Next Article in Journal
Multi-Label Classification from Multiple Noisy Sources Using Topic Models
Previous Article in Journal
The Diffraction Research of Cylindrical Block Effect Based on Indoor 45 GHz Millimeter Wave Measurements
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Subtraction and Division Operations of Simplified Neutrosophic Sets

Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing 312000, China
Information 2017, 8(2), 51; https://doi.org/10.3390/info8020051
Submission received: 4 April 2017 / Revised: 1 May 2017 / Accepted: 2 May 2017 / Published: 4 May 2017
(This article belongs to the Section Information Theory and Methodology)

Abstract

:
A simplified neutrosophic set is characterized by a truth-membership function, an indeterminacy-membership function, and a falsity-membership function, which is a subclass of the neutrosophic set and contains the concepts of an interval neutrosophic set and a single valued neutrosophic set. It is a powerful structure in expressing indeterminate and inconsistent information. However, there has only been one paper until now—to the best of my knowledge—on the subtraction and division operators in the basic operational laws of neutrosophic single-valued numbers defined in existing literature. Therefore, this paper proposes subtraction operation and division operation for simplified neutrosophic sets, including single valued neutrosophic sets and interval neutrosophic sets respectively, under some constrained conditions to form the integral theoretical framework of simplified neutrosophic sets. In addition, we give numerical examples to illustrate the defined operations. The subtraction and division operations are very important in many practical applications, such as decision making and image processing.

1. Introduction

To handle uncertainty, imprecise, incomplete, and inconsistent information, Smarandache [1] proposed the concept of a neutrosophic set from philosophical viewpoint, which is a powerful general formal framework and generalizes the concepts of the classic set, fuzzy set [2], intuitionistic fuzzy set (IFS) [3], interval-valued intuitionistic fuzzy set (IVIFS) [4]. In the neutrosophic set, a truth-membership function ρ(x), an indeterminacy-membership function σ(x), and a falsity-membership function τ(x) are characterized independently, where ρ(x), σ(x), and τ(x) are real standard or nonstandard subsets of ]0, 1+[, such that ρ(x): X → ]0, 1+[, σ(x): X → ]0, 1+[, and τ(x): X → ]0, 1+[. Thus, the sum of ρ(x), σ(x), and τ(x) satisfies the condition 0 ≤ supρ(x) + supσ(x) + supτ(x) ≤ 3+. Then, the obvious advantage of the neutrosophic set is that its components are best fit in the representation of indeterminacy and inconsistent information, because IFSs and IVIFSs cannot represent indeterminacy and inconsistent information. However, the defined range of ρ(x), σ(x), and τ(x) in a neutrosophic set is the non-standard unit interval ]0, 1+[, the neutrosophic set is merely used for philosophical applications, while its engineering applications are difficult. Hence, the defined range of ρ(x), σ(x), and τ(x) can be constrained to the real standard unit interval [0, 1] for convenient engineering applications. Thus, a single valued neutrosophic set (SVNS) [5] and an interval neutrosophic set (INS) [6] were introduced as the subclasses of a neutrosophic set. Further, Ye [7] introduced a simplified neutrosophic set (SNS), which is a subclass of the neutrosophic set and contains a SVNS and an INS, and defined some basic operational laws over SNSs, such as addition and multiplication. Then, Zhang et al. [8] indicated some unreasonable phenomena of the basic operational laws over SNSs in [7] and improved some basic operational laws of INSs. Recently, Smarandache [9] defined the subtraction and division operators in the basic operational laws of neutrosophic single-valued numbers and presented some restrictions for these operations of neutrosophic single-valued numbers and neutrosophic single-valued overnumbers/undernumbers/offnumbers. However, as far as we know, there has not been any investigation on the subtraction and division operations over SNSs (SVNSs and INSs) until now. Since SNSs are the extension of IFSs and IVIFSs, the subtraction and division operations are very important in forming the integral theoretical framework of SNSs. Then, the subtraction and division operations of IFSs and IVIFSs have been investigated by some researchers [10,11,12,13,14]. Motivated by the subtraction and division operations for IFSs and IVIFSs, this paper will try to develop the two new subtraction and division operations for SNSs to form the integral theoretical framework of SNSs.
The remainder of this paper is arranged as follows. Section 2 describes some basic knowledge on SNSs and their basic operations. Section 3 proposes the subtraction operation and division operation over SNSs. Some remarks are contained in Section 4.

