setswhich are the generalization of fuzzy, interval val-ued fuzzy set and neutrosophic sets. Also the intervalneutrosophic set studied in [26].

In this paper, we introduce neutrosophic cubicset and define some new notions such as internal(external) neutrosophic cubic sets. The notion of neu-trosophic cubic set generalizes the concept of cubicset. We also investigate some of the core proper-ties of neutrosophic cubic set. By using these newnotions we then construct a decision making methodcalled neutrosophic cubic method. We finally presentan application which shows that the methods can besuccessfully applied to many problems containinguncertainties.

1.1. Fundamental concepts

In this section, we present basic definitions of fuzzysets [23], interval valued fuzzy sets [18], neutrosophicsets [17], interval valued neutrosophic sets [21] andcubic sets [7]. For more detail of these sets, we referto the earlier studies [1, 5–7, 17, 18, 21, 23].

Definition 1. [23] Let E be a universe. Then a fuzzyset � over E is defined by

X = {�x (x)/x : x ∈ E}where μx is called membership function of X anddefined by �x: E → [0.1]. For each x E, the valueμx(x) represents the degree of x belonging to thefuzzy set X.

Definition 2. [18] Let E be a universe. Then, an inter-val valued fuzzy set A over E is defined by

A = {[A− (x) , A+ (x)

]/x : x ∈ E

}where A− (x) and A+ (x) are referred to as the lowerand 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 cubicset, we mean a structure

� = {〈x, A(x), μ(x)〉 |x ∈ X}in which A is an interval valued fuzzy set (IVF) andμ is a fuzzy set. It is denoted by 〈A, μ〉.Definition 4. [7] Let X be a non-empty set. A cubicset � = 〈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 setin X respectively.

Definition 5. [7] Let X be a non-empty set. A cubicset � = 〈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 = 〈A1, μ1〉 and �2 =〈A2, μ2〉 be cubic sets in X. Then we define

1. (Equality) �1 = �2 if and only if A1 = A2 andμ1 = μ2.

2. (P-order) �1 ⊆p �2 if and only if A1 ⊆ A2 andμ1 ≤ μ2.

3. (R-order) �1 ⊆R �2 if and only if A1 ⊆ A2 andμ1 ≥ μ2.

Definition 7. [7] For any

�j = {⟨x, Aj(x), μj(x)

⟩ |x ∈ X}

where j ∈ �, we define

1. ∪pj∈�

�j ={⟨

x,

(∪

j∈�Aj

)(x),

(∨

j∈�μj

)⟩|x ∈ X

}(P-union).

2. ∩pj∈�

�j ={⟨

x,

(∩

j∈�Aj

)(x),

(∧

j∈�μj

)⟩|x ∈ X

}(P-intersection).

3. ∪Rj∈�

�j ={⟨

x,

(∪

j∈�Aj

)(x),

(∧

j∈�μj

)⟩|x ∈ X

}(R-union).

4. ∩Rj∈�

�j ={⟨

x,

(∩

j∈�Aj

)(x),

(∨

j∈�μj

)⟩|x ∈ X

}(R-intersection).

Definition 8. [17] Let X be an universe. Then a neu-trosophic (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+[ definerespectively the degree of Truth, the degree of inde-terminacy, and the degree of Falsehood of the elementx ∈ X to the set λ with the condition.

−0 ≤ T (x) + I (x) + F (x) ≤ 3+

For two NS,

λ1 = {< x, T1 (x) , I1 (x) , F1 (x) > |x ∈ X}and

λ2 = {< x, T2 (x) , I2 (x) , F2 (x) > |x ∈ X}the operations are defined as follows:

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M. Ali et al. / The theory of neutrosophic cubic sets and their applications in pattern recognition 1959

1. λ1�⊆ λ1 if and only if

T1 (x) ≤ T2 (x) , I1 (x) ≥ I2 (x) , F1 (x) ≥ F2 (x)

