Abstract
Fuzzy Inf. Eng. (2011) 2: 209-222 DOI 10.1007/s12543-011-0078-5 ORIGINAL ARTICLE Xue-hai Yuan· Hong-xing Li · Kai-biao Sun Received: 12 January 2010 / Revised: 29 March 2011/ Accepted: 16 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract The connections between Zadeh fuzzy set and three-valued fuzzy set are established in this paper. The concepts of interval-valued level cut sets on Zadeh fuzzy set are presented and new decomposition theorems and representation theorems of Zadeh fuzzy set are established based on new cut sets. Firstly, four interval-valued level cut sets on Zadeh fuzzy set are deﬁned as three-valued fuzzy sets and it is shown that the interval-valued level cut sets of Zadeh fuzzy set are generalizations of normal cut sets on Zadeh fuzzy set, and have the same properties as those of normal cut sets of Zadeh fuzzy set. Secondly, the new decomposition theorems are established based on these new cut sets. It is pointed out that each kind of interval-valued level cut sets corresponds to two decomposition theorems. Thus eight decomposition theorems are obtained. Finally, the deﬁnitions of three-valued inverse order nested sets and three- valued order nested sets are presented with eight representation theorems based on new nested sets. Keywords Fuzzy sets · Interval-valued level cut sets · Decomposition theorems · Representation theorems 1. Introduction It is well known that the cut sets of fuzzy sets play an important role in fuzzy algebra [1, 2], fuzzy optimization and decision making [3, 4], fuzzy reasoning [5-7] as well Xue-hai Yuan () Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, P.R. China email: yuanxuehai@yahoo.com Hong-xing Li Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, P.R. China Kai-biao Sun Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, P.R. China 210 Xue-hai Yuan· Hong-xing Li · Kai-biao Sun (2011) fuzzy measure [8-12] and so on. The cut sets on Zadeh fuzzy set [13] are described by using the neighborhood relations between fuzzy point and fuzzy set and the four kinds of cut sets on Zadeh fuzzy set are acquired in [14]. Based on these cut sets of fuzzy set, the connections are established between the fuzzy set and classical set. Recently, it is shown that three-valued fuzzy set are intimately connected with some L-fuzzy sets such as intuitionistic fuzzy set and interval valued fuzzy set [15, 16]. Yuan et al [15] deﬁned the cut sets of intuitionistic fuzzy set and interval valued fuzzy set as three-valued fuzzy sets, and established decomposition theorems and representation theorems of intuitionistic fuzzy sets as well as interval-valued fuzzy sets by using these cut sets. Clearly, people often take on some fuzzy concepts when making decision on a complex system. From the point of decision making, the normal cut set A is a subset of universe X and x ∈ A shows that degree of x belonging to A is not less than level a.If level a ∈ [0, 1] is taken as a threshold value, then x is considered as a qualiﬁed element at A(x) ≥ a and x is not considered as a qualiﬁed element at A(x) < a. However, can