Rough convex cones and rough convex fuzzy cones - Semantic Scholar
Soft Comput (2012) 16:2083–2087 DOI 10.1007/s00500-012-0877-6
ORIGINAL PAPER
Rough convex cones and rough convex fuzzy cones Zuhua Liao • Juan Zhou
Published online: 28 June 2012 Ó Springer-Verlag 2012
Abstract Based on the equivalence relation on a linear space, in this paper we introduce the definition of rough convex cones and rough convex fuzzy cones and discuss some of the fundamental properties of such rough convex cones. Keywords Rough sets Rough convex cones Rough convex fuzzy cones
1 Introduction Fuzzy set theory (Zadeh 1965) and rough set theory (Pawlak 1982) are two powerful tools to cope with uncertainty and incompleteness of knowledge in information systems. In order to use these two complementary theories, authors combine them into fuzzy rough set theory (Dubois 1990; Kuroki 1996; Mi 2004; Mordeson 2001; Morsi 1998; Nanda 1992; Pawlak 1985; Radzikowska 2002; Yao 1998). The notion of rough sets was firstly introduced by Pawlak (1985) in 1982, and later Biswas (1994) proved that rough set can be considered as a special case of rough fuzzy sets. After more than 20 years studying, people have achieved a lot of excellent results on rough set theory in reference (Jarvinen 2002; Quafafou 2000; Wu 2002; Yao and Lingras 1998; Yao 1998). Rough set theories are widely applied in machine learning, knowledge discovery, data mining and so on. For example, pawlak and Skowron (2007a, b, c) discussed the applications of rough sets in pattern Z. Liao (&) J. Zhou School of Science, Jiangnan University, Wu Xi 214122, People’s Republic of China e-mail: [email protected]; [email protected] J. Zhou e-mail: [email protected]
recognition and conflict analysis. Many mathematicians have also considered the relations between rough sets and algebraic systems, such as groups, rings, modules, and a lot of work in this field has been done. For example, Bonikowaski (1995), Iwinski (1987) and Pomykala (1988) studied algebraic properties of rough sets. Biswas and Nanda (1994) introduced the notion of rough subgroups. Kuroki (1997) introduced the notion of rough ideals in a semigroup. Davvaz (2004) proposed the notion of rough subrings with respect to an ideal of a ring, in (Davvaz 2006) he also introduced the notion of rough submodule with respect to a submodule of an R-module. Equivalence relation is the key concept in Pawlak’s rough sets and most concepts of the theory are established via equivalence classes. Based on the equivalence relation on a linear space (Liu 2006), in this paper we introduce the definition of rough convex cones and rough convex fuzzy cones and discuss some of the fundamental properties of such rough convex cones. Fuzzy numbers can be considered as a special case of convex fuzzy sets, which is widely applied in fuzzy database, fuzzy knowledge presentation. A deep study of this can see (He 1999), this is the first time to propose the definition of rough convex cones and rough convex fuzzy cones, the possible applications of such rough convex cones can be also used in fuzzy database and fuzzy knowledge presentation, and some results need to left for our further study.
2 Preliminaries 2.1 Pawlak rough sets First, we present the fundamental concepts and properties of the Pawlak’s rough set theory. For an equivalence relation R on a finite set U 6¼ ;; the set of the elements of U
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that are related to x 2 U; is called the equivalence class of x, and is denoted by R(x). A pair (U, R) where U 6¼ ;; and R is an equivalence relation on U, is called an approximation space. Definition 2.1.1 For an approximation space (U, R), by rough approximation in (U, R) we mean a mapping Apr : PðUÞ ! PðUÞ PðUÞ defined for every X 2 PðUÞ by AprðXÞ ¼ ðAprðXÞ; AprðXÞÞ; where AprðXÞ ¼ fx 2 Uj RðxÞ Xg and AprðXÞ ¼ fx 2 UjRðxÞ \ X 6¼ ;g: AprðXÞ is called a lower rough approximation of X in (U, R), where as AprðXÞ is called an upper rough approximation of X in (U, R). Definition 2.1.2 Given an approximation space (U, R), a pair ðA; BÞ 2 PðUÞ PðUÞ is called a rough subset in (U, R) if and only if (A, B) = Apr(X) for some X 2 PðUÞ: 2.2 Convex cones In the following passages, we always suppose V is a linear space over real number filed R. Definition 2.2.1 A nonempty subset A of V is said to be a convex cone if for all a; b 2 A and l1 C 0, l2 C 0 such that l1 a þ l2 b 2 A: Definition 2.2.2 Let A and B be nonempty subsets of V, the sum of A and B is defined: A þ B ¼ fa þ bja 2 A; b 2 Bg; for some l C 0, the product of l and A is defined: lA ¼ flaja 2 Ag: Lemma 2.2.1 Let A be a convex cone of V, for all l [ 0, then we have lA = A. Note: when l C 0, we have lA A: Lemma 2.2.2 Let A and B be convex cones of V, then A \ B and A ? B are convex cones. 2.3 Convex fuzzy cones Definition 2.3.1 A map A : X ! ½0; 1 is said to be a fuzzy subset of X, and the all fuzzy set of X is denoted by F(X). Let A be a fuzzy subset of X, if there a; y ¼ x exists x 2 X; the form AðyÞ ¼ is said to be a 0; y 6¼ x fuzzy point with support x and value a is denoted by xa. Definition 2.3.2
Z. Liao, J. Zhou
S; l1 0; l2 0; such that l1 xa þ l2 yb 2 S; where l1xa ? l2yb = (l1x ? l2y)(a ^ b). Lemma 2.3.1 If A is a fuzzy subset of V, the following statements are equivalent: 1. 2. 3.