2. λ1 = λ1 if and only if,

T1 (x) = T2 (x) , I1 (x) = I2 (x) , F1 (x) = F2 (x)

3. λc1 = {x, F1 (x) , I1 (x) , T1 (x) >: x ∈ X}

4. λ1�∩ λ2 = {< x, min {T1 (x) , T2 (x)} , max

{I1 (x) , I2 (x)} , max {F1 (x) , F2 (x)} >: x ∈ X}5. λ1

�∪ λ2 = {< x, max {T1 (x) , T2 (x)} , max{I1 (x) , I2 (x)} , min {F1 (x) , F2 (x)} >: x ∈ X}

Definition 9. [21] Let X be a non-empty set. Aninterval neutrosophic set (INS) A in X is char-acterized by the truth-membership function AT,the indeterminacy-membership function AL and thefalsity-membership function AF. For each point x ∈X, AT (x) , AI (x) , AF (x) ⊆ [0, 1].

For two INS

A = {< x,

[A−

T (x) , A+T (x)

],[A−

I (x) , A+I (x)

],[

A−F (x) , A+

F (x)]

>: x ∈ X}

and

B = {< x,

[B−

T (x) , B+T (x)

],[B−

I (x) , B+I (x)

],[

B−F (x) , B+

F (x)]

>: x ∈ X}

Then,

1. A⊆B if and only if

A−T (x) ≤ B−

T (x) , A+T (x) ≤ B+

T (x) ,

A−I (x) ≥ B−

I (x) , A+I (x) ≥ B+

I (x) ,

A−F (x) ≥ B−

F (x) , A+F (x) ≥ B+

F (x) for all x ∈ X.

2. A = B if and only if

A−T (x) = B−

T (x) , A+T (x) = B+

T (x) ,

A−I (x) = B−

I (x) , A+I (x) , B+

I (x) ,

A−F (x) = B−

F (x) , A+F (x) = B+

F (x) for all x ∈ X.

3. Ac = {< x,

[A−

F (x) , A+F (x)

],[A−

I (x) , A+I (x)

],[

A−T (x) , A+

T (x)]

>: x ∈ X}

4. A∩B = {< x,

[min

{A−

T (x) , B−T (x)

},

min{

A+T (x) , B+

T (x)}]

,[max

{A−

I (x) , B−I (x)

}, max

{A+

I (x) , B+I (x)

}],[

max{

A−F (x) , B−

F (x)}

, max{

A+F (x) , B+

F (x)}]

>: x ∈ X}

5. A∪B = {< x,

[max

{A−

T (x) , B−T (x)

},

max{

A+T (x) , B+

T (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. Neutrosophic cubic set

In this section, we extend the notion of cubic set toneutrosophic set and introduced neutrosophic cubicset. We also defined internal neutrosophic cubic set(INCS) and external neutrosophic cubic set (ENCS).Some of it is quoted from [1, 5–7, 25].

Definition 10. Let X be an universe. Then a neutro-sophic cubic set (NCS) set � is an object having theform

� = {〈x, A(x), λ(x)〉 |x ∈ X}where A is an interval neutrosophic set in X andλ is a neutrosophic set in X. We simply denoted aneutrosophic cubic set as � = 〈A, λ〉.

Note that the sets of all neutrosophic cubic setsover X will be denoted by CX

N.

Example 1. Suppose that X = {x1, x2, x3} be a uni-verse set. Then an interval neutrosophic set A of X

defined by

A =

⎧⎪⎪⎨⎪⎪⎩

〈[0.22, 0.3], [0.3, 0.4], [0.7, 0.9]〉 /x1+〈[0.1, 0.2], [0.5, 0.9], [0.1, 0.9]〉 /x2+〈[0.7, 0.8], [0.1, 0.1], [0.5, 0.4]〉 /x3

⎫⎪⎪⎬⎪⎪⎭

and a neutrosophic set λ is a set of X defined by

λ ={ 〈0.01, 0.2, 0.4〉 /x1, 〈0.1, 0.02, 0.2〉 /x2,

〈0.3, 0.1, 0.7〉 /3

}

Then � = 〈A, λ〉 is a neutrosophic cubic set in X.