we decide which element is qualiﬁed if an interval-valued level [a , a ] 1 2 (a subset of [0, 1]) is taken as a threshold value? In order to solve this problem, we need to make a comparison between A(x) and [a , a ]. Thus, we need to introduce 1 2 the concept of interval-valued level cut sets. In this paper, we introduce the concepts of interval-valued level cut sets from Zadeh fuzzy sets and establish the new decomposition theorems and representation theorems based on new cut sets. The rest of this paper is organized as follows: we ﬁrst provide the preliminaries in Section 2. Then, we give four kinds of deﬁnitions of interval-valued level cut sets on Zadeh fuzzy sets and study the properties of these cuts in Section 3. In Section 4 and 5, based on the new cut sets, we established eight new decomposition theorems and eight new representation theorems. 2. Preliminary Deﬁnition 2.1 [13] Let X be a set. The mapping A : X → [0, 1] is called a (Zadeh) fuzzy subset of X. Normal cut sets of Zadeh fuzzy set are given as follows. Deﬁnition 2.2 [14, 15, 17] Let A be a fuzzy subset of X andλ ∈ [0, 1]. Then A = {x|x ∈ X, A(x) ≥ λ} and A = {x|x ∈ X, A(x) >λ} are called λ-upper cut set λ λ and strongλ -upper cut set of A, respectively. λ λ A = {x|x ∈ X, A(x) ≤ λ} and A = {x|x ∈ X, A(x)<λ} are called λ-lower cut set and strongλ-lower cut set of A, respectively. A = {x|x ∈ X,λ+ A(x) ≥ 1} and A ={x|x ∈ X,λ+ A(x) > 1} are calledλ-upper [λ] [λ] Q-cut set and strongλ-upper Q-cut set of A, respectively. [λ] [λ] A = {x|x ∈ X,λ+ A(x) ≤ 1} and A = {x|x ∈ X,λ+ A(x) < 1} are calledλ-lower Q-cut set and strongλ-lower Q-cut set of A, respectively. Deﬁnition 2.3 [18] If L is a completely distributive lattice with order-reversing in- volution mapping, then L is called an F-lattice. Fuzzy Inf. Eng. (2011) 2: 209-222 211 Deﬁnition 2.4 [14, 15, 17] Let X be a set and 2 represents the power set of X. Let H :[0, 1] → 2 be a mapping. (1) If (λ <λ ⇒ H(λ ) ⊃ H(λ )), then H is called an inverse order nested set 1 2 1 2 over X. (2) If (λ <λ ⇒ H(λ ) ⊂ H(λ )), then H is called an order nested set over X. 1 2 1 2 3. Interval-valued Level Cut Sets on Zadeh Fuzzy Set t t , a ]|0 ≤ a ≤ a ≤ 1},α = [a , a ],β = [b , b ] and α = [a , a ](t ∈ Let L = {[a 1 2 1 2 1 2 1 2 t 1 2 ¯ ¯ T ) ∈ L. We set in L: α ≤ β ⇔ a ≤ b , a ≤ b ,α ≺ β ⇔ a < b , a < b . 1 1 2 2 1 1 2 2 α<β ⇔ (a ≤ b , a < b )or(a < b , a ≤ b ). 1 1 2 2 1 1 2 2 t t t t α = [ a , a ], α = [ a , a ], t t 1 2 1 2 t∈T t∈T t∈T t∈T t∈T t∈T α = [1− a , 1− a ], 1 = [1, 1], 0 = [0, 0]. 2 1 Then (L,∨,∧, c, 1, 0) is an F-lattice and ¯ ¯ ¯ λ = ∨{α ∈ L| α ≺ λ} = ∨{α ∈ L|α<λ} = ∨{α ∈ L|α ≤ λ}. ¯ ¯ ¯ μ = ∧{α ∈ L| μ ≺ α} = ∧{α ∈ L|μ<α} = ∧{α ∈ L|μ ≤ α}. X X Let X be a set and 3 = {A|A : X →{0, , 1} is a mapping}. Then 3 is an F-lattice according to Zadeh’s operations. Let F (X) be a set of all Zadeh fuzzy subsets of X. Then we have the following deﬁnition. Deﬁnition 3.1 Let A∈F (X) andα = [a , a ] ∈ L. Then 1 2 (1) If A , A ∈ 3 and α α ⎧ ⎧ ⎪ ⎪ ⎪ 1, A(x) ≥ a , ⎪ 1, A(x) > a , 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎨ 1 A (x) = A (x) = α ⎪ , a ≤ A(x) < a , α ⎪ , a < A(x) ≤ a , 1 2 1 2 ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, A(x) < a , 0, A(x) ≤ a , 1 1 then A and A are called interval-valued level upper cut set and interval-valued α α level strong upper cut set of A, respectively. α α X (2) If A , A ∈ 3 and ⎧ ⎧ ⎪ ⎪ ⎪ 1, A(x) ≤ a , ⎪ 1, A(x) < a , 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎨ 1 α α A (x) = , a < A(x) ≤ a , A (x) = , a ≤ A(x) < a , ⎪ ⎪ 1 2 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, A(x) > a , 0, A(x) ≥ a , 2 2 α α then A and A are called interval-valued level lower cut set and interval-valued level strong lower cut set of A, respectively. 212 Xue-hai Yuan· Hong-xing Li · Kai-biao Sun (2011) (3) If A , A ∈ 3 and [α] [α] ⎧ ⎧ ⎪ ⎪ ⎪ 1, A(x)+ a ≥ 1, ⎪ 1, A(x)+ a > 1, 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎨ 1 A (x) = A (x) = [α] ⎪ , a < 1− A(x) ≤ a , [α] ⎪ , a ≤ 1− A(x) < a , 1 2 1 2 ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, A(x)+ a < 1, 0, A(x)+ a ≤ 1, 2 2 then A and A are called interval-valued level upper Q-cut set and interval-valued [α] [α] level strong upper Q-cut set of A, respectively. [α] [α] X (4) If A , A ∈ 3 and ⎧ ⎧ ⎪ ⎪ ⎪ 1, A(x)+ a ≤ 1, ⎪ 1, A(x)+ a < 1, 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎨ 1 [α] [α] A (x) = , a ≤ 1− A(x) < a , A (x) = , a < 1− A(x) ≤ a , ⎪ ⎪ 1 2 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, A(x)+ a > 1, 0, A(x)+ a ≥ 1, 1 1 [α] [α] then A and A are called interval-valued level lower Q-cut set and interval-valued level strong lower Q-cut set of A, respectively. Remark 3.1 It is shown that in Deﬁnition 3.1(1) if an interval-valued level [a , a ]is 1 2 taken as a threshold value of decision making, then we should consider the following cases: (a) We should consider x as a qualiﬁed element at A(x) ≥ a ; (b) We should not consider x as a qualiﬁed element at A(x) < a ; (c) Degree of x as a qualiﬁed element should be uncertainty at a ≤ A(x) < a . 1 2 We have the following properties. Property 3.1 (1) A ⊂ A . [α] α (2)α ≤ β ⇒ A ⊃ A , A ⊃ A ;α ≺ β ⇒ A ⊃ A . α β α β α β (3) A ⊂ B ⇒ A ⊂ B , A ⊂ B . α α α α c c c c (4) (A ) = (A c) , (A ) = (A c) . α α α α (5) (A∪ B) = A ∪ B , (A∪ B) = A ∪ B , (A∩ B) = A ∩ B , α α α α α α α α α (A∩ B) = A ∩ B . α α α (6) (A ) ⊂ ( A ) , (A ) = ( A ) , (A ) = ( A ) , t α t α t α t α t α t α t∈T t∈T t∈T t∈T t∈T t∈T (A ) ⊂ ( A ) . t α t α t∈T t∈T (7) A = A ∩ A , A = A ∩ A , A = A ∪ A , A = A ∪ A . α∨β α β α∨β α β α∧β α β α∧β α β (8) Let α ∈ L,α = α ,β = α . Then t t t t∈T t∈T A = A , A ⊂ A , A ⊃ A , A = A . β α β α α α α α t t t t t∈T t∈T t∈T t∈T In general, A = A = A , A = A = A . λ α α λ α α α≺λ α<λ λ≺α λ<α Fuzzy Inf. Eng. (2011) 2: 209-222 213 (9) A = X, A = ∅. 