A is a convex fuzzy cone; For all x; y 2 V; 8l1 0; l2 0; Aðl1 x þ l2 yÞ AðxÞ ^AðyÞ; 8k 2 ½0; 1Þ; Ak is a convex cone, where Ak ¼ fxjAðxÞ
[ kg: In the following, we introduce the rough sets into convex cones and convex fuzzy cones.
3 Rough convex cones Definition 3.1 Let W be a subspace of V, for a; b 2 V we say a and b are congruence with respect to W, written as a 2 Rw ðbÞ; if a b 2 W: It is easy to see that Rw-congruence relation is equivalence relation. Therefore we can classify the vectors of V according to Rw: a and b are in the same Rw class if and only if a b 2 W; we denote by Rw(a) the equivalence class containing the element a 2 V: Definition 3.2 Let W be a subspace of V, if A is nonempty subset of V, then the sets Rw A ¼ fa 2 VjRw ðaÞ Ag and Rw A ¼ fa 2 VjRw ðaÞ \ A 6¼ ;g are called, respectively, Rw-lower and Rw-upper approximations of A with respect to W. Definition 3.3 Let W be a subspace of V and A be a subset of V, A is called an Rw-lower (resp. Rw-upper) rough convex cone if Rw A (resp. Rw A) is a convex cone of V. If both Rw A and Rw A are convex cones, then A is called Rwrough convex cone. Lemma 3.1 Let W be a subspace of V, for all a; b 2 V; l [ 0; then Rw ðaÞ ¼ a þ W; Rw ðaÞ þ Rw ðbÞ ¼ Rw ða þ bÞ; Rw ðlaÞ ¼ lRw ðaÞ: Note: when l C 0, we have lRw ðaÞ Rw ðlaÞ: Theorem 3.1 Let W be a subspace of V and A be a convex cone of V, then A is an Rw-lower rough convex cone if and only if W A:
Definition 2.3.3 Let A be a fuzzy subset of X, a fuzzy point xa is said to belong to A, if A(x) C a, denoted by xa 2 A:
Proof If A is an Rw-lower rough convex cone, i.e., Rw A is a convex cone, therefore 0 2 Rw A which implies W A: Conversely, for all a; b 2 Rw A; Rw ðaÞ A; Rw ðbÞ A: For all l1 C 0, l2 C 0, then l1 Rw ðaÞ l1 A A; and l2 Rw ðbÞ l2 A A; thus l1 Rw ðaÞ þ l2 Rw ðbÞ A:
Definition 2.3.4 A fuzzy subset S of X is said to be a convex fuzzy cone, if for all fuzzy point xa ; yb 2
Case 1 When l1 [ 0 or l2 [ 0, we have l1 Rw ðaÞ þ l2 Rw ðbÞ ¼ Rw ðl1 a þ l2 bÞ A; i.e., l1 a þ l2 b 2 Rw A:
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Case 2 When l1 = l2 = 0, we have Rw ðl1 a þ l2 bÞ ¼ Rw ð0Þ ¼ W A; i.e., l1 a þ l2 b 2 Rw A: Hence, Rw A is a convex cone, i.e., A is an Rw-lower rough convex cone. Theorem 3.2 Let W be a subspace of V and A be a convex cone of V, then A is an Rw-lower rough convex cone if and only if Rw A ¼ A: Proof Let A be an Rw-lower rough convex cone, by Theorem 3.1, we have got W A; for every a 2 A; then a þ W A; i.e., a 2 Rw A; therefore A Rw A: The opposite side is evident. Theorem 3.3 Let W be a subspace of V, if A is a convex cone of V, then A is an Rw-upper rough convex cone. Proof For all a; b 2 Rw A; then Rw ðaÞ \ A 6¼ ; and Rw ðbÞ \ A 6¼ ;; there exist a0 2 Rw ðaÞ \ A and b0 2 Rw ðbÞ \ A; thus a0 ; b0 2 A; a0 a 2 W and b0 b 2 W: For all l1 C 0, l2 C 0, then ðl1 a0 þ l2 b0 Þ ðl1 a þ l2 bÞ ¼ l1 ða0 aÞ þ l2 ðb0 bÞ 2 W; l1 a0 þ l2 b0 2 Rw ðl1 a þ l2 bÞ: Since A is a convex cone, then l1 a0 þ l2 b0 2 A; therefore l1 a0 þ l2 b0 2 Rw ðl1 a þ l2 bÞ \ A 6¼ ;; thus l1 a þ l2 b 2 Rw A: Hence, Rw A is a convex cone, i.e., A is an Rw-upper rough convex cone. By Theorem 3.1 and Theorem 3.2, we have the following conclusions. Corollary 3.1 Let W be a subspace of V and W A; if A is a convex cone of V, then A is an Rw-rough convex cone. Lemma 3.2 Let W be a subspace of V, if A and B are nonempty subsets of V, then Rw ðA \ BÞ ¼ Rw A \ Rw B; Rw ðA þ BÞ Rw A þ Rw B: Theorem 3.4 Let W be a subspace of V, A and B be nonempty subsets of V, if W A and A ? B = B, then Rw ðA þ BÞ ¼ Rw A þ Rw B: Proof For all a 2 Rw ðA þ BÞ; then Rw ðaÞ A þ B ¼ B; so a 2 Rw B; since Rw ð0Þ ¼ 0 þ W ¼ W A which implies 0 2 Rw A; therefore a ¼ 0 þ a 2 Rw A þ Rw B; so Rw ðA þ BÞ Rw A þ Rw B; as is known from Lemma 3.2, Rw ðA þ BÞ Rw A þ Rw B: Hence, Rw ðA þ BÞ ¼ Rw A þ Rw B: Corollary 3.2 Let W be a subspace of V, A and B be nonempty subsets of V; W A and A ? B = B, if A and B are both Rw-lower rough convex cones, then A ? B and A \ B are Rw-lower rough convex cones. Theorem 3.5 Let W be a subspace of V, A and B be nonempty subsets of V, then Rw A ¼ W þ A; Rw ðA þ BÞ ¼ Rw A þ Rw B:
