Neutrosophic Crisp Points & Neutrosophic Crisp Ideals

Neutrosophic Sets and Systems, Vol. 1, 2013

Yanhui Guo&Abdulkadir Şengür, A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set

Neutrosophic Crisp Points & Neutrosophic Crisp Ideals

A. A. Salama

Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Egypt, E-mail: drsalama44@gmail.com

Abstract. The purpose of this paper is to define the so called "neutrosophic crisp points" and "neutrosophic crisp ideals",

and obtain their fundamental properties. Possible application to GIS topology rules are touched upon.

Keywords: Neutrosophic Crisp Point, Neutrosophic Crisp Ideal.

1 Introduction

Neutrosophy has laid the foundation for a whole family of new mathematical theories, generalizing both

their crisp and fuzzy counterparts. The idea of

"neutrosophic set" was first given by Smarandache [12, 13]. In 2012 neutrosophic operations have been investigated by Salama at el. [4 - 10]. The fuzzy set was introduced by Zadeh [13]. The intuitionstic fuzzy set was introduced by Atanassov [1, 2, 3]. Salama at el. [9] defined intuitionistic fuzzy ideal for a set and generalized the concept of fuzzy ideal concepts, first initiated by Sarker [11]. Here we shall present the crisp version of these concepts.

2 Terminologies

We recollect some relevant basic preliminaries, and in particular the work of Smarandache in [12, 13], and Salama at el. [4 -10].

3 Neutrosophic Crisp Points

One can easily define a natural type of neutrosophic crisp set in X, called "neutrosophic crisp point" in X, corresponding to an elementp∈X:

3.1 Definition

Let X be a nonempty set andp∈X. Then the

neutrosophic crisp pointp_N defined by

{ } { }

c

N p p

p = ,φ, is called a neutrosophic crisp point

(NCP for short) in X, where NCP is a triple ({only one element in X}, the empty set,{the complement of the same element in X}).

Neutrosophic crisp points in X can sometimes be inconvenient when expressing a neutrosophic crisp set in X in terms of neutrosophic crisp points. This situation will

occur ifA= A₁,A₂,A₃ , andp∉A1, whereA1,A2,A3

are three subsets such thatA₁∩A₂=φ,

φ =

∩ ₃

1 A

A ,A₂₁∩A₃=φ. Therefore we define the

vanishing neutrosophic crisp points as follows:

3.2 Definition

Let X be a nonempty set, and p∈Xa fixed element

in X. Then the neutrosophic crisp setpN_N = φ,

{ } { }

p, p c

is called “vanishing neutrosophic crisp point“ (VNCP for short) in X, where VNCP is a triple (the empty set,{only one element in X},{the complement of the same element in X}).

3.1 Example

Let X=

{

a,b,c,d

}

and p=b∈X. Then

{ } {

b

a

c

d

}

p

N

=

,

φ

,

,

,

Now we shall present some

types of inclusions of a neutrosophic crisp point to a neutrosophic crisp set:

3.3 Definition

Letp_N =

{ } { }

p,φ, p c be a NCP in X and

3 2 1,A ,A

A

A= a neutrosophic crisp set in X.

(a) pN is said to be contained in

A

(pN∈A for short)

iff p∈A₁.

(b) Let p_NN be a VNCP in X, and

3 2 1,A ,A

A

A= a

neutrosophic crisp set in X. Then p_NN is said to be

contained in

A

(pN_N∈A for short ) iff p∉A3.

3.1 Proposition

Let

{

D_j :j∈J

}

is a family of NCSs in X. Then

)

(

a

₁

j J j

N D

p ∈ ∩

∈ iff

p

N

∈

D

jfor each

j

∈

J

.

)

(

a

2 j

J j

N D

p _N

∈

∩

∈ iff

p

_N

D

_j

N

∈

for each

j

∈

J

.

)

(

b

1 j

J j

N D

p

∈

∪

∈ iff

∃

j

∈

J

such that

p

_N

∈

D

_j.

Neutrosophic Sets and Systems, Vol. 1, 2013

Yanhui Guo&Abdulkadir Şengür, A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set

)

(

b

_{2 j}

J j

N D

p

N ∈∩_∈ iff

J j∈

∃ such that

p

_N

D

_j

N

∈

.

Proof

Straightforward.

3.2 Proposition

Let

3 2 1,A ,A

A

A= and

3 2 1,B ,B

B

B= be two

neutrosophic crisp sets in X. Then

a) A⊆B iff for each p_N we have

B p A

pN∈ ⇔ N∈ and for each pN_N we have

B p A

pN ∈  N_N ∈ .

b)

A

=

B

iff for each

p

_N we have

B p A

p_N∈  _N∈ and for each

N N

p we

havepN_N ∈A ⇔pN_N∈B.

