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103 Critical Review. Volume XIII, 2016 Subtraction and Division of Neutrosophic Numbers Florentin Smarandache 1 1 University of New Mexico, Gallup, NM, USA [email protected] Abstract In this paper, we define the subtraction and the division of neutrosophic single-valued numbers. The restrictions for these operations are presented for neutrosophic single- valued numbers and neutrosophic single-valued overnumbers / undernumbers / offnumbers. Afterwards, several numeral examples are presented. Keywords neutrosophic calculus, neutrosophic numbers, neutrosophic summation, neutrosophic multiplication, neutrosophic scalar multiplication, neutrosophic power, neutrosophic subtraction, neutrosophic division. 1 Introduction Let = ( 1 , 1 , 1 ) and = ( 2 , 2 , 2 ) be two single-valued neutrosophic numbers, where 1 , 1 , 1 , 2 , 2 , 2 [0, 1] , and 0 1 + 1 + 1 3 and 0 2 + 2 + 2 3. The following operational relations have been defined and mostly used in the neutrosophic scientific literature: 1.1 Neutrosophic Summation ⊕ = ( 1 + 2 1 2 , 1 2 , 1 2 ) (1) 1.2 Neutrosophic Multiplication A ⊗ = ( 1 2 , 1 + 2 1 2 , 1 + 2 1 2 ) (2) 1.3 Neutrosophic Scalar Multiplication ⋋ = (1 − (1 − 1 ) , 1 , 1 ), (3) where ⋋∈ ℝ, and ⋋> 0.

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Page 1: Subtraction and Division of Neutrosophic Numbersfs.unm.edu/CR/SubstractionAndDivision.pdf · Subtraction and Division of Neutrosophic Numbers Florentin Smarandache 1 1 University

103

Critical Review. Volume XIII, 2016

Subtraction and Division

of Neutrosophic Numbers

Florentin Smarandache 1

1 University of New Mexico, Gallup, NM, USA

[email protected]

Abstract

In this paper, we define the subtraction and the division of neutrosophic single-valued

numbers. The restrictions for these operations are presented for neutrosophic single-

valued numbers and neutrosophic single-valued overnumbers / undernumbers /

offnumbers. Afterwards, several numeral examples are presented.

Keywords

neutrosophic calculus, neutrosophic numbers, neutrosophic summation,

neutrosophic multiplication, neutrosophic scalar multiplication, neutrosophic power,

neutrosophic subtraction, neutrosophic division.

1 Introduction

Let 𝐴 = (𝑡1, 𝑖1, 𝑓1) and 𝐵 = (𝑡2, 𝑖2, 𝑓2) be two single-valued neutrosophic

numbers, where 𝑡1, 𝑖1, 𝑓1, 𝑡2, 𝑖2, 𝑓2 ∈ [0, 1] , and 0 ≤ 𝑡1+ 𝑖1+ 𝑓1 ≤ 3 and 0 ≤ 𝑡2+ 𝑖2+ 𝑓2 ≤ 3.

The following operational relations have been defined and mostly used in the

neutrosophic scientific literature:

1.1 Neutrosophic Summation

𝐴⊕ 𝐵 = (𝑡1 + 𝑡2 − 𝑡1𝑡2, 𝑖1𝑖2, 𝑓1𝑓2) (1)

1.2 Neutrosophic Multiplication

A⊗ 𝐵 = (𝑡1𝑡2, 𝑖1 + 𝑖2 − 𝑖1𝑖2, 𝑓1 + 𝑓2 − 𝑓1𝑓2) (2)

1.3 Neutrosophic Scalar Multiplication

⋋ 𝐴 = (1 − (1 − 𝑡1)⋋, 𝑖1

⋋, 𝑓1⋋ ), (3)

where ⋋∈ ℝ, and ⋋> 0.

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104 Florentin Smarandache

Subtraction and Division of Neutrosophic Numbers

Critical Review. Volume XIII, 2016

1.4 Neutrosophic Power

𝐴⋋ = (𝑡1⋋, 1 − (1 − 𝑖1)

⋋, 1 − (1 − 𝑓1)⋋), (4)

where ⋋∈ ℝ, and ⋋> 0.