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Proof On one hand, for all c 2 Rw A; then Rw ðcÞ \ A 6¼ ;; there exists c0 2 Rw ðcÞ \ A 6¼ ;; c c0 2 W; c0 2 A; so c 2 W þ A; therefore Rw A W þ A; on the other hand, for all c 2 W þ A; there exist a 2 W and b 2 A; such that c ¼ a þ b; a 2 W then 0 2 Rw ðaÞ: Since b 2 Rw ðbÞ; then b ¼ 0 þ b 2 Rw ðaÞ þ Rw ðbÞ ¼ Rw ða þ bÞ ¼ Rw ðcÞ; therefore b 2 Rw ðcÞ \ A 6¼ ;; i.e., c 2 Rw A; then Rw A W þ A: Hence, Rw A ¼ W þ A: Rw ðA þ BÞ ¼ W þ ðA þ BÞ ¼ ðW þ AÞ þ ðW þ BÞ ¼ Rw A þ Rw B: Lemma 3.3 (Pawla 1982) Let W be a subspace of V, A and B be nonempty subsets of V, then Rw ðA \ BÞ Rw A \ Rw B; if A B; then Rw ðA \ BÞ ¼ Rw A \ Rw B: Lemma 3.4 Let A and B be nonempty subsets of V, if 0 2 B and A ? B = B, then we have A B: Theorem 3.6 Let A and B be nonempty subsets of V; W A and A ? B = B, if A and B are both Rw-upper rough convex cones, then A \ B is an Rw-upper rough convex cone. Corollary 3.3 Let A and B be nonempty subsets of V; W A and A ? B = B, if A and B are both Rw-rough convex cones, then A \ B and A ? B are both Rw-rough convex cones. Lemma 3.5 Let V1,V2 be subspace of V, A is a nonempty subset of V, then Rv1 A \ Rv2 A Rv1 \v2 A; Rv1 \v2 A Rv1 A\ Rv2 A: Lemma 3.6 Let V1,V2 be subspace of V, A is a nonempty subset of V, if V1 ; V2 A and A ? A = A, then Rv1 A \ Rv2 A ¼ Rv1 \v2 A: Proof For all a 2 Rv1 \v2 A; then Rv1 \v2 ðaÞ A and a 2 A; since V1 A and A þ A ¼ A; Rv1 ðaÞ ¼ a þ V1 A þ A ¼ A; so a 2 Rv1 A: Similarly, we can show a 2 Rv2 A; so a 2 Rv1 A \ Rv2 A; therefore Rv1 \v2 A Rv1 A \ Rv2 A; by Lemma 3.4, we have Rv1 A \ Rv2 A Rv1 \v2 A; thus Rv1 A \ Rv2 A ¼ Rv1 \v2 A: Theorem 3.7 If A is Rv1 lower and Rv2 lower rough convex cone, V1 ; V2 A and A ? A = A, then A is an Rv1 \v2 lower rough convex cone: Lemma 3.7 Let V1,V2 be subspace of V, A is a nonempty subset of V, if Rv1 A ¼ A or Rv2 A ¼ A; then Rv1 \v2 A ¼ Rv1 A \ Rv2 A: Proof It is only needed to prove that ‘‘if Rv1 A ¼ A then Rv1 \v2 A ¼ Rv1 A \ Rv2 A:’’ Assume Rv1 A ¼ A; then Rv1 A \ Rv2 A ¼ A \ Rv2 A ¼ A Rv1 \v2 A; by Lemma 3.4, we have Rv1 \v2 A Rv1 A \ Rv2 A; thus Rv1 \v2 A ¼ Rv1 A \ Rv2 A:
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Theorem 3.8 If A is Rv1 upper and Rv2 upper rough convex cone. And suppose Rv1 A ¼ A or Rv2 A ¼ A; then A is an Rv1 \v2 upperroughconvexcone:
1. 2.
Theorem 3.9 Let V1,V2 be subspace of V, A is a nonempty subset of V, if V1 ; V2 A and A ? A = A, then Rv1 A \ Rv2 A ¼ Rv1 \v2 A; Rv1 \v2 A ¼ Rv1 A \ Rv2 A:
Proof (1) ) (2) Let A be an Rw-rough convex fuzzy cone, i.e., Rw A and Rw A are convex fuzzy cones, by Lemma 2.3.1 ðRw AÞk and ðRw AÞk are convex cones. By Lemma 4.1, we
Proof Firstly, we can have Rv1 A ¼ A; Rv2 A ¼ A: In fact, for all x 2 Rv1 A; Rv1 ðxÞ \ A 6¼ ;; there exists x0 2 Rv1 ðxÞ \ A; x0 2 Rv1 ðxÞ and x0 2 A; then x x0 2 V1 A; x0 2 A; so x ¼ ðx x0 Þ þ x0 A þ A; then Rv1 A A; since A Rv1 A; therefore Rv1 A ¼ A: similarly, Rv2 A ¼ A: By Lemma 3.5 and lemma 3.6, we have Rv1 A \ Rv2 A ¼ Rv1 \v2 A and
show Rw ðAk Þ and Rw ðAk Þ are convex cones, therefore Ak is
Rv1 \v2 A ¼ Rv1 A \ Rv2 A: Corollary 3.4 Let V1,V2 be subspace of V; V1 ; V2 A and A ? A = A, if A is an Rv1 rough and Rv1 rough convex cone, then A is an Rv1 \v2 cone.
