Computer Science and Engineering 2012, 2(7): 129-132 DOI: 10.5923/j.computer.20120207.01
Generalized Neutrosophic Set and Generalized Neutrosophic Topological Spaces A. A. Salama1,* , S. A. Alblowi2 1
Department of M athematics and Computer Science, Faculty of Sciences, Port Said University, Egypt 2 Department of M athematics, King Abdulaziz University,Jeddah Saudi Arabia
Abstract In this paper we introduce definitions of generalized neutrosophic sets. After given the fundamental defin itions
of generalized neutrosophic set operations, we obtain several properties, and discussed the relationship between generalized neutrosophic sets and others. Finally, we extend the concepts of neutrosophic topological space [9], intuition istic fuzzy topological space [5, 6], and fu zzy topological space [4] to the case of generalized neutrosophic sets. Possible applicat ion to GIS topology rules are touched upon.
Keywords Neutrosophic Set, Generalized Neutrosophic Set, Neutrosophic Topology Let T, I,F be real standard or nonstandard subsets of
] 0 ,1 [, with
1. Introduction
− +
Sup_T=t_sup, inf_T=t_inf Neutrosophy has laid the foundation for a whole family of Sup_I=i_sup, inf_I=i_ inf new mathematical theories generalizing both their classical Sup_F=f_sup, inf_F=f_inf and fuzzy counterparts, such as a neutrosophic set theory. n-sup=t_sup+i_sup+f_sup The fuzzy set was introduced by Zadeh [10] in 1965, where n-inf=t_ inf+i_inf+f_ inf, each element had a degree of membership. The intuitionstic T, I, F are called neutrosophic components fuzzy set (Ifs for short) on a universe X was introduced by K. Definiti on [9] Atanassov [1, 2, 3] in 1983 as a generalization of fuzzy set, Let X be a non-empty fixed set. A neutrosophic set ( NS where besides the degree of membership and the degree of A is an object having the form for short) non- membership of each element. After the introduction of the neutrosophic set concept [7, 8,= 9]. In this paper we A x , µ A x ,σ A x , γ A x : x ∈ X Where introduce definitions of generalized neutrosophic sets. After µ A (x ), σ A (x ) and γ A ( x ) wh ich represent the degree of given the fundamental defin itions of generalized neutrosophic set operations, we obtain several properties, member ship function (namely µ A ( x ) ), the degree of and discussed the relationship between generalized neutrosophic sets and others. Finally, we extend the concepts indeterminacy (namely σ A ( x ) ), and the degree of of neutrosophic topological space [9]. non-member ship (namely γ x ) respectively of each
{
( )
( )
A
2. Terminologies
Definiti on.[7, 8]
] 0 ,1 [is nonstandard unit interval. − +
* Corresponding author: drsalama44@gm ail.com (A. A.Salama) Published online at http://journal.sapub.org/computer Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved
( )
element x ∈ X to the set A . Definiti on [9]. The NSS 0N and 1N in X as follows: 0N may be defined as:
We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in [7, 8], Atanassov in[1, 2, 3] and Salama [9]. Smarandache introduced the = ( 01 ) 0N neutrosophic components T, I, F which represent the membership, indeterminacy, and non-membership values = ( 02 ) 0N respectively, where
}
( )
= ( 03 ) 0 N = ( 04 ) 0N
{ x ,0,0,1 : x ∈ X } { x ,0,1,1 : x ∈ X } { x ,0,1,0 : x ∈ X } { x ,0,0,0 : x ∈ X }
1N may be defined as:
= (11 ) 1N