Proof Obvious.

3.4 Proposition

Let A= A₁,A₂,A₃ be a neutrosophic crisp set in X.

Then

{

}

(

p p A

)

(

{

p p A

}

)

A= ∪ _N: _N∈ ∪ ∪ _NN: _NN∈ .

Proof

It is sufficient to show the following equalities:

{

}

(

p p A

)

(

{

p A

}

)

A₁= ∪ }: _N∈ ∪ ∪φ: _NN∈ , A₃=φ

andA3=

(

∩

{

{p }c:pN∈A

}

)

∩

(

∩

{

{p}c:pNN∈A

}

)

,

which are fairly obvious.

3.4 Definition

Let

f

:

X

→

Y

be a function.

(a) Let pN be a nutrosophic crisp point in X. Then

the image of

p

_N under

f

, denoted by

f

(

p

_N

)

, is

defined by f(pN)=

{ } { }

q,φ, qc , whereq= f(p ).

(b) Let

p

_NN be a VNCP in X. Then the image of

NN

p

under

f

, denoted by

f

(

p

_NN

),

is defined by

{ } { }

c

NN q q

p

f( )= φ, , , whereq= f(p ).

It is easy to see that

f

(

p

_N

)

is indeed a NCP in Y,

namely f(p_N)=q_N, whereq= f(p ), and it is

exactly the same meaning of the image of a NCP under the function

f

.

)

(

p

_NN

f

is also a VNCP in Y, namely

, ) (p_NN q_NN

f = where q= f(p ).

3.4 Proposition

Any NCS A in X can be written in the form

NNN NN

NA A A

A= ∪ ∪ , whereA

{

pN pN A

}

N

∈ ∪

= : ,

N N

A=φ and A

{

p_NN p_NN A

}

NNN

∈ ∪

= : . It is easy to show

that, if

3 2 1,A ,A

A

A= , then c

N x A A

A= , ₁,φ, ₁ and

3 2,

,

, A A

x A

NN= φ .

3.5 Proposition

Let f :X→Y be a function and

3 2 1,A ,A

A A=

be a neutrosophic crisp set in X. Then we

have ( ) ( ) ( ) ( )

NNN NN

N

A f A f A f A

f = ∪ ∪ .

Proof

This is obvious from

NNN NN

NA A A

A= ∪ ∪ .

4 Neutrosophic Crisp Ideal Subsets

4.1 Definition

Let X be non-empty set, and L a non–empty family of NCSs. We call L a neutrosophic crisp ideal (NCL for short) on X if

i. A∈L andB⊆AB∈L[heredity],

ii. A∈L andB∈LA∨B∈L[Finite additivity].

A neutrosophic crisp ideal L is called a σ -

neutrosophic crisp ideal if

{

M

}

L

j

j _∈Ν

≤

, implies

L

M

j J

j

∪

∈

∈

(countable additivity).

The smallest and largest neutrosophic crisp ideals on a non-empty set X are

{ }

φ_N and the NSs on X. Also,

c f

,

NCL

L

NC

are denoting the neutrosophic crisp

ideals (NCL for short) of neutrosophic subsets having finite and countable support of X respectively. Moreover, if A is a nonempty NS in X, then

{

B∈NCS:B⊆A

}

is an NCL on X. This is called the principal NCL of all NCSs, denoted by NCL A .

Neutrosophic Sets and Systems, Vol. 1, 2013

Yanhui Guo&Abdulkadir Şengür, A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set 4.1 Remark

i. If X_N∉L, then L is called neutrosophic proper

ideal.

ii. If X_N∈L, then L is called neutrosophic improper

ideal.

iii. ϕN∈L.

4.1 Example

LetX=

{

a,b,c

}

,A =

{ } {

a , a,b,c

} { }

, c ,

{ } { } { }

a, a,c ,

B= C=

{ } { } { }

a,b, c , D=

{ } { } { }

a, c, c ,

{ } { } { }

a, a,b, c ,

E= F=

{ } { } { }

a, a,c,c ,G=

{ } { } { }

a,b,c, c

. Then the family L =

{

φ_N,A,B,D,E,F,G

}

of NCSs is an NCL on X.

4.2 Definition

Let L1 and L2 be two NCLs on X. Then L2 is said to be

finer than L1, or L1 is coarser than L2, if L1

≤

L2. If also L1

≠

L2. Then L2 is said to be strictly finer than L1, or L1 is

strictly coarser than L2.

