Arithmetic operations of non-normal fuzzy sets using gradual numbers

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Arithmetic operations of non-normal fuzzy sets using gradual numbers Hsien-Chung Wu Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan Received 15 March 2019; received in revised form 14 August 2019; accepted 9 October 2019

Abstract This paper will study the arithmetic operations of non-normal fuzzy sets using the concept of gradual numbers that can be regarded as the elements of gradual sets. We shall present the idea in which the fuzzy sets can be formulated as consisting of gradual elements like the usual set consisting of usual elements. When the universal set is taken to be the real number system, the gradual element is also called a gradual number. In this case, the arithmetic operations of non-normal fuzzy sets can be defined by using the gradual numbers. © 2019 Elsevier B.V. All rights reserved. Keywords: Decomposition theorem; Gradual elements; Gradual numbers; Gradual sets; Normal fuzzy sets

1. Introduction Let A be a (crisp) subset of a universal set U . The concept of element in A is well-known by writing x ∈ A. ˜ The Suppose that A˜ is a fuzzy set in U . The main focus of this paper is to consider the elements of fuzzy set A. ˜ gradual elements will play the role to be the elements of A. The motivations are presented below. ˜ • Given any fuzzy set A˜ in R, we are going to propose the concept of belongingness in fuzzy sets like writing aˆ ∈ A, where aˆ will be interpreted as gradual numbers in this paper. This interpretation will be based on a consistent definition. • In the topic of set-valued analysis, given any two subsets A and B of R, the arithmetic A ◦ B is defined by A ◦ B = {a ◦ b : a ∈ A and b ∈ B} , where ◦ ∈ {+, −, ×, /}. The division should avoid dividing by zero. In this paper, given any two fuzzy sets A˜ and B˜ in R, we are going to similarly study the arithmetic operation A˜  B˜ based on the following family   aˆ ◦ bˆ : aˆ ∈ A˜ and bˆ ∈ B˜ , E-mail address: [email protected]. https://doi.org/10.1016/j.fss.2019.10.004 0165-0114/© 2019 Elsevier B.V. All rights reserved.

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where the operation ◦ ∈ {+, −, ×, /} correspond to the operation  ∈ {⊕, , ⊗, }, and aˆ and bˆ are real-valued functions. The concepts of gradual elements and gradual sets based on a universal set U were introduced by Dubois and Prade [4] and Fortin et al. [6], which were inspired by Goetschel [8] and Herencia and Lamata [9]. The gradual element is a function from (0, 1] into U , and the gradual set is a set-valued function from (0, 1] into the hyperspace that consists of all subsets of U . When the universal set U is taken to be R, the gradual element is also called gradual numbers. Boukezzoula et al. [1] used gradual numbers to define the so-called gradual intervals in which the endpoints are assumed to be gradual numbers. More motivated argument can also refer to Dubois and Prade [5]. Let U be a universal set. We denote by P(U ) the collection of all subsets of U . Dubois and Prade [4] considered the gradual set G defined by an assignment function AG : (0, 1] → P(U ) that does not consider the assignment at 0, where the assignment function AG is a set-valued function. The gradual set G with assignment function AG can induce a fuzzy set F (G) with membership function given by μF (G) (x) = sup α · χAG (α) (x),

(1)

α∈(0,1]

where χAG (α) is a characteristic function given by  1 if x ∈ AG (α) χAG (α) (x) = 0 otherwise. Sanchez et al. [16] considered the fuzzy concept A defined by an ordered pair (A , ρA ), where A = {1 = α1 , α2 , · · · , αm = 0} is a finite subset of the unit interval [0, 1] satisfying α1 > α2 > · · · > αm and ρA is a set-valued function ρA : A → P(U ) defined on the finite set A . In this paper, we shall consider the set-valued function G : I → P(U ) from I into P(U ), where I is any subset of [0, 1]. This set-valued function G will also be called as a gradual set (or an extended gradual set). The set-valued function G will cover AG and ρA . • In Dubois and Prade [4], let G1 and G2 be two gradual sets. The intersection and union of G1 and G2 are defined by the assignment functions AG1 ∪G2 (α) = AG1 (α) ∪ AG2 (α) and AG1 ∩G2 (α) = AG1 (α) ∩ AG2 (α). The gradual sets G1 ∪ G2 and G1 ∩ G2 can induce two fuzzy sets F (G1 ∪ G2 ) and F (G1 ∩ G2 ) with membership functions μF (G1 ∪G2 ) and μF (G1 ∩G2 ) , respectively, according to (1). Dubois and Prade [4] also claim that   μF (G1 ∪G2 ) (x) = max μF (G1 ) (x), μF (G2 ) (x) and   μF (G1 ∩G2 ) (x) = min μF (G1 ) (x), μF (G2 ) (x) . • In Sanchez et al. [16], let A and B be two fuzzy concepts with (A , ρA ) and (B , ρB ). The new fuzzy concepts A ∧ B and A ∨ B are defined by A∧B = A∨B = A ∪ B and ρA∧B (α) = ρA (α) ∩ ρB (α) and ρA∨B (α) = ρA (α) ∪ ρB (α), which are similar to the approach of Dubois and Prade [4] by regarding the fuzzy concept A ∧ B as the intersection and the fuzzy concept A ∨ B as the union. Let A˜ and B˜ be two fuzzy subsets of U with membership functions ξA˜ and ξB˜ , respectively. The usual intersection and union of A˜ and B˜ are defined using the min and max functions as follows     ξA∩ (2) ˜ B˜ (x) = min ξA˜ (x), ξB˜ (x) and ξA∪ ˜ B˜ (x) = max ξA˜ (x), ξB˜ (x) .

