Int. J. Appl. Math. Comput. Sci., 2001, Vol.11, No.3, 621{635 621 ROUGH RELATION PROPERTIES Maria do Carmo NICOLETTI∗, Joaquim Quinteiro UCHOA^ ∗∗ Margarete T.Z. BAPTISTINI*** Rough Set Theory (RST) is a mathematical formalism for representing uncer- tainty that can be considered an extension of the classical set theory. It has been used in many different research areas, including those related to induc- tive machine learning and reduction of knowledge in knowledge-based systems. One important concept related to RST is that of a rough relation. This paper rewrites some properties of rough relations found in the literature, proving their validity. Keywords: rough set theory, rough relation, knowledge representation 1. Introduction Rough Set Theory (RST) was proposed by Pawlak (1982) as an extension of the classical set theory, for use when representing incomplete knowledge. Rough sets can be considered sets with fuzzy boundaries|sets that cannot be precisely character- ized using the available set of attributes. During the last few years RST has been approached as a formal tool used in connection with many different areas of research. There have been investigations of the relations between RST and the Dempster-Shafer Theory (Skowron and Grzymala-Busse, 1994; Wong and Lingras, 1989), and between rough sets and fuzzy sets (Pawlak, 1994; Pawlak and Skowron, 1994; Wygralak, 1989). RST has also provided the necessary formalism and ideas for the develop- ment of some propositional machine learning systems (Grzymala-Busse, 1992; Mrózek, 1992; Pawlak, 1984; 1985; Wong et al., 1986). It has also been used for, among many other things, knowledge representation (Orlowska and Pawlak, 1984; Ziarko, 1991), data mining (Aasheim and Solheim, 1996; Deogun et al., 1997), dealing with imperfect data (Grzymala-Busse, 1988; Szladow and Ziarko, 1993), reducing the knowledge rep- resentation (Grzymala-Busse, 1986; Jelonek et al., 1994; Pawlak at el., 1988), helping to solve control problems (Ohrn, 1993; Pawlak, 1997; Sªowi«ski, 1995), and analysing attribute dependencies (Grzymala-Busse and Mithal, 1991; Mrózek, 1989). ∗ Universidade Federal de S~ao Carlos, Departamento de Computa¸c~ao, S~ao Carlos{S~ao Paulo, Brazil, e-mail: [email protected] ∗∗ Universidade Federal de Lavras, Departamento de Ci^encia da Computa¸c~ao, Lavras{Minas Gerais, Brazil, e-mail: [email protected] *** Universidade Federal de S~ao Carlos, Departamento de Matem`atica, S~ao Carlos{S~ao Paulo, Brazil, e-mail: [email protected] 622 M.C. Nicoletti et al. The notions of rough relations and rough functions are based on RST and, as discussed in (Pawlak, 1997, p.139), `are needed in many applications, where experi- mental data are processes, in particular as a theoretical basis for rough controllers'. This paper presents the main concepts related to rough relations, rewrites some of its properties and proves them to be valid. It is organized as follows: Section 2 is a selection of mathematical results that constitute essential background knowledge to what follows. Section 3 presents the basic concepts and notations related to Rough Set Theory (extracted from the various sources listed in the References) as well as some results that are necessary to understand Section 4, where the basic concepts and notations related to rough relations are presented. In Section 5 the main properties of rough relations are established and proved to be valid. 2. Mathematical Prerequisites Some of the results presented in this section have been extracted from (Berztiss, 1975). Definition 1. A binary relation from set A to set B is a subset of A × B. If R is a relation, we write (x; y) 2 R and xRy interchangeably. Definition 2. A subset of A × A is a binary relation in the set A. In particular, the set A × A is the universal relation in A. Definition 3. Let A be a set and R a relation in A. The set of R-relatives of the elements of A is R[A] = fy j for some x in A; xRyg. Definition 4. If R is a relation from A to B, the reversed relation of R, written as R−1, is a relation from B to A such that yR−1x if and only if xRy. Definition 5. A relation R in a set A is 1. reflexive if xRx for all x 2 A, 2. nonreflexive if 9x 2 A such that xR6 x, 3. an identity if it is reflexive and if xRy for x; y 2 A yields x = y, 4. symmetric if xRy for x; y 2 