2. Some Basic Knowledge of SNSs and Their Basic Operations

For the science and engineering applications of neutrosophic sets, Ye [7] introduced the SNS concept, which is a subclass of the neutrosophic set, and gave the following definition of a SNS.
Definition 1
[7]. Let X be a universal of discourse. A SNS N in X is characterized by a truth-membership function ρN(x), an indeterminacy-membership function σN(x), and a falsity-membership function τN(x), where the functions ρN(x), σN(x) and τN(x) are singleton subintervals/subsets in the real standard interval [0, 1], such that ρN(x): X → [0, 1], σN(x): X → [0, 1], and τN(x): X → [0, 1]. Thus, a SNS N is denoted by
N = { x , ρ N ( x ) , σ N ( x ) , τ N ( x ) | x X } .
The SNS is a subclass of the neutrosophic set and contains the concepts of INS and SVNS [7].
If the values of the three functions ρN(x), σN(x) and τN(x) in the SNS N are taken as three real numbers, i.e., ρN(x), σN(x), τN(x) [0, 1], then the SNS N is reduced to the SVNS N. Thus, the sum of ρN(x), σN(x), and τN(x) satisfies the condition 0 ≤ ρN(x) + σN(x) + τN(x) ≤ 3.
For SVNSs N 1 = { x , ρ N 1 ( x ) , σ N 1 ( x ) , τ N 1 ( x ) | x X } and N 2 = { x , ρ N 2 ( x ) , σ N 2 ( x ) , τ N 2 ( x ) | x X } , there are the following relations [5]:
(1)
Complement: N 1 c = { x , τ N 1 ( x ) , 1 σ N 1 ( x ) , ρ N 1 ( x ) | x X } ;
(2)
Inclusion: N1 N2 if and only if ρ N 1 ( x ) ρ N 2 ( x ) , σ N 1 ( x ) σ N 2 ( x ) , and τ N 1 ( x ) τ N 2 ( x ) for any x in X;
(3)
Equality: N1 = N2 if and only if N1 N2 and N2 N1.
Since SVNSs are the special case of INSs, the operational laws of the SVNSs N1 and N2 are introduced as follows [8]:
(1)
N 1 + N 2 = { x , ρ N 1 ( x ) + ρ N 2 ( x ) ρ N 1 ( x ) ρ N 2 ( x ) , σ N 1 ( x ) σ N 2 ( x ) , τ N 1 ( x ) τ N 2 ( x ) | x X } ;
(2)
N 1 × N 2 = { x , ρ N 1 ( x ) ρ N 2 ( x ) , σ N 1 ( x ) + σ N 2 ( x ) σ N 1 ( x ) σ N 2 ( x ) , τ N 1 ( x ) + τ N 2 ( x ) τ N 1 ( x ) τ N 2 ( x ) | x X } ;
(3)
λ N 1 = { x , 1 ( 1 ρ N 1 ( x ) ) λ , σ N 1 λ ( x ) , τ N 1 λ ( x ) | x X } , λ > 0 ;
(4)
N 1 λ = { x , , ρ N 1 λ ( x ) , 1 ( 1 σ N 1 ( x ) ) λ , 1 ( 1 τ N 1 ( x ) ) λ | x X } , λ > 0 .