A is an Rw-rough convex fuzzy cone; 8k 2 ½0; 1Þ; Ak is an Rw-rough convex cone.
Definition 4.2 Let A be a fuzzy subset of V, A is called Rw-lower (resp. Rw-upper) rough convex fuzzy cone if Rw A (resp. Rw A) is a convex fuzzy cone of V. If both Rw A and Rw A are convex fuzzy cones, then A is called an Rw-rough convex fuzzy cone. Definition 4.3 If A is a fuzzy subset of V; k 2 ½0; 1; the sets: Rw ðAk Þ ¼ fx 2 VjRw ðxÞ Ak g; Rw ðAk Þ ¼ fx 2 VjRw ðxÞ \Ak 6¼ ;g are called Rw-lower and Rw-upper approximations of Ak with respect to Rw, respectively, where Ak ¼ fx 2 VjAðxÞ kg: Definition 4.4 Let A be a fuzzy subset of V, the set KerðAÞ ¼ fx 2 VjAðxÞ ¼ 1g; SuppðAÞ ¼ fx 2 VjAðxÞ [ 0g: Theorem 4.1 Let A be a convex fuzzy cone of V, then Ker(A) and Supp(A) are both convex cones. Lemma 4.1 If A is a fuzzy subset of V; k 2 ½0; 1Þ; then Rw ðAk Þ ¼ ðRw AÞk ; Rw ðAk Þ ¼ ðRw AÞk :
Corollary 4.1 Let A be fuzzy subset of V, then KerðRw AÞ ¼ Rw ðKerAÞ; SuppðRw AÞ ¼ Rw ðSuppAÞ: Theorem 4.2 Let A be a fuzzy subset of V, the following statements are equivalent:
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an Rw-rough convex cone, i.e., Rw ðAk Þ and Rw ðAk Þ are
convex cones, by Lemma 4.1, ðRw AÞk and ðRw AÞk are
convex cones, so by Lemma 2.3.1 Rw A and Rw A are convex cones, therefore A is an Rw-rough convex fuzzy cone. Theorem 4.3
If A is a fuzzy convex cone of V and 8k 2
½0; 1Þ; W Ak ; then A is an Rw-rough convex fuzzy cone.
If A is a convex fuzzy cone of V, then Ak is a
Definition 4.1 If A is a fuzzy subset of V, the fuzzy subset Rw A; Rw A are called the Rw-lower and Rw-upper approximations of A with respect to Rw, respectively, the membership functions are defined: Rw AðxÞ ¼ inf fAðyÞjy 2 Rw ðxÞg and Rw AðxÞ ¼ supfAðyÞjy 2 Rw ðxÞg:
convex cone, since W Ak ; by Theorem 3.4, Ak is an
4 Rough convex fuzzy cones
an Rw-rough convex cone. ð2Þ ) ð1Þ 8k 2 ½0; 1Þ; let Ak is
Proof
Rw-rough convex cone, according to Theorem 4.2, therefore A is an Rw-rough convex fuzzy cone. Definition 4.5 (Bhaka 1997) A fuzzy subset A of V is said to have the ‘‘sup property’’, if for any non-empty subset T of V, there exists a 2 T; such that AðaÞ ¼ sup fAðtÞjt 2 Tg: Lemma 4.2 Let A be a fuzzy subset of V, if A has the ‘‘sup property’’ 8k 2 ½0; 1; then (1) Rw ðAk Þ ¼ ðRw AÞk ; (2) Rw ðAk Þ ¼ ðRw AÞk : Proof (1) 8x 2 ðRw AÞk ; Rw AðxÞ k; inf fAðyÞjy 2 Rw ðxÞg k; 8y 2 Rw ðxÞ; AðyÞ k; y 2 Ak ; namely, Rw ðxÞ Ak ; therefore x 2 Rw ðAk Þ: On the other hand, 8x 2 Rw ðAk Þ; Rw ðxÞ Ak ; 8y 2 Rw ðxÞ; y 2 Ak ; namely, inf fAðyÞ jy 2 Rw ðxÞg k; Rw AðxÞ k; therefore x 2 ðRw AÞk : (2) 8x 2 ðRw AÞk ; Rw AðxÞ k; supfAðyÞjy 2 Rw ðxÞg k; as A has the ‘‘sup property’’, there exists x0 2 Rw ðxÞ; such that A(x0 ) = supfAðyÞjy 2 Rw ðxÞg k; x0 2 Ak ; x0 2 Ak \ Rw ðxÞ 6¼ ;; namely, x 2 Rw ðAk Þ: On the other hand, 8x 2 Rw ðAk Þ; Rw ðxÞ \ Ak 6¼ ;; such that x0 2 Rw ðxÞ \ Ak ; Aðx0 Þ k; supfAðyÞjy 2 Rw ðxÞg Aðx0 Þ k; namely, Rw AðxÞ k; x 2 ðRw AÞk : Note: proposition (1) does not need under the condition that fuzzy subset A has the‘‘sup property’’, in general fuzzy subset A, proposition (1) also stands. Theorem 4.4 Let A be a fuzzy subset of V, if A has the ‘‘sup property’’ 8k 2 ½0; 1; W Ak ; then A is an Rw-rough convex fuzzy cone. Acknowledgments This work is supported by Program for Innovative Research Team of Jiangnan University (No:200902).