0 1 c c Proof (a) (A ) = (A c) . α α In fact, letα = [a , a ]. Then 1 2 c c (A ) (x) = 1 ⇔ 1 − A(x) ≥ a ⇔ A(x) ≤ 1 − a ⇔ A c(x) = 0 ⇔ (A c) (x) = α 2 2 α α 1− A (x) = 1. c c (A ) (x) = 0 ⇔ 1 − A(x) < a ⇔ A(x) > 1 − a ⇔ A c(x) = 1 ⇔ (A c) (x) = α 1 1 α α 1− A (x) = 0. c c X c c By (A ) , (A c) ∈ 3 , we have that (A ) = (A c) . α α α α (b) (A ) ⊂ ( A ) . t α t α t∈T t∈T In fact, ( (A ) )(x) = 1 ⇒∃t ∈ T, (A ) )(x) = 1 ⇒∃t ∈ T , (A )(x) ≥ a ⇒ t α t α t 2 t∈T (A )(x) ≥ a ⇒ ( A ) (x) = 1. t 2 t α t∈T t∈T ( A ) (x) = 0 ⇒ A (x) < a ⇒∀t ∈ T, (A )(x) < a ⇒∀t ∈ T, (A ) (x) = t α t 1 t 1 t α t∈T t∈T 0 ⇒ ( (A ) )(x) = 0. t α t∈T By (A ) ,( A ) ∈ 3 , we have that (A ) ⊂ ( A ) . t α t α t α t α t∈T t∈T t∈T t∈T (c) A ⊂ A . β α t∈T In fact, A (x) = 1 ⇒ A(x) > a ⇒∀t ∈ T, A(x) > a ⇒∀t ∈ T, (A )(x) = β α 2 t t∈T 1 ⇒ ( A )(x) = 1. t∈T t t ( A )(x) = 0 ⇒∃t ∈ T, A (x) = 0 ⇒∃t ∈ T, A(x) ≤ a ⇒ A(x) ≤ a ⇒ α α t t 1 1 t∈T t∈T A (x) = 0. By A , A ∈ 3 , we have that A ⊂ A . Proofs of others are similar. β α β α t t t∈T t∈T Obviously, we have the following properties. α α Property 3.2 (1) A ⊂ A . α β α β α (2)α<β ⇒ A ⊂ A , A ⊂ A ,α ≺ β ⇒ A ⊃ A . α α α α (3) A ⊂ B ⇒ B ⊂ A , B ⊂ A . c c c α α c c α α c (4) (A ) = (A ) , (A ) = (A ) . α α α α α α α α α (5) (A∪ B) = A ∩ B , (A∪ B) = A ∩ B , (A∩ B) = A ∪ B , α α α (A∩ B) = A ∪ B . α α α α α α (6) ( A ) = (A ) , ( A ) ⊂ (A ) , ( A ) ⊃ (A ) , t t t t t t t∈T t∈T t∈T t∈T t∈T t∈T α α ( A ) = (A ) . t t t∈T t∈T α∨β β α∧β β α∨β α β α α∧β α β α (7) A = A ∪ A , A = A ∪ A , A = A ∩ A , A = A ∩ A . 214 Xue-hai Yuan· Hong-xing Li · Kai-biao Sun (2011) (8) Let α = α ,β = α . Then t t t∈T t∈T β α α α α α α t t t A ⊃ A , A = A , A = A , A ⊃ A . t∈T t∈T t∈T t∈T In general, λ α α λ α α A = A = A , A = A = A . λ≺α λ<α α≺λ α<λ 0 1 (9) A = ∅, A = X. Property 3.3 (1) A ⊂ A . [α] [α] (2)α<β ⇒ A ⊂ A , A ⊂ A ,α ≺ β ⇒ A ⊂ A . [α] [β] [α] [β] [α] [β] (3) A ⊂ B ⇒ A ⊂ B , A ⊂ B . [α] [α] [α] [α] c c c c (4) (A ) = (A c ) , (A ) = (A ) . [α] [α ] [α] [α ] (5) (A∪ B) = A ∪ B , (A∪ B) = A ∪ B , (A∩ B) = A ∩ B , [α] [α] [α] [α] [α] [α] [α] [α] [α] (A∩ B) = A ∩ B . [α] [α] [α] (6) ( A ) ⊃ (A ) , ( A ) = (A ) , ( A ) = (A ) , t [α] t [α] t [α] t [α] t [α] t [α] t∈T t∈T t∈T t∈T t∈T t∈T ( A ) ⊂ (A ) . t [α] t [α] t∈T t∈T (7) A = A ∪A , A = A ∪A , A = A ∩A , A = A ∩A . [α∨β] [α] [β] [α∨β] [α] [β] [α∧β] [α] [β] [α∧β] [α] [β] (8) Let α = α ,β = α . Then t t t∈T t∈T A = A , A ⊂ A , A ⊃ A , A = A . [α] [α ] [α] [α ] [β] [α ] [β] [α ] t t t t t∈T t∈T t∈T t∈T In general, A = A = A , A = A = A . [λ] [α] [α] [λ] [α] [α] λ≺α λ<α α≺λ α<λ [0] (9) A = ∅, A = X. [1] [α] [α] Property 3.4 (1) A ⊂ A . [β] [α] [β] [α] [β] [α] (2)α<β ⇒ A ⊃ A , A ⊃ A ,α ≺ β ⇒ A ⊂ A . [α] [α] [α] [α] (3) A ⊂ B ⇒ B ⊂ A , B ⊂ A . c c c [α] [α ] c c [α] [α ] c (4) (A ) = (A ) , (A ) = (A ) . [α] [α] [α] [α] [α] [α] [α] [α] [α] (5) (A∪ B) = A ∩ B , (A∪ B) = A ∩ B , (A∩ B) = A ∪ B , [α] [α] (A∩ B) = A ∪ B . [α] [α] [α] [α] [α] [α] [α] (6) ( A ) = (A ) , ( A ) ⊂ (A ) , ( A ) ⊃ (A ) , t t t t t t t∈T t∈T t∈T t∈T t∈T t∈T [α] [α] ( A ) = (A ) . t t t∈T t∈T [α∨β] [β] [α∧β] [β] [α∨β] [α] [β] [α] [α∧β] [α] [β] [α] (7) A = A ∩A , A = A ∩A , A = A ∪A , A = A ∪A . (8) Let α = α ,β = α . Then t t t∈T t∈T [α ] [β] [α ] [β] [α] [α] [α ] [α ] t t t t t A = A , A ⊃ A , A ⊂ A , A = A . t∈T t∈T t∈T t∈T Generally, [λ] [α] [α] [λ] [α] [α] A = A = A , A = A = A . α≺λ α<λ λ≺α λ<α Fuzzy Inf. Eng. (2011) 2: 209-222 215 [0] [1] (9) A = X, A = ∅. Remark 3.2 (1) From the discussions shown as above we can see that each