If the values of the three functions ρN(x), σN(x), and τN(x) in the SNS N are taken as three interval numbers, i.e., ρN(x), σN(x), τN(x) [0, 1], then the SNS N is reduced to the INS N. Thus, the sum of ρN(x), σN(x), and τN(x) satisfies the condition 0 ≤ supρN(x) + supσN(x) + supτN(x) ≤ 3.
For INSs N 1 = { x , ρ N 1 ( x ) , σ N 1 ( x ) , τ N 1 ( x ) | x X } and N 2 = { x , ρ N 2 ( x ) , σ N 2 ( x ) , τ N 2 ( x ) | x X } , there are the following relations [6]:
(1)
Complement: N 1 c = { x , [ inf τ N 1 ( x ) , sup τ N 1 ( x ) ] , [ 1 sup σ N 1 ( x ) , 1 inf σ N 1 ( x ) ] , [ inf ρ N 1 ( x ) , sup ρ N 1 ( x ) ] | x X } ;
(2)
Inclusion: N1 N2 if and only if inf ρ N 1 ( x ) inf ρ N 2 ( x ) , sup ρ N 1 ( x ) sup ρ N 2 ( x ) , inf σ N 1 ( x ) inf σ N 2 ( x ) , sup σ N 1 ( x ) sup σ N 2 ( x ) , inf τ N 1 ( x ) inf τ N 2 ( x ) , and sup τ N 1 ( x ) sup τ N 2 ( x ) for any x in X;
(3)
Equality: N1 = N2 if and only if N1 N2 and N2 N1.
Whereas, the operational laws of the INSs N1 and N2 are introduced as follows [8]:
(1)
N 1 + N 2 = { x , [ inf ρ N 1 ( x ) + inf ρ N 2 ( x ) inf ρ N 1 ( x ) inf ρ N 2 ( x ) , sup ρ N 1 ( x ) + sup ρ N 2 ( x ) sup ρ N 1 ( x ) sup ρ N 2 ( x ) ] , [ inf σ N 1 ( x ) inf σ N 2 ( x ) , sup σ N 1 ( x ) sup σ N 2 ( x ) ] , [ inf τ N 1 ( x ) inf τ N 2 ( x ) , sup τ N 1 ( x ) sup τ N 2 ( x ) ] | x X } ;
(2)
N 1 × N 2 = { x , [ inf ρ N 1 ( x ) inf ρ N 2 ( x ) , sup ρ N 1 ( x ) sup ρ N 2 ( x ) ] , [ inf σ N 1 ( x ) + inf σ N 2 ( x ) inf σ N 1 ( x ) inf σ N 2 ( x ) , sup σ N 1 ( x ) + sup σ N 2 ( x ) sup σ N 1 ( x ) sup σ N 2 ( x ) ] [ inf τ N 1 ( x ) + inf τ N 2 ( x ) inf τ N 1 ( x ) inf τ N 2 ( x ) , sup τ N 1 ( x ) + sup τ N 2 ( x ) sup τ N 1 ( x ) sup τ N 2 ( x ) ] | x X } ;
(3)
λ N 1 = { x , [ 1 ( 1 inf ρ N 1 ( x ) ) λ , 1 ( 1 sup ρ N 1 ( x ) ) λ ] [ inf σ N 1 λ ( x ) , sup σ N 1 λ ( x ) ] , [ inf τ N 1 λ ( x ) , sup τ N 1 λ ( x ) ] | x X } , λ > 0 ;
(4)
N 1 λ = { x , [ inf ρ N 1 λ ( x ) , sup ρ N 1 λ ( x ) ] , [ 1 ( 1 inf σ N 1 ( x ) ) λ , 1 ( 1 sup σ N 1 ( x ) ) λ ] , [ 1 ( 1 inf τ N 1 ( x ) ) λ , 1 ( 1 sup τ N 1 ( x ) ) λ ] | x X } , λ > 0 .
However, there is little any investigation on the subtraction and division operations over SNSs (SVNSs and INSs) in existing literature. Therefore, we should investigate the subtraction and division operations of SNSs to form the integral theoretical framework of SNSs.

