neutrosophic set

neutrosophic set

Let

M

be a subset of a

universe of discourse

U

. Each

element

x

∈

U

has degrees of membership, indeterminacy, and non-membership
in

M

, which are subsets of the hyperreal

interval

]

-

0

,

1

+

[

. The
notation

x

⁢

(

T

,

I

,

F

)

∈

M

means that

•

the degree of membership of

x

in

M

is

T

;

•

the degree of indeterminacy of

x

in

M

is

I

; and

•

the degree of non-membership of

x

in

M

is

F

.

M

is called

neutrosophic set

, whereas

T

,

I

,

F

are called

neutrosophic components

of the element

x

with respect to

M

.

Now let’s explain the previous notations:

A number

ε

is said to be

infinitesimal

if and only if for all positive integers

n

one has

|

ε

|

<

1

n

. Let

ε

>

0

be a such infinitesimal number. The

hyper-real number set

is an

extension

of the

real number

set, which includes classes of

infinite

numbers and classes of infinitesimal numbers.

Generally, for any real number

a

one defines

a

-

which signifies a

monad

, i.e. a set of hyper-real numbers in

non-standard analysis

, as follows:

a

-

=

{

a

-

ε

:

ε

∈

R

*

,

ε

is infinitesimal

}

,

and similarly one defines

a

+

, which is also a monad, as:

a

+

=

{

a

+

ε

:

ε

∈

R

*

,

ε

is infinitesimal

}

.

A

binad

a

+

-

is a union of the above two monads, i.e.

a

+

-

=

-

a

∪

a

+

.

For example: The non-standard finite number

1

+

=

1

+

ε

, where

1

is its

standard part

and

ε

its

non-standard part

, and similarly the non-standard finite number

0

-

=

0

-

ε

, where

0

is its standard part and

ε

its

non-standard part

.

Similarly for

3

+

=

3

+

ε

, etc.

Note that

]

-

0

,

1

+

[

is called the

non-standard unit interval

.

More information on hyperreal intervals

http://www.gallup.unm.edu/

smarandache

/

Introduction

.pdf

is available.

The superior sum of the neutrosophic components is defined as

n

s

⁢

u

⁢

p

=

s

u

p

(

T

)

+

s

u

p

(

I

)

+

s

u

p

(

F

)

∈

]

-

0

,

3

+

[

which may be as high as 3 or

3

+

.

While the inferior sum of the neutrosophic components is defined as

n

i

⁢

n

⁢

f

=

i

n

f

(

T

)

+

i

n

f

(

I

)

+

i

n

f

(

F

)

∈

]

-

0

,

3

+

[

which may be as low as 0 or

0

-

.

The notion of neutrosophic set was introduced by Florentin Smarandache in 1995 as a

generalization

of

fuzzy set

(especially of intuitionistic fuzzy set) when

n

s

⁢

u

⁢

p

=

1

, of

intuitionistic set

when

n

s

⁢

u

⁢

p

<

1

, and of

paraconsistent set

when

n

s

⁢

u

⁢

p

>

1

.

The main distinctions between the neutrosophic set (NS) and intuitionistic fuzzy set (IFS) are the facts that (a) the sum of the scalar neutrosophic components (or their superior sum,

n

s

⁢

u

⁢

p

, if the neutrosophic components are subsets) in NS is not necessarily 1 as in IFS but any number from

0

-

to

3

+

in order to allow the

characterization

of incomplete or paraconsistent information, and (b) in NS one uses the non-standard interval

]

-

0

,

1

+

[

in order to make a

difference

between

absolute membership

, denoted by

1

+

, and

relative membership

, denoted by

1

, while in IFS one only uses the standard interval

[

0

,

1

]

.

An example

:

Let

A

be a neutrosophic set.

One can say, by abuse of

language

, that any element

neutrosophically

belongs to any set, due to the flexibility of degrees of truth/indeterminacy/falsity involved, which each varies between

0

-

and

1

+

.

Thus the element

x

⁢

(

0.1

,

0.2

,

0.3

)

∈

A

means, the degree of membership of

x

in

A

is 0.1, the degree on indeterminacy (undecidability) is 0.2, and the degree of non-membership is 0.3 (as one sees, the sum of components is ¡ 1).

Similarly the element

y

⁢

(

0.6

,

0.2

,

0.5

)

∈

A

, with the sum of components ¿ 1.

Or the element

z

⁢

(

0.7

,

0.1

,

0.2

)

∈

A

, with the sum of components = 1.

More general, the element

w

⁢

(

(

0.20

-

0.30

)

,

(

0.40

-

0.45

)

∪

[

0.50

-

0.51

]

,

{

0.20

,

0.24

,

0.28

}

)

∈

A

, means:

- the degree of membership is between 0.20-0.30 (one cannot find an exact approximation because of various sources used);

- the degree of indeterminacy related to the appurtenance of

w

to

A

is between 0.40-0.45 or between 0.50-0.51 (limits included);

- the degree of non-membership is 0.20 or 0.24 or 0.28.

A remark

:

- In technical applications, where there is no need for distinctions between absolute membership and relative membership, we can use standard subsets instead of non-standard subsets and respectively the unit interval

[

0

,

1

]

instead of the non-standard unit interval

]

-

0

,

1

+

[

.

References

1

F. Smarandache,

A Unifying Field in Logics:

Neutrosophic Logic

. Neutrosophy, Neutrosophic Set,

Neutrosophic Probability

and

Statistics

, third edition, Xiquan, Phoenix, 2003.

http://www.gallup.unm.edu/ smarandache/eBook-Neutrosophics2.pdf

The whole book is also online and can be downloaded here. .

2

F. Smarandache, J. Dezert, A. Buller, M. Khoshnevisan, S. Bhattacharya, S. Singh, F. Liu, Gh. C. Dinulescu-Campina, C. Lucas, C. Gershenson,

Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics

, The University of New Mexico, Gallup Campus, 1-3 December 2001.

\htmladdnormallink

The Proceedings are also online and can be downloaded here.http://arxiv.org/pdf/math.GM/0306384

3

Haibin Wang, Praveen Madiraju, Yanqing Zhang, Rajshekhar Sunderraman,

Interval Neutrosophic Sets

, International Journal of Applied Mathematics and Statistics, Vol. 3, No. M05, 1-18, 2005.

Title

neutrosophic set

Canonical

name

NeutrosophicSet

Date of creation

2013-03-22 15:21:49

Last modified on

2013-03-22 15:21:49

Owner

para0doxa (5174)

Last modified by

para0doxa (5174)

Numerical id

9

Author

para0doxa (5174)

Entry type

Definition

Classification

msc 03E70

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