{ x ,1,0,0
:x ∈X
}
130
A. A. Salama: Generalized Neutrosophic Set and Generalized Neutrosophic Topological Spaces
= (12 ) 1N = (13 ) 1N = (14 ) 1N
{ x ,1,0,1 : x ∈ X } { x ,1,1,0 : x ∈ X } { x ,1,1,1 : x ∈ X }
possible definit ions for subsets ( A ⊆ B )
(A ⊆ B )
∀x ∈ X A ⊆ B ⇔ µ A ( x ) ≤ µ B ( x ) , γ A ( x ) ≥ γ B ( x ) and σ A ( x ) ≥ σ B ( x )
Proposition For any generalized neutrosophic set A the following are holds
3. Generalized Neutrosophic Sets We shall now consider some possible definitions for basic concepts of the generalized neutrosophic set. Definiti on Let X be a non-empty fixed set. A generalized neutrosophic set (G NS for short) A is an object having the form A =
{ x, µ A ( x ) , σ A ( x ) , γ A ( x ) : x ∈ X }
Where
µ A (x ), σ A (x ) and γ A ( x ) which represent the degree of member ship function (namely µ A ( x ) ), the degree of indeterminacy (namely σ A ( x ) ), and the degree of
non-member ship (namely γ A ( x ) ) respectively of each
element x ∈ X to the set A where the functions satisfy the condition µ A ( x ) ∧ σ A ( x ) ∧ ν A ( x ) ≤ 0.5 . Remark A generalized neutrosophic A = < x, µ A ( x ) ,σ A ( x ) , γ A ( x ) >: x ∈ X
{
ordered triple
< µ A ,σ A , γ A >
in
}
can be identified to an
0,1 −
+
on.X, where the trip le
functions satisfy the condition µ A (x ) ∧ σ A (x ) ∧ ν A (x ) ≤ 0.5 Remark For the sake of simp licity, we shall use the symbol A= < x, µ A ,σ A , γ A > for the G NS
{
A = < x, µ A ( x ) ,σ A ( x ) , γ A ( x ) >: x ∈ X
}
Example Every GIFS A a non-empty set X GNS having the form
{
(
is obviously on
)
}
Definiti on Let A = µ A ,σ A , γ A a GNSS on X , then the
(C ( A ) , for short )
maybe
defined as three kinds of comp lements (C 1 ) C ( A) = x,1 − µ A ( x), σ A ( x),1 −ν A ( x) : x ∈ X ,
{ = (C ) C ( A) { x, γ A ,σ A ( x ) , µ A ( x ) : x ∈ X } (C ) C ( A) = { x, γ A ,1 − σ A ( x ) , µ A ( x ) : x ∈ X }
}
2
3
One can define several relat ions and operations between GNSS as fo llo ws: Definiti on
x
0N ⊆ A , 0N ⊆ 0N A ⊆ 1N , 1N ⊆ 1N
Definiti on Let X be a non-empty set, and A= < x, µ A ( x ) , γ A ( x ) , σ A ( x ) > , B= < x, µ B ( x ) ,σ B ( x ) , γ B ( x ) > are GNSS Then A B maybe defined as: ( I1 ) A B =< x, µ A ( x ).µ B ( x ) ,σ A ( x ) .σ B ( x ) ,
γ A ( x ) .γ B ( x ) >
< x, µ A ( x ) ∧ µ B ( x ) , σ A ( x ) ∧ σ B ( x ) , (I 2 ) A B = γ A ( x) ∨ γ B ( x) > < x, µ A ( x ) ∧ µ B ( x ) , σ A ( x ) ∨ σ B ( x ) , (I 3 ) A B = γ A ( x) ∨ γ B ( x) > A B may be defined as: < x, µ A ( x ) ∨ µ B ( x ) , σ A ( x ) ∨ σ B ( x ) , (U1 ) A B =
γ A ( x) ∧ γ B ( x) >
< x, µ A ( x ) ∨ µ B ( x ) , σ A ( x ) ∧ σ B ( x ) , (U 2 ) A B = γ A ( x) ∧ γ B ( x) > < x, µ A ( x ) ,σ A ( x ) ,1 − µ A ( x ) > A = < > A =< x,1 − γ A ( x ) ,σ A ( x ) , γ A ( x ) >
Example.3.2. Let X = {a, b, c, d , e} and
A = < x, µ A ( x ) ,1 − µ A ( x ) + γ A ( x ) , γ A ( x ) >: x ∈ X
complement of the set A
may be defined as
A ⊆ B ⇔ µ A ( x ) ≤ µ B ( x ) , γ A ( x ) ≥ γ and σ A ( x ) ≤ σ B ( x )
Let be a non-empty set, and GNSS A and the form A = x, µ A ( x ) ,σ A ( x ) , γ A ( x ) ,
B in
B = x , µ B ( x ) ,σ B ( x ) , γ B ( x ) , then we may consider two
A = x, µ A , σ A ,ν A given by: X
µ A (x )
ν A (x )
σ A (x)
µ A (x ) ∧ ν A ( x) ∧ σ A (x )
a b c d e
0.6 0.5 0.4 0.3 0.3
0.3 0.3 0.4 0.5 0.6
05 0.6 0.5 0.3 0.4
0.3 0.3 0.4 0.3 0.3
Then the family
G = {O~ , A} is an GNSS on X.
