Approximations Structures Generated By
Trees Vertices
M.Shokry,
Department of Physics and Mathematics, Faculty of Engineering, Tanta University, Tanta-Egypt
Abstract
The purpose of this paper is generate a new topological types of approximations structures on the set of tree vertices by special types of neighborhoods based on outlinked subtrees with first child . Properties of these structures are studied. We studied rough concepts on any path in tree.. We obtain some new rules to find approximations paths, make comparisons between two paths in same tree and study accuracy by different ways.
Keywords: Rough set,Topology and graph theory.
1-Introduction :
The methodology in components of digital images depended on enumerated and classified it to discrete objects so, properties of these classes are useful and important to be built a new mathematical models structure formed by these classes [22] . In most applications of artificial intelligence the universe of discourse is a finite set[9].Relation and graph theory are suitable models for studying data structures of digital images and computer screen enumeratability. Also the theory of abstract topological space is a more general theory for studying digital topology and digital images[20] and discrete data structures [2, 5, 13,17,37]. One of the most important notions in system analysis is the concept of topological structures [12] and their generalizations. Many works have topological approaches appeared recently for example in structural analysis [8], in chemistry [7], and physics [6].
The purpose of the present work is to put a starting point for the applications of abstract topological theory and some types of graph theory approches into rough set analysis. Rough set theory, introduced by Pawlak in 1982 [36], is a mathematical tool that supports also the uncertainty reasoning but qualitatively. In this paper, we shall integrate some ideas in terms of concepts in topology with trees and rough set theory. Topology concepts exist not only in almost all branches of mathematics, but also in many real life applications. We believe that topological structure will be an important base for modification of knowledge extraction and processing.
A tree is a concrete data structure such that the header points to a single cell and adjacent that cell to a chain of pointers. A binary tree structure is a tree structure such that each cell has two pointers, the left child of a cell is the cell to which the first pointer points, the right child is the cell to which the second pointer points.A binary search tree structure (BST) is a binary tree structure in which for every cell, all cells accessible through the left child have lower keys, and all cells accessible through the right child have higher keys. To find a target key t in the binary search tree T, apply BST search(T, t) [25].
Topology is a branch of mathematics, whose concepts exist not only in almost all branches of mathematics, but also in many real life applications. We believe that topological structure will be an important base for modification of knowledge extraction and processing. Topology is method of classification for all subsets of set X to a collection includes the empty set and X, a finite intersections and arbitrary unions of sets in the collection are also in the collections[3].
In this paper we denoted the interior set of points of A by int(A)
A
o (the union of all open subsets of Xwhich contained inA) as lower approximation of set A. Note that A is open iff AA.Also , the closure of A is denoted by cl(A)A (A is the smallest closed subset of X which contains A). Note that A is closed
iff AA.
decreasing of the neighborhood radius coursing the neighborhood to small neighborhoods which increasing the number of sets in the main neighborhood .We used this concepts on graph theory by replace the neighborhood distance by subtrees so, this method classified the main tree to partitions classes this way will affect the accuracy of any path with respect to this partitions which is useful in many life applications.
Definition 1.1[23]
A subtree of a tree T(V,E) is a graph whose vertices and edge sets are subsets of those of T(V,E) and do not contain a cycle .
Definition 1.2[2]
2 is subtopology of 1 if for all open set G12 there exists open set
G 1such that G 1 G.
Proposition1.1[24]
If2 is causer than 1 then 2 is subtopology of 1
Definition 1.3
Let
X
,
be a topological space, then the subsetA
X
is called:Regular open [24] ( briefly r-open ) if
A
A
. Semi-open [21] ( briefly s-open ) ifA
A
.The complement of a r-open (s-open ) set is called r-closed (s-closed ) set .The family of all r-open (s-open) sets of
X
,
is denoted by RO(X) (SO(X)).The family of all r-closed (s-closed ) sets of
X
,
is denoted by RC(X) (SC(X)).The aim of the following example is to illustrate the existence of spaces in which the above classes of near open sets and near closed sets are not coincided and are not the discrete structure.
Example 1.
Let
X
{
a
,
b
,
c
,
d
}
and
X
,
,
{
a
,
b
},
{
d
},
{
a
,
b
,
d
}
. Then the classes of near open sets are
,
,
{
},
{
,
}
)
(
X
X
d
a
b
RO
X
d
a
b
c
d
a
b
c
a
b
d
X
SO
(
)
,
,
{
},
{
,
},
{
,
},
{
,
,
},
,
,
and the classes of near closed sets are
,
,
{
,
,
},
{
,
}
)
(
X
X
a
b
c
c
d
RC
,
,
{
,
,
},
{
,
},
{
,
},
{
},
{
}
)
(
X
X
a
b
c
c
d
a
b
d
c
SC
.2 Fundamentals of the Pawlak’s rough sets
Let U be a finite set and R be an equivalence relation on U. R generates a partition U/R = {Y1,Y2, . . .
,Ym}on U where Y1,Y2, . . . ,Ym are the equivalence classes generated by the equivalence relation R. In the
rough set theory, these are also called elementary sets of R. For any X U, we can describe X by the elementary sets of R and the two sets[36,37]
} :
{ ) ( }, :
{ )
(X U R X R X U R X
R Yi Yi Y i Y i
are called the lower and the upper approximation of X, respectively.
Let be the empty set, -X the complement of X in U, we have the following properties of the Pawlak’s rough sets:
(2) R () = (Normality) (3)
R
(X) X (Contraction)(3) X R (X) (Extension) (4)
R
(
X
Y
)
R
(
X
)
R
(
Y
)
(Multiplication)(5)
R
(X
Y) =R (X)
R
(Y) (Addition)(6) R (R (X)) =R
(X) (Idempotency)(7) R (R (X)) = R (X) (Idempotency) (8)
R
(-X) = -R
(X) (Duality)(9) R (-X) = -R (X) (Duality) (10) X
Y R
(X)
R
(Y) (Monotone)(11) X Y R (X)
R (Y) (Monotone)(12)
R
(-R
(X)) = -R
(X) (Lower-complement relation)(13) R (-
R
(X)) = -R (X) (Upper-complement relation)(14) KU/R,
R
(K) = K (Granularity)(15) K U/R,
R
(K) = K (Granularity3 Main Results
Topological peroperties are a powerful concepts which plays an important part in practical applications and also to solve many real-life problems. Many topological properties have been applied in some methods of data analysis, for example psuedoclosure concepts which used as mathematical tool for data analysis [38,26,27,28,6]
An induced subtree T1(V1,E1 ) consists of subset of vertices V1 V and subset of edges E1 ={xy E, x, y V1
}, E1 E. A subtree is subgraph of a graph T(V , E) written T1 T .In the following we try to construct a new
definition using it in topological concepts to illustrate the comparing of it with graph theory concepts[2].
In Definition1.2 subtopology means that for all open set in subtopology there exists induced open set in the main topology contains it .Since, any not
Indiscrete topology can form subtopology from its induces open sets .Therefore we can find suitable new classes RO(X)1 and SO(X)1 compared with RO(X) and SO(X) .
Definition 3.1
Let (X,) be topological space and (X,1)be subtopology then the following are hold:
(1) RO(X)1 is sub R-open structure of RO(X) if for all open set G1 RO(X)1 there exists open set G RO(X)
such that G 1 G.
(2) SO(X)1 is sub S-open structure of SO(X) if for all open set G1 SO(X)1 there exists open set G SO(X)
such that G 1 G.
(3) RC(X)1 is sub R-closed structure of RC(X) if for all closed set F1 RC(X)1 there exists closed set F
RC(X) such that F 1 F.
(4) SC(X)1 is sub S- closed structure of SC(X) if for all closed set F1 SC(X)1 there exists closed set F SC(X)
such that F 1 F.
(5) In general the collection M of open sets is sub-open structure of the collection N if for all open set G1 M
,
X
RO(X)
1
S
O(X)
1
,
X
,
{
c
,
d
},
{
a
,
c
,
d
},
b
,
c
,
d
,
,
{
},
{
,
}
)
(
X
X
b
c
d
RO
SO
(
X
)
X
,
,
{
b
},
{
c
,
d
},
{
a
,
b
},
{
a
,
c
,
d
},
b
,
c
,
d
,
,
{
,
},
{
,
,
}
)
(
X
X
a
b
a
c
d
RC
,
,
{
,
},
{
,
,
},
{c,
d},
{b},
{a}}
)
(
X
X
a
b
a
c
d
SC
Example 3.1. Let
X
{
a
,
b
,
c
,
d
}
and
X
,
,
{
c
,
d
},
{
b
},
{
b
,
c
,
d
}
. Then the classes of near open sets are, and
We can find a sub topology 1 from 1={x, ,{c,d}} .Since {c,d}1 , {b,c,d}and {c,d}
{b,c,d} so 1.
, , RC(X)1={X,}and
SC(X)1={X,,{a},{b},{a,b}}
.Therefore
S
O(X)
1is sub S-open structure from SO(X),S
c(X)
1is sub S-open structure fromS
c(X)
1 ,1
O(X)
R
is sub R-open structure From RO(X) and ,R
C(X)
1is sub R-open structure from RO(X).The wonderful approaches is studying some topological properties on trees by using spaces more general than topological spaces. Studying separations axioms take us validity to know the methodology of separated some tree vertices to the other vertices under suitable conditions on the formed structures. The closures concepts discussed the closure and the interior of any path P in tree by new approaches [16].
Defination 3.2
If T(V,E) is a tree and xV,the neighborhood of x is defined by
} child
is y , leave not and subtree of
root is x : {y
N
x xy E
Definition 3.3
If T(V,E) is a tree, x and y belong to V, y is called out vertex of x if xy E
The following is an examples for the above concepts and used it to form topological space if we add root of tree with neighborhoods sets .we notice that if the set of vertices is ordering the binary tree corresponding to only one topological space since its root and level vertices is the class of topology base .Also we notice that the result topology is quasi -discrete topology.
Example 3.2
Consider the binary tree T=(V,E)
Fig(3.1)
N4 ={2,6 }, N2={1,3},N6 ={5,7}and N1= N3= N5= N7=.We added the singleton set of root vertices{4} to form a
basis of topology ,
={V, {4},{2,6 },{1,3},{5,7}}.The topology building on tree vertices V is ={V,, {4},{2,6 },{1,3},{5,7},{2,4,6},{1,3,4 },{4,5,7},{1,2,3,6},{2,5,6,7}, {1,3,5,7},{1,2,3,4,6},{2,4,5,6,7},{1,2,4,5,6,7},{1,3,4,5,7 },{1,2,3,4,6,7}}. The set of all closed sets is
F={V,, {4},{2,6 },{1,3},{5,7},{2,4,6},{1,3,4 },{4,5,7},{1,2,3,6},{2,5,6,7}, {1,3,5,7},{1,2,3,4,6},{2,4,5,6,7},{1,2,4,5,6,7},{1,3,4,5,7 },{1,2,3,4,6,7}}.
5
7
3 2
4
1
If chosen any path from the above tree we notice that ,the interior for set of its vertices with respect to topology ((V(P))= int(V(P))))indicated to smallest numbers of vertices used to know the lower approximation of the path and so, the closure of this set
(V(P)= cl(V(P)))) is highest number of vertices used to know the upper approximation of the path .Let P1be a
path from 4 to 1(4-2-1),the lower approximation of P1 is
Int(V(P1)={4},the graph of lower approximation of P1 is G(Int(V(P1)), the upper approximation of P1 is
cl(V(P1)={1,2,3,4,6},the graph of lower approximation of P1 is G(cl(V(P1))
4
G(Int(V(P1))
G(cl(V(P1))
Fig(3.2)
Let P2 be a path from 2 to 7 (2-4-6-7) then the lower and upper approximations of the path are
Int(V(P2)={2,4,6}and cl(V(P2))={2,4,5,6,7}.
G(Int(V(P2)) G(cl(V(P2))
Fig(3.3)
In [18] there are a new definition of lower and upper approximation classes from tree vertices by forming a partitions of tree vertices and add to it the root of the tree also, discussed the accuracy of dependence for any path with another path with respect that this structure and rough inclusion between many paths using some new three type of roughly power set of vertices. We used these concepts to building topological space and form some special classes from it.
Proposition 3.1
Let T(V,E) be a tree. If is topology on T(V,E) generated by the neighborhood Nx then
(RO(V(P) ) is equal (SO(V(P))) where V(P) is set of vertices of any path in tree. Proof
Obviously since the result topology is quasi-discrete topology which all sets are open and closed
Example 3.3
Consider the binary tree
3 2
4
1
6
4
2 6
5
7
2
4
Fig(3.4)
We define the neighborhoods as the out vertices in sub tree adjacent to vertex. Na ={b,c }, Nb={d,e},Nc ={f,g}.
={V,,{a}, {b,c}, {d,e},{f,g},{ a,b,c, }, { a,d,e},{ a,f,g},{ b,c,d,e},{ b,c,f,g},{d,e,f,g}, {a,b,c,d,e},{ a,b,c ,d,e},{ a,b,c,f,g},{ b,c,f,g},{a,d,e,f,g},{ b,c,d,e,f,g}}.
The closed sets
F ={,V ,{a}, {b,c}, {d,e},{f,g},{ a,b,c, },{a,b,c,d,e},{ a,d,e},{ a,f,g},{ b,c,d,e}, {b,c,f,g},{d,e,f,g},{ a,b,c ,d,e},{ a,b,c,f,g},{ b,c,f,g},{a,d,e,f,g},{ b,c,d,e,f,g}}.
(RO(V(P)))={V,,{a},{b,c},{d,e},{f,g},{a,b,c,},{a,d,e},{a,f,g},{b,c,d,e},
{a,b,c,d,e}, { b,c,f,g},{d,e,f,g},{ a,b,c ,d,e},{ a,b,c,f,g},{ b,c,f,g}, {a,d,e,f,g},{ b,c,d,e,f,g}}=(SO(V(P)) ) .
In the following, we introduce a comparison between topological concepts and the consepts of graph theory for trees properties as in [2], we take tree and sub tree as a comparison models of our work. In [2] A.Koza and M.Shokry delete edges from connected graph G=(V,E)and studied change in topological properties , this implies that the induced topology for graph and sub graph satisfies the inclusion properties. In the following, we study this concepts on tree and sub tree vertices when deleting are in the last level so, the set of vertices of tree and sub tree are various, this implies that the inclusion properties not satisfy unless the neighborhoods are fare from deletions place.
Proposition 3.2
Let T(V,E) be a tree and T(V1,E1) be a sub tree resultant from T(V, E ) by eliminate nodes from
last level , V1 V and E1 E. If 1 is topology on T(V,E) generated by the neighborhood Nx and 2 is a topology on T1(V1,E1) generated by the neighborhood Nx1 then (RO(V(P) )2 results by 2 is sub-open structure of (SO(V(P)))1 results by 1 where V(P) is set of vertices of any path in tree.
Proof
Let T(V,E) has (h)-level and x be a node lies in (h-1)-level .If
N
1x
(
RO
(
V(P)
))
2 so Nx1 int(cl(N1x)there are two possible cases:
(1) If x is in a part of T(V1,E1) which has no deletions of adjacent vertices from
h-level in T(V,E) then
N
1x=Nx for all Nx1 .Since 1 is subclasses of (SO(V(P)))1therefore
N
1x(SO(V(P)))1 so, (RO(V(P)) )2 is sub-open structure of (SO(V(P)))1 .(2) If x is in a part of T(V1,E1) which has deletions of adjacent vertices from
h-level in T(V,E) then
N
1x Nx for all Nx1. Since (RO(V(P)) )2 is sub-open structure of 1 , 1is subclasses of (SO(V))1 so, (RO(V(P)) )2 is sub-open structure of (SO(V(P)))1.f
g
e
e1 e2
e3 e4 e6
e5
b
a
d
In the following example, we build two topological structure by above method , illustrated that the condition must putting to satisfy the inclusion property for two induced topological structures.
Example 3.4
Consider the tree in example3.2 if we deleted two nodes from the last level
Fig(3.5)
Na ={b,c }, Nb={d,e}and the root of the tree {a}
The set of subbase S2 = {{a}, {b,c},{d,e}}
2={V,,{a},{b,c},{d,e},{a,b,c},{a,d,e},{b,c,d,e}{a,b,c,d,e}}
RO(V(P)) )2 ={V,,{a,b,c},{b,c},{a,b,c,d,e},{d,e},
{a},{a,d,e},{b,c,d,e}}=(SO(V(P)))2
(RO(V(P)) )2 is sub-open structure of (SO(V(P))) where(SO(V(P))) as in
We can obtain a new measure for the exactness of any path with respect to partition tree in level also, we used various topological properties to get this important quantity. One of the following five factors tell us the best method of determine the lower approximations vertices and upper approximations vertices for obtained any path in tree by approximate methods.
Definition3.4
The accuracy of exactness for any path P in a tree T(V,E) is obtained by many methods:
) (V)) (
) (V(P) int( 1
T cl
) (RO(V)) (
) (RO(V(P)) int(
2
Ro cl
) (SO(V)) (
) (SO(V(P)) int(
3
cl
So
) (SO(V)) (
) (RO(V(P)) int(
4
cl
RS
) (RO(V)) (
) (SO(V(P)) int(
5
cl
SR
where | | is the cardinality.Corollary 3.1
In any quasi-discrete topology the five measure of accuracy of exactness of any set in topology
are equals
T
Ro
So
RS
SR Example 3.4:In Example 3.1 consider the path P = ae1be3d
int(RO(V(P) ))= int((SO(V(P))))= int(V(P)))= {a} and cl((RO(V)))= cl((SO(V(P))))= cl(V(P)))={{a,b,c,d,e} .The accuracy of exactness for any path P
T
Ro
So
RS
SR0.2If we change the method of covering tree vertices(in Definition 3.2) by adding the root of every subtree in its neighborhood system we find many changes in topological properties.
Defination 3.5
If T(V,E) is a tree and xV,the neighborhood of x is defined by
e
e1 e2
e3 e4
b
a
d
} child,
is y , leave not and subtree of
root is x : {y {x}
N
x xy E
Example 3.5
Consider the binary tree
Fig(3.6)
We define the neighborhoods as the out vertices in sub tree adjacent to vertex. Na ={a,b,c }, Nb={b,d,e},Nc ={c,f,g}.
The set of basis = { V, {b}, {c}, { a,b,c }, { c,f,g },{b,d,e}}
={V,, {b}, {c}, { a,b,c },{ c,f,g },{b,d,e},{a,b,c,d,e},{b,d,e,c}, ,{b,c},{ b,c,f,g},{ b,c,d,e,f,g},{a,b,c,f,g}}. The closed sets
F ={,V ,{a}, {d,e},{f,g},{ a,d,e },{a,f,g},{ f,g},{ a,c,e,f,g}, {a,b,d,e},{a,d,e,f,g}{d,e,f,g},{ a,b,d,e,f,g},{ a,c,d,e,f,g}}.
(RO(V(P)))={V,,{c},{c,f,g},{b,d,e}}
(RC(V(P)))={V,, { a,b,d,e,f,g},{ a,b ,d,e},{ a,c,f,g}}
(SO(V(P)) ) ={V,,{b},{c},{a,b,c },{c,f,g},{b,d,e},{b,c},{b,c,f,g},{b,d,e,c,}
{ a,b,c,f,g},{ a,b,c,d,e}}
(SC(V(P))) ={V,, {a,c,d,e,f,g},{a,b, d,e,f,g},{a,b,d,e},{a,c,f,g}, {d,e},{f,g},{a,d,e},{a,f,g}}
consider the path P1 = ae2ce5f
int((V(P1) ))= {c}, cl(V(P1) ))= { a,b,c,f,g}, int(RO(V(P1) ))= {c}
cl(RO(V(P1) ))= { a,c,f,g}, int(SO(V(P1) ))= {c}and cl(SO(V(P1) ))= { a,c,f,g}.
The accuracy of exactness for path P1 is
T 15and
Ro
So
RS
SR0.25In this case we concluded that topology is better way to find lower and upper approximation of path
P1.Consider subtree from above binary tree obtained by removed the leaves vertices,that not lied on the
search path P
Fig(3.7)
Na ={a,b,c }, Nc={c,f}
The set of subbase S2 = {{a,b,c },{c,f}}
1={V,,{c},{a,b,c },{c,f},{a,b,c,f}}
F1={ V,, { a,b ,d,e,f,g},{d,e,f,g},{a,b,d,e,g}{d,e,g}}
Subtopology1 is better than topology for identified lower and upper approximation of path P1
f
g
e
e1 e2
e3 e4 e6
e5
b
a
d
c
f e1 e2
e5
b
a
4 Rough Tree Vertices Properties Generated By Topological Spaces
In this section, we introduce and investigate the concept of approximation space. Also, we introduce the concepts of lower approximation and upper approximation
and study their properties. The reference space in rough set theory is the approximation space whose topology is generated by the tree vertices. This topology belongs to a special class known by Clopen topology, in which every open set is closed[11]. Clopen topology is called quasi-discrete topology in digital geometry; Lin calls it Pawlak space [37].Clopen topology is a kind of approximations.Lin introduced neighborhood system to handle such general situations
[24, 10, 30].
We will express rough vertices set V(P)of any path P, study properties in terms of topological concepts. Let P any path in right or lift subtree,
(V(P))
, (V(P))oand (V(P))b be closure, interior, and boundary points respectively. V(P) is exact path if (V(P))b = , otherwise V(P) is rough. In Pawlak space a subset V(P) V has two possibilities rough or exact. For a general topological space, V(P) V.Definition 4.1
Let
K
V
,
N
x
be an approximation structure on vertices of tree T(V,E) and RO(X) is the R-openstructure associated to K.Then the triple
K
V
,
N
x,(RO(X))
is called a R-open structure approximation space.The importance of topological view of graph theory comes from that topology studied graphs in many approximation way and properties.We used interior and closure concepts
to decrease of the boundary regions for the indefinable path in tree, this used in some problems for rough sets theory,such that vagueness of knowledge and decrease of the boundary regions for the indefinable set in information [1,14,15,4,9,19,31,32,33,34,35]
Definition 4.2
Let
K
V
,
N
x,(RO(X))
be a R-open approximation structure . If P is any path in the tree,V
P
V
(
)
, then the upper approximation (resp.lower approximation )of the path is defined by)
V(P)
(
)
(
) ( ( ))(
T
V
P
RO XU
(resp. T
(
)
(V(P))
0)) ( ( )
(
X RO
P
V
L
).
Definition4.3
Let
K
V
,
N
x,(
RO
(
X
))
be a R-open approximation structure generated by topological space
on rooted tree vertices .If P is a path in T(V,E)
(1) P is totally definable path with respect to
K
V
,
N
x,(
RO
(
X
))
if
L
T(
V
(
P
))
V(P)
U
T(
V
(
P
))
.(2) P is internally definable path with respect to
K
V
,
N
x,(RO(X))
if
V(P)
L
T(
V
(
P
)),
V(P)
U
T(
V
(
P
))
(3) P is externally definable path with respect to
K
V
,
N
x,(RO(X))
if
V(P)
L
T(V(P)),V(P)
U
T(V(P)) (4) P is indefinable if path with respect toK
V
,
N
x,(RO(X))
V(P)
L
T(
V
(
P
)),
V(P)
U
T(
V
(
P
))
.If P is an exact in
K
V
,
N
x,(RO(X))k
and (RO(X)) K is coarser than(RO(X))
'K
then P
is exact with respect to
,
,(RO(X))
'1
K
N
V
K
x .Proof.
Since V(P) is exact in
K
V
,
N
x,(RO(X))k
so V(P) )) ( ()
V(P)
(
0RO X ,
(
V(P)
V(P)
(
( ))
(
BND
'BND
RO X then RO XK K
and V(P) is exact with respect to
) (X '
RO
K
. In other words if V(P) is(RO(X))k exact then V(P) is clopen and consequently RO(X)'K
clopen. Hence V(P) is exact in RO(X)'K.
It is clearly that, there are paths τ' exact but not τ exact. Let us observe that
(
V(P)
)
(
V(P)
)
) ( )
(X ' RO X
RO
K
K
iff
(V(P))
0(V(P))
0) ( )
(X ' RO X
RO
K
K
. The following
proposition gives the condition for τ' exact paths to be τ exact paths, τ is is coarser than τ'.
Proposition 4.2:
If
V
,
N
,(
RO
(
X
))
K
x K is R-open approximation structure generated by tree T(V,E)vertices and
(RO(X))
'K
is coarser than RO(X)K,
,
(RO(X))
' , 1 KN
V
K
x is topologicalapproximation space generated by subtree T1(V1,E1) then each exact path P with V1(P) in
'
K is exact in K
iff(
(
(
)))
) (P
V
U
T RO X =(
U
T(
V
1(
P
)))
RO(X)'Kfor all V1(P)K1 and V(P)K.
Proof
If P is RO(X)'K- exact then
(
(
(
)))
(RO(X))V(P)
' P
V
U
T K and(P)
V
)))
(
(
(
U
TV
1P
RO(X) 1 , hence(
V(P)
)
RO(X)(
V
1(P)
)
(RO(X)) . Conversely if
)
(P)
V
(
)
V(P)
(
RO(X) 1 RO(X)
and P is RO(X)'K –exact path since V (P) in RO(X)'K then P is
)
(
X
RO
–exact path since V1(P) inRO
(
X
)
.Proposition 4.3
Let
K
V
,
N
x,(
RO
(
X
))
be a R-open approximation structure. IfP
1 andP
2 are two paths inT(V,E) then
1)
L
T(
V
(
P
1))
V
(
P
1)
U
T(
V
(
P
1))
.2)
U
T((
V
(
P
1))
(
V
(
P
1)))
(
U
T((
V
(
P
1)))
(
U
T((
V
(
P
2)))
.3)
L
T((
V
(
P
1))
(
V
(
P
1)))
(
L
T((
V
(
P
1)))
(
L
T((
V
(
P
2)))
.4)
IfV
(
P
1)
V
(
P
2)
, thenL
T(
V
(
P
1))
L
T((
V
(
P
2))
.5)
IfV
(
P
1)
V
(
P
2)
, thenU
T(
V
(
P
1))
U
T((
V
(
P
2))
.6)
L
T((
V
(
P
1))
(
V
(
P
1)))
(
L
T((
V
(
P
1)))
(
L
T((
V
(
P
2)))
.7)
U
T((
V
(
P
1))
(
V
(
P
1)))
(
U
T((
V
(
P
1)))
(
U
T((
V
(
P
2)))
.8)
L
T((
V
V
(
P
1))
(
U
T((
V
(
P
1))
.9)
U
T((
V
V
(
P
1))
(
L
T((
V
(
P
1))
.Proof. Obviously.