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The generalization for considering t-norm and t-conorm can refer to Dubois and Prade [2] and Weber [18]. On the other hand, Tan et al. [17] proposed a different generalization for intersection and union of fuzzy sets. The alternative definitions for the intersection and union of fuzzy sets are also widely discussed in the literature by referring to Yager [20,21]. Klement [10] considered the axiomatic approach for operations on fuzzy sets. More detailed properties can refer to the monographs Dubois and Prade [3] and Klir and Yuan [11]. In this paper, we are not going to consider the intersection and union of gradual sets as in Dubois and Prade [4] and Sanchez et al. [16] presented above. We shall study the arithmetic operations ⊕, , ⊗,  of fuzzy sets based on the concepts of gradual sets and gradual numbers. In order to claim the consistency for considering gradual elements in fuzzy sets, a more general decomposition theorem will be established, where the basic properties for decomposition theorem can refer to Fullér and Keresztfalvi [7], Negoita and Ralescu [12], Nguyen [13], Ralescu [14] and Wu [19]. Let A be a subset of U . The element x in A is simply written as x ∈ A. Now we assume that A˜ is a fuzzy subset ˜ In other words, we want to define an element xˆ such that we can of U . The purpose is to consider the elements of A. ˜ Under some suitable settings, we shall see that the element xˆ in A˜ is also a gradual element. reasonably write xˆ ∈ A. When the universal set U is taken to be R, the gradual element is also called a gradual number. In this case, the arithmetic operations ⊕, , ⊗,  of fuzzy sets will be defined by considering the arithmetic operations +, −, ×, / of gradual numbers, which is the main purpose of this paper. In section 2, we present some basic properties of non-normal fuzzy set. On the other hand, a generalized decomposition theorem regarding the non-normal fuzzy set is also established, which will be useful for showing the consistency of gradual elements in fuzzy sets. In section 3, we present the concepts of gradual elements and gradual sets, and study the relationships with fuzzy sets. In section 4, we study the arithmetic operations of fuzzy sets using the concept of gradual numbers. 2. Ranges of non-normal fuzzy sets Let A˜ be a fuzzy subset of a universal set U with membership function denoted by μA˜ . For α ∈ (0, 1], the α-level set of A˜ is denoted and defined by   (3) A˜ α = x ∈ U : μA˜ (x) ≥ α . We also define   A˜ α+ = x ∈ U : μA˜ (x) > α . It is clear to see that if α is larger than the supremum of the membership function μA˜ then the α-level set A˜ α is an empty set. The support of a fuzzy set A˜ is the crisp set defined by A˜ 0+ = {x ∈ U : μA˜ (x) > 0}. We say that A˜ is a normal fuzzy set in U if and only if there exists x ∈ U such that μA˜ (x) = 1. The definition of 0-level set is an important issue in fuzzy sets theory. If the universal set U is endowed with a ˜ i.e., topology τ , then the 0-level set A˜ 0 can be defined as the closure of the support of A,   (4) A˜ 0 = cl A˜ 0+ . If U is not endowed with a topological structure, then the intuitive way for defining the 0-level set is to follow the equality (3) for α = 0. In this case, the 0-level set of A˜ is the whole universal set U . This kind of 0-level set seems not so useful. Therefore, we always endow a topological structure to the universal set U when the 0-level set should be seriously considered. The range of membership function μA˜ is denoted by R(μA˜ ) that is a subset of [0, 1]. We see that the range R(μA˜ ) can be a proper subset of [0, 1] with R(μA˜ ) = [0, 1]. Even though A˜ is a normal fuzzy set in U , it is not necessarily that we must have R(μA˜ ) = [0, 1]. In other words, there may exist α ∈ [0, 1] such that the membership degree μA˜ (x) = α for all x ∈ U . Notice that if α ∈ / R(μA˜ ), we still can consider the α-level set A˜ α . Since R(μA˜ ) = [0, 1], it is possible that the ˜ α-level set Aα can be an empty set for some α ∈ [0, 1]. Therefore, when we study the properties that deal with more

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than two fuzzy sets, we cannot simply present the properties by saying that they hold true for each α ∈ [0, 1], since some of the α-level sets can be empty. In this case, we need to carefully treat the ranges of membership functions. The following useful lemma will be used in the subsequent discussion. Lemma 2.1. (Royden [15, p. 161]) Let U be a topological space, and let K be a compact subset of U . Let f be a real-valued function defined on U . If f is upper semi-continuous, then f assumes its maximum on a compact subset of U ; that is, the supremum is attained in the following sense sup f (x) = max f (x). x∈K

x∈K

Remark 2.2. Let A˜ be a fuzzy set in U with membership function μA˜ . Define α ∗ = sup R(μA˜ ) and α ◦ = inf R(μA˜ ). Then we have the following observations. / R(μA˜ ), we have A˜ α = ∅. It is also obvious that A˜ α = ∅ for α > α ∗ . • For any 0 ≤ α < α ∗ , even though α ∈ • If sup R(μA˜ ) is attained, i.e., sup R(μA˜ ) = max R(μA˜ ), then we have A˜ α ∗ = ∅. If sup R(μA˜ ) is not attained, then A˜ α ∗ = ∅. For example, assume that  1 − x1 , if x ≥ 1 μA˜ (x) = 0, if x < 1. It is clear to see that R(μA˜ ) = [0, 1). In this case, sup R(μA˜ ) is not attained. However, we have sup R(μA˜ ) = 1 = / R(μA˜ ). α ∗ . In this case, the 1-level set A˜ α ∗ = A˜ 1 = ∅, since α ∗ = 1 ∈ • For any 0 ≤ α < α ◦ , even though α ∈ / R(μA˜ ), we have A˜ α = A˜ α ◦ = ∅. Proposition 2.3. Let A˜ be a fuzzy set in U with membership function μA˜ . Define α ∗ = sup R(μA˜ ) and  [0, α ∗ ), if sup R(μA˜ ) is not attained IA˜ = [0, α ∗ ], if sup R(μA˜ ) is attained.

(5)

Then A˜ α = ∅ for all α ∈ IA˜ and A˜ α = ∅ for all α ∈ / IA˜ satisfying R(μA˜ ) ⊆ IA˜ . Proof. The results follow from Remark 2.2 immediately.

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˜ We see that the interval range I ˜ The interval IA˜ presented in Proposition 2.3 is also called an interval range of A. A contains the actual range R(μA˜ ). The role of interval range IA˜ can be used to say A˜ α = ∅ for all α ∈ IA˜ and A˜ α = ∅ for all α ∈ / IA˜ . Remark 2.4. Let A˜ be a fuzzy set in U . We see that the interval range IA˜ of A˜ is always an interval. However, the range R(μA˜ ) of A˜ is not necessarily an interval even though A˜ is a normal fuzzy set in U . If A˜ is a normal fuzzy set in U , then it is clear to see that the interval range IA˜ = [0, 1] according to Proposition 2.3. Suppose that A˜ is a normal fuzzy set in U . Then the well-known decomposition theorem says that the membership function μA˜ can be expressed as μA˜ (x) = sup α · χA˜ α (x) = sup α · χA˜ α (x), α∈[0,1]

α∈(0,1]

where χA˜ α is the characteristic function of the α-level set A˜ α defined by  1 if x ∈ A˜ α χA˜ α (x) = 0 if x ∈ / A˜ α ,

(6)

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which can refer to Fullér and Keresztfalvi [7], Negoita and Ralescu [12], Nguyen [13], Ralescu [14] and Wu [19]. Now we are going to generalize it by considering the countable dense set. This generalized decomposition theorem will be used for studying the set operations of fuzzy sets in U . Recall that Q denotes the set of all rational numbers. It is well-known that the countable set Q is dense in R. In other words, given any x ∈ R, there exists a sequence {qn }∞ n=1 in Q such that qn → r as n → ∞. More precisely, in Q such that qnk ↑ r or a decreasing subsequence {pnk }∞ there exists an increasing subsequence {qnk }∞ k=1 k=1 in Q such that pnk ↓ r. For convenience, we write Q[0, 1] = Q ∩ [0, 1] and Q(0, 1] = Q ∩ (0, 1]. Then Q[0, 1] is dense in [0, 1] and Q(0, 1] is dense in (0, 1]. Therefore, for any α ∈ (0, 1], there exists an increasing sequence {αn }∞ n=1 in Q(0, 1] such that αn ↑ α. Now we are in a position to present the generalized decomposition theorem, which will be used to prove the consistency between gradual sets and gradual elements. Theorem 2.5. (Generalized Decomposition Theorem) Let A˜ be a fuzzy set in U . Then the following statements hold true. (i) The membership function μA˜ can be expressed as μA˜ (x) =

sup

α∈R(μA˜ )

α · χA˜ α (x) = max α · χA˜ α (x) α∈R(μA˜ )

= sup α · χA˜ α (x) = max α · χA˜ α (x), α∈IA˜

α∈IA˜

(7)

where IA˜ is given in (5). (ii) Suppose that the following countable set CA˜ = IA˜ ∩ Q(0, 1] is dense in IA˜ such that, given any 0 < α0 ∈ IA˜ , there exists an increasing sequence {αn }∞ n=1 in CA˜ satisfying αn ↑ α0 . Then the membership function μA˜ can also be expressed as μA˜ (x) = sup α · χA˜ α (x). α∈CA˜

(iii) Suppose that the following countable set DA˜ = R(μA˜ ) ∩ Q(0, 1] is dense in R(μA˜ ) such that, given any 0 < α0 ∈ R(μA˜ ), there exists an increasing sequence {αn }∞ n=1 in DA˜ satisfying αn ↑ α0 . Then the membership function μA˜ can also be expressed as μA˜ (x) = sup α · χA˜ α (x). α∈DA˜

3. Gradual elements and gradual sets ˜ Given any fuzzy sets A˜ in R, we are going to propose the concept of belongingness in fuzzy sets like writing aˆ ∈ A, where aˆ will be interpreted as a gradual number in this section. Let U be a universal set. We denote by P(U ) the collection of all subsets of U , which is also called a power set or hyperspace of U . By referring to Dubois and Prade [4], we propose the slightly different concepts of gradual set and gradual element as follows. Definition 3.1. Let I be a subset of [0, 1]. The gradual element g in U is defined to be an assignment function g : I → U . If U = R, then the gradual element is also called a gradual number. The gradual set G in U (or gradual subset of U ) is defined to be an assignment function G : I → P(U ) from I into P(U ) such that each G(α) is a nonempty subset of U for α ∈ I .