3. Subtraction and Division Operations over SNSs

3.1. Subtraction and Division Operations over SVNSs

For any two given SVNSs A and B, the problem is how to find the unknown SVNS C = AB, which satisfies ρC(x), σC(x), τC(x) [0, 1] and 0 ≤ ρC(x) + σC(x) + τC(x) ≤ 3 for x X. Based on the addition operation of SVNSs for A = C + B, there are ρA(x) = ρC(x) + ρB(x) − ρC(x)ρB(x), σA(x) = σC(x)σB(x), and τA(x) = τC(x)τB(x). Then, we can obtain the following results:
ρ C ( x ) = ρ A ( x ) ρ B ( x ) 1 ρ B ( x )
σ C ( x ) = σ A ( x ) σ B ( x )
τ C ( x ) = τ A ( x ) τ B ( x )
Then, the functions ρC(x), σC(x), τC(x) in C must take values in the interval [0, 1] and satisfy the following conditions:
0 ρ A ( x ) ρ B ( x ) 1 ρ B ( x ) 1   for ρ B ( x ) ρ A ( x ) and ρ B ( x ) 1
0 σ A ( x ) σ B ( x ) 1   for σ B ( x ) σ A ( x ) and σ B ( x ) 0
0 τ A ( x ) τ B ( x ) 1   for τ B ( x ) τ A ( x ) and τ B ( x ) 0
Obviously, the inequalities (4)–(6) hold only if AB, ρB(x) ≠ 1, σB(x) ≠ 0, and τB(x) ≠ 0. Therefore, we can give the following definition of subtraction operation for SVNSs.
Definition 2.
For any two given SVNSs A and B, the subtraction operation of the SVNSs A and B are defined as
A B = { x , ρ A ( x ) ρ B ( x ) 1 ρ B ( x ) , σ A ( x ) σ B ( x ) , τ A ( x ) τ B ( x ) | x X }
which is valid under the conditions A ≥ B, ρB(x) ≠ 1, σB(x) ≠ 0, and τB(x) ≠ 0.
Example 1.
Let us consider two SVNSs A and B in the universe of discourse X = {x}: A = {<x, 0.7, 0.4, 0.3>| x X} and B = {<x, 0.5, 0.6, 0.5>| x X}. Then, we find the unknown SVNS C = A − B.
By using Equation (7), we can obtain that
C = A B = { x , 0.7 0.5 1 0.5 , 0.4 0.6 , 0.3 0.5 | x X } = { x , 0.4 , 0.6667 , 0.6 | x X } .
For any two given SVNSs A and B, the problem is how to find the unknown SVNS D = A/B, which satisfies ρD(x), σD(x), τD(x) [0, 1] and 0 ≤ ρD(x) + σD(x) + τD(x) ≤ 3 for x X. Based on the multiplication operation of SVNSs for A = D × B, there are ρA(x) = ρD(x)ρB(x), σA(x) = σD(x) + σB(x) − σD(x)σB(x), and τA(x) = τD(x) + τB(x) − τD(x)τB(x). Then, we can obtain the following results:
ρ D ( x ) = ρ A ( x ) ρ B ( x )
σ D ( x ) = σ A ( x ) σ B ( x ) 1 σ B ( x )
τ D ( x ) = τ A ( x ) τ B ( x ) 1 τ B ( x )
Then, the functions ρD(x), σD(x), τD(x) in D must take the values in the interval [0, 1] and satisfy the following conditions:
0 ρ A ( x ) ρ B ( x ) 1   for ρ B ( x ) ρ A ( x ) and ρ B ( x ) 0
0 σ A ( x ) σ B ( x ) 1 σ B ( x ) 1   for σ B ( x ) σ A ( x ) and σ B ( x ) 1
0 τ A ( x ) τ B ( x ) 1 τ B ( x ) 1   for τ B ( x ) τ A ( x ) and τ B ( x ) 1
Obviously, the inequalities (11)–(13) hold only if BA, ρB(x) ≠ 0, σB(x) ≠ 1, and τB(x) ≠ 1. Thus, we can give the following definition of division operation for SVNSs A and B.
Definition 3.
For any two given SVNSs A and B, the division operation of the SVNSs A and B is defined as
A / B = { x , ρ A ( x ) ρ B ( x ) , σ A ( x ) σ B ( x ) 1 σ B ( x ) , τ A ( x ) τ B ( x ) 1 τ B ( x ) | x X }
which is valid under the conditions B ≥ A, ρB(x) ≠ 0, σB(x) ≠ 1, and τB(x) ≠ 1.
Example 2.
Let us consider two SVNSs A and B in the universe of discourse X = {x}: A = {<x, 0.6, 0.3, 0.2>| x X} and B = {<x, 0.8, 0.2, 0.1>| x X}. Then, we need to find the unknown SVNS D = A/B.
According to Equation (14), we have that
D = A / B = { x , 0.6 0.8 , 0.3 0.2 1 0.2 , 0.2 0.1 1 0.1 | x X } = { x , 0.75 , 0.125 , 0.1111 | x X } .