A yields yRx, 5. nonsymmetric if 9x; y 2 A such that xRy and yR6 x, 6. antisymmetric if xRy and yRx for x; y 2 A yields x = y, 7. transitive if xRy and yRz for x; y; z 2 A yields xRz. Definition 6. A reflexive, antisymmetric and transitive relation in a set is a partial order relation or a partial ordering in that set. If R is a partial ordering in A, the ordered pair (A; R) is a partially ordered set. Rough relation properties 623 Definition 7. A relation in a set A is an equivalence relation in A if it is reflexive, symmetric and transitive. Definition 8. Let R be an equivalence relation on a set A. Consider an element a of A. The set of R-relatives of a in A, R[fag] is called the R-equivalence class generated by a. When there is no danger of confusion, the symbol R[fag] can be abbreviated to [a]. Proposition 1. Let R be an equivalence relation on A and let a; b 2 A. Then 1. a 2 [a], and 2. if aRb then [a] = [b]. Proposition 2. If Q is an equivalence relation in A, then Q = [1≤i≤nZi × Zi, where Zi; 1 ≤ i ≤ n are equivalence classes in A induced by Q. Proof. We have (a; b) 2 Q , a; b 2 Zi, for some i 2 f1; 2; : : : ; ng, where Zi is an equivalence class in A induced by Q , (a; b) 2 [1≤i≤nZi × Zi for i 2 f1; 2; : : : ; ng. Definition 9. If V and W are relations in A, then W •V is a relation in A defined as W • V = f(a; c) 2 A × A such that (a; b) 2 V and (b; c) 2 W for some b 2 Ag. Proposition 3. If V; W; V1; W1 are relations in A, V1 ⊆ V and W1 ⊆ W , then W1 • V1 ⊆ W • V . Proof. Let (a; c) 2 W1 • V1 ) 9b 2 A such that (a; b) 2 V1 and (b; c) 2 W1. Then (a; b) 2 V and (b; c) 2 W so that (a; c) 2 W • V . 3. Rough Set Theory 3.1. Basic Concepts The basic concept of Rough Set Theory is the notion of an approximation space, which is an ordered pair A = (U; R), where U is a nonempty set of objects, called the universe, and R stands for the equivalence relation on U, called the indiscernibility relation. If x; y 2 U and xRy, then x and y are indistinguishable in A. Each equivalence class induced by R, i.e. each element of the quotient set R~ = U=R, is called an elementary set in A. An approximation space can be alternatively denoted by A = (U; R~). It is assumed that the empty set is also elementary for every approximation space A. A definable set in A is any finite union of elementary sets in A. For x 2 U, let [x]R denote the equivalence class of R, containing x. For each subset X in A; X is characterized by a pair of sets|its lower and upper approximations in A, defined respectively as AA-low(X) = x 2 U j [x]R ⊆ X ; AA-upp(X) = x 2 U j [x]R \ X 6= ;g: 624 M.C. Nicoletti et al. When there is no risk of misunderstanding and for the sake of simplicity, we prefer to use Alow and Aupp instead of AA-low and AA-upp, respectively. The lower approximation of X in A is the greatest definable set in A contained in X, and the upper approximation of X in A is the smallest definable set in A containing X, with relation to the set inclusion. A set X ⊆ U is definable in A iff Alow(X) = Aupp(X). A rough set in A is the family of all subsets of U having the same lower and upper approximations. Another definition found in (Klir and Yuan, 1995) states that `a rough set is a representation of a given set X, by two subsets of the quotient set U=R which approach X as closely as possible from inside and outside, respectively. That is, hAlow(X); Aupp(X)i'. Both the definitions are shown to be equivalent in (Nicoletti and Uch^oa, 1997). 3.2. Some Basic RST Results The results presented in this section are relevant to the proofs that follow. They are stated as propositions and their proofs can be found in related literature. Let A = (U; R) be an approximation space and X ⊆ U. Proposition 4. The following assertions hold: 1. Alow(X) = [Y such that Y is definable and Y ⊆ X, and 2. AA-upp(X) = \Y such that Y is definable and X ⊆ Y . Proposition 5. Alow(X) ⊆ X ⊆ Aupp(X). Proposition 6. Alow(X) = Aupp(X) , X is definable. Proposition 7. Aupp(X [ Y ) = Aupp(X) [ Aupp(Y ). 4. Rough Relations Let A1 = (U1; R1) and A2 = (U2; R2) be two approximation spaces. The product of A1 by A2 is the approximation space denoted by A = (U; S), where U = U1 × U2 2 and the indiscernibility relation S ⊆ (U1 × U2) is defined by ((x1; y1); (x2; y2)) 2 S , (x1; x2) 2 R1 and (y1; y2) 2 R2, x1; x2 2 U1 and y1; y2 2 U2.