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S -Approximation Space based on simply -open sets
M.E.
Department
of
mathematics,faculty
of
Science,T
anta
Ab d
universit
,y
El
Monsef,
M.Sw
ealam
and
M.El
Sayed,
Egypt.
Egypt.
Abstract
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In this paper we introduce new types of lower and upper approximations namely S -lower and S upper approximations for any set based on the concept of simply -open set which generated from general
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relation.Also we study S
rough approximations.Finally we show the relation a monge of these types and
another types of approximations and we study some of thier basic properties.
Key words : Rough sets and topology.
[email protected]
1
Introduction
In 2007 M. El Sayed [2] introduced the concept of simply -open sets.The complement of simply -open set
is also simply -open sets. We use this kind in the development of approximation space as a generalization
the approximation of Pawlak. The rough set introduced by Pawlak [4] concept can be of some importance,
primarily in some branches of arti…cial intelligence, such as inductive reasoning, automatic classi…cation, pattern
recognition, learning algorithms, etc.The standard rough set theory starts from an equivalence relation.
De…nition 1.1 [1] A subset B of a topological space (X; )is called nowhere dense if int(cl(B) = :
De…nition 1.2 [2] A subset B of a topological space (X; ) is called simply open set if
cl( int(B))or B = G [ N , where G is -open set and N is nowhere dense .
int( cl(B))
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We denoted to the class of simply
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closed sets by S
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open sets of a universe set X by S O(X) and the class of simply M
C(X);such that S O(X) = S
C(X).
De…nition 1.3 [6] For the pair (U; R)where U is a universe set and R be a binary relation,let xR be an after
set de…ned by : xR = fy 2 U : xRyg:
De…nition 1.4 [3] Let X be a universe set and R be binary relation.Then the lower and upper approximations
are de…ned by:
LA = [fG; G
A; G is open set g; LA = \fF; F
A; F is closed set g:
De…nition 1.5 [5] If (U; R) is an information system then we can de…ne R
mation of X as follows:
R(X) = [fY 2 U j R; Y
Xg;
R(X) = [fY 2 U j R; Y \ X 6= g:
Proposition 1.1 [6] If (U; R) is an information system and A; B
1. R(A)
A
R(A):
1
U .Then :
lower and R– upper approxi-
2. R( ) = R( ) =
and R(U ) = R(U ) = U:
3. R(A [ B) = R(A) [ R(B) and R(A \ B) = R(A) \ R(B):
4. If A
B then R(A)
5. R(A [ B)
6. R( A) =
R(B) and R(A)
R(A) [ R(B) and R(A \ B)
R(A) and R( A) =
R(B):
R(A) \ R(B):
R(A):
7. R(R(A)) = R(R(A)) = R(A): and R(R(A)) = R(R(A)) = R(A):
Where
2
A is the complement of A:
Main work
In this section we introduce new types of approximations called simply lower approximation and simply upper
approximation for any subset A of space X and for any subset of condition attributes of the original attribute.
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De…nition 2.1 If (X; R; S O(X)) is a S
universe set X and A X. Then the simply
approximation space associated with general relation R over a
-lower and simply -upper approximations are de…ned by:
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B (A) = [ G :G 2 S O(X); G
S
A ;
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BS (A) = \fF; F 2 S C(X); F
Ag; respectively. The accuracy of approximation of A in (X; R; S O(X))
B (A)
by
(A) =
S
B S (A)
where j:j denotes the cardinality of the set, B S (A) 6= . The simply
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-boundary region
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of the set A (brie‡y S b(A)) is S b(A) = BS (A)
B (A):
S
Remark 2.1 If the lower and upper approximations are identical (i.e. B (A) = B S (A)),then the set A is
S
de…nable or simply exact.Otherwise A is unde…nable or simply roughly in X .
Remark 2.2 We can show that
is unde…nable in X:
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o
(A)
1: If
(A) =1 ,then A is de…nable in X.If
(A)
1:Then A
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De…nition 2.2 If (X; R; S O(X)) is a S
universe set X and A X then :
approximation space associated with general relation R over a
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1. If B (A) 6= ; and BS (A) 6= X: Then A is called S - roughly de…nable in (X; R; S O(X)).
S
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2. If B (A) 6= ; and BS (A) = X: Then A is called S -externally unde…nable in (X; R; S O(X)).
S
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3. If B (A) = ; and BS (A) 6= X: Then A is called S -internally unde…nable in (X; R; S O(X)).
S
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4. If B (A) = ; and BS (A) = X: Then A is called S -totally unde…nable in (X; R; S O(X)).
S
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5. If B (A) = BS (A) = A:Then A is S - exact set in (X; R; S O(X)) .
S
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where S - roughly (resp. S -externally, S -internally, S -totally unde…nable and S - exact) denotes to simply
roughly (resp.simply externally, simply internally, simply totally and simply exact) sets
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The following example shows the classi…cation of simply rough sets brie‡y ( S -rough) as follows:
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Example 2.1 Let X = fa; b; c; dg; aR = bR = ; cR = fa; b; cgand dR = fb; cg and the topology associated with
this relation R is = fX; ; fag; fb; cg; fa; b; cgg:The partition which obtained from the class of simply open sets
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is ffag; fdg; fb; cgg:Then the sets A = fag and B = fa; dg are examples of S -roughly de…nable.The correspondM
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ing S -approximations, boundaries and the accuracies are : B (A) = fag; BS (A) = fa; b; cg, S b(A) = fb; cg,
S
(A) =
1
3
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= 0:33; B (B) = fag; Bs (B) = fa; b; cg; S b(B) = fb; cg, (B) =
S
1
3
= 0:33 .
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The sets C=fa; c; dg and D = fa; b; dg are examples of S -externally unde…nable.We have the corresponding
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S -approximations,boundaries and the accuracies are :
B (C) = fa; dg; Bs (C) = X; S b(C) = fb; cg, (C) =
S
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2
4
= 0:5, B (D) = fa; dg; BS (D) = X, S b(D) = fb; cg, (D) =
S
2
4
= 0:5.
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Also if we take the partition ffa; dg; fb; cgg:Then the sets U = fcgand V = fdg are examples of S M
internally unde…nable. The corresponding S -approximations, boundaries and the accuracies are B (U ) =
S
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; B(U ) = fb; cg; S b(U ) = fb; cg, (U ) = 0; B (V ) = ; Bs (V ) = fa; dg; S b(V ) = fa; dg, (V ) = 0.
S
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The sets E = fc; dgand F = fa; bg are examples
of
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S -totally unde…nable.The corresponding S M
approximations, boundaries and the accuracies are : B (E) =
S
; BS (E) = X; S b(E) = X ,
(E) =
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0; B (F ) = ; BS (F ) = X , S b(F ) = X, (F ) = 0:
S
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Remark 2.3 If (X; R; S O(X)) is a S
universe set X and A
X. Then: L(A)
approximation space associated with general relation R over a
L(A)
B
(A):
S
The following example indicates this remark.
Example 2.2 let X = fg; h; i; j; kg;the partition of pawlak UR1 = ffgg; fh; i; j; kgg;the class of subsets of X
UR2 = ffgg; fh; i; j; kg; fg; i; jg; fh; i; j; kggand the relation R2 over X is gR2 = fi; jg; hR2 = fg; i; jg; iR2 = fgg;
jR2 = fh; i; j; kg and kR2 = : Then the topology associated with this relation is
= fX;
; fgg ; fi; jg; fg; i; jg; fh; i; j; kgg :Then the class of lower and upper approximations of the uni-
verse set X are given by R(A) = fX; ; fgg; fh; i; j; kgg; R(X) = fX; ; fgg; fh; i; j; kgg;
L(A) = fX; ; fgg; fh; i; j; kg; fi; jg; fg; i; jgg and L(A) = fX; ; fgg; fh; i; j; kg; fg; h; kg; fh; kgg;
B
S
(A) = fX; ; fgg; fhg; fkg; fg; hg; fi; jg; fh; i; jg; fg; h; kg; fh; i; j; kgfg; i; jg; fg; i; j; kg; fg; h; i; jgg and
B S (A) = fX; ; fgg; fhg; fkg; fg; hg; fi; jg; fh; i; jg; fg; h; kg; fh; i; j; kg; fg; i; jg; fg; i; j; kg; fg; h; i; jgg
Let A = fg; i; j; kg,then we see that A = fg; i; j; kg 2 B (X), but A = fg; i; j; kg 2
= L(X) and fg; i; j; kg 2
=
S
L(A): Where B
A
X.
(A); B S (A); L(A) and L(A) are the family of all lower and uuper approximations for every
S
Proposition 2.1 The complement of all simply open sets in any topological space (X; ) are simply open
sets.Moreover …nite intersection of simply open sets is simply open set.
Proof. Obvious.
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Proposition 2.2 If (X; R; S O(X)) is a S
universe set X and A X.Then:
approximation space associated with general relation R over a
1. A set A is de…nable (resp.simply roughly de…nable,totally unde…nably) if and only if so is AC :
2. A set A is simply externally (simply internally unde…nable) if and only if AC is simply internally ( simply
externally ) unde…nable.
3
where AC denoted to the complement of the set A:
Proof.
1. (i) If A is a simply de…nable.Then A = B S (A) = B
(A). By taken the complement for the sides we
S
have AC = (B S (A))C = (B
(A) )C if and only if B C = B
S
simply de…nable.
(A)C = B S (A)C . Then AC is also
S
(ii)If A is a simply roughly de…nable,then B
(A) 6=
S
exists x 2 X and simply open set G such that x2 G
and B S (A) 6= X since B
A if and only if X
G
X
S
(A) 6= .Then there
A if and only if AC
6= X if and only if B S (AC ) 6= X where X G is simply closed set. Similarly B S (A) 6= X if and only if
there exists x 2 X and F is simply closed set such that x 2 F A if and only if X F X A if and only
if AC 6= if and only if B (AC ) if and only if B (AC ) 6= :Then AC is simply roughly de…nable.
S
S
(A) =
(iii)Since A is totally unde…nable. Then B
for both sides we get
(AC ) = ; B S (AC ) = X; if and only if AC is simply totally
(A) )C = X, (B S (A))C = . Hence B
(B
and B S (A) = X . Then by taken the complement
S
S
S
unde…nable.
2. Similarly as 1.
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Proposition 2.3 If (X; R; S O(X)) is a S
universe set X and A X.Then
1. B
(A)
A
approximation space associated with general relation R over a
B S (A)
S
2.
B S (A [ B) = B S (A) [ B S (B)
3.
B
(A \ B) = B
S
4.
If A
S
(A) \ B
B then B S (A)
(B):
S
(A)
B S (B) and B
5.
B
(A [ B)
S
6.
B
S
(A) [ B
(B):
B
S
S
(B):
S
(AC )]C :
B S (A) = [B
S
(A) = [B S (A)C ]C :
7. B
S
8.
B
(A):
(A)) = B
(B
S
S
S
9. B S (B S (A)) = B S (A):
10. B S (A \ B)
11. B
B S (A) \ B S (B)
( ) = BS ( ) =
and B (X) = B S (X) = X:
S
S
Proof.
1 Obvious.
2 .Obvious.
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3 Let x 2 B
S
(A\B):Then there exists a simply open set G such that x 2 [fG : G
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if and only if x 2 [fG : G
x 2 (B (A) \ B (B)):
S
A\B; G 2 S O(X)g
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A; G 2 S O(X)g and x 2 [fG : G
S
4
B; G 2 S O(X)
if and only if
4 Obvious.
5 Obvious.
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6 Since [B (AC )]C = X
[fG : G
S
X
A; G 2 S O(X)g = \fX
G:X
G
X
(X
A); X
G2
(X
A); X
F 2
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S C(X)g = B S (A):Then [B (AC )]C = B S (A):
S
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7 Let [B S (A)C ]C = X
AC ; 2 S C(X)gC = [fX
\fF : F
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C C
F
X
(A):
(A):Then [B S (A) ] = B
S O(X)g = B
F: X
S
S
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8 Let B
(B
S
(A)) = [fG : G
S
B
(A)
A; G 2 S O(X)g = B S (A): Hence B
S
(B
S
(A)) = B S (A):
S
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9. Let B S (B S (A)) = [fF : F
B S (A)
A; F 2 S C(X)g = B S (A) :Thus B S (B S (A) = B S (A):
10. Obvious.
11. Since X and
are simply exact . Then B
(X) = B S (X) = X; B S ( ) = B
S
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( )= :
S
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Proposition 2.4 If (X; R; S O(X)) is a S
approximation space associated with general relation R over a
universe set X and A; B X. Then the following statement are not hold :
1. B S (A
B)
B S (A)
B
(B):
S
2. B
(A
B)
B
S
(A)
B
S
(B):
S
Proof.
B) = B S (A \ B C )
1. Since B S (A
B S (A) \ ( B (B))C = B S (A)
S
B S (A)
B
B
B S (A) \ B S (B C ) =
(B):Then B S (A
B)
S
(B):
S
2 Since B
(A
B
S
(A \ B C ). Then B (A \ B C ) =
B) = B
S
S
(A) \ B
Therefore B
C
(B )
S
(A
S
B
S
B)
(A) \ (B
(A)
B
S
C
(B)) = B
S
B
(A)
S
(B):
B
(B):
S
S
S
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Proposition 2.5 If (X; R; S O(X)) is a S
universe set X and A X. Then:
approximation space associated with general relation R over a
(A)]C = B S (A)C :
1. [B
S
2. [B S (A)]C = B
(AC ):
S
Proof.
(A)]C = X
1. Since [B
S
C
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[fG : G
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A; G 2 S (X)g = \fX
G:X
G
X
A; X
G 2 S C(X)g =
F :X
F
X
A; X
F 2 S O(X)g =
(A)]C = B S (A)C
B S (A) : Then [B
S
2. Since [B S (A)]C = X
M
\fF : F
(AC ): Then [B S (A)]C = B
B
S
M
A; F 2 S C(X)g = [fX
(AC ):
S
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Proposition 2.6 If (X; R; S O(X)) is a S
universe set X and A; B X. Then :
2. (B
S
S
(A) [ B S (B))C = B S (A)C \ B
S
S
(A) [ B
S
S
S
(B))C = B
3. (B S (A) [ B
4. (B
S
6. (B S (A) \ B
(A)C \ B S (B)C :
(A)C [ B
S
(A)C [ B S (B)C
(B))C = B
S
(A) \ B S (B))C = B S (A)C [ B
S
(A) \ B
Proof.
S
(B)C :
S
S
8. (B
(B)C :
(B))C = B S (A)C \ B S (B)C :
5. (B S (A) \ B S (B))C = B
7. (B
(B)C :
(A)C \ B
1. (B S (A) [ B S (B))C = B
approximation space associated with general relation R over a
S
(B)C :
S
(B))C = B S (A)C [ B S (B)C :
Obvious.
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Proposition 2.7 If (X; R; S O(X)) is a S
universe set X and A; B X. Then
(A)
1. B
approximation space associated with general relation R over a
(A)):
B S (B
S
S
2. B( B S (A)
B S (A):
S
3. B S (A)
B
(B S (A)):
S
Proof.
1. Since B
(A)
A
B S (A): Then B
S
2. Since B
(B
S
(B S (A))
B S (A)
(A))
(B S (A))
B
(B S (A)):Then B S (A)
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(B S (A)):
approximation space associated with general relation R over a
(AC ) = :
S
(A)\ B S (AC ) = :
S
B
S
S
3. B
B S (A):
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Proposition 2.8 If (X; R; S O(X)) is a S
universe set X and A X. Then.
2. B
(A)):
S
(B S (A))
B S (B S (A)):Then B
S
(A) \ B
B S (B
S
3. Since B S (B S (A))
S
(A)
S
S
1. B
B
S
(AC ) \ B S (A) = :
4. B S (A) [ B S (AC ) = X:
6
5. B S (AC ) [ B
6. B
S
(A) = X:
S
(AC ) [ B S (A) = X:
Proof.
Obvious
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Proposition 2.9 If (X; R; S O(X)) is a S
universe set X and A X. Then
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approximation space associated with general relation R over a
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1. A is S - exact () S b(A) = :
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2. A is S -rough () S b(A) 6= :
Proof.
Obvious.
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Proposition 2.10 If (X; R; S O(X)) is a S
a universe set X and A X. Then
approximation space associated with general relation R over
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1. B
S
(A) [ B
(AC ) = [ S b(A)]C :
S
M
2. B S (A)\ B S (AC ) = S b(A):
Proof.
Obvious.
References
[1] Baurboki, N. " GeneralTopology part " Reading Math. Addison Wesley,(1966).
[2] El- Sayed,M. " Relation between some types of continuity in topologiacl spaces" M. Sc. Thesies Tanta
University Faculty of Science 2007.
[3] Medhat,T. " Supra Topological approach for decision making Via granular Computing " ph.D.Thesis Tanta
Egypt (2008).
[4] Pawlak, Z. " Rough sets international Journal of Computer and information Sciences".11,5, 341-356 (1982).
[5] Pawlak, Z."Rough sets theoretical aspects of reasoning about data system theory, Knoweledge Engineering
and problem solvin " vol.9, Kluwer Academic publishers ,Dordrecht, The Netherlands,1991.
[6] Y.Y.Yao ; Line,T.Y. " Generalization of Rough sets Using Modal logic " int.Automation ,Soft Comput.
2(1996) 103-120.
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