3.2. Subtraction and Division Operations of INSs

Based on the aforementioned discussions in the subtraction and division operations of SVNSs, we can give the subtraction and division operations of INSs as the extension of the subtraction and division operations of SVNSs.
If we only consider ρA(x), σA(x), τA(x) [0, 1] and ρB(x), σB(x), τB(x) [0, 1] in two SNSs A and B as interval numbers, i.e., two INSs A and B, the problem is how to find the unknown INS C = AB, which satisfies ρC(x), σC(x), τC(x) [0, 1] and 0 ≤ supρC(x) + supσC(x) + supτC(x) ≤ 3 for x X. Based on the addition operation of INSs for A = C + B, there are ρA(x) = [infρC(x) + infρB(x) − infρC(x)infρB(x), supρC(x) + supρB(x) − supρC(x)supρB(x)], σA(x) = [infσC(x)infσB(x), supσC(x)supσB(x)], and τA(x) = [infτC(x)infτB(x), supτC(x)supτB(x)]. Then, we can obtain
ρ C ( x ) = [ inf ρ A ( x ) inf ρ B ( x ) 1 inf ρ B ( x ) , sup ρ A ( x ) sup ρ B ( x ) 1 sup ρ B ( x ) ]
σ C ( x ) = [ inf σ A ( x ) inf σ B ( x ) , sup σ A ( x ) sup σ B ( x ) ]
τ C ( x ) = [ inf τ A ( x ) inf τ B ( x ) , sup τ A ( x ) sup τ B ( x ) ]
Then, the functions ρC(x), σC(x), τC(x) in C must take the subintervals in the real standard interval [0, 1]:
ρ C ( x ) = [ inf ρ A ( x ) inf ρ B ( x ) 1 inf ρ B ( x ) , sup ρ A ( x ) sup ρ B ( x ) 1 sup ρ B ( x ) ] [ 0 , 1 ] for inf ρ A ( x ) inf ρ B ( x ) , sup ρ A ( x ) sup ρ B ( x ) and ρ B ( x ) [ 1 , 1 ]
σ C ( x ) = [ inf σ A ( x ) inf σ B ( x ) , sup σ A ( x ) sup σ B ( x ) ] [ 0 , 1 ] for inf σ B ( x ) inf σ A ( x ) , sup σ B ( x ) sup σ A ( x ) and σ B ( x ) [ 0 , 0 ]
τ C ( x ) = [ inf τ A ( x ) inf τ B ( x ) , sup τ A ( x ) sup τ B ( x ) ] [ 0 , 1 ] for inf τ B ( x ) inf τ A ( x ) , sup τ B ( x ) sup τ A ( x ) and τ B ( x ) [ 0 , 0 ]
Obviously, Equations (18)–(20) hold only if AB, ρB(x) ≠ [1, 1], σB(x) ≠ [0, 0], and τB(x) ≠ [0, 0]. Thus, we can give the following definition of subtraction operation for INSs A and B.
Definition 4.
If we consider two INSs A and B, the subtraction operation of the INSs A and B is defined as
A B = { x , [ inf ρ A ( x ) inf ρ B ( x ) 1 inf ρ B ( x ) , sup ρ A ( x ) sup ρ B ( x ) 1 sup ρ B ( x ) ] ,   [ inf σ A ( x ) inf σ B ( x ) , sup σ A ( x ) sup σ B ( x ) ] , [ inf τ A ( x ) inf τ B ( x ) , sup τ A ( x ) sup τ B ( x ) ] | x X }
which is valid under the conditions A ≥ B, ρB(x) ≠ [1, 1], σB(x) ≠ [0, 0], and τB(x) ≠ [0, 0].
Example 3.
Let us consider two INSs A and B in the universe of discourse X = {x}: A = {<x, [0.7, 0.9], [0.4, 0.5], [0.3, 0.4]>| x X} and B = {<x, [0.5, 0.7], [0.5, 0.6], [0.5, 0.6]>| x X}. Then, we need to obtain the unknown INS C = A − B.
By using Equation (21), we can obtain that
C = A B = { x , [ 0.7 0.5 1 0.5 , 0.9 0.7 1 0.7 ] , [ 0.4 0.5 , 0.5 0.6 ] , [ 0.3 0.5 , 0.4 0.6 ] | x X } = { x , [ 0.4 , 0.6667 ] , [ 0.8 , 0.8333 ] , [ 0.6 , 0.6667 ] | x X } .
