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A new similarity measure based on falsity value between single valued neutrosophic sets based on the centroid points of transformed single valued neutrosophic numbers with applications to pattern recognition.

1 Introduction

In [1] Atanassov introduced a concept of intuitionistic sets based on the concepts of fuzzy sets [2]. In [3] Smarandache introduced a concept of neutrosophic sets which is characterized by truth function indeterminacy function and falsity function where the functions are completely independent. Neutrosophic set has been a mathematical tool for handling problems involving imprecise, indeterminant and inconsistent data; such as cluster analysis, pattern recognition, medical diagnosis and decision making. In [4] Smarandache et. al introduced a concept of single valued neutrosophic sets. Recently few researchers have been dealing with single valued neutrosophic sets [5-10],

The concept of similarity is fundamentally important in almost every scientific field. Many methods have been proposed for measuring the degree of similarity between intuitionistic fuzzy sets [11-15], Furthermore, in [13-15] methods have been proposed for measuring the degree of similarity between intuitionistic fuzzy sets based on transformed techniques for pattern recognition. But those methods are unsuitable for dealing with the similarity measures of neutrosophic sets since intuitionistic sets are characterized by only a membership function and a non-membership function. Few researchers dealt with similarity measures for neutrosophic sets [16-22], Recently, Jun [18] discussed similarity measures on internalneutrosophic sets, Majumdar et al. [17] discussed similarity and entropy of neutrosophic sets, Broumi et. al. [16] discussed several similarity measures of neutrosophic sets. Ye [9] discussed single-valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine. Deli et. al. [10] discussed multiple criteria decision making method on single valued bipolar neutrosophic set based on correlation coefficient similarity measure, Ulucay et. al. [21] discussed Jaccard vector similarity measure of bipolar neutrosophic set based on multi-criteria decision making and Ulucay et. al. [22] discussed similarity measure of bipolar neutrosophic sets and their application to multiple criteria decision making.

In this paper, we propose methods to transform between single valued neutrosophic numbers based on centroid points. Here, as single valued neutrosophic sets are made up of three functions, to make the transformation functions be applicable to all single valued neutrosophic numbers, we divide them into four according to their truth indeterminacy and falsity values. While grouping according to the truth values, we take into account whether the truth values are greater or smaller than the indeterminancy and falsity values. Similarly, while grouping according to the indeterminancy/falsity values, we examine the indeterminancy/falsity values and their greatness or smallness with respect to their remaining two values. We also propose a new method to measure the degree of similarity based on falsity values between single valued neutrosophic sets. Then we prove some properties of new similarity measure based on falsity value between single valued neutrosophic sets. When we take this measure with respect to truth or indeterminancy we show that it does not satisfy one of the conditions of similarity measure. We also apply the proposed new similarity measures based on falsity value between single valued neutrosophic sets to deal with pattern recognition problems. Later, we define the method based on falsity value to measure the degree of similarity between single valued neutrosophic set based on centroid points of transformed single valued neutrosophic numbers and the similarity measure based on falsity value between single valued neutrosophic sets.

In section 2, we briefly review some concepts of single valued neutrosophic sets [4] and property of similarity measure between single valued neutrosophic sets. In section 3, we define transformations between the single valued neutrosophic numbers based on centroid points. In section 4, we define the new similarity measures based on falsity value between single valued neutrosophic sets and we prove some properties of new similarity measure between single valued neutroshopic sets. We also apply the proposed method to deal with pattern recognition problems. In section 5, we define the method to measure the degree of similarity based on falsity value between single valued neutrosophicset based on the centroid point of transformed single valued neutrosophic number and we apply the measure to deal with pattern recognition problems. Also we compare the traditional and new methods in pattern recognition problems.

