Properties of Rough Approximations - Semantic Scholar
J¨arvinen, J.
Paper: jc9-5-2359 2005/6/14
Properties of Rough Approximations Jouni J¨arvinen Turku Centre for Computer Science (TUCS) Lemmink¨aisenkatu 14 A, FIN-20520 Turku, Finland E-mail: [email protected] [Received 00/00/00; accepted 00/00/00]
In this paper, we present the basic ideas and fundamental concepts of rough set theory, focusing on properties of rough approximations.
Keywords: indiscernibility relation, rough approximation operators, ordered sets and lattices
1. Basic Ideas Rough set theory, introduced by Pawlak [10], is a mathematical formalism dealing with uncertainty and to some extent overlapping fuzzy set theory introduced by Zadeh [12]. In fuzzy set theory vagueness is expressed by a membership function. The rough set theory approach is based on indiscernibility relations and approximations. A major advantage of rough set theory is that it needs no preliminary or additional information about data, such as membership functions in fuzzy set theory. The basic idea of rough set theory is that knowledge about objects is represented by indiscernibility relations. Indiscernibility relations are usually assumed to be equivalences—reflexive, symmetric, and transitive binary relations—interpreted so that two objects are equivalent if we cannot distinguish them by their properties. This means that if we observe objects through knowledge given by an indiscernibility relation, our ability to distinguish objects is blurred—we cannot distinguish individual objects, only their equivalence classes. Let us consider the situation in Fig.1. Let X be a subset of a given universe of discourse U and let be an indiscernibility relation on U. Since the equivalence induces the partition U whose blocks are the equivalence classes of , the objects of the universe U are classified by in threeclasses for any subset X U: (a)
objects
(b) objects (c) objects
that are surely in X; that are surely not in X; that are possibly in X.
Objects in class (a) form the lower approximation of X, and objects of type (a) and (c) together form its upper approximation. The boundary of X consists of objects in class (c). 502
!
Fig. 1. A set X observed through knowledge restricted by an equivalence . We cannot distinguish individual objects, only their equivalence classes.
"
2. Rough Approximations As noted in the previous section, indiscernibility relations are commonly assumed to be equivalences. The literature, however, contains studies in which rough approximations are defined by tolerances—reflexive and symmetric binary relations—e.g., [5] and [11]. In [7] and [8], rough approximations determined by quasi-orders— reflexive and transitive binary relations—and their connection to topological spaces and fuzzy sets are studied. In [9], operators determined by frames of information relations reflecting distinguishability or indistinguishability of objects of an information system are considered. In [13], approximations operators defined by arbitrary binary relations are further considered. Here we adopt a similar viewpoint. Definition 2.1. Let R be a binary relation on U, and let us denote for all x # U, R $ x %'&)( y
#
U * x R y +-,
The upper approximation of X X ./&)( x
#
U is
U * R $ x %10 X
& 2 0/ +
and the lower approximation of X is X 3/&)( x
Journal of Advanced Computational Intelligence and Intelligent Informatics
#
U * R $ x %4 X +-, Vol.9 No.5, 2005
Properties of Rough Approximations
The set B $ X %
&
X.
X 3 is the boundary of X.
The above definitions mean that x # X . if there is an element in X to which x is R-related. Similarly, x # X 3 if all the elements to which x is R-related are in X. Obviously, x # B $ X % if both in X and outside X there are elements to which x is R-related. Note that B $ X %4& 0/ means that for any object x # U, we can, with certainty, decide whether x # X by using knowledge provided by R. The next proposition lists basic properties of rough approximations. For any X U, we denote by X c the com ℘$ U % , we for any plement U X of X. Further, write . & ( X . * X # + and 3 &)( X 3 * X # + . Proposition 2.2. Let R be a binary relation on U. The following assertions then hold: (a) The maps X X 3 and X X . are mutually dual: for any X U, X 3 c & X c . and X . c & X c 3 . (b) For all X
U, B $ X %
&
B$ Xc% .
%. & .. ℘$ U % , $ % 3 & 3 . (d) For all (e) If X Y , then X 3 Y 3 and X . Y . . Proof. (a) x # X 3 c x # X 3 R $ x % 2 X R $ x % 0 X c & 2 0/ x # X c . . Further, X . c & X cc . c & X c 3 cc & X c 3 . (b) B $ X % & X . X 3 & X . 0 X 3 c & X c 3 c 0 X c . & c X . X c 3 & B $ X c % . (c) x # $ % . R $ x % 0 & 2 0/ $ X # % R $ x % 0 X & 2 0/ $ X # % x # X . x# .. 3 R $ x % $ X # % R $ x %4 (d) x # $ % X $ X # % x # X3 x # 3. (e) If X Y , then X 3 0 Y 3 & $ X 0 Y % 3 & X 3 and X . Y . & $ X Y % . & Y . . Thus, X 3 Y 3 and X . Y . . (c) For all
℘$ U % ,
$
In the previous proposition, (a) is interpreted so that if an element does not belong with certainty to a set, it belongs possibly to the complement of that set, and if an element does not belong possibly to a set, then it belongs with certainty to the complement. Assertion (b) means that if we cannot decide whether an element belongs to a set, we cannot decide whether the element is in the set’s complement either. Claim (c) says that elements belong possibly to the union of some sets if they belong possibly to at least one of the sets in question. An element belongs with certainty to the intersection of sets if it is with certainty in all sets; this is stated in (d). Condition (e) simply means that rough approximation operators are order-preserving. Below, we show how properties of binary relations are expressed by rough approximations. Note that a survey of the correspondence theory of classical modal logic is found in [2]. We begin with the following definition: Definition 2.3. A binary relation R on a set U is said to be (a) serial, if R $ x % Vol.9 No.5, 2005
&2
0/ for all x
#
U;
(b) reflexive, if x R x for all x # U; (c) symmetric, if x R y implies y R x for all x y
#
U;
(d) transitive, if x R y and y R z imply x R z for all x y z U;
#
(e) a quasi-order, if R is reflexive and transitive; (f) an equivalence, if R is a symmetric quasi-order. Reflexivity clearly implies seriality. Note also that if a relation is serial, symmetric, and transitive, it is of necessity an equivalence. Proposition 2.4. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is serial;
(b) X 3
X . for all X
U.
Proof. (a) (b): Let x # X 3 . Then R $ x % X, which gives R $ x % 0 X & R $ x % & 2 0, / that is, x # X . . (b) (a): Assume that R is not serial, that is, R $ x % & 0/ for some x # U. This means that x # X 3 and x # X . for this particular x and for any set X U—a contradiction! Each set is bounded by its approximations determined by a reflexive relation, as seen in the next proposition. Proposition 2.5. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is reflexive; (b) X (c) X 3
X . for all X X for all X
U;
U.
Proof. (a) (b): If x # X, then x # R $ x % 0 X & 2 0, / that is, x # X . . (b) (c): Obviously, X c X c . & X 3 c , which is equivalent to X 3 X. (c) (a): If R is not reflexive, then there exists x # U such that $ x x % # R. Let us consider the set X & U ( x + . For all y # U, $ x y% # R implies y # X. Thus, x # X 3 and x # X—a contradiction! Galois connections are found in numerous settings from algebra to computer science and defined in two theoretically equivalent ways—the one adopted here, in which maps are order-preserving, and the other, in which they are order-reversing. Definition 2.6. For two ordered sets $ P % and $ Q % , a pair $ % of mappings : P Q and : Q P is a Galois connection between P and Q if for all p # P and q # Q, p q p q ,
It is well known that a pair $ % is a Galois connection between P and Q if and only if and are orderpreserving, p p for all p # P, and q q for all q # Q. The next proposition presents important properties of Galois connections, e.g., [4].
Journal of Advanced Computational Intelligence and Intelligent Informatics
503
J¨arvinen, J.
Proposition 2.7. Let $ complete lattice $ P % . (a) For all p
#
P, p
%
&
be a Galois connection on a p and p
&
p .
(b) The map p p is a closure operator and the map p p is an interior operator. Proposition 2.8. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is symmetric; (b)
$ . 3 %
is a Galois connection on $ ℘$ U %
%.
Proof. (a) (b): The maps X X . and X X 3 are order-preserving by Proposition 2.2(e). If x # X 3 . , then there exists y # X 3 such that $ x y % # R. By symmetry, x # X. Thus, X 3 . X. This also gives X . 3 c & X c 3 . X c , which is equivalent to X X . 3 . Then, for (b) (a): Assume that R is not symmetric. some x y # U, $ x y % # R, but $ y x % # R. Let us consider the set X & ( x + . For all z # U, $ y z % # R implies z # X. This gives y # X . . Hence, x # X and x # X . 3 —a contradiction! Proposition 2.9. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is transitive;
. X. X3 X3 3
(b) X .
for all X
(c)
for all X
U; U.
Proof. (a) (b): Let x # X . . . This means that there exists y # X . such that $ x y % # R. Because y # X . , there is z # X such that $ y z %4# R. Now, $ x z %4# R by the transitivity of R. Hence, x # X . . (b) (c): Obviously, X 3 3 c & X c . . X c . & X 3 c , which means X 3 X 3 3 . (c) (a): Assume that R is not transitive. Then there exist x y z # U such that $ x y % # R and $ y z % # R, but $ x z % # R. Let us consider the set X & U ( z + . Then for all w # U, $ x w% implies w # X. This means that x # X 3 . Obviously, y # X 3 and hence x # X 3 3 —a contradiction! Propositions 2.5 and 2.9 have the following corollary: Corollary 2.10. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is a quasi-order;
(c) The map X
X . is a closure operator;
(b) The map X
X 3 is an interior operator.
3. Structure of Approximations In this section, we consider ordered sets of lower and upper approximations. As in Section 2, we denote ℘$ U % 3 &)( X 3 * X U + and ℘$ U % . &)( X . * X U + . We begin with the following observation: 504
Proposition 3.1. Let R be any binary relation on U. The ordered sets $ ℘$ U % 3 % and $ ℘$ U % . % are then dually isomorphic complete lattices. Proof. It follows from claims (c) and (d) of Proposition 2.2 that these ordered sets are complete lattices. We must prove that $ ℘$ U % 3 % & $ ℘$ U % . % . It is clear that the map X 3 X c . is from ℘$ U % 3 onto ℘$ U % . . Further, X 3 Y 3 if and only if X c . & X 3 c Y 3 c & Y c . .
The following proposition follows easily from Proposition 2.7: Proposition 3.2. Let R be a symmetric relation on U. The complete lattices $ ℘$ U % 3 % and $ ℘$ U % . % are then isomorphic, that is, $ ℘$ U % 3 % & $ ℘$ U % . % .
Proof. The map ϕ : X 3 X 3 . is from ℘$ U % 3 onto ℘$ U % . , since for all X U, ϕ $ X . 3 % & X . 3 . & X . . Furthermore, X 3 Y 3 implies ϕ $ X 3 % & X 3 . Y 3 . & ϕ $ Y 3 % , and ϕ $ X 3 % & X 3 . Y 3 . & ϕ $ Y 3 % implies X 3 & X3 . 3 Y3 . 3 & Y3 .
Note that Propositions 3.1 and 3.2 give $ ℘$ U % 3 % & $ ℘$ U % . % & $ ℘$ U % 3 % & $ ℘$ U % . % whenever R is symmetric. ℘$ U % is called a complete ring A family of sets # and # for all . Evof sets if ery complete ring of sets forms a completely distributive lattice for the order , e.g., [3]. For quasi-orders, we write the following proposition:
Proposition 3.3. Let R be a quasi-order on U. Then ℘$ U % 3 and ℘$ U % . are complete rings of sets.
we Proof. Let ℘$ U % . In Proposition 2.2, 3 proved that & & $ % . We show that 3 3 $ 3 % 3 . It is clear that, for all X # , X 3 3 and X 3 & X 3 3 $ 3 % 3 . Hence, 3 $ 3 % 3 . 3 Trivially, $ 3% 3. The proof for the other part is analogous.
The next lemma presents properties of approximations determined by equivalences. Lemma 3.4. Let R be an equivalence on U. The following equations then hold for all X U: (a) X 3
(b) X .
. & 3 &
X3 ; X. .
Proof. (a) Because R is reflexive, X 3 X 3 . . Reflexivity and transitivity imply X 3 & X 3 3 and X 3 . & X 3 3 . . Now, X 3 3 . X 3 since R is symmetric. (b) X . & X . cc & X c 3 c & X c 3 . c & X . 3 cc & X . 3 . For equivalences, the set of upper approximations and the set of lower approximations are equal, as stated in the next lemma. Lemma 3.5. Let R be an equivalence on U. ℘$ U % 3 & ℘$ U % . . Proof. By Lemma 3.4, X 3 all X U.
Journal of Advanced Computational Intelligence and Intelligent Informatics
&
X3
.
and X .
&
X.
Then
3
for
Vol.9 No.5, 2005
Properties of Rough Approximations
A complete ring of sets is called a complete field of , Xc # . Every complete field of sets if for all X # sets forms a complete atomic Boolean lattice for the order , e.g., [1].
Name: Jouni J¨arvinen
Affiliation:
Proposition 3.6. Let R be an equivalence on U. Then ℘$ U % 3 and ℘$ U % . are complete fields of sets. Proof. For all X proof since ℘$ U % 3
&
U, X . ℘$ U %
.
c
Turku Centre for Computer Science (TUCS) and Department of Information Technology at the University of Turku
&
X c 3 . This completes the by Lemma 3.5.
It can be easily seen that ( ( x + . * x of atoms of $ ℘$ U % 3 % and $ ℘$ U % .
#
U + forms the set
%.
Address: Lemmink¨aisenkatu 14 A, FIN-20520 Turku, Finland
Brief Biographical History: 1999 Received PhD degree at the Department of Mathematics of the University of Turku, Finland.
4. Conclusions
Main Works: Papers on logical and algebraic foundations of relational reasoning.
In this paper, we have presented some central properties of rough approximation operators. Rough approximation operators may also be studied in a more general setting of complete atomic Boolean lattices [6]. References: [1] R. Balbes and Ph. Dwinger, “Distributive Lattices,” University of Missouri Press, Missouri, 1974. [2] J. van Benthem, “Correspondence Theory,” Handbook of philosophical logic II, pp. 167-247, 1984. [3] G. Birkhoff, “Rings of Sets,” Duke Mathematical Journal 3, pp. 443-454, 1937. [4] B. A. Davey and H. A. Priestley, “Introduction to Lattices and Order,” Cambridge University Press, Cambridge, 2002. [5] J. J¨arvinen, “Approximations and Rough Sets Based on Tolerances,” Lecture Notes in Artificial Intelligence 2005, pp. 182-189, 2001. [6] J. J¨arvinen, “On the Structure of Rough Approximations,” Fundamenta Informaticae 53, pp. 135-153, 2002. [7] J. J¨arvinen and J. Kortelainen, “A Note on Definability in Rough Set Theory,” in: B. De Baets et al., editors, Current Issues in Data and Knowledge Engineering, pp. 272-277, EXIT, Warsaw, 2004. [8] J. J¨arvinen and J. Kortelainen, “A Unifying Study Between Modallike Operators, Topologies, and Fuzzy Sets,” TUCS Technical Report 642, Turku Centre for Computer Science, Turku, Finland, 2004. [9] E. Orłowska, “Introduction: What You Always Wanted to Know About Rough Sets,” in: E. Orłowska, editor, Incomplete Information: Rough Set Analysis, Physica, Heidelberg, pp. 1-20, 1998. [10] Z. Pawlak, “Rough Sets,” International Journal of Computer and Information Sciences 5, pp. 341-356, 1982. [11] J. A Pomykała, “About Tolerance and Similarity Relations in Information Systems,” Lecture Notes in Artificial Intelligence 2475, pp. 175-182, 2002. [12] L. A. Zadeh, “Fuzzy Sets,” Information and Control 8, pp. 338-353, 1965. [13] Y. Y. Yao, T. Y. Lin, “Generalization of Rough Sets using Modal Logics,” Intelligent Automation and Soft Computing 2, pp. 103120, 1996.
Vol.9 No.5, 2005
Journal of Advanced Computational Intelligence and Intelligent Informatics
505
Paper: jc9-5-2359 2005/6/14
Properties of Rough Approximations Jouni J¨arvinen Turku Centre for Computer Science (TUCS) Lemmink¨aisenkatu 14 A, FIN-20520 Turku, Finland E-mail: [email protected] [Received 00/00/00; accepted 00/00/00]
In this paper, we present the basic ideas and fundamental concepts of rough set theory, focusing on properties of rough approximations.
Keywords: indiscernibility relation, rough approximation operators, ordered sets and lattices
1. Basic Ideas Rough set theory, introduced by Pawlak [10], is a mathematical formalism dealing with uncertainty and to some extent overlapping fuzzy set theory introduced by Zadeh [12]. In fuzzy set theory vagueness is expressed by a membership function. The rough set theory approach is based on indiscernibility relations and approximations. A major advantage of rough set theory is that it needs no preliminary or additional information about data, such as membership functions in fuzzy set theory. The basic idea of rough set theory is that knowledge about objects is represented by indiscernibility relations. Indiscernibility relations are usually assumed to be equivalences—reflexive, symmetric, and transitive binary relations—interpreted so that two objects are equivalent if we cannot distinguish them by their properties. This means that if we observe objects through knowledge given by an indiscernibility relation, our ability to distinguish objects is blurred—we cannot distinguish individual objects, only their equivalence classes. Let us consider the situation in Fig.1. Let X be a subset of a given universe of discourse U and let be an indiscernibility relation on U. Since the equivalence induces the partition U whose blocks are the equivalence classes of , the objects of the universe U are classified by in threeclasses for any subset X U: (a)
objects
(b) objects (c) objects
that are surely in X; that are surely not in X; that are possibly in X.
Objects in class (a) form the lower approximation of X, and objects of type (a) and (c) together form its upper approximation. The boundary of X consists of objects in class (c). 502
!
Fig. 1. A set X observed through knowledge restricted by an equivalence . We cannot distinguish individual objects, only their equivalence classes.
"
2. Rough Approximations As noted in the previous section, indiscernibility relations are commonly assumed to be equivalences. The literature, however, contains studies in which rough approximations are defined by tolerances—reflexive and symmetric binary relations—e.g., [5] and [11]. In [7] and [8], rough approximations determined by quasi-orders— reflexive and transitive binary relations—and their connection to topological spaces and fuzzy sets are studied. In [9], operators determined by frames of information relations reflecting distinguishability or indistinguishability of objects of an information system are considered. In [13], approximations operators defined by arbitrary binary relations are further considered. Here we adopt a similar viewpoint. Definition 2.1. Let R be a binary relation on U, and let us denote for all x # U, R $ x %'&)( y
#
U * x R y +-,
The upper approximation of X X ./&)( x
#
U is
U * R $ x %10 X
& 2 0/ +
and the lower approximation of X is X 3/&)( x
Journal of Advanced Computational Intelligence and Intelligent Informatics
#
U * R $ x %4 X +-, Vol.9 No.5, 2005
Properties of Rough Approximations
The set B $ X %
&
X.
X 3 is the boundary of X.
The above definitions mean that x # X . if there is an element in X to which x is R-related. Similarly, x # X 3 if all the elements to which x is R-related are in X. Obviously, x # B $ X % if both in X and outside X there are elements to which x is R-related. Note that B $ X %4& 0/ means that for any object x # U, we can, with certainty, decide whether x # X by using knowledge provided by R. The next proposition lists basic properties of rough approximations. For any X U, we denote by X c the com ℘$ U % , we for any plement U X of X. Further, write . & ( X . * X # + and 3 &)( X 3 * X # + . Proposition 2.2. Let R be a binary relation on U. The following assertions then hold: (a) The maps X X 3 and X X . are mutually dual: for any X U, X 3 c & X c . and X . c & X c 3 . (b) For all X
U, B $ X %
&
B$ Xc% .
%. & .. ℘$ U % , $ % 3 & 3 . (d) For all (e) If X Y , then X 3 Y 3 and X . Y . . Proof. (a) x # X 3 c x # X 3 R $ x % 2 X R $ x % 0 X c & 2 0/ x # X c . . Further, X . c & X cc . c & X c 3 cc & X c 3 . (b) B $ X % & X . X 3 & X . 0 X 3 c & X c 3 c 0 X c . & c X . X c 3 & B $ X c % . (c) x # $ % . R $ x % 0 & 2 0/ $ X # % R $ x % 0 X & 2 0/ $ X # % x # X . x# .. 3 R $ x % $ X # % R $ x %4 (d) x # $ % X $ X # % x # X3 x # 3. (e) If X Y , then X 3 0 Y 3 & $ X 0 Y % 3 & X 3 and X . Y . & $ X Y % . & Y . . Thus, X 3 Y 3 and X . Y . . (c) For all
℘$ U % ,
$
In the previous proposition, (a) is interpreted so that if an element does not belong with certainty to a set, it belongs possibly to the complement of that set, and if an element does not belong possibly to a set, then it belongs with certainty to the complement. Assertion (b) means that if we cannot decide whether an element belongs to a set, we cannot decide whether the element is in the set’s complement either. Claim (c) says that elements belong possibly to the union of some sets if they belong possibly to at least one of the sets in question. An element belongs with certainty to the intersection of sets if it is with certainty in all sets; this is stated in (d). Condition (e) simply means that rough approximation operators are order-preserving. Below, we show how properties of binary relations are expressed by rough approximations. Note that a survey of the correspondence theory of classical modal logic is found in [2]. We begin with the following definition: Definition 2.3. A binary relation R on a set U is said to be (a) serial, if R $ x % Vol.9 No.5, 2005
&2
0/ for all x
#
U;
(b) reflexive, if x R x for all x # U; (c) symmetric, if x R y implies y R x for all x y
#
U;
(d) transitive, if x R y and y R z imply x R z for all x y z U;
#
(e) a quasi-order, if R is reflexive and transitive; (f) an equivalence, if R is a symmetric quasi-order. Reflexivity clearly implies seriality. Note also that if a relation is serial, symmetric, and transitive, it is of necessity an equivalence. Proposition 2.4. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is serial;
(b) X 3
X . for all X
U.
Proof. (a) (b): Let x # X 3 . Then R $ x % X, which gives R $ x % 0 X & R $ x % & 2 0, / that is, x # X . . (b) (a): Assume that R is not serial, that is, R $ x % & 0/ for some x # U. This means that x # X 3 and x # X . for this particular x and for any set X U—a contradiction! Each set is bounded by its approximations determined by a reflexive relation, as seen in the next proposition. Proposition 2.5. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is reflexive; (b) X (c) X 3
X . for all X X for all X
U;
U.
Proof. (a) (b): If x # X, then x # R $ x % 0 X & 2 0, / that is, x # X . . (b) (c): Obviously, X c X c . & X 3 c , which is equivalent to X 3 X. (c) (a): If R is not reflexive, then there exists x # U such that $ x x % # R. Let us consider the set X & U ( x + . For all y # U, $ x y% # R implies y # X. Thus, x # X 3 and x # X—a contradiction! Galois connections are found in numerous settings from algebra to computer science and defined in two theoretically equivalent ways—the one adopted here, in which maps are order-preserving, and the other, in which they are order-reversing. Definition 2.6. For two ordered sets $ P % and $ Q % , a pair $ % of mappings : P Q and : Q P is a Galois connection between P and Q if for all p # P and q # Q, p q p q ,
It is well known that a pair $ % is a Galois connection between P and Q if and only if and are orderpreserving, p p for all p # P, and q q for all q # Q. The next proposition presents important properties of Galois connections, e.g., [4].
Journal of Advanced Computational Intelligence and Intelligent Informatics
503
J¨arvinen, J.
Proposition 2.7. Let $ complete lattice $ P % . (a) For all p
#
P, p
%
&
be a Galois connection on a p and p
&
p .
(b) The map p p is a closure operator and the map p p is an interior operator. Proposition 2.8. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is symmetric; (b)
$ . 3 %
is a Galois connection on $ ℘$ U %
%.
Proof. (a) (b): The maps X X . and X X 3 are order-preserving by Proposition 2.2(e). If x # X 3 . , then there exists y # X 3 such that $ x y % # R. By symmetry, x # X. Thus, X 3 . X. This also gives X . 3 c & X c 3 . X c , which is equivalent to X X . 3 . Then, for (b) (a): Assume that R is not symmetric. some x y # U, $ x y % # R, but $ y x % # R. Let us consider the set X & ( x + . For all z # U, $ y z % # R implies z # X. This gives y # X . . Hence, x # X and x # X . 3 —a contradiction! Proposition 2.9. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is transitive;
. X. X3 X3 3
(b) X .
for all X
(c)
for all X
U; U.
Proof. (a) (b): Let x # X . . . This means that there exists y # X . such that $ x y % # R. Because y # X . , there is z # X such that $ y z %4# R. Now, $ x z %4# R by the transitivity of R. Hence, x # X . . (b) (c): Obviously, X 3 3 c & X c . . X c . & X 3 c , which means X 3 X 3 3 . (c) (a): Assume that R is not transitive. Then there exist x y z # U such that $ x y % # R and $ y z % # R, but $ x z % # R. Let us consider the set X & U ( z + . Then for all w # U, $ x w% implies w # X. This means that x # X 3 . Obviously, y # X 3 and hence x # X 3 3 —a contradiction! Propositions 2.5 and 2.9 have the following corollary: Corollary 2.10. Let R be a binary relation on U. The following assertions are then equivalent: (a) R is a quasi-order;
(c) The map X
X . is a closure operator;
(b) The map X
X 3 is an interior operator.
3. Structure of Approximations In this section, we consider ordered sets of lower and upper approximations. As in Section 2, we denote ℘$ U % 3 &)( X 3 * X U + and ℘$ U % . &)( X . * X U + . We begin with the following observation: 504
Proposition 3.1. Let R be any binary relation on U. The ordered sets $ ℘$ U % 3 % and $ ℘$ U % . % are then dually isomorphic complete lattices. Proof. It follows from claims (c) and (d) of Proposition 2.2 that these ordered sets are complete lattices. We must prove that $ ℘$ U % 3 % & $ ℘$ U % . % . It is clear that the map X 3 X c . is from ℘$ U % 3 onto ℘$ U % . . Further, X 3 Y 3 if and only if X c . & X 3 c Y 3 c & Y c . .
The following proposition follows easily from Proposition 2.7: Proposition 3.2. Let R be a symmetric relation on U. The complete lattices $ ℘$ U % 3 % and $ ℘$ U % . % are then isomorphic, that is, $ ℘$ U % 3 % & $ ℘$ U % . % .
Proof. The map ϕ : X 3 X 3 . is from ℘$ U % 3 onto ℘$ U % . , since for all X U, ϕ $ X . 3 % & X . 3 . & X . . Furthermore, X 3 Y 3 implies ϕ $ X 3 % & X 3 . Y 3 . & ϕ $ Y 3 % , and ϕ $ X 3 % & X 3 . Y 3 . & ϕ $ Y 3 % implies X 3 & X3 . 3 Y3 . 3 & Y3 .
Note that Propositions 3.1 and 3.2 give $ ℘$ U % 3 % & $ ℘$ U % . % & $ ℘$ U % 3 % & $ ℘$ U % . % whenever R is symmetric. ℘$ U % is called a complete ring A family of sets # and # for all . Evof sets if ery complete ring of sets forms a completely distributive lattice for the order , e.g., [3]. For quasi-orders, we write the following proposition:
Proposition 3.3. Let R be a quasi-order on U. Then ℘$ U % 3 and ℘$ U % . are complete rings of sets.
we Proof. Let ℘$ U % . In Proposition 2.2, 3 proved that & & $ % . We show that 3 3 $ 3 % 3 . It is clear that, for all X # , X 3 3 and X 3 & X 3 3 $ 3 % 3 . Hence, 3 $ 3 % 3 . 3 Trivially, $ 3% 3. The proof for the other part is analogous.
The next lemma presents properties of approximations determined by equivalences. Lemma 3.4. Let R be an equivalence on U. The following equations then hold for all X U: (a) X 3
(b) X .
. & 3 &
X3 ; X. .