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Definition 3.2. Let I be a subset of [0, 1]. We say that the gradual set G defined on I is nested if and only if G(α) ⊆ G(β) for α, β ∈ I with α > β. We say that the gradual set G defined on I is strictly nested if and only if G(α) ⊂ G(β) and G(α) = G(β) for α, β ∈ I with α > β. The gradual set is in fact a set-valued function from I into U . We remark that the gradual set defined in Dubois and Prade [4] consider the assignment function G : (0, 1] → P(U ) that does not consider the assignment at 0. In general, the unit interval [0, 1] can be extended to consider as a lattice and consider I to be a sub-lattice. Let A˜ be a fuzzy set in a topological space U with membership function μA˜ and let α ∗ = sup R(μA˜ ). By referring to Proposition 2.3, we can induce a gradual set GA˜ from A˜ by defining the assignment function as follows. • Suppose that sup R(μA˜ ) is not attained. Then the assignment function GA˜ : [0, α ∗ ) → P(U ) is defined on I ≡ [0, α ∗ ) given by GA˜ (α) = A˜ α . • Suppose that sup R(μA˜ ) is attained. Then the assignment function GA˜ : [0, α ∗ ] → P(U ) is defined on I ≡ [0, α ∗ ] given by GA˜ (α) = A˜ α . Therefore, we see that  A˜ α for α ∈ [0, α ∗ ), if sup R(μA˜ ) is not attained GA˜ (α) = A˜ α for α ∈ [0, α ∗ ], if sup R(μA˜ ) is attained,

(8)

˜ We remark that the assignment G ˜ (0) at 0 is not where GA˜ (0) = A˜ 0 is defined to be the closure of the support of A. A needed in the sense of gradual set proposed by Dubois and Prade [4]. Example 3.3. Let A˜ be a fuzzy set in R with membership function given by ⎧ (r − a L )/(a1 − a L ) if a L ≤ r ≤ a1 ⎪ ⎪ ⎪ ⎨1 if a1 < r ≤ a2 μA˜ (r) = U U ⎪ (a − r)/(a − a2 ) if a2 < r ≤ a U ⎪ ⎪ ⎩ 0 otherwise. It is clear to see that the interval range IA˜ of A˜ is the whole unit interval, i.e., IA˜ = [0, 1]. For α ∈ [0, 1], the α-level ˜U set of A˜ is a bounded closed interval denoted by A˜ α = [A˜ L α , Aα ], where L U ˜U A˜ L α = (1 − α)a + αa1 and Aα = (1 − α)a + αa2 .

(9)

Therefore, the gradual set GA˜ : [0, 1] → P(R) induced from the fuzzy set A˜ is given by

  L U ˜U GA˜ (α) = A˜ L α , Aα = (1 − α)a + αa1 , (1 − α)a + αa2 for α ∈ [0, 1]. On the other hand, given a gradual set G : I → P(U ), we can induce a fuzzy set A˜ G in U by using the form of decomposition theorem. The membership function of A˜ G is defined by μA˜ G (x) = sup α · χG(α) (x).

(10)

α∈I

Example 3.4. Given a gradual set G : [0, 1] → P(R) defined by G(α) = [(1 − α)x1 + αx2 , (1 − α)x3 + αx4 ] for α ∈ [0, 1], where x1 < x2 < x3 < x4 . It is clear to see that the gradual set G is strictly nested. Then we can induce a fuzzy set A˜ G in R with membership function given by μA˜ G (x) = sup α · χG(α) (x) = sup α · χ[(1−α)x1 +αx2 ,(1−α)x3 +αx4 ] (x). α∈[0,1]

α∈[0,1]

˜G ˜G In the sequel, we can see that the α-level set A˜ G α of A is equal to G(α), i.e., Aα = G(α) for α ∈ [0, 1].

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Remark 3.5. Let A˜ G be a fuzzy set in U induced by a gradual set G : I → P(U ) with membership function given in (10). Then the interval range IA˜ G of A˜ G has the form of (5). By referring to (10), it is clear to see that sup μA˜ G (x) = sup R(μA˜ G ) = sup I.

x∈U

Let α ∗ = sup I . The domain I of gradual set G is not necessarily an interval of the form [0, α ∗ ] or [0, α ∗ ). The domain I can be the disjoint union of more than two intervals. However, if the domain I of gradual set G happens to be an interval of the form [0, α ∗ ] or [0, α ∗ ), then the interval range IA˜ G of A˜ G is equal to I , i.e., I = IA˜ G . We also remark that if the gradual set is taken from the sense of Dubois and Prade [4] then the interval range IA˜ G of A˜ G cannot be equal to the domain I , since the gradual set proposed by Dubois and Prade [4] does not consider the assignment at 0. The following interesting result will be useful for further discussion. Proposition 3.6. Suppose that the gradual set G is strictly nested. Then the range of membership function μA˜ G is equal to the domain I of gradual set G; that is, R(μA˜ G ) = I . Proof. It is clear to see R(μA˜ G ) ⊆ I . For proving another direction of inclusion, given any α ∗ ∈ I , since G is strictly nested, it says that G(α) ⊂ G(α ∗ ) and G(α) = G(α ∗ ) for α > α ∗ . Therefore, there exists x ∈ G(α ∗ ) such that x∈ / G(α) for all α > α ∗ . Since G(α ∗ ) ⊂ G(α) for α ≤ α ∗ , it follows that x ∈ G(α) for α ≤ α ∗ . From (10), we have   μA˜ G (x) = sup α · χG(α) (x) = max α∈I



= max 0,

 sup

{α∈I :α≤α ∗ }

sup

{α∈I :α>α ∗ }

α · χG(α) (x),

sup

{α∈I :α≤α ∗ }

α · χG(α) (x)

α = α∗,

which says that α ∗ ∈ R(μA˜ G ). This completes the proof.

2

Let  be an index set that can be an uncountable set. We consider a family {gλ : λ ∈ } of gradual elements in U in which the gradual elements gλ for λ ∈  are all defined on the same domain I ⊆ [0, 1]. This family of gradual elements can induce a gradual set G in U defined on I by G(α) = {gλ (α) : λ ∈ } for α ∈ I.

(11)

Based on this gradual set G in U , we can also induce a fuzzy set A˜ G in U with membership function defined in (10). In this case, we can also say that the family {gλ : λ ∈ } of gradual elements induces the fuzzy set A˜ G . This kind of consideration will be used to study the arithmetics of fuzzy sets using gradual numbers. Example 3.7. We take the index set  = [0, 1] and the domain I = [0, 1]. Let f1 (α) = (1 − α)x1 + αx2 and f2 (α) = (1 − α)x3 + αx4 for α ∈ [0, 1], where x1 < x2 < x3 < x4 . Then f1 (α) ≤ f2 (α) for all α ∈ [0, 1]. For λ ∈  = [0, 1], we define gλ : [0, 1] → R by gλ (α) = λf1 (α) + (1 − λ)f2 (α). Then we have a family {gλ : λ ∈ [0, 1]} of gradual numbers. According to (11), we can induce a gradual set G : [0, 1] → P(R) given by G(α) = {gλ (α) : λ ∈ [0, 1]} = [f1 (α), f2 (α)] = [(1 − α)x1 + αx2 , (1 − α)x3 + αx4 ] that is a bounded closed interval for α ∈ [0, 1]. This gradual set G is the same as the gradual set in Example 3.4. Therefore, we can induce the same fuzzy set in Example 3.4.

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The gradual set G is a set-valued function defined on I ⊆ [0, 1] with function value G(α) to be a subset of U for each α ∈ I . According to the topic of set-valued analysis, the selector (or selection function) of G is a single-valued function g : I → U defined on I by g(α) ∈ G(α). In this case, we may write g ∈ G. It is clear to see that the selector of gradual set G (i.e. set-valued function G) is a gradual element in U . In some sense, we may say that the gradual set consists of gradual elements, which is similar to say that the (usual) set consists of (usual) elements. Given a subset A of U , the concept of elements of A can be realized in the usual sense by simply writing a ∈ A when a is assumed to be an element of A. For the fuzzy set A˜ in U , we plan to consider the concept of element of A˜ ˜ where the definition of aˆ will be presented below. by also simply writing aˆ ∈ A, Given a fuzzy set A˜ in a topological space U , we can induce a gradual set GA˜ as given in (8). Therefore we have ˆ ∈ GA˜ (α) = A˜ α for α ∈ I . We also see that the selector aˆ is a gradual element in U . the selector aˆ of GA˜ given by a(α) ˜ The formal definition is given This gradual element aˆ can be regarded as an element of A˜ by simply writing aˆ ∈ A. below. Definition 3.8. Let A˜ be a fuzzy set in a topological space U . We say that an element aˆ is in A˜ if and only if aˆ is a gradual element aˆ : IA˜ → U defined on IA˜ satisfying a(α) ˆ ∈ A˜ α for each α ∈ IA˜ , where IA˜ is given in (5). In this case, ˜ we also write aˆ ∈ A. Example 3.9. Consider the fuzzy set A˜ given in Example 3.3. The α-level set is given by

  U ˜ , A A˜ α = (1 − α)a L + αa1 , (1 − α)a U + αa2 ≡ A˜ L α α for α ∈ [0, 1]. We also see that IA˜ = [0, 1]. We define the gradual number aˆ : [0, 1] → R by a(α) ˆ = (1 − α)a L + αa1 = A˜ L α ˜ In general, for λ ∈ [0, 1], we define for α ∈ [0, 1]. Then aˆ ∈ A. ˜U aˆ λ (α) = λA˜ L α + (1 − λ)Aα . Then aˆ λ ∈ A˜ for λ ∈ [0, 1]. Since aˆ ∈ A˜ is a gradual element in U , this gradual element aˆ can also be regarded as a gradual set G given by G(α) = {a(α)} ˆ that is a singleton set for α ∈ IA˜ . According to (10), we can induce a fuzzy set A˜ G in U with membership function given by μA˜ G (x) = sup α · χG(α) (x) = sup α · χ{a(α)} (x) = ˆ α∈IA˜

=

α∈IA˜

⎧ ⎨ 0, ⎩

sup

α,

sup

{α∈IA˜ :α>0}

α · χ{a(α)} (x) ˆ

ˆ =x if there is no α ∈ IA˜ with α > 0 satisfying a(α) otherwise.

{α∈IA˜ :α>0,a(α)=x} ˆ

Therefore, the formal definition of membership function of element aˆ ∈ A˜ is proposed below. ˜ the membership function of aˆ is defined by Definition 3.10. Let A˜ be a fuzzy set in U . Given any aˆ ∈ A, ⎧ ˆ =x if there is no α ∈ IA˜ with α > 0 satisfying a(α) ⎨ 0, μaˆ (x) = α, otherwise. sup ⎩

(12)

{α∈IA˜ :α>0,a(α)=x} ˆ

In the sequel, we are going to claim that Definition 3.8 is well-defined. A fuzzy set A˜ in U can be regarded as ˜ Therefore, according to (11), this family A˜ of gradual a family consisting of elements (i.e., gradual elements) in A. elements can induce a gradual set G given by   G(α) = a(α) ˆ : aˆ ∈ A˜ for α ∈ IA˜ , (13)

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where IA˜ is given in (5). Also, according to (8), the fuzzy set A˜ can induce another gradual set GA˜ given by GA˜ (α) = A˜ α for α ∈ IA˜ . On the other hand, according to (10), the gradual set G in (13) can also induce another fuzzy set A˜ G ˜ in U . In order to claim the consistency of Definition 3.8, we need to show G(α) = GA˜ (α) for all α ∈ IA˜ and A˜ G = A, which will be presented below. Proposition 3.11. Let A˜ be a fuzzy set in a topological space U . Then the following statements hold true. (i) The gradual set G induced by the family A˜ of gradual elements satisfies G(α) = A˜ α = GA˜ (α) for each α ∈ IA˜ , where IA˜ is given in (5). ˜ (ii) Let A˜ G be a fuzzy set in U induced by the gradual set G in part (i). Then A˜ G = A. ˜ we see that G(α) ⊆ A˜ α for α ∈ I ˜ . On the Proof. To prove part (i), according to (13) and the definition of aˆ ∈ A, A other hand, given any fixed α ∈ IA˜ and any x ∈ A˜ α , we define a function aˆ on IA˜ by  x, if β = α a(β) ˆ = ˜ y for some y ∈ Aβ , if β = α. ˜ This shows that a(α) Then it is clear to see that aˆ ∈ A. ˆ = x ∈ G(α), i.e., A˜ α ⊆ G(α). Therefore we obtain G(α) = A˜ α for α ∈ IA˜ . To prove part (ii), from (10), the membership function of A˜ G is given by μA˜ G (x) = sup α · χG(α) (x). α∈IA˜

Using part (i) of the Generalized Decomposition Theorem 2.5, the membership function of A˜ can be expressed as μA˜ (x) = sup α · χA˜ α (x). α∈IA˜

Since G(α) = A˜ α for each α ∈ IA˜ by part (i), we obtain A˜ G = A˜ by referring to their membership functions. This completes the proof. 2 4. Arithmetics using gradual numbers Let A˜ and B˜ be a fuzzy set in R with membership function μA˜ and μB˜ , respectively. Let ∗ αA = sup R(μA˜ ) and αB∗ = sup R(μB˜ ).

By referring to (5), the interval ranges IA˜ and IB˜ are given by  ∗ ), if sup R(μ ) is not attained [0, αA A˜ IA˜ = ∗ ], if sup R(μ ) is attained [0, αA A˜ and

 IB˜ =

[0, αB∗ ), if sup R(μB˜ ) is not attained [0, αB∗ ], if sup R(μB˜ ) is attained

˜ We see that A˜ α = ∅ for α ∈ IA˜ and A˜ α = ∅ for α ∈ / IA˜ by Proposition 2.3, where the same situation also apples to B. ˜ ˆ ˜ ˆ According to Definition 3.8, for aˆ ∈ A and b ∈ B, the gradual numbers aˆ and b are real-valued functions defined on ˆ IA˜ and IB˜ , respectively. More precisely, we have a(α) ˆ ∈ A˜ α for α ∈ IA˜ , and b(α) ∈ B˜ α for α ∈ IB˜ .

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It is clear to see that I ∩ ≡ IA˜ ∩ IB˜ = ∅ and is given by ⎧ ∗ ) ∩ [0, α ∗ ), if sup R(μ ) and sup R(μ ) are not attained [0, αA ⎪ B A˜ B˜ ⎪ ⎪ ⎨ [0, α ∗ ] ∩ [0, α ∗ ), if sup R(μ ) is attained and sup R(μ ) is not attained A B A˜ B˜ I∩ = ∗ ) ∩ [0, α ∗ ], if sup R(μ ) is not attained and sup R(μ ) is attained ⎪ [0, α ⎪ A B A˜ B˜ ⎪ ⎩ ∗ ] ∩ [0, α ∗ ], if sup R(μ ) and sup R(μ ) are attained [0, αA ˜ ˜ B A B

(14)

˜ the arithmetic aˆ ◦ bˆ is defined by For any two gradual numbers aˆ ∈ A˜ and bˆ ∈ B, ˆ ˆ (aˆ ◦ b)(α) = a(α) ˆ ◦ b(α) for all α ∈ I ∩ , where ◦ is an arithmetic operation +, −, × or / for real numbers. When ◦ = / is taken, we need to ˆ assume that b(α) = 0 for each α. It is clear to see that aˆ ◦ bˆ is also a gradual number defined on I ∩ . In the topic of set-valued analysis, given any two subsets A and B of R, the arithmetic A ◦ B is defined by A ◦ B = {a ◦ b : a ∈ A and b ∈ B} ,

(15)

where ◦ ∈ {+, −, ×, /}. The division should avoid dividing by zero. We are going to follow this similar concept to define the arithmetic operation A˜  B˜ of fuzzy sets A˜ and B˜ for  ∈ {⊕, , ⊗, }. Consider the following family   aˆ ◦ bˆ : aˆ ∈ A˜ and bˆ ∈ B˜ (16) that consists of gradual numbers, where the arithmetic operation  ∈ {⊕, , ⊗, } corresponds to the arithmetic operation ◦ ∈ {+, −, ×, /}. Then the above family (16) can induce a gradual set G : I ∩ → P(R) given by   ˆ G(α) = a(α) ˆ ◦ b(α) : aˆ ∈ A˜ and bˆ ∈ B˜ . (17) Using this gradual set and referring to (10), we can induce a fuzzy set C˜ G in R with membership function given by μC˜ G (x) = sup α · χG(α) (x). α∈I ∩

(18)

In this case, we define A˜  B˜ = C˜ G . From Remark 3.5, we see that the interval range IC˜ G of C˜ G is equal to I ∩ , i.e., IC˜ G = I ∩ . Given any two fuzzy sets A˜ and B˜ in R, by referring to (13), we can induce two gradual sets A and B corresponding ˜ More precisely, we have to A˜ and B.     ˆ A(α) = a(α) ˆ : aˆ ∈ A˜ for α ∈ IA˜ , and B(α) = b(α) : bˆ ∈ B˜ for α ∈ IB˜ . (19) Since A˜ α = ∅ for α ∈ IA˜ and B˜ α = ∅ for α ∈ IB˜ by Proposition 2.3, part (i) of Proposition 3.11 says that A(α) = A˜ α for α ∈ IA˜ , and B(α) = B˜ α for α ∈ IB˜ .

(20)

According to (15), (17), (19) and (20), we see that G(α) = A(α) ◦ B(α) = A˜ α ◦ B˜ α for α ∈ I ∩ .

(21)

From the basic properties of fuzzy sets, we have A˜ β ⊆ A˜ α and B˜ β ⊆ B˜ α for α, β ∈ I ∩ = IA˜ ∩ IB˜ with α < β. Using the equalities (21), we obtain G(β) ⊆ G(α) for α, β ∈ I ∩ with α < β.

(22)

This also says that the family {G(α) : α ∈ I ∩ } is nested. Example 4.1. Consider the fuzzy set A˜ in Example 3.3. Now we consider another fuzzy set B˜ in R with membership function given by

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⎧ (r − bL )/(b1 − bL ) if bL ≤ r ≤ b1 ⎪ ⎪ ⎪ ⎨ 1 if b1 < r ≤ b2 μB˜ (r) = U U U ⎪ ⎪ (b − r)/(b − b2 ) if b2 < r ≤ b ⎪ ⎩ 0 otherwise. Then IB˜ = [0, 1]. We also see that I ∩ = [0, 1]. According to (18), the membership functions of A˜ ⊕ B˜ is given by μA⊕ ˜ B˜ (x) = sup α · χG(α) (x), α∈[0,1]

where G(α) = A˜ α + B˜ α

  = (1 − α)a L + αa1 , (1 − α)a U + αa2 + (1 − α)bL + αb1 , (1 − α)bU + αb2  

   = (1 − α) a L + bL + α (a1 + b1 ) , (1 − α) a U + bU + α (a2 + b2 ) for α ∈ [0, 1] by referring to (21). It is not hard to see that the α-level set of A˜ ⊕ B˜ is given by   A˜ ⊕ B˜ = G(α) = A˜ α + B˜ α α

for α ∈ [0, 1]. ˜ Consider the following families Now, given any A˜ in R, we want to study the meaning of −A˜ and 1/A.     −aˆ : aˆ ∈ A˜ and 1/aˆ : aˆ ∈ A˜ with a(α) ˆ = 0 for all α ∈ IA˜

(23)

that consists of gradual numbers. Then the above two families (23) can induce two gradual sets G(−) : IA˜ → P(R) and G(/) : IA˜ → P(R) given by     G(−) (α) = −a(α) ˆ : aˆ ∈ A˜ and G(/) (α) = 1/a(α) ˆ : aˆ ∈ A˜ , (−) (/) respectively. Using these two gradual sets and referring to (10), we can induce two fuzzy sets D˜ G and D˜ G in R with membership function given by

μD˜ G(−) (x) = sup α · χG(−) (α) (x) and μD˜ G(/) (x) = sup α · χG(/) (α) (x), α∈IA˜

α∈IA˜

respectively. In this case, we define (−) (/) −A˜ = D˜ G and 1/A˜ = D˜ G . (−) (/) From Remark 3.5, we see that the interval ranges ID˜ G(−) of D˜ G and ID˜ G(/) of D˜ G are equal to IA˜ . Since

    −A˜ α = −x : x ∈ A˜ α and 1/A˜ α = 1/x : x ∈ A˜ α with x = 0 ,

we can similarly obtain G(−) (α) = −A˜ α and G(/) (α) = 1/A˜ α for α ∈ IA˜ . The following results can be obtained. ˜ If μ ˜ (x) = 0 for each x ∈ R, Proposition 4.2. Given any two fuzzy sets A˜ and B˜ in R, we have A˜  B˜ = A˜ ⊕ (−B). B ˜ ˜ ˜ ˜ then we have A  B = A ⊗ (1/B).

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By referring to (17), for ◦ ∈ {+, ×}, since ˆ ˆ a(α) ˆ ◦ b(α) = b(α) ◦ a(α) ˆ for all α ∈ I ∩ , it follows that the commutativity A˜  B˜ = B˜  A˜ holds true for  ∈ {⊕, ⊗}. Using Proposition 4.2, we see that ˜ = −B˜ ⊕ A. ˜ A˜  B˜ = A˜ ⊕ (−B) If μB˜ (x) = 0 for each x ∈ R, then ˜ = (1/B) ˜ ⊗ A. ˜ A˜  B˜ = A˜ ⊗ (1/B) Let C˜ be another fuzzy set in R. For any cˆ ∈ C˜ and ◦1 , ◦2 ∈ {+, −}, since the associativity     ˆ ˆ ˆ = a(α) ˆ ◦1 b(α) ˆ a(α) ˆ ◦1 b(α) ◦2 c(α) ◦2 c(α) holds true for all α ∈ I ∩ ≡ IA˜ ∩ IB˜ ∩ IC , it follows that the associativity     A˜ 1 B˜ 2 C˜ = A˜ 1 B˜ 2 C˜ also holds true for 1 , 2 ∈ {⊕, }, where the arithmetic operations 1 and 2 correspond to the arithmetic operations ◦1 and ◦2 . In this case, we can simply write     ˜ A˜ 1 B˜ 2 C˜ = A˜ 1 B˜ 2 C˜ ≡ A˜ 1 B˜ 2 C. The associativity for ⊗ can be similarly realized as follows     ˜ A˜ ⊗ B˜ ⊗ C˜ = A˜ ⊗ B˜ ⊗ C˜ ≡ A˜ ⊗ B˜ ⊗ C. In general, let A˜ (1) , · · · , A˜ (n) be fuzzy sets in R, and let Ii ≡ IA˜ (i) be the interval range of A˜ (i) for i = 1, · · · , n. We also define I ∩ = I1 ∩ · · · ∩ In = ∅. Consider the arithmetic operation i ∈ {⊕, , ⊗, } corresponding to the arithmetic operation ◦i ∈ {+, −, ×, /} for i = 1, · · · , n − 1. Then we can induce a gradual set G : I ∩ → P(R) given by    G(α) = aˆ 1 ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n   = aˆ 1 (α) ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n . Using this gradual set and referring to (10), we can induce a fuzzy set A˜ G in R with membership function given by μA˜ G (x) = sup α · χG(α) (x). α∈I ∩

(24)

In this case, we define A˜ (1) 1 · · · n−1 A˜ (n) = A˜ G , where the operations ⊗ and  corresponding to the operations × and / have the highest priority for performing arithmetic calculations. By referring to (21) and (22), we have ∩ ˜ (n) G(α) = A˜ (1) α ◦1 · · · ◦n−1 Aα for α ∈ I ,

(25)

G(β) ⊆ G(α) for α, β ∈ I ∩ with α < β.

(26)

and

This also says that the family {G(α) : α ∈ I ∩ } is nested.

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Remark 4.3. For the fuzzy set A˜ G given in (24), it is clear to see that sup R(μA˜ G ) = sup I ∩ . Let ∗ αG = sup R(μA˜ G ) = sup I ∩ .

(27)

By referring to (5), the interval range of A˜ G is given by  ∗ ), if sup I ∩ is not attained [0, αG IA˜ G = ∗ ], if sup I ∩ is attained. [0, αG ∩ Using (14) inductively and Remark 3.5, we see that IA˜ G = I ∩ . Then the α-level sets A˜ G α = ∅ for all α ∈ I and G ∩ A˜ α = ∅ for all α ∈ /I .

Definition 4.4. We say that A˜ is a canonical fuzzy set in U if and only if A˜ α = A˜ β for α = β. It is clear to see that if A˜ is a canonical fuzzy set then the family of its α-level sets is strictly nested. Proposition 4.5. Let A˜ (1) , · · · , A˜ (n) be fuzzy sets in R. Suppose that some of A˜ (1) , · · · , A˜ (n) are canonical fuzzy sets in R. Then the family {G(α) : α ∈ I ∩ } given in (25) is strictly nested, and we also have   R μA˜ (1) 1 ···n−1 A˜ (n) = R(μA˜ G ) = IA˜ G = I ∩ . Proof. Since some of A˜ (1) , · · · , A˜ (n) are canonical fuzzy sets in R, it follows that G(α) = G(β) for α < β. From (26), we see that the family {G(α) : α ∈ I ∩ } is strictly nested. Using Proposition 3.6 and Remark 4.3, we obtain the desired equalities. This completes the proof. 2 Suppose that the parentheses are included in the expression A˜ (1) 1 · · · n−1 A˜ (n) . Then the terms within the parentheses have the highest priority for performing arithmetic calculations. Example 4.6. Suppose that we want to calculate the following expression        A˜ (1)  A˜ (2) ⊗ A˜ (3)  A˜ (4) ⊕ A˜ (5)  A˜ (6) ⊗ A˜ (7) ⊕ A˜ (8)  A˜ (9)  A˜ (10) .

(28)

Then we can induce a gradual set G : I ∩ → P(R) given by         G(α) = aˆ 1 − aˆ 2 ∗ aˆ 3 − aˆ 4 + aˆ 5 − aˆ 6 ∗ aˆ 7 + aˆ 8 − aˆ 9 − aˆ 10 (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , 10        (2) (3) (4) (5) (6) (7) (8) (9) (10) ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ − A  A ⊕ A × A + A − A − A × A − A . = A˜ (1) α α α α α α α α α α The membership function of expression (28) is given in the form of (24). Now we want to study the α-level sets of the arithmetic expression A˜ (1) 1 · · · n−1 A˜ (n) . Theorem 4.7. Let A˜ (1) , · · · , A˜ (n) be fuzzy sets in R such that sup I ∩ is attained. Suppose that, for any fixed x ∈ R, the function ηx (α) = α · χG(α) (x) is upper semi-continuous on I ∩ . Then the α-level set of A˜ G = A˜ (1) 1 · · · n−1 A˜ (n) is given by   (1) (n) ˜ ˜ A A˜ G A =  · · ·  = {x ∈ R : μA˜ G (x) ≥ α} 1 n−1 α α   = G(α) = aˆ 1 (α) ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n ˜ (n) = A˜ (1) α ◦1 · · · ◦n−1 Aα for every 0 < α ∈ I ∩ , and

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  ˜ (1) 1 · · · n−1 A˜ (n) A˜ G 0+ = A

0+



=

=





A˜ (1) 1 · · · n−1 A˜ (n)



{α∈I ∩ :α>0}

  aˆ 1 (α) ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n .

α



=

G(α)

{α∈I ∩ :α>0}

{α∈I ∩ :α>0}

Proof. From Remark 4.3, since sup I ∩ is attained, it follows that I ∩ is a closed interval. We also note that A˜ G α = ∅ ∩ ∩ G ˜ / G(β) for all β ∈ I ∩ for all α ∈ I . Given any fixed 0 < α ∈ I , suppose that x ∈ Aα . Now we also assume that x ∈ with β ≥ α. We want to lead to a contradiction. Under this assumption, we see that β · χG(β) (x) < α for all β ∈ I ∩ . Since ηx (β) = β · χG(β) (x) is upper semi-continuous on the closed interval I ∩ by the assumption, the supremum of function ηx is achieved by Lemma 2.1. This says that μA˜ G (x) = sup ηx (β) = sup β · χG(β) (x) = max β · χG(β) (x) = β ∗ · χG(β ∗ ) (x) < α β∈I ∩

β∈I ∩

β∈I ∩

∩ for some β ∗ ∈ I ∩ , which violates x ∈ A˜ G α . Therefore there exists β0 ∈ I with β0 ≥ α such that x ∈ G(β0 ) ⊆ G(α) G ˜ by (26), which shows the inclusion Aα ⊆ G(α). The following inclusion is obvious   G(α) ⊆ x ∈ R : sup β · χG(β) (x) ≥ α = {x ∈ R : μ ˜ G (x) ≥ α} = A˜ G α . A

β∈I ∩

Therefore, using (25), we obtain the desired equalities. This completes the proof. 2 Corollary 4.8. Let A˜ (1) , · · · , A˜ (n) be normal fuzzy sets in R. Suppose that, for any fixed x ∈ R, the function ηx (α) = α · χG(α) (x) is upper semi-continuous on [0, 1]. Then the α-level set of A˜ G = A˜ (1) 1 · · · n−1 A˜ (n) is given by   (1) (n) ˜ ˜ = A  · · ·  = {x ∈ R : μA˜ G (x) ≥ α} A˜ G A 1 n−1 α α   = G(α) = aˆ 1 (α) ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n ˜ (n) = A˜ (1) α ◦1 · · · ◦n−1 Aα for every α ∈ (0, 1], and   ˜ (1) 1 · · · n−1 A˜ (n) A˜ G 0+ = A

0+

=

=

 

A˜ (1) 1 · · · n−1 A˜ (n)

α∈(0,1]



   aˆ 1 (α) ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n .