2 Preliminaries

Definition 2.1 [3] Let U be a universe of discourse. The neutrosophic set A is an object having the farm A = {(x: [T.sub.A(x)], [I.sub.A(x)], [F.sub.A(X)]), x [member of] U] where the functions T,I,F:U [right arrow]][-.sup.0], [1.sup.+][ respectively the degree of membership, the degree of indeterminacy and degree of non-membership of the element x [member of] U to the set A with the condition:

[0.sup.-] [less than or equal to] [T.sub.A(x)] + [I.sub.A(x)] + [F.sub.A(x)] [less than or equal to] [3.sup.+]

Definition 2.2 [4] Let U be a universe of discourse. The single valued neutrosophic set A is an object having the farm A = {(x: [T.sub.A(x)], [I.sub.A(x)], [F.sub.A(x))], x [member of] U] where the functions T,I,F: U [right arrow] [0, 1] respectively the degree of membership, the degree of indeterminacy and degree of non-membership of the element x [member of] U to the set A with the condition:

0 [less than or equal to] [T.sub.A(x)] + [I.sub.A(x)] + [F.sub.A(x)] [less than or equal to] 3

For convenience we can simply use x = (T,I,F) to represent an element x in SVNS, and element x can be called a single valued neutrosophic number.

Definition 2.3 [4] A single valued neutrosophic set A is equal to another single valued neutrosophic set B, A = B if[for all]x [member of] U,

[T.sub.A(x)] = [T.sub.B(x)-] [I.sub.A(x)] = [I.sub.B(x)], [F.sub.A(x)] = [F.sub.B(x)].

Definition 2.4 [4] A single valued neutrosophic set A is contained in another single valued neutrosophic set B, A [subset or equal to] B if [for all]x [member of] U,

[T.sub.A{x)] [less than or equal to] [T.sub.B(x)], [I.sub.A(x)] [less than or equal to] [I.sub.B(x)], [F.sub.A(x)] [greater than or equal to] [F.sub.B(x)].

Definition 2.5 [16] (Axiom of similarity measure)

A mapping S (A, B): [NS.sub.(x)] x [NS.sub.(x)] [right arrow] [0,1], where [NS.sub.(x)] denotes the set of all NS inx = {[x.sub.1],..., [x.sub.n]}, is said to be the degree of similarity between A and B if it satisfies the following conditions:

[sP.sub.1]) 0 [less than or equal to] S(A,B) [less than or equal to] 1

[sp.sub.2]) S(A,B) = 1 if and only if A = B, [for all] A, B [member of] NS [sP.sub.3]) S(A,B) = S(B,A)

[sp.sub.4]) If A [subset or equal to] B [subset or equal to] C for all A, B, C [member of] NS, then S(A,B) [greater than or equal to] S(A, C) and S(B,C) [greater than or equal to S(AC).

3 The Transformation Techniques between Single Valued Neutrosophic Numbers

In this section we propose transformation techniques between a single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote the single valued neutrosophic numbers to represent an element [x.sub.i] in the single valued neutrosophic set A, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the center of a triangle (SLK) which was obtained by the transformation on the three-dimensional Z - Y - M plane.

First we transform single valued neutrosophic numbers according to their distinct [T.sub.A], [I.sub.A], [F.sub.A] values in three parts.

3.1 Transformation According to the Truth Value

In this section, we group the single valued neutrosophic numbers after the examination of their truth values [T.sub.A]'s greatness or smallness against [I.sub.A] and [F.sub.A] values. We will shift the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] values on the Z - axis and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] values on the Y - axis onto each other. We take the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] value on the M - axis. The shifting on the Z and Y planes are made such that we shift the smaller value to the difference of the greater value and 2, as shown in the below figures.

1. First Group

For the single valued neutrosophic numbers

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2. Second Group

For the single valued neutrosophic numbers

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. Third Group

For the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4. Fourth Group

For the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 3.1.1 Transform the following single valued neutrosophic numbers according to their truth values.

<0.2, 0.5, 0.7>, <0.9, 0.4, 0.5>, <0.3, 0.2, 0.5>, <0.3, 0.2, 0.4>.