kind of interval-valued level cut set has completely similar properties. (2) From the results in [14, 15, 17], we know that the interval-valued level cut sets on Zadeh fuzzy sets have the same properties as those of normal cut sets on them. 4. Decomposition Theorems of Zadeh Fuzzy Set Based on Interval-valued Level Cut Sets We ﬁrst present deﬁnition as follows. Deﬁnition 4.1 Let f : L× 3 →F (X) be mapping(i = 1, 2,···, 8).Forα = [a , a ] ∈ i 1 2 L, A ∈ 3 ,f satisﬁes ⎧ ⎧ ⎪ ⎪ ⎪ a , A(x) = 1, ⎪ 1, A(x) = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎨ 1 f (α, A)(x) = f (α, A)(x) = 1 ⎪ a , A(x) = , 2 ⎪ a , A(x) = , 1 2 ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, A(x) = 0, a , A(x) = 0, ⎧ ⎧ ⎪ ⎪ ⎪ 0, A(x) = 1, ⎪ a , A(x) = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎨ 1 f (α, A)(x) = a , A(x) = , f (α, A)(x) = a , A(x) = , 3 ⎪ 4 ⎪ 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ a , A(x) = 0, 1, A(x) = 0, ⎧ ⎧ ⎪ ⎪ 1− a , A(x) = 1, 1, A(x) = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎨ 1 f (α, A)(x) = f (α, A)(x) = ⎪ 1− a , A(x) = , ⎪ 1− a , A(x) = , 5 6 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, A(x) = 0, 1− a , A(x) = 0, ⎧ ⎧ ⎪ ⎪ ⎪ 0, A(x) = 1, ⎪ 1− a , A(x) = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎨ 1 f (α, A)(x) = f (α, A)(x) = ⎪ 1− a , A(x) = , ⎪ 1− a , A(x) = , 7 2 8 1 ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 1− a , A(x) = 0, 1, A(x) = 0. Then we have the following decomposition theorems. Theorem 4.1 Let A∈F (X). Then (1) A = f (α, A ) = f (α, A ); 1 α 2 α ¯ ¯ α∈L α∈L (2) A = f (α, A ) = f (α, A ); 1 α 2 α ¯ ¯ α∈L α∈L (3) Let H : L → 3 be a mapping such that A ⊂ H(α) ⊂ A . Then α α (I) A = f (α, H(α)) = f (α, H(α); 1 2 ¯ ¯ α∈L α∈L (II)α ≺ β ⇒ H(α) ⊃ H(β); (III) A = H(α), A = H(α). λ λ α≺λ λ≺α 216 Xue-hai Yuan· Hong-xing Li · Kai-biao Sun (2011) Proof (1) ( f (α, A ))(x) = f (α, A )(x)=(∨{a |α = [a , a ] ∈ L, 1 α 1 α 2 1 2 ¯ ¯ α∈L α∈L ¯ ¯ A (x) = 1}) ∨ (∨{b |β = [b , b ] ∈ L, A (x) = })=(∨{a |α = [a , a ] ∈ L, A(x) ≥ α 1 1 2 β 2 1 2 a })∨ (∨{b |β = [b , b ] ∈ L, b ≤ A(x) < b }) = A(x). Then 2 1 1 2 1 2 A = f (α, A ). 1 α α∈L Similarly, we have ( f (α, A ))(x) = f (α, A )(x)=(∧{a |α = [a , a ] 2 α 2 α 1 1 2 ¯ ¯ α∈L α∈L ¯ ¯ ∈ L, A(x) < a })∧ (∧{b |β = [b , b ] ∈ L, b ≤ A(x) < b }) = A(x). 1 2 1 2 1 2 Then A = f (α, A ). 2 α α∈L Proof of (2) is similar. (3) (I) By A ⊂ H(α) ⊂ A , we have the following. α α f (α, A ) ⊂ f (α, H(α)) ⊂ f (α, A ), f (α, A ) ⊂ f (α, H(α)) ⊂ f (α, A ). 1 α 1 1 α 2 α 2 2 α Then A = f (α, A ) ⊂ f (α, H(α)) ⊂ f (α, A ) = A, 1 α 1 1 α ¯ ¯ ¯ α∈L α∈L α∈L A = f (α, A ) ⊂ f (α, H(α)) ⊂ f (α, A ) = A. 2 α 2 2 α ¯ ¯ ¯ α∈L α∈L α∈L Thus A = f (α, H(α)) = f (α, H(α). 