Rough convex cones and rough convex fuzzy cones Thanks for the reviewers giving me so many important suggestions which improves the quality of the paper.
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ORIGINAL PAPER
Rough convex cones and rough convex fuzzy cones Zuhua Liao • Juan Zhou
Published online: 28 June 2012 Ó Springer-Verlag 2012
Abstract Based on the equivalence relation on a linear space, in this paper we introduce the definition of rough convex cones and rough convex fuzzy cones and discuss some of the fundamental properties of such rough convex cones. Keywords Rough sets Rough convex cones Rough convex fuzzy cones
1 Introduction Fuzzy set theory (Zadeh 1965) and rough set theory (Pawlak 1982) are two powerful tools to cope with uncertainty and incompleteness of knowledge in information systems. In order to use these two complementary theories, authors combine them into fuzzy rough set theory (Dubois 1990; Kuroki 1996; Mi 2004; Mordeson 2001; Morsi 1998; Nanda 1992; Pawlak 1985; Radzikowska 2002; Yao 1998). The notion of rough sets was firstly introduced by Pawlak (1985) in 1982, and later Biswas (1994) proved that rough set can be considered as a special case of rough fuzzy sets. After more than 20 years studying, people have achieved a lot of excellent results on rough set theory in reference (Jarvinen 2002; Quafafou 2000; Wu 2002; Yao and Lingras 1998; Yao 1998). Rough set theories are widely applied in machine learning, knowledge discovery, data mining and so on. For example, pawlak and Skowron (2007a, b, c) discussed the applications of rough sets in pattern Z. Liao (&) J. Zhou School of Science, Jiangnan University, Wu Xi 214122, People’s Republic of China e-mail: [email protected]; [email protected] J. Zhou e-mail: [email protected]
recognition and conflict analysis. Many mathematicians have also considered the relations between rough sets and algebraic systems, such as groups, rings, modules, and a lot of work in this field has been done. For example, Bonikowaski (1995), Iwinski (1987) and Pomykala (1988) studied algebraic properties of rough sets. Biswas and Nanda (1994) introduced the notion of rough subgroups. Kuroki (1997) introduced the notion of rough ideals in a semigroup. Davvaz (2004) proposed the notion of rough subrings with respect to an ideal of a ring, in (Davvaz 2006) he also introduced the notion of rough submodule with respect to a submodule of an R-module. Equivalence relation is the key concept in Pawlak’s rough sets and most concepts of the theory are established via equivalence classes. Based on the equivalence relation on a linear space (Liu 2006), in this paper we introduce the definition of rough convex cones and rough convex fuzzy cones and discuss some of the fundamental properties of such rough convex cones. Fuzzy numbers can be considered as a special case of convex fuzzy sets, which is widely applied in fuzzy database, fuzzy knowledge presentation. A deep study of this can see (He 1999), this is the first time to propose the definition of rough convex cones and rough convex fuzzy cones, the possible applications of such rough convex cones can be also used in fuzzy database and fuzzy knowledge presentation, and some results need to left for our further study.
2 Preliminaries 2.1 Pawlak rough sets First, we present the fundamental concepts and properties of the Pawlak’s rough set theory. For an equivalence relation R on a finite set U 6¼ ;; the set of the elements of U
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that are related to x 2 U; is called the equivalence class of x, and is denoted by R(x). A pair (U, R) where U 6¼ ;; and R is an equivalence relation on U, is called an approximation space. Definition 2.1.1 For an approximation space (U, R), by rough approximation in (U, R) we mean a mapping Apr : PðUÞ ! PðUÞ PðUÞ defined for every X 2 PðUÞ by AprðXÞ ¼ ðAprðXÞ; AprðXÞÞ; where AprðXÞ ¼ fx 2 Uj RðxÞ Xg and AprðXÞ ¼ fx 2 UjRðxÞ \ X 6¼ ;g: AprðXÞ is called a lower rough approximation of X in (U, R), where as AprðXÞ is called an upper rough approximation of X in (U, R). Definition 2.1.2 Given an approximation space (U, R), a pair ðA; BÞ 2 PðUÞ PðUÞ is called a rough subset in (U, R) if and only if (A, B) = Apr(X) for some X 2 PðUÞ: 2.2 Convex cones In the following passages, we always suppose V is a linear space over real number filed R. Definition 2.2.1 A nonempty subset A of V is said to be a convex cone if for all a; b 2 A and l1 C 0, l2 C 0 such that l1 a þ l2 b 2 A: Definition 2.2.2 Let A and B be nonempty subsets of V, the sum of A and B is defined: A þ B ¼ fa þ bja 2 A; b 2 Bg; for some l C 0, the product of l and A is defined: lA ¼ flaja 2 Ag: Lemma 2.2.1 Let A be a convex cone of V, for all l [ 0, then we have lA = A. Note: when l C 0, we have lA A: Lemma 2.2.2 Let A and B be convex cones of V, then A \ B and A ? B are convex cones. 2.3 Convex fuzzy cones Definition 2.3.1 A map A : X ! ½0; 1 is said to be a fuzzy subset of X, and the all fuzzy set of X is denoted by F(X). Let A be a fuzzy subset of X, if there a; y ¼ x exists x 2 X; the form AðyÞ ¼ is said to be a 0; y 6¼ x fuzzy point with support x and value a is denoted by xa. Definition 2.3.2
Z. Liao, J. Zhou
S; l1 0; l2 0; such that l1 xa þ l2 yb 2 S; where l1xa ? l2yb = (l1x ? l2y)(a ^ b). Lemma 2.3.1 If A is a fuzzy subset of V, the following statements are equivalent: 1. 2. 3.