1. <0.2, 0.5, 0.7} single valued neutrosophic number belongs to the first group.

The center is calculated by the formula, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we have [C.sub.A(x)] = <0.566, 0.633, 0.7}.

ii. <0.9, 0.4, 0.5> single valued neutrosophic number is in the second group.

The center for the values of the second group is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and for <0.9, 0.4, 0.5>, [C.sub.A(x)] = <0.7, 0.633, 0.5>.

iii. <0.3, 0.2, 0.5> single valued neutrosophic number belongs to the third group.

The formula for the center of <0.3, 0.2, 0.5} is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and therefore we have [C.sub.A(x)] = <0.7, 0.7, 0.5>.

iv. <0.3, 0.2, 0.4}single valued neutrosophic number is in the third group and the center is calculated to be [C.sub.A(x)] = <0.733, 0.7, 0.4>.

Corollary 3.1.2 The corners of the triangles obtained using the above method need not be single valued neutrosophic number but by definition, trivially their centers are.

Note 3.1.3 As for the single valued neutrosophic number<1, ber<l, 1, 1> there does not exist any transformable triangle in the above four groups, we take its transformation equal to itself.

Corollary 3.1.4 If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the transformation gives the same center in all four groups. Also, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the center in the first group is equal to the one in the third group and if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center in the second group is equal to the center in the fourth group. Similarly, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the center in the first group is equal to the center in the fourth group and if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center in the second group is equal to the one in the third group.

3.2 Transformation According to the Indeterminancy Value

In this section we group the single valued neutrosophic numbers after the examination of their indeterminancy values [1.sub.A]'s greatness or smallness against [T.sub.A] and [F.sub.A] values. We will shift the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] values on the Z - axis and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] values on the Y - axis onto each other. We take the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] value on the M - axis. The shifting on the Z and Y planes are made such that we shift the smaller value to the difference of the greater value and 2, as shown in the below figures.

1. First Group

For the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We transformed the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the center of the SKL triangle, namely [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2. Second Group

For the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. Third Group

For the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4. Fourth Group

For the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 3.2.1: Transform the single neutrosophic numbers of Example 3.1.3,

<0.2, 0.5, 0.7>, <0.9, 0.4, 0.5>, <0.3, 0.2, 0.5>, <0.3, 0.2, 0.4> according to their indeterminancy values.

i. <0.2, 0.5, 0.7} single valued neutrosophic number is in the third group. The center is given by the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and so [C.sub.A(x)] = <0.766, 0.633, 0.7>.

ii. <0.9, 0.4, 0.5> single valued neutrosophic number is in the first group.

By

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have [C.sub.A(x)] = <0.733, 0.633, 0.5>.

iii. <0.3, 0.2, 0.5) single valued neutrosophic number belongs to the first group and the center is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

iv. <0.3, 0.2, 0.4> single valued neutrosophic number is in the first group.

Using

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have [C.sub.A(x)] = <0.666, 0.7, 0.4>.

Corollary 3.2.2 The corners of the triangles obtained using the above method need not be single valued neutrosophic numbers but by definition, trivially their centers are.

Note 3.2.3 As for the single valued neutrosophic number <1, 1, 1} there does not exist any transformable triangle in the above four groups, we take its transformation equal to itself.

Corollary 3.2.4 If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the transformation gives the same center in all four groups. Also if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the center in the first group is equal to the center in the third group, and if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the center in the second group is the same as the one in the fiurth group. Similarly, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the center in the first group is equal to the one in the fourth and in the case that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center in the second group is equal to the center in the third.

3.3 Transformation According to the Falsity Value

In this section, we group the single valued neutrosophic numbers after the examination of their indeterminancy values [F.sub.A]'s greatness or smallness against [I.sub.A] and [F.sub.A] values. We will shift the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] values on the Z - axis and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] values on the Y - axis onto each other. We take the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] value on the M - axis. The shifting on the Z and Y planes are made such that we shift the smaller value to the difference of the greater value and 2, as shown in the below figures.