1 2 ¯ ¯ α∈L α∈L (II)α ≺ β ⇒ H(α) ⊃ A ⊃ A ⊃ H(β). α β (III) At α ≺ λ,we have H(α) ⊃ A ⊃ A . It follows that H(α) ⊃ A . On the α λ λ α≺λ other hand, from H(α) ⊂ A and Property 3.1(8), we have the next. H(α) ⊂ A = A . Thus A = H(α). α λ λ α≺λ α≺λ α≺λ Similarly, we have A = H(α). λ≺α Obviously, we have the theorems as follows. Theorem 4.2 Let A∈F (X). Then α α (1) A = f (α, A ) = f (α, A ); 3 4 ¯ ¯ α∈L α∈L α α (2) A = f (α, A ) = f (α, A ); 3 4 ¯ ¯ α∈L α∈L X α α (3) Let H : L → 3 be a mapping such that A ⊂ H(α) ⊂ A . Then (I) A = f (α, H(α)) = f (α, H(α); 3 4 ¯ ¯ α∈L α∈L (II)α ≺ β ⇒ H(α) ⊂ H(β); (III) A = H(α), A = H(α). λ≺α α≺λ Theorem 4.3 Let A∈F (X). Then Fuzzy Inf. Eng. (2011) 2: 209-222 217 (1) A = f (α, A ) = f (α, A ); 5 [α] 6 [α] ¯ ¯ α∈L α∈L (2) A = f (α, A ) = f (α, A ]); 5 [α] 6 [α ¯ ¯ α∈L α∈L (3) Let H : L → 3 be a mapping such that A ⊂ H(α) ⊂ A . Then [α] [α] (I) A = f (α, H(α)) = f (α, H(α); 5 6 ¯ ¯ α∈L α∈L (II)α ≺ β ⇒ H(α) ⊂ H(β); (III) A = H(α), A = H(α). [λ] [λ] λ≺α α≺λ Theorem 4.4 Let A∈F (X). Then [α] [α] (1) A = f (α, A ) = f (α, A ); 7 8 ¯ ¯ α∈L α∈L [α] [α] (2) A = f (α, A ) = f (α, A ); 7 8 ¯ ¯ α∈L α∈L X [α] [α] (3) Let H : L → 3 be a mapping such that A ⊂ H(α) ⊂ A . Then (I) A = f (α, H(α)) = f (α, H(α); 7 8 ¯ ¯ α∈L α∈L (II)α ≺ β ⇒ H(α) ⊃ H(β); [λ] [λ] (III) A = H(α), A = H(α). α≺λ λ≺α Remark 4.1 (1) From Theorem 4.1-Theorem 4.4, we know that the new decompo- sition theorems of Zadeh fuzzy sets have been established based on interval-valued level cut sets. (2) If we use αA and α◦ A to denote f (α, A) and f (α, A) respectively, i.e., αA 1 2 f (α, A),α◦ A f (α, A), then Theorem 4.1 can be rewritten as: 1 2 A = αA = α◦ A, A = αH(α) = α◦ H(α), ¯ ¯ ¯ ¯ α∈L α∈L α∈L α∈L which are consistent with the normal decomposition theorems of Zadeh fuzzy sets [14, 15, 17]. (3) Theorem 4.2-Theorem 4.4 can be expressed in the same forms. Therefore we can conclude that each kind of interval-valued level cut set correspondences to two decomposition theorems. 5. Representation Theorems of Zadeh Fuzzy Set Based on Interval-valued Level Cut Sets We ﬁrst give the following deﬁnitions. Deﬁnition 5.1 Let H : L → 3 be a mapping. We set T (H)∈F (X) and T (H) = f (α, H(α))(i = 1, 3, 5, 7); T (H) = f (α, H(α))(i = 2, 4, 6, 8). i i i i ¯ ¯ α∈L α∈L Deﬁnition 5.2 Let H : L → 3 be a mapping. If α ≺ β ⇒ H(α) ⊃ H(β), then H is called a three-valued inverse order nested set over X. 218 Xue-hai Yuan· Hong-xing Li · Kai-biao Sun (2011) Ifα ≺ β ⇒ H(α) ⊂ H(β), then H is called a three-valued order nested set over X. Lemma 5.1 Let H : L → 3 be a mapping. Then (1) T (H) = T (H)⇔∃A∈F (X) such that A ⊂ H(α) ⊂ A ,∀α ∈ L. 1 2 α α α α (2) T (H) = T (H)⇔∃A∈F (X) such that A ⊂ H(α) ⊂ A ,∀α ∈ L. 3 4 (3) T (H) = T (H)⇔∃A∈F (X) such that A ⊂ H(α) ⊂ A ,∀α ∈ L. 5 6 [α] [α] [α] [α] (4) T (H) = T (H)⇔∃A∈F (X) such that A ⊂ H(α) ⊂ A ,∀α ∈ L. 7 8 Proof (1) “⇐” is clear. “⇒” Let A = T (H) = T (H). Then 1 2 ¯ ¯ A(x) = (∨{a |α = [a , a ] ∈ L, H(α)(x) = 1})∨ (∨{b | β = [b , b ] ∈ L, 2 1 2 1 1 2 H(β)(x) = }), A(x) = (∧{a |α = [a , a ] ∈ L, H(α)(x) = 0})∧ 1 1 2 (∧{b | β = [b , b ] ∈ L, H(β)(x) = }). 2 1 2 Let α = [a , a ] ∈ L.At H(α)(x) = 1, we have A(x) ≥ a . Thus A (x) = 1. At 1 2 2 α A (x) = 0, we have A(x) < a . It follows that H(α)(x) = 0. By A , H(α) ∈ 3 ,we α 1 α have H(α) ⊂ A . Similarly, we have A ⊂ H(α). Therefore A ⊂ H(α) ⊂ A ,∀α ∈ L. α α Similar are proofs of others . Remark 5.1 Let H :[0, 1] → 2 be a mapping and T (H) = aH(a), T (H) = a◦ H(a). 