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On relationship between modified sets, topological spaces and rough sets
Fuzzy Sets and Systems 61 (1994) 91-95 North-Holland
9l
On relationship between modified sets, topological spaces and rough sets Jari Kortelainen Mikkeli Polytechnic, Institute of Business, 50100 Mikkeli, Finland Received April 1993 Revised July 1993
Abstract: In this paper we define modifiers by relations. Especially, weakening and substantiating modifiers are defined by a so called accessibility relation which is a reflexive relation on a non-empty set X. After presenting some main results we prove that this type of modifiers will satisfy the Kuratowski Closure Axioms. This means that modifiers in fact induce topological spaces. Also rough sets are considered to be a special case of modified sets.
Keywords: Closure operators; fuzzy sets; modifiers; modifier logic; rough sets.
It was discussed e.g. in [11] that modal logics can be characterized algebraically by topological Boolean algebras. Also connections between modal logic and modifier logic are discussed in [8] and [9]. So, the main idea of this paper is to show that weakening modifiers can be thought of as closure operators and rough sets can be thought of as modified sets, because rough sets are defined by equivalence relations. The power-set of X, ~9(X), consists of only ordinary subsets of X. After these ideas and notions we present the following definitions. Definition 1.1. Let ~ ( X ) be the ordinary power-set of a non-empty set X. We say that mapping ~ : ~ ( X ) - - ) ~ ( X ) is modifier, if for any A c X, ~ ( A ) = {y e X ] 3x e X , ( x , y ) e R ~ and x e A } .
1. Introduction
(1.1) It has been discussed by many authors that modifiers are pointwise functions with domain [0, 1] (the real unit interval). However, in [3] and [7] we can find some criticism of that approach. In this paper we think that modifiers are set-functions which are defined by a so called accessibility relation. Accessibility relations model vagueness or uncertainty when collecting elements into subsets of a non-empty set X. So, more formally, we connect two elements of X, x and y, together by a relation R c X × X, and we say that R is an accessibility relation, whenever x is collected into a subset of X, then also y might be collected into that same set. So, in this paper we assume that accessibility relations are fixed subsets of X x X and they do not depend on subsets of X. By Kripke's basic idea, we think that accessibility relations are reflexive. Indeed, we demand that all elements of A are also possible elements of A. Correspondence to: Dr. J. Kortelainen, Institute of Business, Mikkeli Mikkeli, Finland.
Polytechnic, Paamajankuja
4, 50100
If R~c is an accessibility relation (reflexive relation), then we say that ~ is a weakening modifier. Now, Yg(A) can be determined so that for all y e X we first find pairs (x,y)eR~, and we collect y e X into Y((A) if for some pair (x, y) also x e A. Clearly Y((A) is the same set as e.g. the max-min-relation composition of R~ and A, but we still prefer (1.1). Definition 1.2. Let ~ be a weakening modifier. If for any A c X we assign ~*(A) = ~(A), then we say that ~ * is dual of ~ . Usually we say that ~ * is substantiating modifier. The overbar denotes the complementation. Example 1.3. Let X = { 1 , 2 , 3 , 4 } and A = {1, 3}. We define an accessibility relation R~e by relation table 1.
0165-0114/94/$07.00 ~ 1994---Elsevier Science B.V. All rights reserved SSDI: 0165-0114(93)E0186-V
92
J. Kortelainen / Modified sets, topological spaces and rough sets
valid. Formula (2.4) can be now proved by (2.3) and the properties of complement. Also (2.5) can be proved clearly by the properties of complement. []
Table 1 R~e
1
2
3
4
1 2 3 4
1 0 0 0
1 1 0 0
1 1 1 0
0 0 0 1
Proposition 2.2. Let 9( be a weakening modifier and A, B c X. If A c B, then
So, R~e= ((1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3), (4, 4)}. Now, 1 • 9((A), because e.g. (1, 1) • R~c and 1 • A, 2 • 9((A), because (1, 2) • R~e and 1 • A, 3 • 9((A), because e.g. (1, 3) • R~e and 1 • A, 4 ¢ 9((A), because only (4, 4) • R~e, but 4 ¢ A, So, 9 ( ( A ) = {1, 2, 3}. Similarly, if we want to determine 9(*(A), we first determine the complement of A, .4 = {2, 4}. Element 1 is not in 9((.zi), because only (1, 1) • R~e but 1 ~ A. Certainly elements 2 and 4 are in 9((A). Also element 3 is in 9((A), because (2, 3) • R~e and 2 • / i . Taking the complement of 9((.zi) we find 9(*(A) = {1}.
L Some main results Our interest in this chapter is to describe the basic nature of weakening and substantiating modifiers. So, the main results, which are provable by Definitions 1.1 and 1.2, are as [ollows.
Proposition 2.1. Let 9( be a weakening modifier, X a non-empty set and R~ an accessibility ,elation on X. Then VA c X,
9((A) c 9((B),
(2.6)
9(*(A) c 9(*(B).
(2.7)
Proof. We prove (2.6) as follows. Let y • 9((A). Now there exists an x • A such that (x, y) • R~e. Also x • B because A c B . This means by Definition 1.1 that y • 9((B). So, (2.6) holds. Formula (2.7) can be proved by (2.6) and Definition 1.3. [] Proposition 2.3. Let 9( and 9( be weakening modifiers. If R~e c Rx, then for any A c X, 9((A) c 9((A),
(2.8)
9~*(A) c 9(*(A).
(2.9)
Proof. We prove first (2.8). Let y c 9((A). Now there exists an x • A such that ( x , y ) • R ~ . Because R~ c R~, pair (x, y) • R~c. This means that y • :~(A) by Definition 1.1. Formula (2.9) can be now proved by (2.8) and Definition 1.3. [] Now we like to find results on associating modifiers with combined sets.
Proposition 2.4. Let 9( be a weakening modifier and R~e an accessibility relation. Then for any A, B c X ,
~(0) = O,
(2.1)
~ * ( X ) = X,
(2.2)
4 c 9((A),
(2.3)
9((A n B) c 9((A) n 9((B),
(2.10)
~ * ( a ) c A,
(2.4)
9(*(A ) U 9(*(B) = 9(*(A U B).
(2.11)
[9(*)*(A) = 9((A)
(2.5)
Proof. We prove (2.1) as follows. Let y • 9((0). Now, there exists an x • 0 such that pair Ix, Y ) • R~ by Definition 1.1, but this is clearly impossible and (2.1) holds. Also (2.2) is valid because X = ~. Formula (2.3) can be proved as [ollows. Let x • A . Now, pair (x,x) •R~e because R~e is a reflexive relation. This means that V x • X , x • A ~ x • 9 ( ( A ) . Thus, (2.3) is
Proof. We prove (2.10) as follows. For any A, B c X , A A B c A and A A B c B . Now, by Proposition 2.2 both 9((A n B) c 9((A) and 9((A n B) c 9((B). This means that (2.10) holds. Formula (2.11) can be proved the same way as (2.10). So, for any A, B c X , A c A U B and B c A O B. By Proposition 2.2 both 9(*(A) c 9(*(A U B) and 9(*(B) c 9(*(A U B). This means that also (2.11) holds. []
93
J. Kortelainen / Modifiedsets, topologicalspaces and rough sets
Proposition 2.5. Let ~( be a weakening modifier and R~e an accessibility relation. Then for any A, B c X , ~ ( A U B) = ~(A) U ~ ( a ) ,
(2.12)
~*(A
(2.13)
n
B)
=
~*(A)
n
~*(B).
holds. We prove (2.15) by Definition 1.2 and (2.14) as follows: ~*(~*(A)) = ~(~(A)) = =
Proof. We prove first (2.12) as follows. Clearly A~AUB and B c A O B for any A, B c X . Now, by Proposition 2.2 both Y((A) ~ ~ ( A O B) and ~ ( B ) ~ ~ ( A O B). This means that (1)
~(A)
U
~ ( B ) ~ ~ ( A U B).
Suppose now that y e ~ ( A O B). There exists x e A O B such that pair (x, y) e R~e. Without violating generality we can say that x e A. So, y e ~(A) and especially y e ~(A) O ~(B). Clearly (2)
~(A
B) c ~ ( A )
U
U
~(B).
Now, results (1) and (2) proves (2.12). Formula (2.13) can be proved by (2.12), Definition 1.2 and De Morgan Laws as follows: ~*(A n B) = ~ ( A n B) =
B)
u
=
u
=
n
=
n
[]
Finally, the last Proposition of this chapter discusses a property of transitive accessibility relations.
Proposition 2.6. Let Y( be a weakening modifier and R~ a transitive accessibility relation. Then for any A ~ X , ~(Yg(A)) = ~(A),
(2.14)
~ * ( ~ * ( A ) ) = ~*(A).
(2.15)
Proof. We prove (2.14) as follows. Clearly ~ ( A ) c ~ ( ~ ( A ) ) , because ~ is a weakening modifier. Now, let z e ~g(~(A)). There exists a y e ~(A) such that pair (y, z) e R~e. Because y e Y((A), there exists an x e A such that pair (x, y ) e R ~ . Because R~e is a transitive accessibility relation, pair (x, z ) e R ~ . This means that z e Y((A) and ~tf(~(A)) c ~lf(A). So, (2.14)
= X*(A).
[]
3. Modifiers inducing topological spaces and rough sets In this chapter we show that certain modifiers are in fact closure operators and rough sets could be understood as modified sets.
Proposition 3.1. Let Y( be a weakening modifier and R~ a transMve accessibility relation. Then there exists a unique topology f f ~e on X such that ~ ( A ) will be the ff ~-closure of subset A c X. Proof. Let ~ be a weakening modifier and R~e a transitive accessibility relation. Now, ~ will satisfy the Kuratowski Closure Axioms, because for any A, B ~ X, (K1)
~(0)=0,
(K2)
A c ~(A),
(K3)
~ ( A U B) = ~ ( A ) U ~ ( B ) ,
(K4)
~ ( ~ ( A ) ) = ~(A),
by (2.2), (2.4), (2.17) and (2.20).
[]
Proposition 3.2. Let ~ and ~ be weakening modifiers such that R~ c R~. Let both R~e and R~ also be transMve accessibility relations. Then and ~ induce topologies J-~ and ff'x such that 5r~ is finer than fix. Proof. Because R~ecR~c and ~ and X are closure operators, for any A ~ X, ~ ( A ) ~ X(A) and this means that topology ~-~ is finer than ff,Jc (see e.g. [1] p. 29). []
Example 3.3. Let X be a non-empty set and R~ a fuzzy accessibility relation on X such that for any oc e [0, 1), the strong aMevel set R ~ is also transitive. In this case, for any A c X, ~ ( A ) = {y c X l 3x e X , (x, y) e R ~ and x eA}.
94
J. Kortelainen I Modified sets, topological spaces and rough sets
In the topology, which is induced by 9(~, any Yg,~(A) is an open set. So, we mark this topology by 9-~-" denoting that any Yg,(A) is a member of an ordinary ( 1 - tr)-level class of ~ e . If a, f l e [0, 1) are such that fl ~< tx, then topology 9-~-~ is finer than 9-~ t~ by Proposition 3.2. We may now call Y( a fuzzifier because ~ induces a fuzzy class of sets. This should not be confused with classes of fuzzy sets.
Proposition 3.4. Let ~ be a weakening modifier and R~ an equivalence accessibility relation. Then for any A c X, Yf(A)=U{Y6X/R~c
[ Y N A 4:0},
(3.1)
where X /R~c denotes the family of all equivalence classes of R~c. Proof. Suppose y ~ Y((A). Now there exists an x c A such that (x, y ) ~ R~e. Because R~e is an equivalence relation, x and y are in the same equivalence class, say Y. Clearly Y n A 4:0 and
Clearly ~*(A) consists of all those equivalence classes which are inside A. Thus, (3.2) holds. [] Proposition 3.4 and 3.5 show that the upper approximation of A c X can be understood as weakened set and the lower approximation of A m X can be understood as substantiated set.
4. Concluding remarks In this paper we defined modifiers by so called accessibility relations which are reflexive relations on a non-empty set X. If we demand that accessibility relations must also be transitive, then our modifiers in fact induce topological spaces. Also, if we demand that accessibility relations must be equivalence relations, then our modifiers induce the rough sets. If we need to operate with fuzzy sets, then clearly
~(A)(Y) = max min{/~A(X), /,tR.,(x, y)},
Y ~ U {YEX/R~e I Y A A 4:0}. This means that (1)
generalizes Definition 1. l.
9~(A) c U { Y e X / R ~ e
I YOA4:0}.
Now, suppose there is an equivalence class Y such that y ~ Y and Y O A 4: ~. Now, there exists an x e A such that x and y are in the same equivalence class. Thus, 3x ~ A and (x, y) ~ R~ which means that y ~ gg(A). So (2)
U { Y e X / R ~ J Y 0 A 4: I~} ~ Y((A).
The results (1) and (2) prove (3.1).
[]
Proposition 3.5. Let Y( be a weakening modifier and R~e an equivalence accessibility relation. Then for any A c X, ~£*(A) = U { Y 6X/R~e
x ~ X. (4.1)
I YEA},
(3.2)
where X/R~e denotes all equivalence classes of R~e.
Proof. Because ~ is a weakening modifier, then for any A ~ X,
g((A) = {y 6 X I 3x c X, (x, y) ~ R~e and x e A} by Definition 1.1. Now, by Definition 1.2 we conclude that
gg*(A)= {y 6 X [Vx 6 X , (x, y) e R ~ e ~ x ~A}.
References [1] N. Bourbaki, Elements of Mathematics: General Topology, 2nd printing (Springer-Verlag, Berlin, 1989). [2] D. Dubois, H. Prade and X. Mo, Semantic and algorithmic aspects of approximate reasoning in possibility theory, Rapport IRIT/90-53/R (1990). [3] H. Hellendoorn, The generalized modus ponens considered as a fuzzy relation, Fuzzy Sets and Systems 46 (1992) 29-48. [4] I.M. James, General Topology and Homotopy Theory (Springer-Verlag, Berlin, 1984). [5] J. Kortelainen, On modified set-operations, in: P. Eklund and J.K. Mattila (Eds.), FiF '93, Fuzziness in Finland '93, /~bo Akademi Reports on Computer Science and Mathematics, Ser. B. No. 15 (1993) 48-58. [6] S.A. Kripke, Semantical considerations on modal logic, Acta Philosophica Fennica Fasc. XVI (1963). [7] G. Lakoff, Hedges: A study in meaning criteria and the logic of fuzzy concepts, The Journal of Philosophical Logic 2 (1973) 458-508. [8] J.K. Mattila, Calculus of modifier operators in fuzzy logic, Publications of the Institute of Applied Mathematics No 14, University of Turku, Turku, Finland (1989). [9] J.K. Mattila, Modelling fuzziness by Kripke structures, in: Terano, Sugeno, Mukaidono and Shigemasu (Eds.),
J. Kortelainen / Modified sets, topological spaces and rough sets Fuzzy Engineering Towards Human Friendly Systems Vol. 2, Proceedings of IFES '91, Yokohama, Japan (1991). [10] T. Patronis and P. Stavrinos, Fuzzy equivalence and resulting topology, Fuzzy Sets and Systems 46 (1992) 237-243.
95
[11] H. Rasiowa, An Algebraic Approach to Non-Classical Logics (North-Holland, Amsterdam, 1974). [12] Z. Pawlak, Rough sets: A new approach to vagueness, In: Zadeh and Kacprzyk (Eds.), Fuzzy Logic for Management of Uncertainty (John Wiley, New York, 1992).
9l
On relationship between modified sets, topological spaces and rough sets Jari Kortelainen Mikkeli Polytechnic, Institute of Business, 50100 Mikkeli, Finland Received April 1993 Revised July 1993
Abstract: In this paper we define modifiers by relations. Especially, weakening and substantiating modifiers are defined by a so called accessibility relation which is a reflexive relation on a non-empty set X. After presenting some main results we prove that this type of modifiers will satisfy the Kuratowski Closure Axioms. This means that modifiers in fact induce topological spaces. Also rough sets are considered to be a special case of modified sets.
Keywords: Closure operators; fuzzy sets; modifiers; modifier logic; rough sets.
It was discussed e.g. in [11] that modal logics can be characterized algebraically by topological Boolean algebras. Also connections between modal logic and modifier logic are discussed in [8] and [9]. So, the main idea of this paper is to show that weakening modifiers can be thought of as closure operators and rough sets can be thought of as modified sets, because rough sets are defined by equivalence relations. The power-set of X, ~9(X), consists of only ordinary subsets of X. After these ideas and notions we present the following definitions. Definition 1.1. Let ~ ( X ) be the ordinary power-set of a non-empty set X. We say that mapping ~ : ~ ( X ) - - ) ~ ( X ) is modifier, if for any A c X, ~ ( A ) = {y e X ] 3x e X , ( x , y ) e R ~ and x e A } .
1. Introduction
(1.1) It has been discussed by many authors that modifiers are pointwise functions with domain [0, 1] (the real unit interval). However, in [3] and [7] we can find some criticism of that approach. In this paper we think that modifiers are set-functions which are defined by a so called accessibility relation. Accessibility relations model vagueness or uncertainty when collecting elements into subsets of a non-empty set X. So, more formally, we connect two elements of X, x and y, together by a relation R c X × X, and we say that R is an accessibility relation, whenever x is collected into a subset of X, then also y might be collected into that same set. So, in this paper we assume that accessibility relations are fixed subsets of X x X and they do not depend on subsets of X. By Kripke's basic idea, we think that accessibility relations are reflexive. Indeed, we demand that all elements of A are also possible elements of A. Correspondence to: Dr. J. Kortelainen, Institute of Business, Mikkeli Mikkeli, Finland.
Polytechnic, Paamajankuja
4, 50100
If R~c is an accessibility relation (reflexive relation), then we say that ~ is a weakening modifier. Now, Yg(A) can be determined so that for all y e X we first find pairs (x,y)eR~, and we collect y e X into Y((A) if for some pair (x, y) also x e A. Clearly Y((A) is the same set as e.g. the max-min-relation composition of R~ and A, but we still prefer (1.1). Definition 1.2. Let ~ be a weakening modifier. If for any A c X we assign ~*(A) = ~(A), then we say that ~ * is dual of ~ . Usually we say that ~ * is substantiating modifier. The overbar denotes the complementation. Example 1.3. Let X = { 1 , 2 , 3 , 4 } and A = {1, 3}. We define an accessibility relation R~e by relation table 1.
0165-0114/94/$07.00 ~ 1994---Elsevier Science B.V. All rights reserved SSDI: 0165-0114(93)E0186-V
92
J. Kortelainen / Modified sets, topological spaces and rough sets
valid. Formula (2.4) can be now proved by (2.3) and the properties of complement. Also (2.5) can be proved clearly by the properties of complement. []
Table 1 R~e
1
2
3
4
1 2 3 4
1 0 0 0
1 1 0 0
1 1 1 0
0 0 0 1
Proposition 2.2. Let 9( be a weakening modifier and A, B c X. If A c B, then
So, R~e= ((1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3), (4, 4)}. Now, 1 • 9((A), because e.g. (1, 1) • R~c and 1 • A, 2 • 9((A), because (1, 2) • R~e and 1 • A, 3 • 9((A), because e.g. (1, 3) • R~e and 1 • A, 4 ¢ 9((A), because only (4, 4) • R~e, but 4 ¢ A, So, 9 ( ( A ) = {1, 2, 3}. Similarly, if we want to determine 9(*(A), we first determine the complement of A, .4 = {2, 4}. Element 1 is not in 9((.zi), because only (1, 1) • R~e but 1 ~ A. Certainly elements 2 and 4 are in 9((A). Also element 3 is in 9((A), because (2, 3) • R~e and 2 • / i . Taking the complement of 9((.zi) we find 9(*(A) = {1}.
L Some main results Our interest in this chapter is to describe the basic nature of weakening and substantiating modifiers. So, the main results, which are provable by Definitions 1.1 and 1.2, are as [ollows.
Proposition 2.1. Let 9( be a weakening modifier, X a non-empty set and R~ an accessibility ,elation on X. Then VA c X,
9((A) c 9((B),
(2.6)
9(*(A) c 9(*(B).
(2.7)
Proof. We prove (2.6) as follows. Let y • 9((A). Now there exists an x • A such that (x, y) • R~e. Also x • B because A c B . This means by Definition 1.1 that y • 9((B). So, (2.6) holds. Formula (2.7) can be proved by (2.6) and Definition 1.3. [] Proposition 2.3. Let 9( and 9( be weakening modifiers. If R~e c Rx, then for any A c X, 9((A) c 9((A),
(2.8)
9~*(A) c 9(*(A).
(2.9)
Proof. We prove first (2.8). Let y c 9((A). Now there exists an x • A such that ( x , y ) • R ~ . Because R~ c R~, pair (x, y) • R~c. This means that y • :~(A) by Definition 1.1. Formula (2.9) can be now proved by (2.8) and Definition 1.3. [] Now we like to find results on associating modifiers with combined sets.
Proposition 2.4. Let 9( be a weakening modifier and R~e an accessibility relation. Then for any A, B c X ,
~(0) = O,
(2.1)
~ * ( X ) = X,
(2.2)
4 c 9((A),
(2.3)
9((A n B) c 9((A) n 9((B),
(2.10)
~ * ( a ) c A,
(2.4)
9(*(A ) U 9(*(B) = 9(*(A U B).
(2.11)
[9(*)*(A) = 9((A)
(2.5)
Proof. We prove (2.1) as follows. Let y • 9((0). Now, there exists an x • 0 such that pair Ix, Y ) • R~ by Definition 1.1, but this is clearly impossible and (2.1) holds. Also (2.2) is valid because X = ~. Formula (2.3) can be proved as [ollows. Let x • A . Now, pair (x,x) •R~e because R~e is a reflexive relation. This means that V x • X , x • A ~ x • 9 ( ( A ) . Thus, (2.3) is
Proof. We prove (2.10) as follows. For any A, B c X , A A B c A and A A B c B . Now, by Proposition 2.2 both 9((A n B) c 9((A) and 9((A n B) c 9((B). This means that (2.10) holds. Formula (2.11) can be proved the same way as (2.10). So, for any A, B c X , A c A U B and B c A O B. By Proposition 2.2 both 9(*(A) c 9(*(A U B) and 9(*(B) c 9(*(A U B). This means that also (2.11) holds. []
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J. Kortelainen / Modifiedsets, topologicalspaces and rough sets
Proposition 2.5. Let ~( be a weakening modifier and R~e an accessibility relation. Then for any A, B c X , ~ ( A U B) = ~(A) U ~ ( a ) ,
(2.12)
~*(A
(2.13)
n
B)
=
~*(A)
n
~*(B).
holds. We prove (2.15) by Definition 1.2 and (2.14) as follows: ~*(~*(A)) = ~(~(A)) = =
Proof. We prove first (2.12) as follows. Clearly A~AUB and B c A O B for any A, B c X . Now, by Proposition 2.2 both Y((A) ~ ~ ( A O B) and ~ ( B ) ~ ~ ( A O B). This means that (1)
~(A)
U
~ ( B ) ~ ~ ( A U B).
Suppose now that y e ~ ( A O B). There exists x e A O B such that pair (x, y) e R~e. Without violating generality we can say that x e A. So, y e ~(A) and especially y e ~(A) O ~(B). Clearly (2)
~(A
B) c ~ ( A )
U
U
~(B).
Now, results (1) and (2) proves (2.12). Formula (2.13) can be proved by (2.12), Definition 1.2 and De Morgan Laws as follows: ~*(A n B) = ~ ( A n B) =
B)
u
=
u
=
n
=
n
[]
Finally, the last Proposition of this chapter discusses a property of transitive accessibility relations.
Proposition 2.6. Let Y( be a weakening modifier and R~ a transitive accessibility relation. Then for any A ~ X , ~(Yg(A)) = ~(A),
(2.14)
~ * ( ~ * ( A ) ) = ~*(A).
(2.15)
Proof. We prove (2.14) as follows. Clearly ~ ( A ) c ~ ( ~ ( A ) ) , because ~ is a weakening modifier. Now, let z e ~g(~(A)). There exists a y e ~(A) such that pair (y, z) e R~e. Because y e Y((A), there exists an x e A such that pair (x, y ) e R ~ . Because R~e is a transitive accessibility relation, pair (x, z ) e R ~ . This means that z e Y((A) and ~tf(~(A)) c ~lf(A). So, (2.14)
= X*(A).
[]
3. Modifiers inducing topological spaces and rough sets In this chapter we show that certain modifiers are in fact closure operators and rough sets could be understood as modified sets.
Proposition 3.1. Let Y( be a weakening modifier and R~ a transMve accessibility relation. Then there exists a unique topology f f ~e on X such that ~ ( A ) will be the ff ~-closure of subset A c X. Proof. Let ~ be a weakening modifier and R~e a transitive accessibility relation. Now, ~ will satisfy the Kuratowski Closure Axioms, because for any A, B ~ X, (K1)
~(0)=0,
(K2)
A c ~(A),
(K3)
~ ( A U B) = ~ ( A ) U ~ ( B ) ,
(K4)
~ ( ~ ( A ) ) = ~(A),
by (2.2), (2.4), (2.17) and (2.20).
[]
Proposition 3.2. Let ~ and ~ be weakening modifiers such that R~ c R~. Let both R~e and R~ also be transMve accessibility relations. Then and ~ induce topologies J-~ and ff'x such that 5r~ is finer than fix. Proof. Because R~ecR~c and ~ and X are closure operators, for any A ~ X, ~ ( A ) ~ X(A) and this means that topology ~-~ is finer than ff,Jc (see e.g. [1] p. 29). []
Example 3.3. Let X be a non-empty set and R~ a fuzzy accessibility relation on X such that for any oc e [0, 1), the strong aMevel set R ~ is also transitive. In this case, for any A c X, ~ ( A ) = {y c X l 3x e X , (x, y) e R ~ and x eA}.
94
J. Kortelainen I Modified sets, topological spaces and rough sets
In the topology, which is induced by 9(~, any Yg,~(A) is an open set. So, we mark this topology by 9-~-" denoting that any Yg,(A) is a member of an ordinary ( 1 - tr)-level class of ~ e . If a, f l e [0, 1) are such that fl ~< tx, then topology 9-~-~ is finer than 9-~ t~ by Proposition 3.2. We may now call Y( a fuzzifier because ~ induces a fuzzy class of sets. This should not be confused with classes of fuzzy sets.
Proposition 3.4. Let ~ be a weakening modifier and R~ an equivalence accessibility relation. Then for any A c X, Yf(A)=U{Y6X/R~c
[ Y N A 4:0},
(3.1)
where X /R~c denotes the family of all equivalence classes of R~c. Proof. Suppose y ~ Y((A). Now there exists an x c A such that (x, y ) ~ R~e. Because R~e is an equivalence relation, x and y are in the same equivalence class, say Y. Clearly Y n A 4:0 and
Clearly ~*(A) consists of all those equivalence classes which are inside A. Thus, (3.2) holds. [] Proposition 3.4 and 3.5 show that the upper approximation of A c X can be understood as weakened set and the lower approximation of A m X can be understood as substantiated set.
4. Concluding remarks In this paper we defined modifiers by so called accessibility relations which are reflexive relations on a non-empty set X. If we demand that accessibility relations must also be transitive, then our modifiers in fact induce topological spaces. Also, if we demand that accessibility relations must be equivalence relations, then our modifiers induce the rough sets. If we need to operate with fuzzy sets, then clearly
~(A)(Y) = max min{/~A(X), /,tR.,(x, y)},
Y ~ U {YEX/R~e I Y A A 4:0}. This means that (1)
generalizes Definition 1. l.
9~(A) c U { Y e X / R ~ e
I YOA4:0}.
Now, suppose there is an equivalence class Y such that y ~ Y and Y O A 4: ~. Now, there exists an x e A such that x and y are in the same equivalence class. Thus, 3x ~ A and (x, y) ~ R~ which means that y ~ gg(A). So (2)
U { Y e X / R ~ J Y 0 A 4: I~} ~ Y((A).
The results (1) and (2) prove (3.1).
[]
Proposition 3.5. Let Y( be a weakening modifier and R~e an equivalence accessibility relation. Then for any A c X, ~£*(A) = U { Y 6X/R~e
x ~ X. (4.1)
I YEA},
(3.2)
where X/R~e denotes all equivalence classes of R~e.
Proof. Because ~ is a weakening modifier, then for any A ~ X,
g((A) = {y 6 X I 3x c X, (x, y) ~ R~e and x e A} by Definition 1.1. Now, by Definition 1.2 we conclude that
gg*(A)= {y 6 X [Vx 6 X , (x, y) e R ~ e ~ x ~A}.
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