We can easily generalize the operations of generalized intersection and union in definit ion 3.4 to arb itrary family of GNSS as fo llo w: Definiti on Let {Aj : j ∈ J } be a arbitrary family of NSS in X , then
Aj maybe defined as:
x, ∧ µ Aj ( x ) , ∧ σ Aj ( x ) , ∨γ Aj ( x ) 1) Aj = j∈J
j∈J
Computer Science and Engineering 2012, 2(7): 129-132
x, ∧ µ Aj ( x ) , ∨σ Aj ( x ) , ∨γ Aj ( x ) 2) Aj = Aj maybe defined as:
1) ∪ A j = 2) ∪ A j =
Definiti on Let A and B are generalized neutrosophic sets then A B may be defined as AB= x, µ A ∧ γ B , σ A ( x ) σ B ( x ) , γ A ∨ µ B ( x ) Proposition For all A , B two generalized neutrosophic sets then the following are true i) C ( A B ) = C ( A ) C ( B ) ii) C ( A B ) = C ( A ) C ( B )
4. Generalized Neutrosophic Topological Spaces Here we extend the concepts of and intuitionistic fuzzy topological space [5, 7], and neutrosophic topological Space [ 9] to the case of generalized neutrosophic sets. Definiti on A generalized neutrosophic topology (GNT for short) an a non empty set X is a family τ of generalized neutrosophic subsets in X satisfying the following axio ms
1
2
In this case the pair
for any G1 ,G 2 ∈ τ ,
∀{G i : i ∈ J } ⊆ τ
( X ,τ )
is called a generalized
neutrosophic topological space (G NTS for short) and any neutrosophic set in τ is known as neuterosophic open set ( NOS for short) in X . The elements of τ are called open generalized neutrosophic sets, A generalized neutrosophic set F is closed if and only if it C (F) is generalized neutrosophic open. Remark A generalized neutrosophic topological spaces are very natural generalizations of intuitionistic fu zzy topological spaces allow more general functions to be members of intuit ionistic fuzzy topology. Example Let X = x and
= A
= B = D = C
}
G NTSS on X as follows
x, ∨ µ A , ∨σ A , ∧ν A j j j j∈J
(GNT1 ) O N ,1N ∈τ , (GNT2 ) G G ∈τ (GNT3 ) G i ∈τ
Example Let ( X ,τ 0 ) be a fuzzy topological space in Changes [4] sense such that τ 0 is not indiscrete suppose now that τ 0 = {0 N ,1N } ∪ V j : j ∈ J then we can construct two
{
x, ∨ µ A , ∧σ A , ∧ν A j j j j∈J
131
{}
{ x ,0.5,0.5,0.4 : x ∈ X } { x ,0.4,0.6,0.8 : x ∈ X } { x ,0.5,0.6,0.4 : x ∈ X } { x ,0.4,0.5,0.8 : x ∈ X }
Then the family τ = {O n ,1n , A , B ,C , D } of G Ss in X is generalized neutrosophic topology on X
τ 0 = {0 N ,1N } ∪ {< x, V j , σ ( x),0 >: j ∈ J } τ 0 = {0 N ,1N } ∪ {< x,V j ,0,σ ( x),1 − V j >: j ∈ J }
Proposition Let ( X ,τ ) be a GNT on X , then we can also construct several GNTSS on X in the following way : a) τ o,1 = {[ ]G : G ∈ τ }, b) τ o , 2 = { G : G ∈ τ }, Proof a) (GNT1 ) and (GNT2 ) are easy. (GNT3 ) Let [ ]G j : j ∈ J , G j ∈ τ ⊆ τ 0,1 .Since
{
{
}
}{
}{
}
∪ G j = x,∨ µG j ,∨σ G j ,∧γ G j or x,∨ µG j ,∧σ G j ,∧γ G j or x,∨ µG j ,∧σ G j ,∨γ G j ∈τ ,
we have ∪ [ ]G j = x,∨ µG j ,∨σ G j ,∧(1 − µG j ) or x,∨ µG j ,∨σ G j , (1 − ∨ µG j ) ∈τ 0,1
( )
{
}{
}
This similar to (a) Definiti on Let ( X ,τ 1 ), ( X ,τ 2 ) be two generalized neutrosophic topological spaces on X . Then τ 1 is said be contained inτ 2 (in symbols τ 1 ⊆ τ 2 ) if G ∈ τ 2 for each G ∈ τ 1 . In this case, we also say that τ 1 is coarser thanτ 2 . Proposition Let {τ j : j ∈ J } be a family of NTSS on X . Then ∩ τ j is A generalized neutrosophic topology on X .Furthermo re,
∩τ
j
is the coarsest NT on X containing all.