If we only consider ρA(x), σA(x), τA(x) [0, 1] and ρB(x), σB(x), τB(x) [0, 1] in two SNSs A and B as interval numbers, i.e., two INSs A and B, the problem is how to find the unknown INS D = A/B, which satisfies ρD(x), σD(x), τD(x) [0, 1] and 0 ≤ supρD(x) + supσD(x) + supτD(x) ≤ 3 for x X. Based on the multiplication operation of INSs for A = D × B, there are ρA(x) = [infρD(x)infρB(x), supρD(x)supρB(x)], σA(x) = [infσD(x) + infσB(x) − infσD(x)infσB(x), supσD(x) + supσB(x) − supσD(x)supσB(x)], and τA(x) = [infτD(x) + infτB(x) − infτD(x)infτB(x), supτD(x) + supτB(x) − supτD(x)supτB(x)]. Then, we can obtain the following:
ρ D ( x ) = [ inf ρ A ( x ) inf ρ B ( x ) , sup ρ A ( x ) sup ρ B ( x ) ]
σ D ( x ) = [ inf σ A ( x ) inf σ B ( x ) 1 inf σ B ( x ) , sup σ A ( x ) sup σ B ( x ) 1 sup σ B ( x ) ]
τ D ( x ) = [ inf τ A ( x ) inf τ B ( x ) 1 inf τ B ( x ) , sup τ A ( x ) sup τ B ( x ) 1 sup τ B ( x ) ]
Then, the functions ρD(x), σD(x), τD(x) in D must take the subintervals in the real standard interval [0, 1]:
ρ D ( x ) = [ inf ρ A ( x ) inf ρ B ( x ) , sup ρ A ( x ) sup ρ B ( x ) ] [ 0 , 1 ] for inf ρ B ( x ) inf ρ A ( x ) , sup ρ B ( x ) sup ρ A ( x ) and ρ B ( x ) [ 0 , 0 ]
σ D ( x ) = [ inf σ A ( x ) inf σ B ( x ) 1 inf σ B ( x ) , sup σ A ( x ) sup σ B ( x ) 1 sup σ B ( x ) ] [ 0 , 1 ] for inf σ B ( x ) inf σ A ( x ) , sup σ B ( x ) sup σ A ( x ) and σ B ( x ) [ 1 , 1 ]
τ D ( x ) = [ inf τ A ( x ) inf τ B ( x ) 1 inf τ B ( x ) , sup τ A ( x ) sup τ B ( x ) 1 sup τ B ( x ) ] [ 0 , 1 ] for inf τ B ( x ) inf τ A ( x ) , sup τ B ( x ) sup τ A ( x ) and τ B ( x ) [ 1 , 1 ]
Obviously, the inequalities (25)–(27) hold only if BA, ρB(x) ≠ [0, 0], σB(x) ≠ [1, 1], and τB(x) ≠ [1, 1]. Thus, we can give the following definition of division operation for the INSs A and B.
Definition 5.
If we consider two INSs A and B, the division operation of the SNSs A and B is defined as
A / B = { x , [ inf ρ A ( x ) inf ρ B ( x ) , sup ρ A ( x ) sup ρ B ( x ) ] , [ inf σ A ( x ) inf σ B ( x ) 1 inf σ B ( x ) , sup σ A ( x ) sup σ B ( x ) 1 sup σ B ( x ) ] , [ inf τ A ( x ) inf τ B ( x ) 1 inf τ B ( x ) , sup τ A ( x ) sup τ B ( x ) 1 sup τ B ( x ) ] | x X }
which is valid under the conditions B ≥ A, ρB(x) ≠ [0, 0], σB(x) ≠ [1, 1], and τB(x) ≠ [1, 1].
Example 4.
Let us consider two INSs A and B in the universe of discourse X = {x}, which are given by A = {<x, [0.4, 0.6], [0.3, 0.5], [0.2, 0.4]>| x X} and B = {<x, [0.6, 0.8], [0.2, 0.3], [0.1, 0.3]>| x X}. Then, we need to obtain the unknown INS D = A/B.
According to Equation (28), we can yield that
D = A / B = { x , [ 0.4 0.6 , 0.6 0.8 ] , [ 0.3 0.2 1 0.2 , 0.5 0.3 1 0.3 ] , [ 0.2 0.1 1 0.1 , 0.4 0.3 1 0.3 ] | x X } = { x , [ 0.6667 , 0.75 ] , [ 0.125 , 0.2857 ] , [ 0.1111 , 0.1429 ] | x X } .