2 Remarks

Actually, the neutrosophic scalar multiplication is an extension of

neutrosophic summation; in the last, one has ⋋= 2.

Similarly, the neutrosophic power is an extension of neutrosophic

multiplication; in the last, one has ⋋= 2.

Neutrosophic summation of numbers is equivalent to neutrosophic union of

sets, and neutrosophic multiplication of numbers is equivalent to neutrosophic

intersection of sets.

That's why, both the neutrosophic summation and neutrosophic

multiplication (and implicitly their extensions neutrosophic scalar

multiplication and neutrosophic power) can be defined in many ways, i.e.

equivalently to their neutrosophic union operators and respectively

neutrosophic intersection operators.

In general:

𝐴⊕ 𝐵 = (𝑡1 ∨ 𝑡2, 𝑖1 ∧ 𝑖2, 𝑓1 ∧ 𝑓2), (5)

or

𝐴⊕ 𝐵 = (𝑡1 ∨ 𝑡2, 𝑖1 ∨ 𝑖2, 𝑓1 ∨ 𝑓2), (6)

and analogously:

𝐴⊗ 𝐵 = (𝑡1 ∧ 𝑡2, 𝑖1 ∨ 𝑖2, 𝑓1 ∨ 𝑓2) (7)

or

𝐴⊗ 𝐵 = (𝑡1 ∧ 𝑡2, 𝑖1 ∧ 𝑖2, 𝑓1 ∨ 𝑓2), (8)

where "∨" is the fuzzy OR (fuzzy union) operator, defined, for 𝛼, 𝛽 ∈ [0, 1], in

three different ways, as:

𝛼 1∨𝛽 = 𝛼 + 𝛽 − 𝛼𝛽, (9)

or

𝛼 2∨𝛽 = 𝑚𝑎𝑥{𝛼, 𝛽}, (10)

or

𝛼 3∨𝛽 = 𝑚𝑖𝑛{𝑥 + 𝑦, 1}, (11)

etc.

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105

Critical Review. Volume XIII, 2016

Florentin Smarandache

Subtraction and Division of Neutrosophic Numbers

While "∧" is the fuzzy AND (fuzzy intersection) operator, defined, for 𝛼, 𝛽 ∈

[0, 1], in three different ways, as:

𝛼 ∧1𝛽 = 𝛼𝛽, (12)

or

𝛼 ∧2𝛽 = 𝑚𝑖𝑛{𝛼, 𝛽}, (13)

or

𝛼 ∧3𝛽 = 𝑚𝑎𝑥{𝑥 + 𝑦 − 1, 0}, (14)

etc.

Into the definitions of 𝐴⊕𝐵 and 𝐴⊗𝐵 it's better if one associates 1∨ with ∧

1,

since 1∨ is opposed to ∧

1, and 2

∨ with ∧

2, and 3

∨ with ∧

3, for the same reason. But other

associations can also be considered.

For examples:

𝐴⊕ 𝐵 = (𝑡1 + 𝑡2 − 𝑡1𝑡2, 𝑖1 + 𝑖2 − 𝑖1𝑖2, 𝑓1𝑓2), (15)

or

𝐴⊕ 𝐵 = (𝑚𝑎𝑥{𝑡1, 𝑡2},𝑚𝑖𝑛{𝑖1, 𝑖2},𝑚𝑖𝑛{𝑓1, 𝑓2}), (16)

or

𝐴⊕ 𝐵 = (𝑚𝑎𝑥{𝑡1, 𝑡2},𝑚𝑎𝑥{𝑖1, 𝑖2},𝑚𝑖𝑛{𝑓1, 𝑓2}), (17)

or

𝐴⊕ 𝐵 = (𝑚𝑖𝑛{𝑡1 + 𝑡2, 1}, 𝑚𝑎𝑥{𝑖1 + 𝑖2 − 1, 0},𝑚𝑎𝑥{𝑓1 + 𝑓2 −

1, 0}). (18)

where we have associated 1∨ with ∧

1, and 2

∨ with ∧

2, and 3

∨ with ∧

3 .