1. First Group

For the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then

as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) into the single valued neutrosophic number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. Second Group

For the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then

as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. Third Group

For the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4. Fourth Group

For the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then as shown in the figure below, we transformed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the single valued neutrosophic numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the center of the SKL triangle, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 3.3.1: Transform the single neutrosophic numbers of Example 3.1.3.

<0.2, 0.5, 0.7>, <0.9, 0.4, 0.5>, <0.3, 0.2, 0.5>, <0.3, 0.2, 0.4> according to their falsity values.

i. <0.2, 0.5, 0.7> single valued neutrosophic number belongs to the second group. So, the center is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = <0.766, 0.7, 0.7>.

ii. <0.9, 0.4, 0.5> single valued neutrosophic number is in the third group. Using the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we see that [C.sub.A(x)] = <0.766, 0.7, 0.5>.

iii. <0.3, 0.2, 0.5> single valued neutrosophic number is in the second group. As

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the center of the triangle is [C.sub.A(x)] = <0.633, 0.7, 0.5>.

iv. <0.3, 0.2, 0.4> single valued neutrosophic number belongs to the second group.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and so we have [C.sub.A(x)] = <0.666, 0.733, 0.4>.

Corollary 3.3.2The corners of the triangles obtained using the above method need not be single valued neutrosophic numbers but by definition, trivially their centers are single valued neutrosophic values.

Note 3.3.3 As for the single valued neutrosophic ber<l, 1, 1} there does not exist any transformable triangle in the above four groups, we take its transformation equal to itself.

Corollary 3.3.4 If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the transformation gives the same center in all four groups. Also, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the center in the first group is equal to the one in the fourth group, and if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the center in the second group is the same as the center in the third. Similarly, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the centers in the first and third groups are same and lastly, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the center in the second group is equal to the one in the fourth group.

4. A New Similarity Measure Based on Falsity Value Between Single Valued Neutrosophic Sets

In this section, we propose a new similarity measure based on falsity value between single valued neutrosophic sets.

Definition 4.1 Let A and B two single valued neutrosophic sets in x = {[x.sub.1], [x.sub.2],..., [x.sub.n]}.

Let A = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

B = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The similarity measure based on falsity value between the neutrosophic numbers A([x.sub.i]) and B([x.sub.i]) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, we use the values

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since we use the falsity values [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in all these three values, we name this formula as "similarity measure based on falsity value between single valued neutrosophic numbers".

Property 4.2: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: By the definition of Single valued neutrosophic numbers, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Property 4.3: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. i) First we show [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then by Definition 2.3, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

ii) Now we show if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By Definition 2.3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Property 4.4: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Property 4.5: If A [subset or equal to] B [subset or equal to] C,

i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof:

By the single valued neutrosophic set pro pe rty, if A [subset or equal to] B [subset or equal to] C, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Using (1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Similarly, by (2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Using (4) and (5) together, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by (1) and (3),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

ii. The proof of the latter part can be similarly done as the first part.

Corollary 4.6: Suppose we make similar definitions to Definition 4.1, but this time based on truth values or indeterminancy values. If we define a truth based similarity measure, or namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or if we define a measure based on indeterminancy values like

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

these two definitions don't provide the conditions of Property 4.5. For instance, for the truth value

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

when we take the single valued neutrosophic numbers [A.sub.(x)] = <0,0.1,0>, [B.sub.(x)] = <1,0.2,0> and [C.sub.(x)] = <1,0.3,0>, we see S{[A.sub.(x)], [B.sub.(x)]) = 0.233 and S([A.sub.(x)], [C.sub.(x)]) = 0.244. This contradicts with the results of Property 4.5.