A is a convex fuzzy cone; For all x; y 2 V; 8l1 0; l2 0; Aðl1 x þ l2 yÞ AðxÞ ^AðyÞ; 8k 2 ½0; 1Þ; Ak is a convex cone, where Ak ¼ fxjAðxÞ
[ kg: In the following, we introduce the rough sets into convex cones and convex fuzzy cones.
3 Rough convex cones Definition 3.1 Let W be a subspace of V, for a; b 2 V we say a and b are congruence with respect to W, written as a 2 Rw ðbÞ; if a b 2 W: It is easy to see that Rw-congruence relation is equivalence relation. Therefore we can classify the vectors of V according to Rw: a and b are in the same Rw class if and only if a b 2 W; we denote by Rw(a) the equivalence class containing the element a 2 V: Definition 3.2 Let W be a subspace of V, if A is nonempty subset of V, then the sets Rw A ¼ fa 2 VjRw ðaÞ Ag and Rw A ¼ fa 2 VjRw ðaÞ \ A 6¼ ;g are called, respectively, Rw-lower and Rw-upper approximations of A with respect to W. Definition 3.3 Let W be a subspace of V and A be a subset of V, A is called an Rw-lower (resp. Rw-upper) rough convex cone if Rw A (resp. Rw A) is a convex cone of V. If both Rw A and Rw A are convex cones, then A is called Rwrough convex cone. Lemma 3.1 Let W be a subspace of V, for all a; b 2 V; l [ 0; then Rw ðaÞ ¼ a þ W; Rw ðaÞ þ Rw ðbÞ ¼ Rw ða þ bÞ; Rw ðlaÞ ¼ lRw ðaÞ: Note: when l C 0, we have lRw ðaÞ Rw ðlaÞ: Theorem 3.1 Let W be a subspace of V and A be a convex cone of V, then A is an Rw-lower rough convex cone if and only if W A:
Definition 2.3.3 Let A be a fuzzy subset of X, a fuzzy point xa is said to belong to A, if A(x) C a, denoted by xa 2 A:
Proof If A is an Rw-lower rough convex cone, i.e., Rw A is a convex cone, therefore 0 2 Rw A which implies W A: Conversely, for all a; b 2 Rw A; Rw ðaÞ A; Rw ðbÞ A: For all l1 C 0, l2 C 0, then l1 Rw ðaÞ l1 A A; and l2 Rw ðbÞ l2 A A; thus l1 Rw ðaÞ þ l2 Rw ðbÞ A:
Definition 2.3.4 A fuzzy subset S of X is said to be a convex fuzzy cone, if for all fuzzy point xa ; yb 2
Case 1 When l1 [ 0 or l2 [ 0, we have l1 Rw ðaÞ þ l2 Rw ðbÞ ¼ Rw ðl1 a þ l2 bÞ A; i.e., l1 a þ l2 b 2 Rw A:
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Case 2 When l1 = l2 = 0, we have Rw ðl1 a þ l2 bÞ ¼ Rw ð0Þ ¼ W A; i.e., l1 a þ l2 b 2 Rw A: Hence, Rw A is a convex cone, i.e., A is an Rw-lower rough convex cone. Theorem 3.2 Let W be a subspace of V and A be a convex cone of V, then A is an Rw-lower rough convex cone if and only if Rw A ¼ A: Proof Let A be an Rw-lower rough convex cone, by Theorem 3.1, we have got W A; for every a 2 A; then a þ W A; i.e., a 2 Rw A; therefore A Rw A: The opposite side is evident. Theorem 3.3 Let W be a subspace of V, if A is a convex cone of V, then A is an Rw-upper rough convex cone. Proof For all a; b 2 Rw A; then Rw ðaÞ \ A 6¼ ; and Rw ðbÞ \ A 6¼ ;; there exist a0 2 Rw ðaÞ \ A and b0 2 Rw ðbÞ \ A; thus a0 ; b0 2 A; a0 a 2 W and b0 b 2 W: For all l1 C 0, l2 C 0, then ðl1 a0 þ l2 b0 Þ ðl1 a þ l2 bÞ ¼ l1 ða0 aÞ þ l2 ðb0 bÞ 2 W; l1 a0 þ l2 b0 2 Rw ðl1 a þ l2 bÞ: Since A is a convex cone, then l1 a0 þ l2 b0 2 A; therefore l1 a0 þ l2 b0 2 Rw ðl1 a þ l2 bÞ \ A 6¼ ;; thus l1 a þ l2 b 2 Rw A: Hence, Rw A is a convex cone, i.e., A is an Rw-upper rough convex cone. By Theorem 3.1 and Theorem 3.2, we have the following conclusions. Corollary 3.1 Let W be a subspace of V and W A; if A is a convex cone of V, then A is an Rw-rough convex cone. Lemma 3.2 Let W be a subspace of V, if A and B are nonempty subsets of V, then Rw ðA \ BÞ ¼ Rw A \ Rw B; Rw ðA þ BÞ Rw A þ Rw B: Theorem 3.4 Let W be a subspace of V, A and B be nonempty subsets of V, if W A and A ? B = B, then Rw ðA þ BÞ ¼ Rw A þ Rw B: Proof For all a 2 Rw ðA þ BÞ; then Rw ðaÞ A þ B ¼ B; so a 2 Rw B; since Rw ð0Þ ¼ 0 þ W ¼ W A which implies 0 2 Rw A; therefore a ¼ 0 þ a 2 Rw A þ Rw B; so Rw ðA þ BÞ Rw A þ Rw B; as is known from Lemma 3.2, Rw ðA þ BÞ Rw A þ Rw B: Hence, Rw ðA þ BÞ ¼ Rw A þ Rw B: Corollary 3.2 Let W be a subspace of V, A and B be nonempty subsets of V; W A and A ? B = B, if A and B are both Rw-lower rough convex cones, then A ? B and A \ B are Rw-lower rough convex cones. Theorem 3.5 Let W be a subspace of V, A and B be nonempty subsets of V, then Rw A ¼ W þ A; Rw ðA þ BÞ ¼ Rw A þ Rw B:
