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242 Z. Ma, J. Mi / Information Sciences 370–371 (2016) 239–255
The above mentioned three types of approximation operators are equivalent to each other if R is an equivalence relation.
We also have shown that these different types of operators are different if R is a non-equivalence relation. And these types
of approximation operators are different pathways to extend the classical rough set. In other words, when R is not an
equivalence relation, they will generate the corresponding binary relations based generalized rough sets. A basic unit is a
set which can express the minimal ensemble. From an intuitive point of view, the basic units are the minimal units among
all of the precise knowledge. We also think the basic units are rudiments of information units.
Proposition 3. For any x ∈ U , R
s
(x ) = { y ∈ U| xRy } is a basic unit if and only if R is an equivalence relation on U.
Proof. “⇐ ”It is obvious from Definition 1 .
“⇒ ”For any x ∈ U , if R
s
(x ) = { y ∈ U| xRy } is a basic unit, we need to prove that R is an equivalence relation on U .
∀ x ∈ U , if R
s
( x ) is a basic unit, then R
s
( x ) = ∅ from Definition 1 . Suppose y ∈ R
s
( x ), ( x , y ) ∈ R , we have { x , y } ⊆R
s
( x ), namely
x ∈ R
s
( x ), ( x , x ) ∈ R . Thus R is a reflexive relation on U .
Suppose ( x , y ) ∈ R , namely y ∈ R
s
( x ), since R is a reflexive relation on U , y ∈ R
s
( y ), y ∈ R
s
( x ) ∩ R
s
( y ) = ∅ . Moreover, R
s
( x )
and R
s
( y ) are basic units on U , we have R
s
(x ) = R
s
(y ) , x ∈ R
s
(x ) = R
s
(y ) holds from Proposition 1 , namely ( y , x ) ∈ R holds.
Thus R is a symmetric relation on U .
Suppose ( x , y ) ∈ R and ( y , z ) ∈ R , we have y ∈ R
s
( x ) and y ∈ R
s
( y ) from the above conclusion. Moreover R
s
( x ) and R
s
( y )
are basic units on U , so R
s
(x ) = R
s
(y ) . Therefore, z ∈ R
s
(y ) = R
s
(x ) , namely ( x , z ) ∈ R holds. Thus R is a transitive relation
on U .
Finally, we conclude that R is an equivalence relation on U .
Proposition
3 also shows, when R is a non-equivalence relation on U , there exists x ∈ U , R
s
( x ) is not a basic unit. In this
sense, it is more reasonable to define the approximation operators by means of the basic unit than successor neighborhood
of x . In general, the precision of the latter definition is lower than one of the former’s.
Example 2. Let U = { a, b, c, d, e, f, g} . R = { (a, b) , (c, d) , (d, e ) , (e, f ) , ( f, g) } is a binary relation on U . Then U
R
=
{{ a, b} , { c, d, e, f, g}} . R
s
(a ) = { b} , R
s
(b) = ∅ , R
s
(c) = { d } , R
s
(d ) = { e } , R
s
(e ) = { f } , R
s
( f ) = { g} , R
s
(g) = ∅ . And let X =
{ a, b, c, d } , thus,
R
0
(X ) = { a, b, c, g} , R
0
(X ) = { a, c} , P
0
(X ) = 2 is no sense;
R
1
(X ) = { b, d} , R
1
(X ) = { b, d} , P
1
(X ) = 1 is no sense because of X = { a, b, c, d};
R
2
(X ) = { a, b, c, d} , R
2
(X ) = { a, b, c, d, e } , P
2
(X ) = 0 . 8 .
Example 2 illustrates that the 0-type and 1-type approximation operators already lose the sense of “low” and “up”,
namely the lower approximation of X do not have to be contained in set X , and X do not have to be contained in the upper
approximation of X . The 2-type approximation operators remain the sense, but it is more rough to describe information of
U from the point of view of classification. So, we hope to find a more reasonable definition.
The essence of the classical rough sets lies in approximation. The difference of the upper and lower approximations is
called the boundary. Now, we introduce the following definition.
Definition 2. ∀ X ⊆U , we call
BR
L
(X ) = { x ∈ X |∃ y ∈∼ X, xRy ∨ yRx } ,
BR
H
(X ) = { x ∈∼ X |∃ y ∈ X, xRy ∨ yRx }
the lower and upper boundaries of set X based on binary relation R , respectively. For any X , it is apparent that BR
L
( X ) = U
and BR
H
( X ) = U ; BR
L
(X ) = ∅ if and only if BR
H
(X ) = ∅ . And BR (X ) = BR
L
(X ) ∪ BR
H
(X ) is called the boundary of set X based
on binary relation R , which is called the boundary for short.
Obviously, the boundary
BR (X ) = { x ∈ U|∃ y ∈ U, xRy ∨ yRx, (x ∈ X ∧ y ∈∼ X ) ∨ (x ∈∼ X ∧ y ∈ X ) } .
Proposition 4. For any X , Y ⊆U , the boundary BR ( X ) satisfies the following properties:
(1) BR (∅ ) = ∅ ;
(2) BR (U) = ∅ ;
(3) BR (X ) = BR (∼ X ) ;
(4) BR ( X ∩ Y ) ⊆BR ( X ) ∪ BR ( Y );
(5) BR ( X ∪ Y ) ⊆BR ( X ) ∪ BR ( Y );
(6) ∀ x ∈ X , R
s
( x ) ∪ R
p
( x ) ⊆X ⇒ BR (X) = ∅ ;
(7) if X is a precise set, then BR (X ) = ∅ ;
(8) if X is a basic unit, then BR (X ) = ∅ ;
(9) BR (X ) = X ⇔ X = ∅ .