Similarly, for the indeterminancy values,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if we take the single valued neurosophic numbers [A.sub.(x)] = <0.1,0,1>, [B.sub.(x)] = <0.2,1, l> and [C.sub.(x)] = <0.3,1,1>, we have S([A.sub.(x)], [B.sub.(x)]) = 0.233 and S([A.sub.(x)], [C.sub.(x)]) = 0.244.

These results show that the definition 4.1 is only valid for the measure based on falsity values.

Defintion 4.7 As

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

The similarity measure based on the falsity value between two single valued neutrosophic sets A and B is;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, [S.sub.NS] (A, B) [member of] [0,1] and [w.sub.i]'s are the weights of the [x.sub.i]'s with the property [[summation].sup.n.sub.i=1] [w.sub.i] = 1 . Also,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 4.8 Let us consider three patterns [P.sub.1], [P.sub.2], [P.sub.3] represerted by single valued neutrosophic sets [[??].sub.1], and [[??].sub.2] in X = {[x.sub.1], [x.sub.2]} respectively, where [[??].sub.1] = {<[x.sub.1], 0.2,0.5,0.7}, <[x.sub.2],0.9,0.4,0.5>} and [[??].sub.2] = {<[x.sub.1], 0.3,0.2,0.5>, <[x.sub.2],0.3,0.2,0.4>}. We want to classify an unknown pattern represented by a single valued neutrosophic set [??] in X = {[x.sub.1], [x.sub.2]} into one of the patterns [[??].sub.1], [[??].sub.2]; where [??] = {([x.sub.1], 0.4,0.4,0.1}, <[x.sub.2],0.6,0.2,0.3>}.

Let [w.sub.i] be the weight of element [w.sub.i], where [w.sub.i] = 1/2 1 [less than or equal to] i [less than or equal to] 2,

[S.sub.NS] ([[??].sub.1], [??]) = 0.711

and

[S.sub.NS] ([[??].sub.1], [??]) = 0.772.

We can see that [S.sub.NS]([[??].sub.2], [??]) is the largest value among the values of [S.sub.NS]([[??].sub.1], [??]) and [S.sub.NS] ([[??].sub.2], [??]).

Therefore, the unknown pattern represented by single valued neutrosophic set [??] should be classified into the pattern [P.sub.2].

5. A New Similarity Measure Based on Falsity Measure Between Neutrosophic Sets Based on the Centroid Points of Transformed Single Valued Neutrosophic Numbers

In this section, we propose a new similarity measure based on falsity value between single valued neutrosophic sets based on the centroid points of transformed single valued neutrosophic numbers.

Definition 5.1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Taking the similarity measure as defined in the fourth section, and letting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the centers of the triangles obtained by the transformation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the third section respectively, the similarity measure based on falsity value between single valued neutrosophic sets A and B based on the centroid points of transformed single valued neutrosophic numbers is

[S.sub.NSC](A, B) = [n.summation over (t=1)] ([w.sub.i]xS([C.sub.A(xi)], [C.sub.B(xi)])),

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here again, [w.sub.i]'s are the weights of the [x.sub.i]'s with the property [[summation].sup.n.sub.i=1] [w.sub.i] = 1.

Example 5.2: Let us consider two patterns [P.sub.1] and [P.sub.2] represented by single valued neutrosophic sets [[??].sub.1], [[??].sub.2] in X = {[x.sub.1],[x.sub.2]} respectively in Example 4.8, where

[[??].sub.1] = {<[x.sub.1], 0.2,0.5, 0.7>, <[x.sub.2], 0.9, 0.4,0.5>}

and

[[??].sub.2] = {<[x.sub.1], 0.3, 0.2, 0.5>, <[x.sub.2], 0.3,0.2,0.4>}.

We want to classify an unknown pattern represented by single valued neutrosophic set [??] in X = [[x.sub.1], [x.sub.2]] into one of the patterns [[??].sub.1], [[??].sub.2], where

[??] = {<[x.sub.1], 0.4, 0.4,0.1>, <[x.sub.2], 0.6, 0.2, 0.3>}.