Proof. (a) Because R is reflexive, X 3 X 3 . . Reflexivity and transitivity imply X 3 & X 3 3 and X 3 . & X 3 3 . . Now, X 3 3 . X 3 since R is symmetric. (b) X . & X . cc & X c 3 c & X c 3 . c & X . 3 cc & X . 3 . For equivalences, the set of upper approximations and the set of lower approximations are equal, as stated in the next lemma. Lemma 3.5. Let R be an equivalence on U. ℘$ U % 3 & ℘$ U % . . Proof. By Lemma 3.4, X 3 all X U.
Journal of Advanced Computational Intelligence and Intelligent Informatics
&
X3
.
and X .
&
X.
Then
3
for
Vol.9 No.5, 2005
Properties of Rough Approximations
A complete ring of sets is called a complete field of , Xc # . Every complete field of sets if for all X # sets forms a complete atomic Boolean lattice for the order , e.g., [1].
Name: Jouni J¨arvinen
Affiliation:
Proposition 3.6. Let R be an equivalence on U. Then ℘$ U % 3 and ℘$ U % . are complete fields of sets. Proof. For all X proof since ℘$ U % 3
&
U, X . ℘$ U %
.
c
Turku Centre for Computer Science (TUCS) and Department of Information Technology at the University of Turku
&
X c 3 . This completes the by Lemma 3.5.
It can be easily seen that ( ( x + . * x of atoms of $ ℘$ U % 3 % and $ ℘$ U % .
#
U + forms the set
%.
Address: Lemmink¨aisenkatu 14 A, FIN-20520 Turku, Finland
Brief Biographical History: 1999 Received PhD degree at the Department of Mathematics of the University of Turku, Finland.
4. Conclusions
Main Works: Papers on logical and algebraic foundations of relational reasoning.
In this paper, we have presented some central properties of rough approximation operators. Rough approximation operators may also be studied in a more general setting of complete atomic Boolean lattices [6]. References: [1] R. Balbes and Ph. Dwinger, “Distributive Lattices,” University of Missouri Press, Missouri, 1974. [2] J. van Benthem, “Correspondence Theory,” Handbook of philosophical logic II, pp. 167-247, 1984. [3] G. Birkhoff, “Rings of Sets,” Duke Mathematical Journal 3, pp. 443-454, 1937. [4] B. A. Davey and H. A. Priestley, “Introduction to Lattices and Order,” Cambridge University Press, Cambridge, 2002. [5] J. J¨arvinen, “Approximations and Rough Sets Based on Tolerances,” Lecture Notes in Artificial Intelligence 2005, pp. 182-189, 2001. [6] J. J¨arvinen, “On the Structure of Rough Approximations,” Fundamenta Informaticae 53, pp. 135-153, 2002. [7] J. J¨arvinen and J. Kortelainen, “A Note on Definability in Rough Set Theory,” in: B. De Baets et al., editors, Current Issues in Data and Knowledge Engineering, pp. 272-277, EXIT, Warsaw, 2004. [8] J. J¨arvinen and J. Kortelainen, “A Unifying Study Between Modallike Operators, Topologies, and Fuzzy Sets,” TUCS Technical Report 642, Turku Centre for Computer Science, Turku, Finland, 2004. [9] E. Orłowska, “Introduction: What You Always Wanted to Know About Rough Sets,” in: E. Orłowska, editor, Incomplete Information: Rough Set Analysis, Physica, Heidelberg, pp. 1-20, 1998. [10] Z. Pawlak, “Rough Sets,” International Journal of Computer and Information Sciences 5, pp. 341-356, 1982. [11] J. A Pomykała, “About Tolerance and Similarity Relations in Information Systems,” Lecture Notes in Artificial Intelligence 2475, pp. 175-182, 2002. [12] L. A. Zadeh, “Fuzzy Sets,” Information and Control 8, pp. 338-353, 1965. [13] Y. Y. Yao, T. Y. Lin, “Generalization of Rough Sets using Modal Logics,” Intelligent Automation and Soft Computing 2, pp. 103120, 1996.
Vol.9 No.5, 2005
Journal of Advanced Computational Intelligence and Intelligent Informatics
505