Two NCLs said to be comparable, if one is finer than the other. The set of all NCLs on X is ordered by the relation: L1 is coarser than L2; this relation is induced the

inclusion in NCSs.

The next Proposition is considered as one of the useful result in this sequel, whose proof is

clear.L_j = A_j1,A_j2,A_j3 .

4.1 Proposition

Let

{

Lj:j∈J

}

be any non - empty family of

neutrosophic crisp ideals on a set X. Then 

J

j j

L

∈

and



J

j j

L

∈

are neutrosophic crisp ideals on X, where

3 2 1,_{j J} j ,_{j J} j

j J j j J j

A A A L

∈ ∈ ∈ ∈

∪ ∩ ∩ =

∩ or

3 2 1,_{j J} j ,_{j J} j

j J j j J

j∈∩ L = ∈∩ A ∪∈ A ∪∈ A and

3 2 1,_{j J} j ,_{j J} j

j J j j J j

A A A L

∈ ∈ ∈

∈ = ∪ ∪ ∩

∪ or

.

,

,

3 2

1 _{j J} j _{j J} j

j J j j J

j∈

L

=

∪

∈

A

∩

∈

A

∩

∈

A

∪

In fact, L is the smallest upper bound of the sets of the Lj in the ordered set of all neutrosophic crisp ideals on

X.

4,2 Remark

The neutrosophic crisp ideal defined by the single neutrosophic set

φ

_N is the smallest element of the ordered set of all neutrosophic crisp ideals on X.

4.2 Proposition

A neutrosophic crisp setA= A₁,A₂,A₃ in the

neutrosophic crisp ideal L on X is a base of L iff every member of L is contained in A.

Proof

(Necessity) Suppose A is a base of L. Then clearly every member of L is contained in A.

(Sufficiency) Suppose the necessary condition holds. Then the set of neutrosophic crisp subsets in X contained in A coincides with L by the Definition 4.3.

4.3 Proposition

A neutrosophic crisp ideal L1, with

baseA= A₁,A₂,A₃ , is finer than a fuzzy ideal L2 with

base

3 2 1,B ,B

B

B= , iff every member of B is

contained in A.

Proof

Immediate consequence of the definitions.

4.1 Corollary

Two neutrosophic crisp ideals bases A, B, on X, are equivalent iff every member of A is contained in B and vice versa.

4.1 Theorem

Let = A_j ,A_j ,A_j :j∈J

3 2 1

η be a non-empty

collection of neutrosophic crisp subsets of X. Then there exists a neutrosophic crisp ideal

   

 

∪ ⊆ ∈

=

∈J j

j A

A NCS A

L(η) : on X for some finite

collection

{

A_j: j=1,2,...,n⊆η

}

.

Proof

It’s clear.

4.3 Remark

The neutrosophic crisp ideal L(

η

) defined above is said to

be generated by

η

and

η

is called sub-base of L(

η

).

Neutrosophic Sets and Systems, Vol. 1, 2013

Yanhui Guo&Abdulkadir Şengür, A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set 4.2 Corollary

Let L1 be a neutrosophic crisp ideal on X and A

∈

NCSs, then there is a neutrosophic crisp ideal L2 which is

finer than L1 and such that A

∈

L2 iff A∪B∈L2 for each

B

∈

L1.

Proof It’s clear.

4.2 Theorem

If L =

{

φN , A1,A2,, A3

}

is a neutrosophic

crisp ideals on X, then:

i)

[ ]

L =

{

φN, A1,A2,,A3c

}

is a

neutrosophic crisp ideals on X.

ii) L =

{

φN, A3,A2,,A1c

}

is a

neutrosophic crisp ideals on X.

Proof

Obvious.

4.3 Theorem

LetA= A₁,A₂,A₃ ∈L1, and

, ,

, _{2 3 2}

1 B B L

B

B= ∈ where

L

₁ and

L

₂are

neutrosophic crisp ideals on X, then A*B is a neutrosophic crisp set:

3 3 2 2 1

1 B,A B ,A B

A B

A∗ = ∗ ∗ ∗ where

{

1 1 2 2 3 3

}

1

1 B A B,A B ,A B

A ∗ =∪ ∩ ∩ ∩ ,

{

1 1 2 2 3 3

}

2

2 B A B,A B ,A B

A ∗ =∩ ∩ ∩ ∩ and

{

1 1 2 2 3 3

}

3

3 B A B,A B ,A B

A ∗ =∩ ∩ ∩ ∩ .

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Received: October 01, 2013. Accepted: December 01, 2013.