We make the classification using the measure in Definition 5.1, namely

[S.sub.NSC](A, B) = [[summation].sup.n.sub.i=1] ([w.sub.i] x S([C.sub.A(xi)], [C.sub.B(xi)])).

Also we find the [C.sub.A(xi)], [C.sub.B(xi)] centers according to the truth values.

Let [w.sub.i] be the weight of element [x.sub.i], [w.sub.i] = 1/2; 1 [less than or equal to] i [less than or equal to] 2.

[[??].sub.1][x.sub.1] = <0.2, 0.5, 0.7> transformed based on falsity value in Example 3.1.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.1][x.sub.2] = <0.9, 0.4, 0.5> transformed based on falsity value in Example 3.1.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.2][x.sub.1] = <0.3, 0.2, 0.5> transformed based on falsity value in Example 3.1.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.1][x.sub.2] = <0.3, 0.2, 0.4> transformed based on falsity value in Example 3.1.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] transformed based on falsity value in Section 3.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (second group)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] transformed based on truth falsity in Section 3.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (second group)

[S.sub.NSC]([[??].sub.1], [??]) = 0.67592

[S.sub.NSC]([[??].sub.2], [??]) = 0.80927

Therefore, the unknown pattern Q, represented by a single valued neutrosophic set based on truth value is classified into pattern [P.sub.2].

Example 5.3: Let us consider two patterns [P.sub.1] and [P.sub.2] of example 4.8, represented by single valued neutrosophic sets [[??].sub.1], [[??].sub.2], in X = {[x.sub.1],[x.sub.2]) respectively, where

[[??].sub.1] = {<[X.sub.1], 0.2, 0.5,0.7}, <[x.sub.2],0.9, 0.4, 0.5>}

and

[[??].sub.2] = {<[x.sub.1], 0.3, 0.2, 0.5}, <[x.sub.2],0.3, 0.2, 0.4>}.

We want to classify an unknown pattern represented by the single valued neutrosophic set [??] in X = {[x.sub.1], [x.sub.2]} into one of the patterns [[??].sub.1], [[??].sub.2], where

[??] = {<[x.sub.1], 0.4, 0.4, 0.1}, <[x.sub.2], 0.6, 0.2,0.3>}.

We make the classification using the measure in Definition 5.1, namely

[S.sub.NSC](A, B) = [[summation].sup.n.sub.i=1] ([w.sub.i]xS([C.sub.A(xi)], [C.sub.B(xi)])).

Also we find the [C.sub.A(xi)], [C.sub.B(xi)] centers according to the indeterminacy values.

Let [w.sub.i] be the weight of element [x.sub.i], [w.sub.i] = 1/2; 1 [less than or equal to] [less than or equal to] 2.

[[??].sub.1][x.sub.1] = <0.2, 0.5, 0.7> transformed based on falsity value in Example 3.2.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.1][x.sub.2] = <0.9, 0.4, 0.5> transformed based on falsity value in Example 3.2.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.2][x.sub.1] = <0.3, 0.2, 0.5> transformed based on falsity value in Example 3.2.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.2][x.sub.2] = <0.3, 0.2, 0.4> transformed based on falsity value in Example 3.2.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] transformed based on falsity value in Section 3.2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](second group)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]transformed based on truth falsity in Section 3.2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (first group)

[S.sub.NSC]([[??].sub.1], [??]) = 0.67592

[S.sub.NSC]([[??].sub.2], [??]) = 0.80927

Therefore, the unknown patternQ, represented by a single valued neutrosophic set based on indeterminacy value is classified into pattern [P.sub.2].

