Reach Us +441474556909

Some Operations and Properties of Neutrosophic Cubic Soft Set

Pramanik S1*, Dalapati S2, Alam S2 and Roy TK2

1Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, West Bengal, India

2Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India

*Corresponding Author:
Surapati Pramanik
Department of Mathematics, Nandalal Ghosh B.T. College
Panpur, Narayanpur, North 24 Parganas-743126
West Bengal, India
Tel: 0332580 1826
E-mail: [email protected]

Received Date: April 01, 2017; Accepted Date: April 21, 2017; Published Date: April 30, 2017

Citation: Pramanik S, Dalapati S, Alam S, et al. Some Operations and Properties of Neutrosophic Cubic Soft Set. Glob J Res Rev. 2017, 4:2.

 
Visit for more related articles at Global Journal of Research and Review

Abstract

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.

Keywords

Neutrosophic cubic soft set; Neutrosophic soft set; Cubic set; Internal neutrosophic Cubic soft set; External neutrosophic cubic soft set

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 [6-9], intuitionistic fuzzy soft set [10-13], interval valued intuitionistic fuzzy soft set [14], neutrosophic soft set [15-18], 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, equation. Ali et al. [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.

In this paper we discuss some new operations and new approach of internal and external neutrosophic cubic soft sets, and P-union, R-union, P-intersection, R-intersection. We also prove some theorems related to neutrosophic cubic soft sets.

Rest of the paper is presented as follows. Section 2 presents some basic definition of neutrosophic sets, interval-valued neutrosophic sets, soft sets, cubic set, neutrosophic cubic sets and their basic operation. Section 3 is devoted to presents some new theorems related to neutrosophic cubic soft sets. Section 4 presents conclusions and future scope of research.

Preliminaries

In this section, we recall some well-established definitions and properties which are related to the present study.

Definition 1: Neutrosophic set [1]

Let U be the space of points with generic element in U denoted by u. A neutrosophic set λ in U is defined as λ={<u, tλ (u), iλ (u), fλ (u)>:u∈U} , where tλ (u):U →]- 0, 1+ [,iλ (u):U →]- 0, 1+ [, and fλ (u):U →]- 0, 1+ [ and − 0 ≤ tλ (u)+ iλ (u)+fλ (u) ≤3+.

Definition 2: Interval value neutrosophic set [31]

Let U be the space of points with generic element in U denoted by u. An interval neutrosophic set A in U is characterized by truthmembership function tA, the indeterminacy function iA and falsity membership function fA. For each u∈U, tA (u), iA (u), fA (u) ⊆ [0, 1] and A is defined as

A={<u, [ t+A (u), t+A (u)], [ iA (u), i+A (u)], [ fA (u), f+A (u)]:u∈U}.

Definition 3: Neutrosophic cubic set [32]

Let U be the space of points with generic element in U denoted by u∈U. A neutrosophic cubic set in U defined as equation ={< u, A (u), 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 equation = <A, λ>. We use Cequation(U) 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 equation is said to be neutrosophic cubic soft set iff equation is the mapping from K to the set of all neutrosophic cubic sets in U (i.e., Cequation(U) ).

i.e. equation : K→Cequation(U) , where K is any subset of parameter set E and U is the initial universe set.

Neutrosophic cubic soft set is defined by

equation

Where, A(ei) is the interval valued neutrosophic soft set and λ(ei) is the neutrosophic soft set.

Definition: Internal neutrosophic cubic soft set (INCSS)

A neutrosophic cubic soft set equation is said to be INCSS if for all ei∈ K E

equation

Definition: External neutrosophic cubic soft set (ENCSS)

A neutrosophic cubic soft set equation is said to be ENCSS if for all ei∈ K E

equation,

equation

Some theorem related to these topics

Theorem 1

Let equation be a neutrosophic cubic soft set in U which is not an ENCSS. Then there exists at least one ei ∈ K ⊆ E for which there exists some u∈U such that

equation

Proof

From the definition of ENCSSs, we have

equation

equation , for all u∈U, corresponding to each ei∈ K ⊆ E.

But given that equation is not an ENCSS, so at least one ei ∈ K ⊆ E.

There exists some u∈U such that equation, equation. Hence the proof is complete.

Theorem 2

Let equationbe a NCSS in U. If equation is both an INCSS and ENCSS in U for all u∈U, corresponding to each ei ∈ K, then, equation , equation ,where

equation

Proof

Suppose equation be both an INCSS and ENCSS corresponding to each ei ∈ K and for all u∈U. We have equation, equation, equation,

Again by definition of ENCSS corresponding to each ei K ∈ and for all u∈U, we have

equation, equation, equation

Since equationis both an ENCSS and INCSS, so only possibility is that, equation , equation

Hence proved.