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Definition 11. Let � = 〈A, λ〉 ∈ CXN. If A−

T (x) ≤T (x) ≤ A+

T (x), A−I (x) ≤ I(x) ≤ A+

I (x) and A−F (x) ≤

F (x) ≤ A+F (x) for all x ∈ X, then � is called an inter-

nal neutrosophic cubic set (INCS).

Example 2. Let � = 〈A, λ〉 ∈ CXN. If A(x) = 〈〈[0.2,

0.4], [0.3, 0.5], [0.3, 0.5]〉〉 and λ(x) =〈0.3, 0.4, 0.4〉 for all x ∈ X, then � = 〈A, λ〉 isa INCS.

Definition 12. Let � = 〈A, λ〉 ∈ CXN. If T(x) /∈

(A−T (x), A+

T (x)), I(x) /∈ (A−I (x), A+

I (x)) and F(x) /∈(A−

F (x), A+F (x)) for all x ∈ X, then � is called an

external neutrosophic cubic set (ENCS).

Example 3. Let � = 〈A, λ〉 ∈ CXN.

If λ(x) = 〈0.5, 0.2, 0.7〉 in the above Example 2,then � = 〈A, λ〉 is a ENCS.

Theorem 1. Let � = 〈A, λ〉 ∈ CXN which is not exter-

nal neutrosophic cubic set. Then, there exist x ∈ X

such that

A−T (x) ≤ T (x) ≤ A+

T (x), A−I (x) ≤ I(x) ≤ A+

I (x)

or A−F (x) ≤ F (x) ≤ A+

F (x).

Theorem 2. Let � = 〈A, λ〉 ∈ CXN. If � is both INCS

and ENCS. Then

T(x) ∈ (U(AT ) ∪ L(AT )),

I(x) ∈ (U(AI ) ∪ L(AI ))

and F(x) ∈ (U(AF ) ∪ L(AF )) for all x ∈ X, where

U(AT ) = {A+T (x)|x ∈ X},

L(AT ) = {A−T (x)|x ∈ X},

U(AI ) = {A+I (x)|x ∈ X},

L(AI ) = {A−I (x)|x ∈ X},

U(AF ) = {A+F (x)|x ∈ X}

and L(AF ) = {A−F (x)|x ∈ X}.

Proof. Suppose that � = 〈A, λ〉 is both INCS andENCS. Then by Definitions (3.1.3) and (3.1.4), wehave

A−T (x) ≤ T (x) ≤ A+

T (x),

A−I (x) ≤ I(x) ≤ A+

I (x),

A−F (x) ≤ F (x) ≤ A+

F (x)

And

T(x) /∈ (A−T (x), A+

T (x)),

I(x) /∈ (A−I (x), A+

I (x)),

F(x) /∈ (A−F (x), A+

F (x)),

For all x ∈ X. Thus

T(x) = A−T (x) or A+

T (x) = T(x),

I(x) = A−I (x) or A+

I (x) = I(x),

F(x) = A−F (x) or A+

F (x) = F(x).

Hence

T(x) ∈ (U(AT ) ∪ L(AT )),

I(x) ∈ (U(AI ) ∪ L(AI )),

F(x) ∈ (U(AF ) ∪ L(AF )).

Definition 13. Let � = 〈A, λ〉 ∈ CXN. If

A−T (x) ≤ T (x) ≤ A+

T (x),

A−I (x) ≤ I(x) ≤ A+

I (x)

and F(x) /∈ (A−F (x), A+

F (x))or

A−T (x) ≤ T (x) ≤ A+

T (x),

A−F (x) ≤ F (x) ≤ A+

F (x)

and I(x) /∈ (A−I (x), A+

I (x))or

A−F (x) ≤ F (x) ≤ A+

F (x),

A−I (x) ≤ I(x) ≤ A+

I (x)

and T(x) /∈ (A−T (x), A+

T (x)) for all x ∈ X, then � iscalled 2

3 -INCS or 13 -ENCS.