Let's associate them in different ways:

𝐴⊕ 𝐵 = (𝑡1 + 𝑡2 − 𝑡1𝑡2, 𝑚𝑖𝑛{𝑖1, 𝑖2},𝑚𝑖𝑛{𝑓1, 𝑓2}), (19)

where 1∨ was associated with ∧

2 and ∧

3; or:

𝐴⊕ 𝐵 = (𝑚𝑎𝑥{𝑡1, 𝑡2}, 𝑖1, 𝑖2, 𝑚𝑎𝑥{𝑓1 + 𝑓2 − 1, 0}), (20)

where 2∨ was associated with ∧

1 and ∧

3; and so on.

Similar examples can be constructed for 𝐴⊗ 𝐵.

3 Neutrosophic Subtraction

We define now, for the first time, the subtraction of neutrosophic number:

𝐴⊖ 𝐵 = (𝑡1, 𝑖1, 𝑓1) ⊖ (𝑡2, 𝑖2, 𝑓2) = (𝑡1−𝑡2

1−𝑡2,𝑖1

𝑖2,𝑓1

𝑓2) = 𝐶, (21)

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Critical Review. Volume XIII, 2016

for all 𝑡1, 𝑖1, 𝑓1, 𝑡2, 𝑖2, 𝑓2 ∈ [0, 1], with the restrictions that: 𝑡2 ≠ 1, 𝑖2 ≠ 0, and

𝑓2 ≠ 0.

So, the neutrosophic subtraction only partially works, i.e. when 𝑡2 ≠ 1, 𝑖2 ≠ 0,

and 𝑓2 ≠ 0.

The restriction that

(𝑡1−𝑡2

1−𝑡2,𝑖1

𝑖2,𝑓1

𝑓2) ∈ ([0, 1], [0, 1], [0, 1]) (22)

is set when the classical case when the neutrosophic number components

𝑡, 𝑖, 𝑓 are in the interval [0, 1].

But, for the general case, when dealing with neutrosophic overset / underset

/offset [1], or the neutrosophic number components are in the interval [Ψ, Ω],

where Ψ is called underlimit and Ω is called overlimit, with Ψ ≤ 0 < 1 ≤ Ω, i.e.

one has neutrosophic overnumbers / undernumbers / offnumbers, then the

restriction (22) becomes:

(𝑡1−𝑡2

1−𝑡2,𝑖1

𝑖2,𝑓1

𝑓2) ∈ ([Ψ, Ω], [Ψ, Ω], [Ψ, Ω]). (23)

3.1 Proof

The formula for the subtraction was obtained from the attempt to be

consistent with the neutrosophic addition.

One considers the most used neutrosophic addition:

(𝑎1, 𝑏1, 𝑐1) ⊕ (𝑎2, 𝑏2, 𝑐2) = (𝑎1 + 𝑎2 − 𝑎1𝑎2, 𝑏1𝑏2, 𝑐1𝑐2), (24)

We consider the ⊖ neutrosophic operation the opposite of the ⊕ neutro-

sophic operation, as in the set of real numbers the classical subtraction − is

the opposite of the classical addition +.

Therefore, let's consider:

(𝑡1, 𝑖1, 𝑓1) ⊖ (𝑡2, 𝑖2, 𝑓2) = (𝑥, 𝑦, 𝑧), (25)

⊕ (𝑡2, 𝑖2, 𝑓2) ⊕ (𝑡2, 𝑖2, 𝑓2)

where 𝑥, 𝑦, 𝑧 ∈ ℝ.

We neutrosophically add ⊕ (𝑡2, 𝑖2, 𝑓2) on both sides of the equation. We get:

(𝑡1, 𝑖1, 𝑓1) = (𝑥, 𝑦, 𝑧) ⊕ (𝑡2, 𝑖2, 𝑓2) = (𝑥 + 𝑡2 − 𝑥𝑡2, 𝑦𝑖2, 𝑧𝑓2). (26)

Or,

{

𝑡1 = 𝑥 + 𝑡2 − 𝑥𝑡2, whence 𝑥 =

𝑡1−𝑡1

1−𝑡2 ;

𝑖1 = 𝑦𝑖2, whence 𝑦 =𝑖1

𝑖2 ;

𝑓1 = 𝑧𝑓2, whence 𝑧 =𝑓1

𝑓2 .