Definition

Let equation be two neutrosophic cubic soft sets in U and K1, K2 be any two subsets of K. Then, we define the following:

1. equation

2. Ifequation and . are two NCSSs then we define P- order asequation iff the following conditions are satisfied:

i. K1 ⊆ K2, and

ii. equation

A(ei) B(ei) and 1 (e ) 2 (e ) u U ⊆ λ ⊆λ ∀ ∈ corresponding to each

equation

3. If equation are two neutrosophic cubic soft sets, then we
define the R-order as equation iff the following conditions are satisfied:

i. K1 ⊆ K2 and

ii. equation for all ei ∈ K1 iff equationcorresponding to each ei ∈ K1.

Definition

Let equation be two NCSSs in U and K1, K2 be any two subsets of parameter set K. Then we define P-union as equation, where K3 ∈ K1∪ K2,

equation

equation

Definition

Let equation be two NCSSs in U and K1, K2 be any two subsets of parameter set K. The P-intersection of equation is denoted by equation where equation and equation defined asequation=equation

Here, equation

Definition: Compliment

The compliment of equation denoted by equation is defined by

equation

Where,

equation

Some properties of P-union and P-intersection

equation

Proof 1:

equation

Here,

equation

Hence the proof.

Proof 2:

equation

Hence the proof.

Definition: R-union and R-intersection

Let equation be two NCSSs over U. Then R-union is denoted as equation, where K3 = K1∪ K2 and equation . Then R-union is defined as

equation

Here equation defined as

equation

R-intersection is denoted as equation

K2.Then R-intersection is defined as:

equation

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 equation the following properties hold

equation

Proof:equation

equation

Hence the proof.

ii. equation

Proof: If equation

equation

equation (1)

equation (2)

Hence the proof.

Theorem 4

Let equation be a NCSS over U,

If equation is an INCSS then equation is also an INCSS.

If equation is an ENCSS then equation is also an ENCSS.

Proof

equation

Theorem 5

Let equation and equation be any two INCSSs then

equation is an INCSS.

equation is an INCSS.

Proof

Since equation are INCSSs, so for equation we have

equation

Also for equation we have

equation

Now by the definition of P-union equation is an INCSS.

equation

ii. Now, equation and by definition,

equation

Theorem 6

Let equation be any two INCSSs over U having the conditions:

equation

Proof

Since equation are INCSSs in U.

So for equation , we have

equation

Also for equation we have

equation

equation

Since equation are INCSSs so from the given condition and definition of INCSS we can write,

equation

If ei ∈ I - J or ei ∈ J - I then, the result is trivial.

Thus equation is an INCSS.

Theorem 7

Let equation be any two INCSSs over U satisfying the condition: equation

Then equationis an INCSS.

Proof

Since equation are INCSSs in U,

we have, equation and equation

Also for equation we have, equation, equation and equation

equation

equation

defined as

equation

Given condition that equationequation Thus from given condition and definition of INCSSs

equation

Hence equation is an INCSS.

Theorem 8

Let equation be any two ENCSSs then

equation

Proof

Since equation and are ENCSSs, we have

equation

equation

equation

Now by definition of P-union, equation.

equation

Here equation defined as

equation

Thus equation

For ei ∈ I - J or ei ∈ J - I these results are trivial.

Hence the proof.

ii. Since equation are ENCSSs, then

equation

and equation

equation

equation

equation

we have

equation

Here equation defined as

equation

Thus equation is an ENCSS.

Theorem 9

Let equation be any two INCSSs in U such that

equation

Proof:

Since equation are INCSSs in U.

we have, equation

equation

equation

equation

Definition

Let equationand equation are two NCSSs in U. We defined new NCSSs by interchanging the neutrosophic part of the two NCSSs. We denoted its by equation and defined byequation, equation respectively.

Theorem 10

equation are ENCSSs andequation are INCSSs in U. Thenequation is an INCSS in U.

Proof

Since equationand equationare ENCSSs,

we have

equation

equation By the definition of ENCSSs and INCSSs all the possibility are as under:

equation

Case 1

If equation then from i(a)., and ii(a). We have

equation

Case 2

equation then from i(b) and ii(b). , we have

equation

Case 3

equation, then from i(a) and i(b)., we have

equation

in all the three cases.

equation is an INCSS in U.

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.

References

 

Select your language of interest to view the total content in your interested language

Viewing options

Post your comment

Share This Article

Flyer image
journal indexing image
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh