Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets

The conventional concept of α-level sets of fuzzy sets will be treated as the upper α-level sets. In this paper, the concept of lower α-level sets of fuzzy sets will be introduced, which can also be regarded as a dual concept of upper α-level sets of fuzzy sets. We shall also introduce the concept of dual fuzzy sets. Under these settings, we can establish the so-called dual decomposition theorem. We shall also study the dual arithmetics of fuzzy sets in R and establish some interesting results based on the upper and lower α-level sets.


Introduction
The properties of fuzzy numbers and the arithmetic of fuzzy quantities (or fuzzy numbers) have been studied for a long time.The interesting issue for studying the additive inverse of a fuzzy number may refer to Hong and Do [1], Vrba [2] and Wu [3,4].Also, Anzilli and Facchinetti [5], Bodjanova [6], Dubois and Prade [7] investigated the median, mean and variance of fuzzy numbers.Wang et al. [8] studied the two-dimensional discrete fuzzy numbers.Mitchell and Schaefer [9] and Yager and Filev [10] studied the orderings of fuzzy numbers.On the other hand, Deschrijver [11,12] studied the arithmetic operators in interval-valued fuzzy set.Ban and Coroianu [13] investigated the approximation of fuzzy numbers.Guerra and Stefanini [14] studied the approximation of arithmetic of fuzzy numbers.Holčapek and Štěpnička [15] studied a new framework for arithmetics of extensional fuzzy numbers.Stup ňanová [16] used a probabilistic approach to study the arithmetics of fuzzy numbers.Wu [17] used the decomposition and construction of fuzzy sets to study the arithmetic operations on fuzzy quantities.In this paper, we shall study the dual arithmetic of fuzzy sets by considering the dual membership function.
The α-level set of a fuzzy set will be called the upper α-level set.In this paper, we shall define the so-called lower α-level set.The well-known (primal) decomposition theorem says that the membership function of a fuzzy set can be expressed in terms of the characteristic function of (upper) α-level sets.In this paper, we are going to establish the so-called dual decomposition theorem by showing that the membership function can be expressed in terms of lower α-level sets.
On the other hand, the concept of dual fuzzy set will be proposed by considering theso-called dual membership function.Based on the dual membership functions, we shall also study the so-called dual arithmetic of fuzzy sets in R. The definition of arithmetic operations is based on the supremum and minimum of membership functions.Inspired by its form, we shall define the so-called dual arithmetic operations based on the infimum and maximum of dual membership functions.A duality relation is also established between the arithmetics and dual arithmetics.
In Section 2, we introduce the concept of lower α-level sets and present some interesting properties that will be used in the subsequent discussion.In Section 3, we introduce the concept of dual fuzzy set and present some interesting results based on the lower α-level sets.In Section 4, we establish the so-called dual decomposition theorem.In Section 5, we introduce the dual arithmetics of fuzzy sets in R and establish some interesting results based on the upper and lower α-level sets.

Lower and Upper Level Sets
Let Ã be a fuzzy subset of a universal set U with membership function denoted by ξ Ã.For α ∈ (0, 1], the α-level set of Ã is denoted and defined by Ãα = {x ∈ U : ξ Ã(x) ≥ α} . ( For α ∈ [0, 1), we also define The support of a fuzzy set Ã within a universal set U is the crisp set defined by The definition of 0-level set is an important issue in fuzzy sets theory.If the universal set U is endowed with a topology τ, then the 0-level set Ã0 can be defined as the closure of the support of Ã, i.e., Ã0 = cl Ã0+ . ( If U is not endowed with a topological structure, then the intuitive way for defining the 0-level set is to follow the equality (1) for α = 0.In this case, the 0-level set of Ã 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.
Let Ã be a fuzzy set in U with membership function ξ Ã.The range of ξ Ã is denoted by R(ξ Ã) that is a subset of [0, 1].We see that the range R(ξ Ã) can be a proper subset of [0, 1] with R(ξ Ã) = [0, 1].Define Remark 1.We have the following observations.
Therefore we have the following interesting and useful result.
Remark 2. Recall that Ã is called a normal fuzzy set in U if and only if there exists x ∈ U such that ξ Ã(x) = 1.
In this case, we have I * Ã = [0, 1].However, the range R(ξ Ã) is not necessarily equal to [0, 1] even though Ã is normal, since the membership function of Ã is not necessarily a continuous function.
The 0-level set Ã0 of Ã is also called the proper domain of Ã, since ξ Ã(x) = 0 for all x ∈ Ã0 .For α ∈ I * Ã, the α-level set Ãα of Ã may be called the upper α-level set (or upper α-cut) of Ã.We also see that Next we shall consider the so-called lower α-level set (or lower α-cut) of Ã.

Definition 1.
Let Ã be a fuzzy set in a topological space U with proper domain Ã0 .For α ∈ [0, 1], the following set We remark that the lower α-level set α Ã is considered in the proper domain Ã0 rather than the whole universal set U. In general, it is clear to see that Next, we present some interesting observations.We first recall that the notation x ∈ A \ B means x ∈ A and x ∈ B.

Remark 3.
Let Ã be a fuzzy set in U with range R(ξ Ã).Recall the notations Then we have the following observations.

It is also obvious that
Regarding the lower α-level sets, from the first observation of Remark 3, we have the following interesting and useful result.Proposition 2. Let Ã be a fuzzy set in U with membership function ξ Ã. Define α • Ã = inf R(ξ Ã) and . However, the range R(ξ Ã) is not necessarily equal to [0, 1] when Ã is normal, since the membership function of Ã is not necessarily a continuous function.

Based on the interval I •
Ã in Proposition 2, we present some basic properties of lower α-level sets, which will be used in the further study.Proposition 3. Let Ã be a fuzzy set in U. Then we have the following results.
Then we obtain the desired equalities and inclusions.
Based on the interval I * Ã in Proposition 1, we can similarly obtain the following results.
Proposition 4. Let Ã be a fuzzy set in U. Then we have the following results.
Let f : S → R be a real-valued function defined on a convex subset S of a real vector space U. Recall that f is quasi-convex on S if and only if, for each x, y ∈ S, the following inequality is satisfied: for each 0 < λ < 1.It is well-known that f is quasi-convex on S if and only if the set {x ∈ S : f (x) ≤ α} is convex for each α ∈ R. We also recall that f is quasi-concave on S if and only if − f is quasi-convex on S.More precisely, the real-valued function f is quasi-concave on S if and only if for each 0 < λ < 1.We also have that f is quasi-concave on S if and only if the set {x ∈ S : f (x) ≥ α} is convex for each α ∈ R.
Let U be a vector space endowed with a topology, and let Ã be a fuzzy subset of U with membership function ξ Ã.It is well-known that the membership function ξ Ã is quasi-concave if and only if the α-level set Ãα is a convex subset of U for each α ∈ (0, 1].In this case, the union 0<α≤1 Ãα is also a convex subset of U.This says that the upper zero-level set Ã0 = cl 0<α≤1 Ãα is a closed and convex subset of U. In particular, if U = R then the convex set Ãα is reduced to be an interval for α ∈ [0, 1]. Let f : (U, τ U ) → R be a real-valued function defined on a topological space (U, τ U ). Recall that f is upper semi-continuous on U if and only if {x ∈ U : f (x) ≥ λ} is a closed subset of U for all λ ∈ R, and f is lower semi-continuous on U if and only if {x ∈ U : f (x) ≤ λ} is a closed subset of U for all λ ∈ R. It is clear to see that if f is upper semi-continuous on U then − f is lower semi-continuous on U, and if f is lower semi-continuous on U then − f is upper semi-continuous on U. Definition 2. Let U be a vector space endowed with a topology, and let Ã be a fuzzy subset of U with membership function ξ Ã.We denote by F cc (U) the family of all fuzzy subsets of U such that each Ã ∈ F cc (U) satisfies the following conditions.

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The membership function ξ Ã is upper semi-continuous and quasi-concave on U.

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The upper 0-level set Ã0 is a compact subset of U.
In particular, if U = R then each Ã ∈ F cc (R) is called a fuzzy interval.If the fuzzy interval ã is normal and the upper 1-level set ã1 is a singleton set {a}, where a ∈ R, then ã is also called a fuzzy number with core value a. Usually, we write the upper case Ã to denote the fuzzy interval, and write the lower case ã to denote the fuzzy number.
The upper semi-continuity and quasi-concavity says that each upper α-level set Ãα is a closed and convex subset of U for α ∈ [0, 1].Recall that, in a topological space, each closed subset of a compact set is a compact set.Since Ãα ⊆ Ã0 for α ∈ (0, 1], it follows that each upper α-level set Ãα is also a compact set for α ∈ (0, 1].
Suppose that Ã is a fuzzy interval.Then the upper 0-level set Ã0 is a closed and bounded subset of R. Also, the convexity, boundedness and closedness of each upper α-level set Ãα says that it is a bounded closed interval for α ∈ [0, 1].More precisely, we have In particular, if ã is a fuzzy number, then ãα = [ ãL α , ãU α ] for all α ∈ [0, 1] and ãL 1 = ãU 1 for α = 1, i.e., the upper 1-level set ã1 = { ãL 1 = ãU 1 = a} is a singleton set, where a is the core value.
Example 1.Let Ã be a fuzzy interval.Then the upper α-level set Ãα is a closed interval given by Ãα Since Ãα ⊆ Ãβ for β < α, if we further assume that the end-points ÃL α and ÃU α are continuous functions with respect to α on This also says that the membership function ξ Ã of Ã is lower semi-continuous.Therefore we conclude that the membership function of Ã is continuous.We also see that the lower 1-level set is Suppose that the minimum min R(ξ Ã) exists, i.e., Then the lower α • Ã-level set of Ã consists of two points as We also see that

Dual Fuzzy Sets
Let Ã be a fuzzy set in U with membership function ξ Ã defined on U. Recall that the membership function of complement fuzzy set of Ã is denoted and defined by ξ Ãc = 1 − ξ Ã that is also defined on U. Since the 0-level set Ã0 is treated as the proper domain of Ã, we consider the restrict function Then we use the notation Ã to denote the dual fuzzy set of Ã.The membership function of Ã is given in ( 6) that is also called a dual membership function.We also remark that ξ Ã = ξ Ãc , since their domains are different.

Remark 5.
The relationships between the α-level sets of Ã and Ã are given below Let us recall that notation I * Ã in Proposition 1 and the notation I • Ã in Proposition 2. Then we have the following interesting results.Proposition 5. Let Ã be a fuzzy set in U with the dual fuzzy set Ã .For α, β ∈ [0, 1] with α + β = 1, we have the following properties.
Proof.We first have The equalities (7) also say that the maximum max R(ξ Ã) exists if and only if the minimum min R(ξ Ã ) exists.To prove part (i), suppose that α ∈ I * Ã.We have two cases.
For the converse, suppose that β ∈ I • Ã .Then we can similarly show that α ∈ I * Ã.Part (ii) follows from Remark 5 and part (i) immediately.This completes the proof.
Let Ã ∈ F cc (U).Then the membership function ξ Ã is upper semi-continuous and quasi-concave on Ã0 .It is also clear to see that the membership function ξ Ã = 1 − ξ Ã of dual fuzzy set Ã is lower semi-continuous and quasi-convex on Ã0 .This says that the lower α-level set α Ã is a closed and convex subset of Ã0 for α ∈ (0, 1].
Example 2. Let Ã be a fuzzy interval with dual fuzzy set Ã .Since the upper 0-level set Ã0 is bounded and the lower α-level set α Ã is a closed and convex subset of R for α ∈ (0, 1], it follows that α Ã is also a bounded closed interval given by Using part (ii) of Proposition 5, we also have This shows that From part (iv) of Proposition 3, for α ∈ I • Ã with α > 0, we have Since β Ã ⊆ α Ã for β < α, if we further assume that the end-points β Ã L and β Ã U are continuous functions with respect to β on [0, 1], then α− Ã = ( α Ã L , α Ã U ) is an open interval.In this case, from Remark 3, for α ∈ I * Ã with α > 0, the upper α-level set Ã α is given by This also says that the membership function of Ã is upper semi-continuous.Therefore, we conclude that the membership functions of Ã and Ã are continuous.

Dual Decomposition Theorems
Let A be a subset of U. The characteristic function χ A of A is defined to be Now we define the so-called dual characteristic function χ A of A as follows It is clear to see that Let Ã be a normal fuzzy set in U.The well-known (primal) decomposition theorem says that the membership function ξ Ã can be expressed as where χ Ãα is the characteristic function of the α-level set Ãα .If Ã is not normal, then we can also show that The (primal) decomposition theorem says that the membership function can be expressed in terms of upper α-level sets.In the sequel, we are going to show that the membership function can also be expressed in terms of lower α-level sets as the following form where χ α Ã is the dual characteristic function of lower α-level set α Ã. Proposition 6.Let Ã be a fuzzy set in a vector space U that is also endowed with a topology.Given any fixed x ∈ U, we have the following results.
(i) Suppose that the minimum min R(ξ Ã) exists.Then the function (ii) Suppose that the maximum max R(ξ Ã) exists.Then the function η Proof.To prove part (i), from Proposition 2, we see that I • Ã is a closed interval.We need to show that the following set Ã is also closed.Therefore we remain to show that F r is closed for each r ∈ (0, 1).Now, for each α ∈ cl(F r ), there exists a sequence {α n } ∞ n=1 in F r such that α n → α, i.e., α n ≤ r and x ∈ α n Ã for all n.Then we have We also see that there exists a subsequence {α This says that α ∈ F r , since α ≤ r.

•
Suppose that α n k ↓ α.Since x ∈ α n k Ã for all k, using part (i) of Proposition 3, we have x ∈ α Ã.This says that α ∈ F r , since α ≤ r.
Therefore, we conclude that cl(F r ) ⊆ F r , i.e., F r is closed.
To prove part (ii), from Proposition 1, we see that I * Ã is a closed interval.We need to show that the following set singleton set is also closed.Therefore we remain to show that F r is closed for each r ∈ (0, 1).Now, for each α ∈ cl(F r ), since r > 0, we have α > 0. Therefore, there exists a sequence {α n } ∞ n=1 such that α n → α and α n ∈ F r for all n, i.e., α n ≥ r and x ∈ Ãα n for all n.Then we have α ≥ r.We also see that there exists a subsequence {α , α ≤ α n k for all k, then x ∈ Ãα , since Ãα n k ⊆ Ãα for all k by part (i) of Proposition 4. This says that α ∈ F r , since α ≥ r.

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If α n k ↑ α, Then since x ∈ Ãα n k for all k by part (ii) of Proposition 4, we have x ∈ Ãα .This says that α ∈ F r , since α ≥ r.
Therefore, we conclude that cl(F r ) ⊆ F r , i.e., F r is closed.This completes the proof.
Theorem 1. (Dual Decomposition Theorem) Let Ã be a fuzzy set in U with proper domain Ã0 .For x ∈ Ã0 , the membership degree ξ Ã(x) can be expressed in terms of lower α-level sets as follows where I • Ã is given in (5).
This shows that the equalities in (8) are satisfied.Now we assume α 0 > 0. Then Since α 0 ∈ I • Ã, the above supremum is attained.It means that The above arguments are still valid when I • Ã is replaced by R(ξ Ã).Therefore we obtain the desired equalities.This completes the proof.Remark 6.The decomposition theorem for dual fuzzy set Ã based on the upper α-level sets of Ã is given by According to Theorem 1, the dual decomposition theorem for Ã based on the lower α-level sets of Ã is given by Next we are going to present the dual decomposition theorem on a countable set.We write , where Q denotes the set of all rational numbers.It well-known that the countable set Q is dense in R.This means that, given any r ∈ R, there exist two sequences {p n } ∞ n=1 and {q n } ∞ n=1 in the countable set Q such that p n ↑ r and q n ↓ r as n → ∞.

Dual Arithmetics of Fuzzy Sets
Let Ã and B be fuzzy numbers in R; that is, Ã and B are normal fuzzy sets in R satisfying some elegant structures such that their α-level sets turn into the bounded closed intervals in R. Then we have the following well-known equality where the upper α-level sets are considered.For convenience, we use the same notation • to denote the operations for the α-level sets Ãα • Bα and the real numbers a • b.In this paper, we shall consider the general fuzzy sets in R rather than the fuzzy numbers to establish the similar equality based on the lower α-level sets.
Let denote any one of the four basic arithmetic operations ⊕, , ⊗, between fuzzy sets Ã and B in R. The membership function of Ã B is defined by for all z ∈ R, where the operation • ∈ {+, −, ×, /}, respectively.Since the 0-level sets Ã0 and B0 are the proper domain of Ã and B, respectively, i.e., ξ Ã(x) = 0 for x ∈ Ã0 and ξ B(x) = 0 for x ∈ B0 , we have Inspired by the above expression ( 13), we define a new operation between Ã and B using the dual membership functions as follows We need to emphasize that inf However, this operation Ã B is reasonable, since we consider the proper domains as shown in (13).Then we have Therefore we say that Ã B is the dual arithmetic of Ã B. This means that, instead of calculating Ã B, we can alternatively calculate Ã B and use the duality (14) to recover Ã B. We are going to study the lower α-level sets of dual arithmetic Ã B and establish the relationships between Ã B and Ã B. Let Ã and B be two fuzzy sets in R with membership functions ξ Ã and ξ B, respectively.Let From Proposition 1 and (3), we see that Ãα = ∅ for α ∈ I * Ã, where I * Ã is given by Similarly, we also see that Bα = ∅ for α ∈ I * B, where I * B is given by For further discussion, we need a simple lemma.
Lemma 1.Let f be a real-valued function defined on A, and let k be a constant.Then Proof.We have Another equality can be similarly obtained.This completes the proof.Proposition 7. Let Ã and B be two fuzzy sets in R. Then the following statements hold true. with We also have (ii) We have with We also have α Proof.To prove part (i), let α * Ã and α * B be defined in (15).Then ξ Ã(x) ≤ α * Ã and ξ B(y) ≤ α * B for all x ∈ Ã0 and y ∈ B0 .It follows that min {ξ Ã(x), ξ B(y)} ≤ min α * Ã, α * B for all x ∈ Ã0 and y ∈ B0 , which implies for all z ∈ U.This says that min{α * Ã, α * B} is an upper bound of function ξ Ã B. Suppose that η ≥ ξ Ã B(z) for all z ∈ U. Then η ≥ min{ξ Ã(x), ξ B(y)} for all x ∈ Ã0 , y ∈ B0 and z ∈ U. Using Lemma 1, we have This says that min{α * Ã, α * B} is a least upper bound of function ξ Ã B. By the definition of supremum, we obtain the desired equality (16).The interval I * Ã B follows from Proposition 1 immediately.
To prove part (ii), we first note that Since for all x ∈ Ã0 and y ∈ B0 , which implies This says that max 1 − α * Ã, 1 − α * B is a greatest lower bound of function ξ Ã B. By the definition of infimum, we obtain the desired equality (17).The interval I • Ã B follows from Proposition 2 immediately.This completes the proof.
We write I * ,∩ = I * Ã ∩ I * B. Then I * ,∩ = ∅ is given by From part (i) of Proposition 7 by referring to ( 16), we see that Let S be a nonempty subset in a topological space (U, τ).Recall that S is compact if and only if, for every sequence {x n } ∞ n=1 in S, there exists a convergent subsequence {x n k } ∞ k=1 in S. If the limit of {x n k } ∞ k=1 is denoted by x 0 , then x 0 is in S. In particular, if U = R n , then S is compact if and only if S is closed and bounded.We need a useful lemma.Lemma 2. (Royden ([18] 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.
(i) If f is lower semi-continuous, then f assumes its minimum on a compact subset of U; that is, the infimum is attained in the following sense inf (ii) 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 If α ∈ I * ,∩ then Ãα = ∅ or Bα = ∅.Therefore, in order to consider the operation we need to take α ∈ I * ,∩ .We also remark that if Ã and B are normal fuzzy sets then Theorem 3. Let Ã and B be two fuzzy sets in R with the dual fuzzy sets Ã and B , respectively.Suppose that the arithmetic operations ∈ {⊕, , ⊗} correspond to the operations • ∈ {+, −, * }.Then the following statements hold true.
We have the following inclusion (iii) Suppose that the membership functions of Ã and B are upper semi-continuous.Then (iv) Suppose that the membership functions of Ã and B are upper semi-continuous, and that the supports Ã0+ and B0+ are bounded.Then Proof.To prove part (i), since To prove part (ii), for α ∈ I * ,∩ with α > 0 and z α ∈ Ãα • Bα , since Ãα = ∅ and Bα = ∅, there exist x α ∈ Ãα and y α ∈ Bα such that z α = x α • y α for • ∈ {+, −, * }, where ξ Ã(x α ) ≥ α and ξ B(y α ) ≥ α.Therefore, we have which says that z α ∈ ( Ã B) α .This shows that Ãα • Bα ⊆ ( Ã B) α for α ∈ I * ,∩ with α > 0. Now, for α = 0 and z 0 ∈ Ã0 • B0 , there also exist x 0 ∈ Ã0 and y 0 ∈ B0 such that Then we see that z n → x 0 • y 0 = z 0 , since the binary operation • ∈ {+, −, * } is continuous.We also have This shows that Ã0 • B0 ⊆ ( Ã B) 0 .Therefore we conclude that Ãα • Bα ⊆ ( Ã B) α for α ∈ I * ,∩ .To prove part (iii), in order to prove another direction of inclusion, we further assume that the membership functions of Ã and B are upper semi-continuous; that is, the nonempty α-level sets Ãα and Bα are closed subsets of R for all α ∈ I * ,∩ .Given any α ∈ I * ,∩ with α > 0 and z α ∈ ( Ã B) α , we have sup Since z α is finite, it is clear to see that F ≡ {(x, y) : We also see that the function g(x, y) = x • y is continuous on R 2 .Since the singleton set {z α } is a closed subset of R, it follows that the inverse image F = g −1 ({z α }) of {z α } is also a closed subset of R 2 .This says that F is a compact subset of R 2 .Now we want to show that the function f (x, y) = min{ξ Ã(x), ξ B(y)} is upper semi-continuous, i.e., we want to show that {(x, y) : f (x, y) ≥ α} is a closed subset of R 2 for any α ∈ R.
We do not consider the operation in Theorem 3. The reasons is that the case of zero denominator should be avoided.We also remark that the arguments in the proof of Theorem 3 are still available for the operation by carefully excluding the zero denominator.In order not to complicate the proof of Theorem 3, we omit the case of operation .
Let Ã and B be two fuzzy sets in R with the dual fuzzy sets Ã and B , respectively.We define From Proposition 2 and (5), we see that α Ã = ∅ for α ∈ I • Ã , where I • Ã = ∅ is given by Similarly, we also see that α B = ∅ for α ∈ I • B , where I • B is given by We write From part (ii) of Proposition 7 by referring to (17), we see that Let (U, τ U ) be a topological space, and let A be a subset of U. Then the subset A can be endowed with a topology τ A such that (A, τ A ) is a topological subspace of (U, τ U ).In other words, the subset C of A is a τ A -closed subset of A if and only if C = A ∩ D for some τ U -closed subset D of U. In this case, we say that f : (A, τ A ) → R is upper semi-continuous on A if and only if {x ∈ A : f (x) ≥ λ} is a τ A -closed subset of A for all λ ∈ R. We also see that if f is upper semi-continuous on A then − f is lower semi-continuous on A, and if f is lower semi-continuous on A then − f is upper semi-continuous on A. We have the following observations.

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Suppose that Ã is a fuzzy set in U such that its membership function ξ Ã is upper semi-continuous on U. Then ξ Ã is also upper semi-continuous on the proper domain Ã0 .Indeed, the set Suppose that Ã is a fuzzy set in U such that its membership function ξ Ã is upper semi-continuous on the proper domain Ã0 .Then it is clear to see that the dual membership function ξ Ã = 1 − ξ Ã of Ã is lower semi-continuous on Ã0 .
Therefore the function f (x, y) is indeed lower semi-continuous on Ã0 × B0 .By Lemma 2, the function f assumes minimum on the compact subset F of Ã0 × B0 ; that is, from (24), we have min In other words, there exists i.e., ξ Ã (x α ) ≤ α and ξ B (y α ) ≤ α.Therefore, we obtain x α ∈ α Ã and y α ∈ α B , which says that . This completes the proof.
The related results regarding the mixed lower and upper α-level sets are presented below.Recall that the 0-level set ( Ã B) 0 is the proper domain of the membership function ξ Ã B of Ã B. Theorem 5. Let Ã and B be two fuzzy sets in R. Consider that the arithmetic operations ∈ {⊕, , ⊗} correspond to the operations • ∈ {+, −, * }.Suppose that the membership functions of Ã and B are upper semi-continuous.Then we have the following results.
We further assume that the supports Ã0+ and B0+ are bounded.Then the above 0-level set ( Ã B) 0 can be replaced by Ã0 • B0 .
This completes the proof.
Theorem 6.Let Ã and B be two fuzzy sets in R with the dual fuzzy set Ã and B , respectively.Consider the dual arithmetic operations ∈ { , , } correspond to the operations • ∈ {+, −, * }.Suppose that the membership functions of Ã and B are upper semi-continuous on Ã0 and B0 , respectively.Then we have the following results.