(27)

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3.2 Checking the Subtraction

With 𝐴 = (𝑡1, 𝑖1, 𝑓1), 𝐵 = (𝑡2, 𝑖2, 𝑓2), and 𝐶 = (𝑡1−𝑡2

1−𝑡2,𝑖1

𝑖2,𝑓1

𝑓2),

where 𝑡1, 𝑖1, 𝑓1, 𝑡2, 𝑖2, 𝑓2 ∈ [0, 1], and 𝑡2 ≠ 1, 𝑖2 ≠ 0, and 𝑓2 ≠ 0, we have:

𝐴⊖ 𝐵 = 𝐶. (28)

Then:

𝐵 ⊕ 𝐶 = (𝑡2, 𝑖2, 𝑓2) ⊕ (𝑡1−𝑡2

1−𝑡2,𝑖1

𝑖2,𝑓1

𝑓2) = (𝑡2 +

𝑡1−𝑡2

1−𝑡2− 𝑡2 ⋅

𝑡1−𝑡2

1−𝑡2, 𝑖2,

𝑖1

𝑖2, 𝑓2,

𝑓1

𝑓2) = (

𝑡2−𝑡22+𝑡1−𝑡2−𝑡1𝑡2+𝑡2

1−𝑡2, 𝑖1, 𝑓1) =

(𝑡1(1−𝑡2)

1−𝑡2, 𝑖1, 𝑓1) = (𝑡1, 𝑖1, 𝑓1). (29)

𝐴⊖ 𝐶 = (𝑡1, 𝑖1, 𝑓1) ⊖ (𝑡1−𝑡2

1−𝑡2,𝑖1

𝑖2,𝑓1

𝑓2) = (

𝑡1−𝑡1−𝑡21−𝑡2

1−𝑡1−𝑡21−𝑡2

,𝑖1𝑖1𝑖2

,𝑓1𝑓1𝑓2

) =

(

𝑡1−𝑡1𝑡2−𝑡1+𝑡21−𝑡2

1−𝑡2−𝑡1+𝑡21−𝑡2

, 𝑖2, 𝑓2) = (−𝑡1𝑡2+𝑡2

1−𝑡2, 𝑖2, 𝑓2) =

(𝑡2(−𝑡1+1)

1−𝑡2, 𝑖2, 𝑓2) = (𝑡2, 𝑖2, 𝑓2). (30)

4 Division of Neutrosophic Numbers

We define for the first time the division of neutrosophic numbers:

𝐴⊘ 𝐵 = (𝑡1, 𝑖1, 𝑓1) ⊘ (𝑡2, 𝑖2, 𝑓2) = (𝑡1

𝑡2,𝑖1−𝑖2

1−𝑖2,𝑓1−𝑓2

1−𝑓2) = 𝐷, (31)

where 𝑡1, 𝑖1, 𝑓1, 𝑡2, 𝑖2, 𝑓2 ∈ [0, 1] , with the restriction that 𝑡2 ≠ 0, 𝑖2 ≠ 1 , and

𝑓2 ≠ 1.

Similarly, the division of neutrosophic numbers only partially works, i.e. when

𝑡2 ≠ 0, 𝑖2 ≠ 1, and 𝑓2 ≠ 1.

In the same way, the restriction that

(𝑡1

𝑡2,𝑖1−𝑖2

1−𝑖2,𝑓1−𝑓2

1−𝑓2) ∈ ([0, 1], [0, 1], [0, 1]) (32)

is set when the traditional case occurs, when the neutrosophic number

components t, i, f are in the interval [0, 1].