4. Conclusions

In this paper, we proposed the subtraction and division operations of the SNSs (SVNSs and INSs) with corresponding constrained conditions. Meantime, numerical examples were provided to show the subtraction and division operations over SNSs (SVNSs and INSs). The subtraction and division operations are very important in forming the integral theoretical framework of SNSs and may have many practical applications, such as decision making and image processing.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (No. 71471172).

Conflicts of Interest

The author declares no conflict of interest.

References

  1. Smarandache, F. A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic; American Research Press: Rehoboth, NM, USA, 1999. [Google Scholar]
  2. Zadeh, L.A. Fuzzy Sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
  3. Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
  4. Atanassov, K.T.; Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 1989, 31, 343–349. [Google Scholar] [CrossRef]
  5. Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multisp. Multistruct. 2010, 4, 410–413. [Google Scholar]
  6. Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Hexis: Phoenix, AZ, USA, 2005. [Google Scholar]
  7. Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst. 2014, 26, 2459–2466. [Google Scholar]
  8. Zhang, H.Y.; Wang, J.Q.; Chen, X.H. Interval neutrosophic sets and their application in multicriteria decision making problems. Sci. World J. 2014. [Google Scholar] [CrossRef] [PubMed]
  9. Smarandache, F. Subtraction and division of neutrosophic numbers. Crit. Rev. 2016, 13, 103–110. [Google Scholar]
  10. Atanassov, K.T. Remark on operations “subtraction” over intuitionistic fuzzy sets. Notes Intuit. Fuzzy Sets 2009, 15, 20–24. [Google Scholar]
  11. Atanassov, K.T. On Intuitionistic Fuzzy Sets Theory; Springer: Berlin, Germany, 2012. [Google Scholar]
  12. Atanassov, K.T.; Riecan, B. On two operations over intuitionistic fuzzy sets. J. Appl. Math. Stat. Inform. 2006, 2, 145–148. [Google Scholar] [CrossRef]
  13. Chen, T.Y. Remarks on the subtraction and division operations over intuitionistic fuzzy sets and interval-valued fuzzy sets. Int. J. Fuzzy Syst 2007, 9, 169–172. [Google Scholar]
  14. Lu, Z.K.; Ye, J. Decision-making method for clay-brick selection based on subtraction operational aggregation operators of intuitionistic fuzzy values. Open Cybern. Syst. J. 2016, 10, 283–291. [Google Scholar] [CrossRef]

Share and Cite

MDPI and ACS Style

Ye, J. Subtraction and Division Operations of Simplified Neutrosophic Sets. Information 2017, 8, 51. https://doi.org/10.3390/info8020051

AMA Style

Ye J. Subtraction and Division Operations of Simplified Neutrosophic Sets. Information. 2017; 8(2):51. https://doi.org/10.3390/info8020051

Chicago/Turabian Style

Ye, Jun. 2017. "Subtraction and Division Operations of Simplified Neutrosophic Sets" Information 8, no. 2: 51. https://doi.org/10.3390/info8020051

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop