Equivalent Structures of Interval Sets and Fuzzy Interval Sets Bao Qing Hu,1,3,∗ Heung Wong,2 Ka-fai Cedric Yiu2 1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, People’s Republic of China 3 Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, People’s Republic of China Ideas of interval sets come from the lower and upper approximations of rough sets to study a unified structure of rough sets and their generalizations. Starting from the interval sets and their operations, this paper summarizes and analyzes other sets that have similarities with the interval sets or fuzzy interval sets. Our conclusions are that interval sets are mathematically equivalent to shadowed sets and flou sets, respectively, and fuzzy interval sets are mathematically equivalent to C 2017 Wiley Periodicals, interval-valued fuzzy sets and intuitionistic fuzzy sets, respectively. Inc. 1. INTRODUCTION The concept of rough sets was proposed by Pawlak in 1982.1 Subsequently, rough sets were extended to rough fuzzy sets,2 fuzzy rough sets,2 generalized rough fuzzy sets,3 generalized fuzzy rough sets,3,4 generalized fuzzy rough sets based on triangle norm,5 generalized fuzzy rough sets based on logic operators,6–8 intervalvalued fuzzy rough sets,9 generalized interval-valued fuzzy rough sets,10–12 and so on. In the extension process, equivalence relations with regard to attributes are extended to fuzzy equivalence relations, general fuzzy relations, and interval-valued fuzzy relations; classic sets with regard to objects are extended to fuzzy sets and interval-valued fuzzy sets; operators are extended to interval-valued fuzzy t-norm, interval-valued fuzzy R-implication, and interval-valued implication.6,11 . No matter what kind of rough sets is considered, there are lower and upper approximations in the conceptual formulation. All kinds of concepts are approximately expressed by their lower approximations or upper approximations. After studying rough set and its various expansions, Yao introduced the concepts of ∗ Author to whom all correspondence should be addressed; e-mail: bqhu@whu.edu.cn INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 00, 1–25 (2017) 2017 Wiley Periodicals, Inc. View this article online at wileyonlinelibrary.com. • DOI 10.1002/int.21940 C 2 HU, WONG, AND YIU interval sets from approximation structures of rough sets.13,14 Particularly in 14 , Yao clearly described the interval sets by the lower and upper approximations of rough sets. Interval set features of a variety of rough sets, for which the lower and upper bounds are their lower and upper approximations, respectively, show that researches on rough sets can be unified to the interval set. This is one of the motivations for the research on interval sets. There are many researches on interval sets now, such as concept of interval sets and their algebraic structures,13–21 extension of interval sets,20 comparison of interval sets,15,22–25 reasoning researches of interval sets,,24–28 applications of interval sets,29–31 and so on. In our studies, we find a number of similar concepts such as interval-valued fuzzy sets, flou sets, shadowed sets, and intuitionistic fuzzy sets, and so on. What relationships are there between interval sets and these concepts? This paper wants to delve into this problem. In the course of study, we show that interval sets are mathematically equivalent to shadowed sets and flou sets, respectively, and the fuzzy interval sets are mathematically equivalent to interval-valued fuzzy sets and intuitionistic fuzzy sets, respectively. The rest of this paper is organized as follows. Section 2 studies (fuzzy) interval sets and their algebraic structures. In this section, first, L-fuzzy interval sets are defined and their operations are discussed, such that interval sets and fuzzy interval sets are their special cases. And also we give measurement methods of interval sets and discuss interval sets based on partition. In Section 3, we study the relationships between interval sets and other similar sets, such as interval-valued fuzzy sets, shadowed sets, flou sets, intuitionistic fuzzy sets, rough sets, and three-way decisions (3WD). The last section concludes this paper. 2. INTERVAL SETS AND FUZZY INTERVAL SETS In this paper, I = [0, 1] or I = ([0, 1], ∨, ∧, c, 0, 1), where ∧, ∨, and c are, respectively, defined by x ∨ y = max {x, y} , x ∧ y = min {x, y} , and x c = 1 − x. (L, ∨, ∧, N, 0, 1) denotes a fuzzy lattice32 or complete De Morgan algebra,33 where 0 and 1 are the minimum and maximum in L, respectively. Further, ≤ is its order relation and N is an involution negator over L, i.e. 1. N(N(x)) = x, ∀x ∈ L and 2. x ≤ y ⇒ N(y) ≤ N(x), ∀x, y ∈ L. International Journal of Intelligent Systems DOI 10.1002/int INTERVAL SETS AND FUZZY INTERVAL SETS 3 Let FL (X) be a family of all L-fuzzy sets over X, i.e., FL (X) = {A|A : X → L}.34 While L = [0, 1], F (X) denotes a class of all fuzzy sets over X, i.e., F (X) = {A|A : X → [0, 1]}.35 While L = {0, 1}, P (X) denotes a family of all subsets of X. In the following, we assume that ∪, ∩, ·N , and ⊆ are union, intersection, complement, and order relation over FL (X), respectively, i.e. ∀x ∈ L (A ∩ B)(x) = A(x) ∧ B(x), (A ∪ B)(x) = A(x) ∨ B(x), AN (x) = N(A(x)), A ⊆ B ⇔ A(x) ≤ B(x), and A − B = A ∩ B N for A, B ∈ FL (X). While L = [0, 1], N(x) = 1 − x, AN is written as Ac , i.e., Ac (x) = 1 − A(x), ∀x ∈ [0, 1]. 