Example 4. Let � = 〈A, λ〉 ∈ CXN, where A(x) =

〈〈[0.1, 0.4], [0.2, 0.5], [0.3, 0.6]〉〉 and λ(x) = 〈0.3,

0.7, 0.4〉 for all x ∈ X. Then � = 〈A, λ〉 is 23 -INCS.

Definition 14. Let � = 〈A, λ〉 ∈ CXN. If

A−T (x) ≤ T (x) ≤ A+

T (x),

I(x) /∈ (A−I (x), A+

I (x))

and F(x) /∈ (A−F (x), A+

F (x))or

A−F (x) ≤ F (x) ≤ A+

F (x),

T(x) /∈ (A−T (x), A+

T (x))

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and I(x) /∈ (A−I (x), A+

I (x))or

A−I (x) ≤ I(x) ≤ A+

I (x),

T(x) /∈ (A−T (x), A+

T (x))

and F(x) /∈ (A−F (x), A+

F (x)), for all x ∈ X, then � iscalled 1

3 -INCS or 23 -ENCS.

Example 5. Let � = 〈A, λ〉 ∈ CXN, where A(x) =

〈〈[0.1, 0.3]T , [0.2, 0.4]I , [0.4, 0.6]F 〉〉 and λ(x)=〈0.2,

0.7, 0.7〉 for all x ∈ X. Then � = 〈A, λ〉 is 13 -INCS.

Theorem 3. Let � = 〈A, λ〉 ∈ CXN. Then,

i. 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 abovedefinitions.

Definition 15. Let �1 = 〈A1, λ1〉 and �2 = 〈A2, λ2〉be neutrosophic cubic sets in X. Then

1. (Equivality) �1 = �2 if and only if A1 = A2and λ1 = λ2,

2. (P − order) �1 ⊆p �2 if and only if A1⊆A2

and λ1�⊆ λ2,

3. (R − order) �1 ⊆R �2 if and only if A1⊆A2

and λ1�⊇ λ2.

Definition 16. Let �j = ⟨Aj, λj

⟩ ∈ CXN, where j ∈

{1, 2, . . . , n}, we define

1. ∪pj∈I

�j ={⟨

x,

(∪

j∈IAj

)(x),

(�∪

j∈Iλj

)(x)

⟩|x ∈ X

}(P-union)

2. ∩pj∈I

�j ={⟨

x,

(∩

j∈IAj

)(x),

(�∩

j∈Iλj

)(x)

⟩|x ∈ X

}(P-intersection)

3. ∪Rj∈I

�j ={⟨

x,

(∪

j∈IAj

)(x),

(�∩

j∈Iλj

)(x)

⟩|x ∈ X

}(R-union)

4. ∩Rj∈I

�j ={⟨

x,

(∩

j∈IAj

)(x),

(�∪

j∈Iλj

)(x)

⟩|x ∈ X

}(R-intersection)

Definition 17. Let � = 〈A, λ〉. The complement of� = 〈A, λ〉 is defined as

�c ={⟨

x, Ac(x), λc(x) : x ∈ X⟩}

Theorem 4. Let �j = ⟨Aj, λj


{1, 2, . . . , n}, the following holds.

1.

(∪pj∈�

�j

)c

= ∩pj∈�

(�j)C and

(∩pj∈�

�j

)c

=

∪pj∈�

(�j)C,

2.

(∪Rj∈�

�j

)c

= ∩Rj∈�

(�j)C and

(∩Rj∈�

�j

)c

=∪Rj∈�

(�j)C.

Theorem 5. Let �j = ⟨Aj, λj


{1, 2, 3, 4}. Then

1. If �1 ⊆p �2 and �2 ⊆p �3, then �1 ⊆p �3.2. If �1 ⊆p �2, then �c

2 ⊆p �c1.