1 2 a∈[0,1] a∈[0,1] Then we have the following equivalent conditions from [14, 15, 17]. Condition 1: H is an inverse order nested set, i.e., a < b ⇒ H(a) ⊃ H(b); Condition 2: T (H) = T (H); 1 2 Condition 3: ∃A∈F (X) such that A ⊂ H(α) ⊂ A . α α Therefore, the class U(X) of them in paper [15, 17] satisﬁes the following condi- tion: U(X) ={H | The mapping H :[0, 1] → 2 is an inverse order nested set} ={H | The mapping H :[0, 1] → 2 satisﬁes T (H) = T (H)}. 1 2 However, the following example will show that Condition 1 is not equivalent to the Condition 2 in this paper. Example5.1 Let X = {x }.Forα = [a , a ], we set H(α) as follows: 1 2 ⎪ ≤ , 1, a ⎪ 2 ⎪ 2 1 1 H(α)(x) = , < a < 1, 2 2 0, a = 1. 2 Fuzzy Inf. Eng. (2011) 2: 209-222 219 Then H is a three-valued inverse order nested set, but T (H)(x) = 1, T (H)(x) = 0. 1 2 Therefore, before establishing representation theorems, we describe U(X) by use of the Condition 2 as above. Thus we set U (X) = {H|H : L → 3 is a mapping and T (H) = T (H)}. i j Remark 5.2 By Lemma 5.1 and Theorem 4.1-Theorem 4.4, we have the following: 2 8 H ∈U (X)(or H ∈U (X)) ⇒ H is a three-valued inverse order nested set over X; 1 7 4 6 H ∈U (X)(or H ∈U (X)) ⇒ H is a three-valued order nested set over X. 3 5 2 8 We set operations in U (X) and U (X) as follows: 1 7 H :( H )(α) = H (α); H :( H )(α) = H (α), γ γ γ γ γ γ γ∈Γ γ∈Γ γ∈Γ γ∈Γ γ∈Γ γ∈Γ c c c c ¯ ¯ ¯ H : H (α) = (H(α )) ; X(α) ≡ X,∅(α)≡∅,∀α ∈ L, 4 6 and operations inU (X) andU (X) as follows: 3 5 H :( H )(α) = H (α); H :( H )(α) = H (α), γ γ γ γ γ γ γ∈Γ γ∈Γ γ∈Γ γ∈Γ γ∈Γ γ∈Γ c c c c H : H (α) = (H(α )) ; X(α) = ∅,∅(α) = X,∀α ∈ [0, 1],∀α ∈ L. Then we have the next lemma. ¯ ¯ Lemma 5.2 (1) (U (X), , , c, X,∅)(i = 1, j = 2 or i = 7, j = 8) is an F-lattice. (2) (U (X), , , c, X,∅)(i = 3, j = 4 or i = 5, j = 6) is an F-lattice. Proof Let H ∈U (X)(γ ∈ Γ). Then T (H ) = T (H ). Thus there exists A ∈ γ 1 γ 2 γ γ F (X)(γ ∈ Γ) such that (A ) ⊂ H (α) ⊂ (A ) ,∀α ∈ L. By Property 3.1(6), we have γ α γ γ α the next. ( A ) = (A ) ⊂ H (α) ⊂ (A ) ⊂ ( A ) , γ α γ α γ γ α γ α γ∈Γ γ∈Γ γ∈Γ γ∈Γ γ∈Γ ( A ) ⊂ (A ) ⊂ H (α) ⊂ (A ) = ( A ) . γ α γ α γ γ α γ α γ∈Γ γ∈Γ γ∈Γ γ∈Γ γ∈Γ 2 2 Thus H ∈U (X), H ∈U (X). γ γ 1 1 γ∈Γ γ∈Γ Let H ∈U (X). Then T (H) = T (H). Then there exists A∈F (X) such that 1 2 A ⊂ H(α) ⊂ A ,∀α ∈ L. α α c c c c c c c c 2 Then (A ) = (A c) ⊂ (H )(α) = (H(α )) ⊂ (A c) = (A ) . Thus H ∈U (X). α α α α From the discussions as above, we can easily show that (U (X), , , c, X,∅)is an F-lattice. 220 Xue-hai Yuan· Hong-xing Li · Kai-biao Sun (2011) Proofs of others are clear. From Lemma 5.2, we easily obtain the theorems below. Theorem 5.1 Let T : U (X)→F (X) be a mapping (i=1,2) as follows, T (H) = f (α, H(α); T (H) = f (α, H(α)). 1 1 2 2 ¯ ¯ α∈L α∈L Then (1) T (T ) is surjection; 1 2 (2) T (H) = H(α), T (H) = H(α); 1 λ 1 λ α≺λ λ≺α c c (3) T ( H ) = T (H ); T ( H ) = T (H ); T (H ) = (T (H)) . 