Example5.4: Let us consider in example 4.8, two patterns [P.sub.1] and [P.sub.2] represented by single valued neutrosophic sets [[??].sub.1], [[??].sub.2] in X = {[x.sub.1], [x.sub.2]} respectively, where

[[??].sub.1] = {<[x.sub.1], 0.2,0.5, 0.7), <[x.sub.2], 0.9, 0.4,0.5>}

and

[[??].sub.2] = {<[x.sub.1], 0.3, 0.2, 0.5), <[x.sub.2], 0.3,0.2,0.4>}.

We want to classify an unknown pattern represented by single valued neutrosophic set [??] inx = {[x.sub.1], [x.sub.2]} into one of the patterns [[??].sub.1], [[??].sub.2], where

[??]= {<[x.sub.1], 0.4, 0.4,0.1), <[x.sub.2],0.6,0.2, 0.3>}.

We make the classification using the measure in Definition 5.1, namely

[S.sub.NSC](A,B) = [[summation].sup.n.sub.i=1] ([w.sub.i]xS([C.sub.A(xi)], [C.sub.B(xi)])).

Also we find the [C.sub.A(xi)], [C.sub.B(xi)] centers according to the falsity values.

Let [w.sub.i] be the weight of element [x.sub.i], [w.sub.i] = 1/2; 1 [less than or equal to] i [less than or equal to] 2.

[[??}.sub.1][x.sub.1] = <0.2, 0.5, 0.7}transformed based on falsity value in Example 3.3.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.1][X.sub.2] = <0.9, 0.4, 0.5} transformed based on falsity value in Example 3.3.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.2][x.sub.1] = <0.3, 0.2, 0.5} transformed based on falsity value in Example 3.3.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[??].sub.2][x.sub.2] = <0.3, 0.2, 0.4} transformed based on falsity value in Example 3.3.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] transformed based on falsity value in Section 3.3

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](first group)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]transformed based on truthfalsity in Section 3.3

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](third group)

[S.sub.NSC]([[??].sub.1], [??]) = 0.7091

[S.sub.NSC]([[??}.sub.2], [??]) = 0.8148

Therefore, the unknown pattern Q, represented by a single valued neutrosophic set based on falsity value is classified into pattern [P.sub.2].

In Example 5.2, Example 5.3 and Example 5.4, all measures according to truth, indeterminancy and falsity values give the same exact result.

Conclusion

In this study, we propose methods to transform between single valued neutrosophic numbers based on centroid points. We also propose a new method to measure the degree of similarity based on falsity values between single valued neutrosophic sets. Then we prove some properties of new similarity measure based on falsity value between single valued neutrosophic sets. When we take this measure with respect to truth or indeterminancy we show that it does not satisfy one of the conditions of similarity measure. We also apply the proposed new similarity measures based on falsity value between single valued neutrosophic sets to deal with pattern recognition problems.

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[22] Ulucay, V., Deli, I, andM. Sahin Similarity measure of bipolar neutrosophic sets and their application to multiple criteria decision making, Neural Comput & Applic, DOI 10. 1007/S00521-016-2479-1 (2016) 1-10

Received: January 30, 2017. Accepted: February 15, 2017.

(1) Memet Sahin, (1) Necati Olgun, Vakkas Ulucay, (1) Abdullah Kargin and (2) Florentin Smarandache

(1) Department of Mathematics, Gaziantep University, Gaziantep, Turkey. E-mail: mesahin@gantep.edu.tr

(2) Department of Mathematics, Gaziantep University, Gaziantep, Turkey. E-mail: olgun@gantep.edu.tr

(3) Department of Mathematics, Gaziantep University, Gaziantep, Turkey. E-mail: vulucay27@gmail.com

(4) Department of Mathematics, Gaziantep University, Gaziantep, Turkey. E-mail: abdullalikargin27@gmail.com

(5) Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA

E-mail: fsmarandaclie@gmail.com
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Author:Sahin, Memet; Olgun, Necati; Ulucay, Vakkas; Kargin, Abdullah; Smarandache, Florentin
Publication:Neutrosophic Sets and Systems
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Date:Mar 1, 2017
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