τ j,s
Proof. Obvious Definiti on The complement of A (C (A) for short) of NOS . A is called a generalized neutrosophic closed set (G NCS for short) in X . Now, we define generalized neutrosophic closure and interior operations in generalized neutrosophic topological spaces: Definiti on
( ) ( ) ( )
< x , µ A x , γ A x ,σ A x > Let ( X ,τ ) be G NTS and A = be a G NS in X . Then the generalized neutrosophic closer and generalized neutrosophic interio r of Aare defined by G NCl ( A) = ∩{K : K is an NCS in X and A ⊆ K} G NInt ( A) = ∪{G : G is an NOS in X and G ⊆ A}.It can
be also shown that It can be also shown that NCl ( A) is NCS and NInt ( A) is a G NOS in X A is in X if and only if G NCl ( A) . A is G N
Proposition
in X if and only if G NInt ( A) = A .
132
A. A. Salama: Generalized Neutrosophic Set and Generalized Neutrosophic Topological Spaces
For any generalized neutrosophic set
A
in
( x ,τ )
we
have (a) G NCl (C ( A) = C (GNInt ( A), (b) G NInt (C ( A)) = C (GNCl ( A)). Proof. Let A = {< x, µ A , σ A ,υ A >: x ∈ X } and suppose that the family o f generalized neutrosophic subsets contained in A are indexed by the family if G NSS
Definitions.
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K. Atanassov, intuitionistic fuzzy sets, in V.Sgurev, ed.,Vii ITKRS Session, Sofia(June 1983 central Sci. and Techn. Library, Bulg. Academy of Sciences( 1984)).
[2]
K. Atanassov, intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986)87-96.
[3]
K. Atanassov, Review and new result on intuitionistic fuzzy sets , preprint IM -MFAIS-1-88, Sofia, 1988.
[4]
C.L. Chang, Fuzzy Topological Spaces, J. M ath. Anal. Appl. 24 (1968)182-1 90.
[5]
Dogan Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems. 88(1997)81-89.
[6]
Reza Saadati, Jin HanPark, On the intuitionistic fuzzy topological space, Chaos, Solitons and Fractals 27(2006)331-344 .
[7]
Florentin Smarandache , Neutrosophy and Neutrosophic Logic , First International Conference on Neutrosophy , Neutrosophic Logic , Set, Probability, and Statistics University of New M exico, Gallup, NM 87301, USA(2002) ,
[email protected]
[8]
F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability. American Research Press, Rehoboth, NM , 1999.
GNInt (GNInt ( A)) = GNInt ( A) ∧ GNInt ( B),
[9]
GNCl ( A ∪ B ) = GNCl ( A) ∨ GNCl ( B ), GNInt (1N ) = 1N , GNCl (O N ) = O N ,
A.A. Salama and S.A. AL-Blowi , NEUTROSOPHIC SET and NEUTROSOPHIC TOPOLOGICAL SPACES, IOSR Journal of M ath. ISSN:2278-5728.Vol.(3) ISSUE4PP31-35(2012)
[10] L.A. Zadeh, Fuzzy Sets, Inform and Control 8(1965)338-353.
contained in A are indexed by the family A = < x, µGi , σ G i ,υGi >: i ∈ J . Then we see that
{
{
}
}
GNInt ( A) = < x,∨ µ Gi ,∨σ G i ,∧υ Gi > and hence
{
}
C (GNInt ( A)) = < x,∧ µ Gi ,∨σ G i ,∨υ Gi > . Since
µGi ≤ µ A and we obtaining C ( A) . i.e C ( A) and
ν Gi ≥ ν A for each
{
i∈J
,
}
GNCl (C ( A)) = < x,∧υ Gi ,∨σ G i ,∨ µ Gi > . Hence
GNCl (C ( A) = C (GNInt ( A), follows immed iately This is analogous to (a). Proposition
Let ( x ,τ ) be a G NTS and A , B be two neutrosophic
sets in X . Then the following properties hold: GNInt ( A) ⊆ A, A ⊆ GNCl ( A), A ⊆ B ⇒ GNInt ( A) ⊆ GNInt ( B ), A ⊆ B ⇒ GNCl ( A) ⊆ GNCl ( B ),
Proof (a), (b) and (e) are obvious (c) follows fro m (a) and