3. If �1 ⊆p �2 and �1 ⊆p �3, then �1 ⊆p (�2 ∩p

�3).4. If �1 ⊆p �2 and �3 ⊆p �2, then (�1 ∪p

�3) ⊆p �2.5. If �1 ⊆p �2 and �3 ⊆p �4, then

(�1 ∪p �3) ⊆p (�2 ∪p �4) and(�1 ∩p �3) ⊆p (�2 ∩p �4).

6. If �1 ⊆R �2 and �2 ⊆R �3, then �1 ⊆R �3.7. If �1 ⊆R �2, then �c

2 ⊆R �c1.

8. If �1 ⊆R �2 and �1 ⊆R �3, then �1 ⊆R

(�2 ∩R �3).9. If �1 ⊆R �2 and �3 ⊆R �2, then (�1 ∪R

�3) ⊆R �2.10. �1 ⊆R �2 and �3 ⊆R �4, then (�1 ∪R �3) ⊆R

(�2 ∪R �4) and (�1 ∩R �3) ⊆R (�2 ∩R �4)

Proof. Straightforward.

Theorem 6. Let � = 〈A, λ〉 ∈ CXN.

a. (�c)c = �.b. If � is an INCS, then �c is an INCS.c. If � is an ENCS, then �c is an ENCS.

Proof. It is easy.

Definition 18. Let �1 = 〈A1, λ1〉 and �2 = 〈A2, λ2〉be neutrosophic cubic sets in X. Then, distance mea-sure between �1 = 〈A1, λ1〉 and �2 = 〈A2, λ2〉 isgiven by

d (�1, �2)

= 1

9n

∑n

i=1

(∣∣∣A−1T

(xi) − A−2T

(xi)∣∣∣

+∣∣∣A−

1I(xi) − A−

2I(xi)

∣∣∣

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+∣∣∣A−

1F(xi) − A−

2F(xi)

∣∣∣+

∣∣∣A+1T

(xi) − A+2T

(xi)∣∣∣

+∣∣∣A+

1I(xi) − A+

2I(xi)

∣∣∣+

∣∣∣A+1F

(xi) − A+2F

(xi)∣∣∣ + |T1 (xi) − T2 (xi)|

+ |I1 (xi) − I2 (xi)| + |F1 (xi) − F2 (xi)|)It is adapted from [25].

Theorem 7. Let �1 = 〈A1, λ1〉 and �2 = 〈A2, λ2〉be two neutrosophic cubic sets. The functiond(�1, �2) = CX

N → R+ given by Definition 18 ismetrics, where R+ is the set of all non-negative realnumbers.

Proof. The proof is straightforward.

Theorem 8. Let �1 = 〈A1, λ1〉 and �2 = 〈A2, λ2〉 beneutrosophic cubic sets. Then,

i. d(�1, �2) ≥ 0;ii. d(�1, �2) = d(�2, �1);

iii. d(�1, �2) = 0 iff �1 = �2;iv. d(�1, �2) + d(�2, �3) ≥ d(�1, �3).

Proof. The proof is straightforward.

3. An application in pattern recognitionproblem

In this section we proposed an decisin makingmethod based on similarity measures of two NCS inpattern recognition problems which is adapted from[25]. In this method, we assume that if similaritybetween the ideal pattern and sample pattern is greaterthan or equal to 0.5, then the sample pattern belongsto the family of ideal pattern in consideration. Thealgorithm of this method is as follows:

Algorithm:Step 1. Construct an ideal NCS � = 〈A, λ〉 on X.Step 2. Construct NCSs �j = ⟨

Aj, λj

⟩, j =

1, 2, 3, . . . , n, on X for the sample patterns whichare to recognized.

Step 3. Calculate the distances d(�, �j), j =1, 2, 3, . . . , n.