But, for the case when dealing with neutrosophic overset / underset /offset

[1], when the neutrosophic number components are in the interval [Ψ, Ω],

where Ψ is called underlimit and Ω is called overlimit, with Ψ ≤ 0 < 1 ≤ Ω, i.e.

one has neutrosophic overnumbers / undernumbers / offnumbers, then the

restriction (31) becomes:

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Critical Review. Volume XIII, 2016

(𝑡1

𝑡2,𝑖1−𝑖2

1−𝑖2,𝑓1−𝑓2

1−𝑓2) ∈ ([Ψ, Ω], [Ψ, Ω], [Ψ, Ω]). (33)

4.1 Proof

In the same way, the formula for division ⊘ of neutrosophic numbers was

obtained from the attempt to be consistent with the neutrosophic

multiplication.

We consider the ⊘ neutrosophic operation the opposite of the ⊗

neutrosophic operation, as in the set of real numbers the classical division ÷

is the opposite of the classical multiplication ×.

One considers the most used neutrosophic multiplication:

(𝑎1, 𝑏1, 𝑐1) ⊗ (𝑎2, 𝑏2, 𝑐2)

= (𝑎1𝑎2, 𝑏1 + 𝑏2 − 𝑏1𝑏2, 𝑐1 + 𝑐2 − 𝑐1𝑐2), (34)

Thus, let's consider:

(𝑡1, 𝑖1, 𝑓1) ⊘ (𝑡2, 𝑖2, 𝑓2) = (𝑥, 𝑦, 𝑧), (35)

⨂(𝑡2, 𝑖2, 𝑓2) ⨂(𝑡2, 𝑖2, 𝑓2)

where 𝑥, 𝑦, 𝑧 ∈ ℝ.

We neutrosophically multiply ⨂ both sides by (𝑡2, 𝑖2, 𝑓2). We get

(𝑡1, 𝑖1, 𝑓1) = (𝑥, 𝑦, 𝑧)⨂(𝑡2, 𝑖2, 𝑓2)

= (𝑥𝑡2, 𝑦 + 𝑖2 − 𝑦𝑖2, 𝑧 + 𝑓2 − 𝑧𝑓2). (36)

Or,

{

𝑡1 = 𝑥𝑡2, whence 𝑥 =

𝑡1

𝑡2; :

𝑖1 = 𝑦 + 𝑖2 − 𝑦𝑖2, whence 𝑦 = 𝑖1−𝑖2

1−𝑖2 ;

𝑓1 = 𝑧 + 𝑓2 − 𝑧𝑓2, whence 𝑧 =𝑓1−𝑓2

1−𝑓2 .

(37)

4.2 Checking the Division

With 𝐴 = (𝑡1, 𝑖1, 𝑓1), 𝐵 = (𝑡2, 𝑖2, 𝑓2), and 𝐷 = (𝑡1

𝑡2,𝑖1−𝑖2

1−𝑖2,𝑓1−𝑓2

1−𝑓2),

where 𝑡1, 𝑖1, 𝑓1, 𝑡2, 𝑖2, 𝑓2 ∈ [0, 1], and 𝑡2 ≠ 0, 𝑖2 ≠ 1, and 𝑓2 ≠ 1, one has:

𝐴 ∗ 𝐵 = 𝐷. (38)

Then:

𝐵

𝐷= (𝑡2, 𝑖2, 𝑓2)× (

𝑡1

𝑡2,𝑖1−𝑖2

1−𝑖2,𝑓1−𝑓2

1−𝑓2) = (𝑡2 ⋅

𝑡1

𝑡2, 𝑖2 +

𝑖1−𝑖2

1−𝑖2− 𝑖2 ⋅

𝑖1−𝑖2

1−𝑖2, 𝑓2 +

𝑓1−𝑓2

1−𝑓2− 𝑓2 ⋅

𝑓1−𝑓2

1−𝑓2) =

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Subtraction and Division of Neutrosophic Numbers

(𝑡1,𝑖2−𝑖2

2+𝑖1−𝑖2−𝑖1𝑖2+𝑖22

1−𝑖2,𝑓2−𝑓2

2+𝑓1−𝑓2−𝑓1𝑓2+𝑓22

1−𝑓2) =

(𝑡1,𝑖1(1−𝑖2)