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Proof On one hand, for all c 2 Rw A; then Rw ðcÞ \ A 6¼ ;; there exists c0 2 Rw ðcÞ \ A 6¼ ;; c c0 2 W; c0 2 A; so c 2 W þ A; therefore Rw A W þ A; on the other hand, for all c 2 W þ A; there exist a 2 W and b 2 A; such that c ¼ a þ b; a 2 W then 0 2 Rw ðaÞ: Since b 2 Rw ðbÞ; then b ¼ 0 þ b 2 Rw ðaÞ þ Rw ðbÞ ¼ Rw ða þ bÞ ¼ Rw ðcÞ; therefore b 2 Rw ðcÞ \ A 6¼ ;; i.e., c 2 Rw A; then Rw A W þ A: Hence, Rw A ¼ W þ A: Rw ðA þ BÞ ¼ W þ ðA þ BÞ ¼ ðW þ AÞ þ ðW þ BÞ ¼ Rw A þ Rw B: Lemma 3.3 (Pawla 1982) Let W be a subspace of V, A and B be nonempty subsets of V, then Rw ðA \ BÞ Rw A \ Rw B; if A B; then Rw ðA \ BÞ ¼ Rw A \ Rw B: Lemma 3.4 Let A and B be nonempty subsets of V, if 0 2 B and A ? B = B, then we have A B: Theorem 3.6 Let A and B be nonempty subsets of V; W A and A ? B = B, if A and B are both Rw-upper rough convex cones, then A \ B is an Rw-upper rough convex cone. Corollary 3.3 Let A and B be nonempty subsets of V; W A and A ? B = B, if A and B are both Rw-rough convex cones, then A \ B and A ? B are both Rw-rough convex cones. Lemma 3.5 Let V1,V2 be subspace of V, A is a nonempty subset of V, then Rv1 A \ Rv2 A Rv1 \v2 A; Rv1 \v2 A Rv1 A\ Rv2 A: Lemma 3.6 Let V1,V2 be subspace of V, A is a nonempty subset of V, if V1 ; V2 A and A ? A = A, then Rv1 A \ Rv2 A ¼ Rv1 \v2 A: Proof For all a 2 Rv1 \v2 A; then Rv1 \v2 ðaÞ A and a 2 A; since V1 A and A þ A ¼ A; Rv1 ðaÞ ¼ a þ V1 A þ A ¼ A; so a 2 Rv1 A: Similarly, we can show a 2 Rv2 A; so a 2 Rv1 A \ Rv2 A; therefore Rv1 \v2 A Rv1 A \ Rv2 A; by Lemma 3.4, we have Rv1 A \ Rv2 A Rv1 \v2 A; thus Rv1 A \ Rv2 A ¼ Rv1 \v2 A: Theorem 3.7 If A is Rv1 lower and Rv2 lower rough convex cone, V1 ; V2 A and A ? A = A, then A is an Rv1 \v2 lower rough convex cone: Lemma 3.7 Let V1,V2 be subspace of V, A is a nonempty subset of V, if Rv1 A ¼ A or Rv2 A ¼ A; then Rv1 \v2 A ¼ Rv1 A \ Rv2 A: Proof It is only needed to prove that ‘‘if Rv1 A ¼ A then Rv1 \v2 A ¼ Rv1 A \ Rv2 A:’’ Assume Rv1 A ¼ A; then Rv1 A \ Rv2 A ¼ A \ Rv2 A ¼ A Rv1 \v2 A; by Lemma 3.4, we have Rv1 \v2 A Rv1 A \ Rv2 A; thus Rv1 \v2 A ¼ Rv1 A \ Rv2 A:
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Theorem 3.8 If A is Rv1 upper and Rv2 upper rough convex cone. And suppose Rv1 A ¼ A or Rv2 A ¼ A; then A is an Rv1 \v2 upperroughconvexcone:
1. 2.
Theorem 3.9 Let V1,V2 be subspace of V, A is a nonempty subset of V, if V1 ; V2 A and A ? A = A, then Rv1 A \ Rv2 A ¼ Rv1 \v2 A; Rv1 \v2 A ¼ Rv1 A \ Rv2 A:
Proof (1) ) (2) Let A be an Rw-rough convex fuzzy cone, i.e., Rw A and Rw A are convex fuzzy cones, by Lemma 2.3.1 ðRw AÞk and ðRw AÞk are convex cones. By Lemma 4.1, we
Proof Firstly, we can have Rv1 A ¼ A; Rv2 A ¼ A: In fact, for all x 2 Rv1 A; Rv1 ðxÞ \ A 6¼ ;; there exists x0 2 Rv1 ðxÞ \ A; x0 2 Rv1 ðxÞ and x0 2 A; then x x0 2 V1 A; x0 2 A; so x ¼ ðx x0 Þ þ x0 A þ A; then Rv1 A A; since A Rv1 A; therefore Rv1 A ¼ A: similarly, Rv2 A ¼ A: By Lemma 3.5 and lemma 3.6, we have Rv1 A \ Rv2 A ¼ Rv1 \v2 A and
show Rw ðAk Þ and Rw ðAk Þ are convex cones, therefore Ak is
Rv1 \v2 A ¼ Rv1 A \ Rv2 A: Corollary 3.4 Let V1,V2 be subspace of V; V1 ; V2 A and A ? A = A, if A is an Rv1 rough and Rv1 rough convex cone, then A is an Rv1 \v2 cone.