1 γ 1 γ 1 γ 1 γ 1 1 γ∈Γ γ∈Γ γ∈Γ γ∈Γ Proof (1) Let A ∈F (X) and H(α) = A ,∀α ∈ L. Then T (H) = A. From Theorem α 1 4.1, we know that (2) is obvious. (3) Let H ∈U (X)(γ ∈ Γ). By Theorem 4.1 and Property 3.1, we obtain the next T ( H ) = ( H )(α) = H (α) 1 γ λ γ γ γ∈Γ λ≺α γ∈Γ λ≺α γ∈Γ = H (α) = T (H ) = ( T (H )) , γ 1 γ λ 1 γ λ λ≺α γ∈Γ γ∈Γ γ∈Γ T ( H ) = ( H )(α) = H (α) 1 γ λ γ γ γ∈Γ α≺λ γ∈Γ α≺λ γ∈Γ = H (α) = T (H ) = ( T (H )) . γ 1 γ λ 1 γ λ γ∈Γ α≺λ γ∈Γ γ∈Γ From Theorem 4.1, we have T ( H ) = T (H ), T ( H ) = T (H ). 1 γ 1 γ 1 γ 1 γ γ∈Γ γ∈Γ γ∈Γ γ∈Γ Similarly, c c c c c c T (H ) = H (α) = (H(α ) ) = ( H(α )) 1 λ c c α≺λ α≺λ λ ≺α c c c = ( H(α)) = (T (H) ) = (T (H) ) . 1 λ 1 λ λ ≺α c c It follows that T (H ) = (T (H)) . 1 1 Corollary 5.1 We set inU (X): H ∼ H ⇔ T (H ) = T (H ). Then∼ is an equiva- 1 2 1 1 1 2 2 2 lent relation overU (X) and factor setU (X)/ ∼ is isomorphic withF (X). 1 1 Therefore, a Zadeh fuzzy set can also be seen as an equivalent class of an L-inverse order nested set. Similarly, we have the following theorems: Theorem 5.2 Let T : U (X)→F (X) be a mapping (i=3,4) as follows, T (H) = f (α, H(α); T (H) = f (α, H(α)). 3 3 4 4 ¯ ¯ α∈L α∈L Fuzzy Inf. Eng. (2011) 2: 209-222 221 Then (1) T (T ) is surjection; 3 4 (2) T (H) = H(α) = H(α), T (H) = H(α) = H(α); 3 3 λ≺α λ<α α≺λ α<λ c c (3) T ( H ) = T (H ); T ( H ) = T (H ); T (H ) = (T (H)) . 3 γ 3 γ 3 γ 3 γ 3 3 γ∈Γ γ∈Γ γ∈Γ γ∈Γ Theorem 5.3 Let T : U (X)→F (X) be a mapping (i=5,6) as follows, T (H) = f (α, H(α); T (H) = f (α, H(α)). 5 5 6 6 ¯ ¯ α∈L α∈L Then (1) T (T ) is surjection; 5 6 (2) T (H) = H(α), T (H) = H(α); 5 [λ] 5 [λ] λ≺α α≺λ c c (3) T ( H ) = T (H ); T ( H ) = T (H ); T (H ) = (T (H)) . 5 γ 5 γ 5 γ 5 γ 5 5 γ∈Γ γ∈Γ γ∈Γ γ∈Γ Theorem 5.4 Let T : U (X)→F (X) be a mapping (i=7,8) as follows, T (H) = f (α, H(α); T (H) = f (α, H(α)). 7 7 8 8 ¯ ¯ α∈L α∈L Then (1) T (T ) is surjection; 7 8 [λ] [λ] (2) T (H) = H(α) = H(α), T (H) = H(α) = H(α); 7 7 α≺λ α<λ λ≺α λ<α c c (3) T ( H ) = T (H ); T ( H ) = T (H ); T (H ) = (T (H)) . 7 γ 7 γ 7 γ 7 γ 7 7 γ∈Γ γ∈Γ γ∈Γ γ∈Γ Remark 5.3 From Theorem 5.1-Theorem 5.4, we know that the representation the- orems of Zadeh fuzzy sets have been established based on interval-valued level cut sets. 7. Conclusion In this paper, the concepts of interval-valued level cut sets are presented on Zadeh fuzzy set. Besides, we have shown that new cut sets have the same properties as those normal cut sets of Zadeh fuzzy set. Finally, based on these new cut sets, new decomposition theorems and new representation theorems of Zadeh fuzzy set are established. These jobs have established connections between three-valued fuzzy set and Zadeh fuzzy set. Acknowledgements This research is supported in part by National Natural Science Foundation of China (No.90818025, No.61074044). References 1. Mordeson J N, Bhutani K R, Rosenfeld A (2005) Fuzzy group theory. Springer, New York 222 Xue-hai Yuan· Hong-xing Li · Kai-biao Sun (2011) 2. Seselja B, Tepavcevic A (2003) Completion of ordered structures by cuts of fuzzy sets: an overview. Fuzzy Sets and Systems 136: 1-19 3. Lai Y J, Hwang C L (1992) Fuzzy mathematical programming-methods and applications. Springer- verlag, Berlin 4. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Jun 1, 2011
Keywords: Fuzzy sets; Interval-valued level cut sets; Decomposition theorems; Representation theorems