α

=



G(α)

α∈(0,1]

α∈(0,1]

Proof. Since Ii = [0, 1] for all i = 1, · · · , n, it follows that I ∩ = [0, 1]. The desired results follow from Theorem 4.7 immediately. 2 Without assuming the upper semi-continuity, we also have the following interesting results. Theorem 4.9. Let A˜ (1) , · · · , A˜ (n) be fuzzy sets in R such that sup I ∩ is attained. Suppose that, given any α ∈ I ∩ with ∩ α > 0 and any increasing convergent sequence {αm }∞ m=1 in I with αm > 0 for all m and αm ↑ α, the following inclusion is satisfied ∞    (n) ˜ ˜ (n) ◦ · · · ◦ A˜ (1) ⊆ A˜ (1) A 1 n−1 α α α ◦1 · · · ◦n−1 Aα . m

m

m=1

Then the α-level set of A˜ G = A˜ (1) 1 · · · n−1 A˜ (n) is given by

(29)

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  ˜ (1) 1 · · · n−1 A˜ (n) = G(α) A˜ G α = A α   = aˆ 1 (α) ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n ˜ (n) = A˜ (1) α ◦1 · · · ◦n−1 Aα for every 0 < α ∈ I ∩ , and   (1) (n) ˜ ˜ = A  · · ·  A˜ G A 1 n−1 0+ 

=

0+

=



G(α)

{α∈I ∩ :α>0}

  aˆ 1 (α) ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n .

{α∈I ∩ :α>0}

Proof. Using the equalities (25), we see that the inclusion (29) is satisfied if and only if the following inclusion is satisfied ∞ 

G(αm ) ⊆ G(α).

(30)

m=1

Given any fixed x ∈ R, we define the following set   Fρ = α ∈ I ∩ : α · χG(α) (x) ≥ ρ . ∗ ], which also says that F = ∅ for ρ ≤ α ∗ and F = ∅ From Remark 4.3, since sup I ∩ is attained, we have I ∩ = [0, αG ρ ρ G ∗ for ρ > αG by referring to (24) and (27); that is to say, Fρ = ∅ for ρ ∈ I ∩ and Fρ = ∅ for ρ ∈ / I ∩ . Next we are going to prove that Fρ is a closed set, i.e., cl(Fρ ) = Fρ for each ρ ∈ I ∩ with ρ > 0. Now, for each α ∈ cl(Fρ ), there exists a ∩ sequence {αm }∞ m=1 in Fρ such that αm → α. Therefore we have αm ∈ I with αm ≥ ρ > 0 and x ∈ G(αm ) for all m, which also says that α > 0, since

α = lim αm ≥ ρ > 0. m→∞

∞ Therefore there exists a subsequence {αmk }∞ k=1 of {αm }m=1 such that αmk ↑ α or αmk ↓ α as k → ∞.

• Suppose that αmk ↓ α. Then αmk ≥ α for all k. This says that x ∈ G(αmk ) ⊆ G(α) by (26). ∗ ], it says that α ∈ I ∩ . Since x ∈ G(α ) • Suppose that αmk ↑ α. Since Fρ ⊆ I ∩ , i.e., cl(Fρ ) ⊆ cl(I ∩ ) = I ∩ = [0, αG mk for all k, using (30), it follows that x ∈ G(α). Therefore we conclude that x ∈ G(α) for both cases. This also says that α · χG(α) (x) ≥ ρ, i.e., α ∈ Fρ . Therefore we obtain the inclusion cl(Fρ ) ⊆ Fρ , which means that Fρ is a closed set for each ρ ∈ I ∩ with ρ > 0. In order to apply Theorem 4.7, we need to show that the function ηx is upper semi-continuous on I ∩ . It is equivalently to claim that the set Fρ is closed for each ρ ∈ R. We consider the following cases. • For each ρ ∈ I ∩ with ρ > 0, the set Fρ is closed as shown above. • For ρ ∈ / I ∩ , the set Fρ = ∅ is empty, which is also closed. ∗ ] is closed. • For ρ ≤ 0, the set Fρ = I ∩ = [0, αG We conclude that the function ηx is indeed upper semi-continuous on I ∩ . Therefore, the desired results follow from Theorem 4.7 immediately. 2 Corollary 4.10. Let A˜ (1) , · · · , A˜ (n) be normal fuzzy sets in R. Suppose that, given any α ∈ (0, 1] and any increasing convergent sequence {αm }∞ m=1 in (0, 1] with αm ↑ α, the following inclusion is satisfied ∞   m=1

 (n) ˜ (1) ˜ ˜ (n) ◦ · · · ◦ A˜ (1) A 1 n−1 αm αm ⊆ Aα ◦1 · · · ◦n−1 Aα .

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Then the α-level set of A˜ G = A˜ (1) 1 · · · n−1 A˜ (n) is given by   ˜ (1) 1 · · · n−1 A˜ (n) = G(α) A˜ G α = A α   = aˆ 1 (α) ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n ˜ (n) = A˜ (1) α ◦1 · · · ◦n−1 Aα for every α ∈ (0, 1], and   ˜ (1) 1 · · · n−1 A˜ (n) A˜ G 0+ = A

0+

=

=



G(α)

α∈(0,1]

   aˆ 1 (α) ◦1 · · · ◦n−1 aˆ n (α) : aˆ i ∈ A˜ (i) for i = 1, · · · , n . α∈(0,1]

Proof. Since I ∩ = [0, 1], the desired results follow from Theorem 4.9 immediately.