Step 4. If d(�, �j) ≤ 0.5 then the pattern �j is tobe recognized to belong to the ideal Pattern � and ifd(�, �j) > 0.5 then the pattern�j is to be recognizednot to belong to the ideal Pattern �.

Now we give an example which is adapted from[25].

Example 6. Here a fictitious numerical exampleis given to illustrate the application of similaritymeasures between two NCSs in pattern recognitionproblem. In this example we take three sample pat-terns which are to be recognized. LetX = {x1, x2, x3}be the universe. Also let � be NCS set of the idealpattern and �j = ⟨

Aj, λj

⟩, j = 1, 2, 3 be the NCSs

of three sample patterns.Step 1. Construct an ideal NCS � = 〈A, λ〉 on X

as;

� =⟨⎧⎪⎪⎨

⎪⎪⎩〈[0.2, 0.4], [0.3, 0.5], [0.3, 0.5]〉 /x1+〈[0.5, 0.7], [0.0, 0.5], [0.2, 0.3]〉 /x2+〈[0.6, 0.8], [0.0, 0.1], [0.3, 0.4]〉 /x3

⎫⎪⎪⎬⎪⎪⎭ ,

{ 〈0.1, 0.2, 0.4〉 /x1, 〈0.1, 0.2, 0.2〉 /x2,

〈0.3, 0.1, 0.7〉 /x3

}⟩

Step 2. Construct NCSs �j = ⟨Aj, λj

⟩, j =

1, 2, 3 on X for the sample patterns as;

�1 =⟨⎧⎪⎪⎨

⎪⎪⎩〈[0.2, 0.5], [0.4, 0.5], [0.3, 0.5]〉 /x1+〈[0.7, 0.7], [0.1, 0.3], [0.1, 0.3]〉 /x2+〈[0.6, 0.8], [0.5, 0.6], [0.3, 0.4]〉 /x3

⎫⎪⎪⎬⎪⎪⎭ ,

{〈0.5, 0.6, 0.4〉 /x1, 〈0.2, 0.2, 0.2〉 /x2,

〈0.3, 0.1, 0.7〉 /x3

}⟩

�2 =⟨⎧⎪⎪⎨

⎪⎪⎩〈[0.3, 0.7], [0.3, 0.5], [0.3, 0.9]〉 /x1+〈[0.6, 0.7], [0.1, 0.8], [0.2, 0.3]〉 /x2+〈[0.6, 0.9], [0.9, 1.0], [0.3, 0.4]〉 /x3

⎫⎪⎪⎬⎪⎪⎭ ,

{〈0.2, 0.5, 0.2〉 /x1, 〈0.1, 0.5, 0.7〉 /x2,

〈0.7, 0.1, 0.7〉 /x3

}⟩

�3 =⟨⎧⎪⎪⎨

⎪⎪⎩〈[0.8, 0.9], [0.1, 0.2], [0.8, 0.9]〉 /x1+〈[0.1, 0.2], [0.8, 0.9], [0.3, 0.9]〉 /x2+〈[0.5, 0.9], [0.1, 1.0], [0.4, 0.7]〉 /x3

⎫⎪⎪⎬⎪⎪⎭ ,

{〈0.9, 0.8, 0.9〉 /x1, 〈0.9, 0.8, 0.9〉 /x2,

〈0.9, 0.9, 0.1〉 /x3

}⟩

AU

THO

R C

OP

Y


Step 3. Calculate the distances d(�, �j), j =1, 2, 3 as;

d(�, �j) = 0.10

d(�, �j) = 0.19

d(�, �j) = 0.51

Step 4. Since d(�, �1) ≤ 0.5, d(�, �2) ≤ 0.5 andd(�, �3) > 0.5, the sample patterns whose corre-sponding NCS sets are represented by �1 and �2 arerecognized as similar patterns of the family of idealpattern whose NCS set is represented by � and thepattern whose NCS is represented by �3 does notbelong to the family of ideal pattern �.

4. Conclusion

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

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The theory of neutrosophic cubic sets and their applications in pattern recognition