1−𝑖2,𝑓1(1−𝑓2)

1−𝑓2) = (𝑡1, 𝑖1, 𝑓1) = 𝐴. (39)

Also:

𝐴

𝐷=

(𝑡1,𝑖1,𝑓1)

(𝑡1𝑡2,𝑖1−𝑖21−𝑖2

,𝑓1−𝑓21−𝑓2

)= (

𝑡1𝑡1𝑡2

,𝑖1−

𝑖1−𝑖21−𝑖2

1−𝑖1−𝑖21−𝑖2

,𝑓1−

𝑓1−𝑓21−𝑓2

1−𝑓1−𝑓21−𝑓2

) =

(𝑡2,

𝑖1−𝑖1𝑖2−𝑖1+𝑖21−𝑖2

1−𝑖2−𝑖1+𝑖21−𝑖2

,

𝑓1−𝑓1𝑓2−𝑓1+𝑓21−𝑓2

1−𝑓2−𝑓1+𝑓21−𝑓2

) = (𝑡2,

𝑖2(−𝑖1+1)

1−𝑖21−𝑖11−𝑖2

,

𝑓2(−𝑓1+1)

1−𝑓21−𝑓11−𝑓2

) =

(𝑡2,𝑖2(1−𝑖1)

1−𝑖1,𝑓2(1−𝑓1)

1−𝑓1) = (𝑡2, 𝑖2, 𝑓2) = 𝐵. (40)

5 Conclusion

We have obtained the formula for the subtraction of neutrosophic numbers ⊖

going backwords from the formula of addition of neutrosophic numbers ⊕.

Similarly, we have defined the formula for division of neutrosophic numbers

⊘ and we obtained it backwords from the neutrosophic multiplication ⨂.

We also have taken into account the case when one deals with classical

neutrosophic numbers (i.e. the neutrosophic components t, i, f belong to [0, 1])

as well as the general case when 𝑡, 𝑖, 𝑓 belong to [𝛹, 𝛺], where the underlimit

𝛹 ≤ 0 and the overlimit 𝛺 ≥ 1.

6 References

[1] Florentin Smarandache, Neutrosophic Overset, Neutrosophic Underset, and

Neutrosophic Offset. Similarly for Neutrosophic Over-/Under-/Off- Logic,

Probability, and Statistics, 168 p., Pons Editions, Bruxelles, Belgique, 2016;

https://hal.archives-ouvertes.fr/hal-01340830 ;

https://arxiv.org/ftp/arxiv/papers/1607/1607.00234.pdf

[2] Florentin Smarandache, Neutrosophic Precalculus and Neutrosophic Calculus,

EuropaNova, Brussels, Belgium, 154 p., 2015;

https://arxiv.org/ftp/arxiv/papers/1509/1509.07723.pdf

Arabic) مبادئ التفاضل والتكامل النيوتروسوفكي و حساب التفاضل والتكامل النيوتروسوفكي

translation) by Huda E. Khalid and Ahmed K. Essa, Pons Editions, Brussels, 112

p., 2016.

[3] Ye Jun, Bidirectional projection method for multiple attribute group decision

making with neutrosophic numbers, Neural Computing and Applications, 2015,

DOI: 10.1007/s00521-015-2123-5.

[4] Ye Jun, Multiple-attribute group decision-making method under a neutrosophic

number environment, Journal of Intelligent Systems, 2016, 25(3): 377-386.

Page 8: Subtraction and Division of Neutrosophic Numbersfs.unm.edu/CR/SubstractionAndDivision.pdf · Subtraction and Division of Neutrosophic Numbers Florentin Smarandache 1 1 University

110 Florentin Smarandache

Subtraction and Division of Neutrosophic Numbers

Critical Review. Volume XIII, 2016

[5] Ye Jun, Fault diagnoses of steam turbine using the exponential similarity measure

of neutrosophic numbers, Journal of Intelligent & Fuzzy Systems, 2016, 30: 1927–

1934.