Let
K
V
,
N
x,(
RO
(
X
))
be R-open approximation structure.If
P
1 andP
2 are two paths in T(V,E) then1)
L
T((
V
(
P
1))
(
V
(
P
2)))
(
L
T((
V
(
P
1)))
(
L
T((
V
(
P
2)))
.2)
U
T((
V
(
P
1))
(
V
(
P
2)))
(
U
T((
V
(
P
1)))
(
U
T((
V
(
P
2)))
.Proof:
1)
We need to show that))
(
(
))
(
(
)
))
(
))
(
((
2 )) ( ( 1 ( ( ))2
1 ( ( ))
0 0 0
P
V
P
V
P
V
P
V
RO X RO X RO X
. Now,))
(
(
))
(
(
)
))
(
))
(
((
V
P
1
V
P
2
V
P
1
V
V
P
2
then)))
(
(
(
))
(
(
)
))
(
))
(
((
2 )) ( ( 1 21 ( ( ))
0
P
V
V
P
V
P
V
P
V
RO X
oRO X
(
(
)
)
(
(
(
2))
)) ( ( 1 ( ( ))
0
P
V
V
P
V
RO X
oRO X . ) 2 ( ( )) (
1 ( ( )) 2
1
))
(
))
)
( ( ))(
(
))
(
V(
P
)
)
(
((
0 X RO o X RO XRO
V
P
V
P
V
P
V
))
(
(
))
(
(
)
)
P
V(
(
))
(
(
2 )) ( ( 1 ( ( ))2 ( ( ))
1 ( ( ))
V
P
V
P
P
V
oRO X oRO XX RO o
X
RO .Then
))) ( (( ( ))) ( (( ( ))) ( ( )) (
((
V
P
V
P
L
V
P
L
V
P
L
T 1 2 T 1 T 22)
We need to show that))
(
(
))
(
(
)))
(
(
))
(
(
2 )) ( ( 1 )) ( ( 2 1 )) ((
V
P
V
P
P
V
P
V
X RO X RO XRO
. ) )) ( ( ( )) ( ( )) ( ((
(
))
(
(
))
))
(
(
))
(
(
V
P
1V
P
2V
P
1V
V
P
2X RO X RO X RO
K
,
)
)
(
(
))
(
(
))
(
(
))
(
(
1 2 0( ( )))) ( ( 2 )) ( ( 1 )) (
(
V
P
V
P
V
P
V
P
V
RO XX RO X
RO X
RO
,)
))
(
(
(
))
(
(
))
(
(
))
(
(
2 0)) ( ( 1 )) ( ( 2 )) ( ( 1 )) ( (
P
V
V
P
V
P
V
P
V
RO XX RO X
RO X
RO
(
(
(
(
))
)
)) ( ( 2 ) ( 1 X ROP
V
V
P
V
o
( ( ))1
(
(
(
))
(
V
(P
)V
V
P
2 RO X
(
V
(P
1)
(
V
(
P
2))
(RO(X)) .Therefore)))
(
((
(
)))
(
((
(
)))
(
(
))
(
((
V
P
1V
P
2U
V
P
1U
V
P
2U
T
T
T .Definition4.4
The higher degree of accuracy for exactness to any path P is
)) (
(
)) (
(
)
P
(
V
U
L
)
P
(
V
L
U
T T
T T HT
where | | isthe cardinality.
Example 4.1:
Consider the tree
Fig(4.1)
Na ={ b,c },Nb={ d,e},Nc=Ne=Nd=, = {V,,{ b,c},{d,e},{b,c,d,e}}.Consider the paths P1=a e1b e2e,
V(P1)={a,b,e}, the path P2= ae1 be4 d ,V(P2) ={a,b,d}. The higher accuracy of exactness to paths P1,P2
under this building is zero.The subtopology as in Corollary3.3 is of the form 1 = {V,,{a},{b},{d,e},{a,b},{a,e,d},{b,e,d},{a,b,d,e}}
The higher accuracy of exactness to path P1 is 0.5 by 1.
Conclusion
The concept of neighborhoods suggested in this paper can help in representing
Political relation between neighbors countries in its general case , problems in genetic reactions and may be apply in shortest path problems by classifying set of shortest path vertices between two approximation sets closures' and interior with respect to neighborhoods structure.
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