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Algebraic and Topological Aspects of Rough Set Theory

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Algebraic and Topological Aspects of Rough Set Theory Algebraic and Topological Aspects of Rough Set Theory Milan Vlach Kyoto College of Graduate Studies for Informatics 7 Monzen-cho, Tanaka, Sakyo-ku, Kyoto, 606-8225 Japan m [email protected] Charles University, School of Mathematics and Physics Malostranske´ nam´ estˇ ´ı 25, 118 00 Praha 1, Czech Republic [email protected] Abstract—The main purpose of this talk is to show how some can expect that various notions and results of rough set theory widely known and well established algebraic and topological have their counterparts in other well established and more notions are closely related to notions and results introduced and developed areas of mathematics. Discovering and studying rediscovered in the rough set literature. relationships between the rough set theory and another field cannot do any harm and often may be useful because such I. INTRODUCTION relationships can enrich both fields and may help to identify Let V be a real vector space, that is, V is a nonempty set some underlying fundamental concepts and results. on which we have a structure consisting of two operations, The main purpose of this talk is to show how some widely addition of elements of V and multiplication of elements from known and well established algebraic and topological notions V by real numbers. The properties of these two operations are closely related to notions and results introduced and make it possible to define various classes of subsets of V. For rediscovered in the rough set literature. example, a set A in V is said to be convex when it has the following property: If x 2 A; y 2 A and 0 < α < 1, then II. PRELIMINARIES (1 − α)x + αy 2 A. Thus some sets in V are convex, some Throughout this paper we assume a modest familiarity with nonconvex, and nonconvex subsets can be approximated by the standard concepts and basic facts of the ordinary set theory. convex ones for a number of purposes. Convex sets in real Mostly we follow the terminology and notation of the book vector spaces are extremely useful, for example, in the theory by P. R. Halmos [23]. of optimization and its various applications. However, no one If A is a set and a is an element of A, then we write a 2 A; has ever claimed that the theory of convex sets extends the otherwise, we write a 62 A: Sets are considered equal if they classical theory of sets or is an alternative to it. have the same elements. If sets A and B are equal, we write Consider now a much more simple structure hU;Ei where U A = B: If they are not equal, we write A 6= B: The empty is a nonempty set and E is an equivalence relation on U. Using set is denoted by ; and a set composed of one element only this simple structure, one can introduce, in different ways, the is called a singleton. notion of a rough set in hU;Ei. Some authors define rough Sets can have other sets as elements. To avoid possible ter- sets as subsets of U with a certain property, some define them minological monotony, we sometimes use the word collection as certain pairs of subsets of U, and some as certain collections synonymously with the word set. The collection of all subsets of subsets of U. Often, the terminology in young fields is not of a set A is denoted by P(A); and it is called the power set well established, and this seems to hold for the theory of rough of A. sets as well. I am convinced, that in the standard set-theoretic If A is a subset of B, that is, if every element of A is also framework, a rough set in hU;Ei should be defined not as a an element of B, then we write A ⊂ B: If A ⊂ B and if there subset of U but as a subset of the power set of U; and every is some element in B that does not belong to A, then A is subset of U should be a member of exactly one rough set. called a proper subset of B1 In any case, when the notion of a rough set is introduced, the The union of sets A and B, denoted by A [ B, is the basic concepts from the ordinary set theory, like the relations set of all elements that are either in A or in B. The unions of membership, equality, and subset, are employed in essential of more than two sets are denoted in the following way. way. Nevertheless, surprisingly often, one can read in the Sn For example, i=1 Ai = A1 [ A2 [···[ An: Similarly, literature that some subsets of U are rough, or that a rough S1 S Ai and Ai: set theory is an extension of classical set theory. i=1 i2I Equivalence relations appear naturally and have an impor- 1Warning: Some author use the symbols A ⊆ B and A ⊂ B to denote tant role in almost every field of mathematics. Therefore, we that ”A is a subset of B” and ”A is a proper subset of B;” respectively. 23 The intersection of sets A and B, denoted by A \ B, is the A binary relation B on a set X is called reflexive if xBx set of all elements that are in both A and B. If the intersection for every x 2 X. A reflexive binary relation on X is called of sets A and B is the empty set, then A and B are said to tolerance if it is symmetric, a preorder if it transitive, an be disjoint. For the intersections of more than two set, we use equivalence if it is transitive and symmetric. Tn T1 T the notation i=1 Ai; i=1 Ai; i2I Ai: Let us recall that partitions and equivalence relations are If A and B are sets, then the relative complement of B in closely connected. Let E be the mapping from the set of all A, denoted by A n B; is the set of those elements in A that do partitions of X into the set of binary relations on X defined not belong to B: The relative complement of B in A is also as follows. If D is a partition of X, then E(D) is the binary called the difference of sets A and B: A natural extension is the relation such that (x; y) 2 E(D) just in case x and y belong symmetric difference, which is the union of A n B with B n A: to the same member of partition D. It can easily be shown We shall denote it by A∆B; that is, A∆B = (AnB)[(BnA): that E(D) is an equivalence relation on X. If we are dealing with subsets of a particular set, we On the other hand, every equivalence relation in X induces often call such a particular set the universe of discourse and a partition of X as follows. If E is an equivalence relation on simplify the notation for the relative complements of subsets X, then an equivalence class with respect to E is defined as with respect the universe of discourse. For example, if U is a subset A of X with the following two properties: (i) every the universe of discourse and A is a subset of U, then the two elements in A are equivalent, and (ii) each element of X difference U n A is called the complement of A. that is equivalent to some element of A also belongs to A: By a partition of a nonempty set we mean a collection of It turns out that the collection of all equivalence classes with nonempty subsets that are disjoint from each other and whose respect to E is a partition of X. By the equivalence class of union is the whole set. If P and Q are partitions of the same an element x of X, we understand the equivalence class to set, then the partition P is called a refinement of the partition which x belongs. The equivalence class of x will be denoted Q if every member of P is a subset of one of the members by x=E; and the collection of all equivalence classes of E of Q. will be denoted by D(E): Notice that x=E = E(x): The order of elements in a set is important in many For every equivalence E on X and every partition D of X, situations. If so, then we need an additional structure and we have notation. To indicate that we are dealing with ordering of D(E(D)) = D and E(D(E)) = E: elements of a finite set, we use parentheses instead of brackets and use the words ordered tuples or finite sequences. Two III. APPROXIMATION SPACES n-tuples (x1; x2; : : : ; xn) and (y1; y2; : : : ; yn) are said to be Many theoretical results in mathematics and many applica- equal if xi = yi for 1 ≤ i ≤ n: The ordered 2-tuples are tions of mathematics in practice are based on the possibility called ordered pairs. of approximating subsets of a fixed set by other subsets. If A and B are sets, then the Cartesian product of A and B, Sometimes approximations are needed because the complexity which is denoted by A × B, is the set of all ordered pairs of objects under investigation necessitates simplification and (a; b) such that a 2 A and b 2 B.
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