A is an Rw-rough convex fuzzy cone; 8k 2 ½0; 1Þ; Ak is an Rw-rough convex cone.
Definition 4.2 Let A be a fuzzy subset of V, A is called Rw-lower (resp. Rw-upper) rough convex fuzzy cone if Rw A (resp. Rw A) is a convex fuzzy cone of V. If both Rw A and Rw A are convex fuzzy cones, then A is called an Rw-rough convex fuzzy cone. Definition 4.3 If A is a fuzzy subset of V; k 2 ½0; 1; the sets: Rw ðAk Þ ¼ fx 2 VjRw ðxÞ Ak g; Rw ðAk Þ ¼ fx 2 VjRw ðxÞ \Ak 6¼ ;g are called Rw-lower and Rw-upper approximations of Ak with respect to Rw, respectively, where Ak ¼ fx 2 VjAðxÞ kg: Definition 4.4 Let A be a fuzzy subset of V, the set KerðAÞ ¼ fx 2 VjAðxÞ ¼ 1g; SuppðAÞ ¼ fx 2 VjAðxÞ [ 0g: Theorem 4.1 Let A be a convex fuzzy cone of V, then Ker(A) and Supp(A) are both convex cones. Lemma 4.1 If A is a fuzzy subset of V; k 2 ½0; 1Þ; then Rw ðAk Þ ¼ ðRw AÞk ; Rw ðAk Þ ¼ ðRw AÞk :
Corollary 4.1 Let A be fuzzy subset of V, then KerðRw AÞ ¼ Rw ðKerAÞ; SuppðRw AÞ ¼ Rw ðSuppAÞ: Theorem 4.2 Let A be a fuzzy subset of V, the following statements are equivalent:
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an Rw-rough convex cone, i.e., Rw ðAk Þ and Rw ðAk Þ are
convex cones, by Lemma 4.1, ðRw AÞk and ðRw AÞk are
convex cones, so by Lemma 2.3.1 Rw A and Rw A are convex cones, therefore A is an Rw-rough convex fuzzy cone. Theorem 4.3
If A is a fuzzy convex cone of V and 8k 2
½0; 1Þ; W Ak ; then A is an Rw-rough convex fuzzy cone.
If A is a convex fuzzy cone of V, then Ak is a
Definition 4.1 If A is a fuzzy subset of V, the fuzzy subset Rw A; Rw A are called the Rw-lower and Rw-upper approximations of A with respect to Rw, respectively, the membership functions are defined: Rw AðxÞ ¼ inf fAðyÞjy 2 Rw ðxÞg and Rw AðxÞ ¼ supfAðyÞjy 2 Rw ðxÞg:
convex cone, since W Ak ; by Theorem 3.4, Ak is an
4 Rough convex fuzzy cones
an Rw-rough convex cone. ð2Þ ) ð1Þ 8k 2 ½0; 1Þ; let Ak is
Proof
Rw-rough convex cone, according to Theorem 4.2, therefore A is an Rw-rough convex fuzzy cone. Definition 4.5 (Bhaka 1997) A fuzzy subset A of V is said to have the ‘‘sup property’’, if for any non-empty subset T of V, there exists a 2 T; such that AðaÞ ¼ sup fAðtÞjt 2 Tg: Lemma 4.2 Let A be a fuzzy subset of V, if A has the ‘‘sup property’’ 8k 2 ½0; 1; then (1) Rw ðAk Þ ¼ ðRw AÞk ; (2) Rw ðAk Þ ¼ ðRw AÞk : Proof (1) 8x 2 ðRw AÞk ; Rw AðxÞ k; inf fAðyÞjy 2 Rw ðxÞg k; 8y 2 Rw ðxÞ; AðyÞ k; y 2 Ak ; namely, Rw ðxÞ Ak ; therefore x 2 Rw ðAk Þ: On the other hand, 8x 2 Rw ðAk Þ; Rw ðxÞ Ak ; 8y 2 Rw ðxÞ; y 2 Ak ; namely, inf fAðyÞ jy 2 Rw ðxÞg k; Rw AðxÞ k; therefore x 2 ðRw AÞk : (2) 8x 2 ðRw AÞk ; Rw AðxÞ k; supfAðyÞjy 2 Rw ðxÞg k; as A has the ‘‘sup property’’, there exists x0 2 Rw ðxÞ; such that A(x0 ) = supfAðyÞjy 2 Rw ðxÞg k; x0 2 Ak ; x0 2 Ak \ Rw ðxÞ 6¼ ;; namely, x 2 Rw ðAk Þ: On the other hand, 8x 2 Rw ðAk Þ; Rw ðxÞ \ Ak 6¼ ;; such that x0 2 Rw ðxÞ \ Ak ; Aðx0 Þ k; supfAðyÞjy 2 Rw ðxÞg Aðx0 Þ k; namely, Rw AðxÞ k; x 2 ðRw AÞk : Note: proposition (1) does not need under the condition that fuzzy subset A has the‘‘sup property’’, in general fuzzy subset A, proposition (1) also stands. Theorem 4.4 Let A be a fuzzy subset of V, if A has the ‘‘sup property’’ 8k 2 ½0; 1; W Ak ; then A is an Rw-rough convex fuzzy cone. Acknowledgments This work is supported by Program for Innovative Research Team of Jiangnan University (No:200902).
Rough convex cones and rough convex fuzzy cones Thanks for the reviewers giving me so many important suggestions which improves the quality of the paper.
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