2

∩ Suppose that, for any increasing sequence {αm }∞ m=1 in I with αm > 0 for all m, the following inclusion is satisfied ∞  ∞  ∞      (1) (n) (1) (n) ˜ ˜ ˜ ˜ Aα ◦1 · · · ◦n−1 Aα ⊆ (31) Aα ◦1 · · · ◦n−1 Aα . m

m=1

m

m

m=1

m

m=1

We are going to claim that the inclusion (29) is satisfied if and only if the inclusion (31) is satisfied. Indeed, according to the basic properties of fuzzy sets, given any γ > 0 and any increasing convergent sequence {γm }∞ m=1 in Ii with  (i) (i) ∩ = I ∩ · · · ∩ I with α > 0 and ˜ ˜ = A . Therefore, given any α ∈ I γm > 0 for all m and γm ↑ γ , we have ∞ A γ 1 n m=1 γm ∩ any increasing convergent sequence {αm }∞ m=1 in I with αm > 0 for all m and αm ↑ α, we have ∞ 

˜ (i) A˜ (i) αm = Aα ,

(32)

m=1

which says that the inclusion (29) is satisfied if and only if the inclusion (31) is satisfied. Therefore, we can also obtain the interesting results from Theorem 4.9 when the inclusion (31) is considered. 5. Arithmetics of fuzzy intervals using gradual numbers We say that A˜ is a fuzzy interval if and only if A˜ is a fuzzy set in R with membership function μA˜ satisfying the following conditions. • The membership function μA˜ is upper semi-continuous and quasi-concave on R. • The 0-level set A˜ 0 is a compact subset of R; that is, a closed and bounded subset of R. If the fuzzy interval A˜ is normal and the 1-level set A˜ 1 is a singleton set {a}, where a ∈ R, then A˜ is also called a fuzzy number with core value a. Let A˜ be a fuzzy interval. Its α-level sets are bounded closed intervals given by  ∅ if α ∈ / IA˜ A˜ α = L U  A˜ α , A˜ α if α ∈ IA˜ , ˜ which explains the terminology called fuzzy interval (ref. Dubois, Prade, where IA˜ denotes the interval range of A, ˜U ˜L ˜U Fortin and Fargier [4–6]). In particular, if A˜ is a fuzzy number, then A˜ α = [A˜ L α , Aα ] for all α ∈ [0, 1] and A1 = A1 U L for α = 1, i.e., the upper 1-level set A˜ 1 = {A˜ 1 = A˜ 1 = a} is a singleton set, where a is the core value. Let A˜ and B˜ be any two fuzzy intervals. According to the previous discussion, the membership function μC˜ G of ˜ A  B˜ = C˜ G is given by

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μC˜ G (x) = sup α · χG(α) (x), α∈I ∩

where the corresponding gradual set G : I ∩ → P(R) is given by   ˆ G(α) = a(α) ˆ ◦ b(α) : aˆ ∈ A˜ and bˆ ∈ B˜ . According to (21), we see that

  ˜U ˜ L ˜ U for α ∈ I ∩ . G(α) = A˜ α ◦ B˜ α = A˜ L α , Aα ◦ Bα , Bα

(33)

We remark that if the operation of division is taken then we assume that the α-level set B˜ α does not contain zero for all α ∈ I ∩ . The conventional arithmetics of A˜ and B˜ is based on the extension principle, which is denoted by A˜ EP B˜ and defined by   μA sup min μA˜ (a), μB˜ (b) ˜ EP B˜ (x) = {(a,b):x=a◦b}

for EP ∈ {⊕EP , EP , ⊗EP , EP }. It is well-known that the α-level set of A˜ EP B˜ is given by

    ˜U ˜ L ˜ U for α ∈ I ∩ . A˜ EP B˜ = A˜ α ◦ B˜ α = A˜ L α , Aα ◦ Bα , Bα α

(34)

˜ Now, we are going to apply Theorem 4.9 to claim A˜  B˜ = A˜ EP B. Theorem 5.1. Let A˜ and B˜ be two fuzzy intervals. We also assume that A˜ and B˜ are normal. Then A˜  B˜ = A˜ EP B˜ with α-level sets given by 

     U L ˜U ˜ ˜ , A , B A˜  B˜ = A˜ EP B˜ = A˜ L ◦ B for α ∈ (0, 1] α α α α α

and



where 

A˜  B˜

A˜  B˜

 0

α

    = A˜ EP B˜ = cl A˜  B˜

 0+

0

  = A˜ EP B˜

0+

0+



=

  = cl A˜ EP B˜

0+

,

  U L ˜U ˜ ˜ , A , B A˜ L ◦ B α α α α .

{α∈I ∩ :α>0}

Proof. Since A˜ and B˜ are assumed to be normal, it follows that IA˜ = IB˜ = I ∩ = [0, 1]. ˜ it is well-known that, given α ∈ (0, 1] and any sequence {αm }∞ in (0, 1] satisfying αm ↑ α, the For any fuzzy set A, m=1 equality ∞ 

A˜ αm = A˜ α

m=1

holds true. Since A˜ EP B˜ is also a fuzzy set, it means ∞  

A˜ EP B˜



m=1

Using (34), we obtain

αm

  = A˜ EP B˜ . α

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which says that the assumption of Theorem 4.9 is satisfied. Therefore, using (33) and (34), we have  

    ˜L ˜U ˜ ˜ ˜U A˜  B˜ = G(α) = A˜ L for α ∈ (0, 1] α , Aα ◦ Bα , Bα = A EP B α

and



A˜  B˜

 0+

α

  = A˜ EP B˜

0+

=

 {α∈I ∩ :α>0}

G(α) =



  U L ˜U ˜ ˜ , A , B A˜ L ◦ B α α α α .

{α∈I ∩ :α>0}

˜ and the proof is complete. 2 It follows that A˜  B˜ = A˜ EP B, ˜U ˜ L ˜U For A˜ and B˜ in Theorem 5.1, we further assume that A˜ L α , Aα , Bα and Bα are continuous with respective to α on L ˜ [0, 1]; that is to say, the function f (α) = Aα is continuous on [0, 1]. Then the 0-level sets of the arithmetic operations can be written as follows 

     ˜U ˜L ˜U A˜  B˜ = A˜ EP B˜ = A˜ 0 ◦ B˜ 0 = A˜ L 0 , A0 ◦ B0 , B0 , 0

0

˜ respectively. where A˜ 0 and B˜ 0 are the 0-level sets of A˜ and B, 6. Conclusions The conventional arithmetic operations of fuzzy sets are based on the extension principle. In this paper, the new arithmetic operations of fuzzy sets are proposed using the concept of gradual numbers. When the fuzzy sets are taken to be the normal fuzzy intervals, we have shown that these two arithmetic operations are equivalent as presented in Theorem 5.1. The concept of belongingness in fuzzy sets like writing aˆ ∈ A˜ is proposed in Definition 3.8. Based on this concept, we can treat the arithmetic operations A˜  B˜ of fuzzy sets for  ∈ {⊕, , ⊗, } to be the arithmetic operations of crisp sets using the following family   aˆ ◦ bˆ : aˆ ∈ A˜ and bˆ ∈ B˜ . Moreover, Proposition 3.11 supports Definition 3.8. The gradual set G is a set-valued function G : I → P(U ) defined on a subset I of unit interval [0, 1], and the gradual element is a single-valued function g : I → U . The gradual element is also called a gradual number when g is a real-valued function, i.e., U = R. The gradual set proposed by Dubois and Prade [4] is given by G : (0, 1] → P(U ) that does not consider the assignment at 0. However, in this paper, the domain I of the gradual set G can be any subsets of [0, 1], which also says that I is not necessarily an interval and the assignment at 0 may be included in I . The comparisons are presented below • The gradual set G can induce a fuzzy set A˜ G . In Remark 3.5, since the interval range IA˜ G of A˜ G contains 0 and is identical to the domain I of the gradual set G, the gradual set proposed by Dubois and Prade [4] cannot be used in this case. • The concept of belongingness in fuzzy sets is a key issue of this paper, which is presented in Definition 3.8. We can see that the interval range IA˜ of A˜ is involved in the definition. Since the interval range IA˜ is an interval of form [0, α ∗ ] or [0, α ∗ ) in which the assignment at 0 is always needed. Therefore, the gradual set proposed by Dubois and Prade [4] cannot be used in Definition 3.8. Since we can consider the (gradual) elements in fuzzy sets like the (usual) elements in crisp sets, we expect to apply the mathematical properties from the topic of set-valued analysis to study the mathematical properties of fuzzy sets, which can be treated as the future research.

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