2.1. Fuzzy Interval Sets and Their Operations Yao introduced the concept of interval set on finite set.13 The following interval set is not limited in finite universe. DEFINITION 1. 13,19 Let X be a universe and Al , Au ⊆ X, Al ⊆ Au . Then A = [Al , Au ] = {A ⊆ X|Al ⊆ A ⊆ Au } is called an interval set of X. The family of all interval sets of X is shown by I (P (X)). Naturally, the concept of interval set can be generalized to fuzzy sets (L-fuzzy sets). DEFINITION 2. Let X be a universe and Al , Au ∈ FL (X) with Al ⊆ Au . Then A = [Al , Au ] = A ∈ FL (X)|Al ⊆ A ⊆ Au is called an L-fuzzy interval set of X. The family of all L-fuzzy interval sets of X is shown by I (FL (X)). Specially if L = [0, 1], an L-fuzzy interval set of X is called as a fuzzy interval set of X and the family of all fuzzy interval sets of X is shown by I (F (X)). For two L-fuzzy interval sets A = [Al , Au ] and B = [Bl , Bu ], we define A B = {A ∩ B|A ∈ A, B ∈ B}, A B = {A ∪ B|A ∈ A, B ∈ B}, A\B = {A − B|A ∈ A, B ∈ B}, ¬A = [X, X]\A. International Journal of Intelligent Systems DOI 10.1002/int 4 HU, WONG, AND YIU It follows from Definition 2 that A B = [Al ∩ Bl , Au ∩ Bu ], A B = [Al ∪ Bl , Au ∪ Bu ], A\B = [Al − Bu , Au − Bl ], ¬A = [(Au )N , (Al )N ]. Inclusion relation of L-fuzzy interval sets is defined as follows: [Al , Au ] [Bl , Bu ] ⇔ Al ⊆ Bl and Au ⊆ Bu ⇔ (∀A ∈ [Al , Au ], ∃B ∈ [Bl , Bu ], s.t. A ⊆ B) and (∀B ∈ [Bl , Bu ], ∃A ∈ [Al , Au ], s.t. A ⊆ B) (i) For L-fuzzy interval sets Ai = [A(i) l , Au ], i ∈ (any index set), we define (i) , ∪ Al , ∪ A(i) u i∈ i∈ i∈ i∈ (i) (i) (i) ] = A , A . Ai = [Al , A(i) ∩ ∩ u u l (i) Ai = [Al , A(i) u ]= i∈ i∈ i∈ i∈ The following is immediate from the definition of interval sets and their operations. PROPOSITION 1. L-interval sets algebra (I (FL (X)), , , ¬, ∅, X ) is a fuzzy lattice. Specially, interval sets algebra (I (P (X)), , , ¬, ∅, X )23 and fuzzy interval sets algebra(I (F (X)), , , ¬, ∅, X ) are fuzzy lattices. The operations and are different from intersection ∩ and union ∪ of classic sets, respectively. For two interval sets A = [Al , Au ] and B = [Bl , Bu ], then A ∩ B = [Al , Au ] ∩ [Bl , Bu ] = A ∪ B = [Al , Au ] ∪ [Bl , Bu ]. [Al ∪ Bl , Au ∪ Bu ], Al ∪ Bl ⊆ Au ∩ Bu [∅, ∅], otherwise A ∩ B is an interval set. But A ∪ B is not necessarily an interval set, which can be illustrated by the following example. International Journal of Intelligent Systems DOI 10.1002/int INTERVAL SETS AND FUZZY INTERVAL SETS 5 Figure 1. Inclusion relationship of power sets for classic sets EXAMPLE 1. Let X = {a, b, c} and consider classic inclusion relation of P (X), as shown in Figure 1. Then A = [{a}, {a, b, c}] = {{a}, {a, b}, {a, c}, {a, b, c}} and B = [{c}, {a, b, c}] = {{c}, {a, c}, {b, c}, {a, b, c}} are interval sets on X. But A ∪ B = {{a}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} is not an interval set. DEFINITION 3. Let X be a universe and Al , Au ⊆ FL (X). Then [∅, Au ] = {A ⊆ X|A ⊆ Au } and [Al , X] = {A ⊆ X|Al ⊆ A} are called a u-interval set and l-interval set, respectively, written as (Au ] and [Al ), respectively. The family of all u-interval sets and l-interval sets of X are denoted as Iu (FL (X)) and Il (FL (X)), respectively. It follows from Definition 3 that if (Au ] ∈ Iu (FL (X)), [Al ) ∈ Il (FL (X)), and Al ⊆ Au , then [Al , Au ] ∈ I (FL (X)). DEFINITION 4. An interval set [Al , Au ] of X is called definable, if Al ⊆ A ⊆ Au implies A = Al or A = Au . Specially, a definable interval set (Au ](Au = ∅) of X is called an atom interval set of X. International Journal of Intelligent Systems DOI 10.1002/int 6 HU, WONG, AND YIU Semantically speaking, the real extension of concept shown by a definable interval set [Al , Au ] could only be Al or Au . For example, [{a}, {a, b}] is a definable interval set and ({a}] is an atom interval set of X. It is clear (X] = P (X) = ({a}]. a∈X 2.2. Implication of Interval Sets Logic implication is an important operation in logical algebra. In the following, we discuss implication operation over L-fuzzy interval sets. First, we consider implication over L-fuzzy sets as follows: For two L-fuzzy sets A, B ∈ FL (X), we define A → B = ∪ C ∈ FL (X) : A ∩ C ⊆ B . For two L-fuzzy interval sets A = [Al , Au ] and B = [Bl , Bu ] on X, we define A → B = [Cl , Cu ] ∈ I (FL (X)) : [Al , Au ] [Cl , Cu ] [Bl , Bu ] . PROPOSITION 2. For two L-fuzzy interval sets A = [Al , Au ] and B = [Bl , Bu ] on X, the following holds: A → B = [(Al → Bl ) ∩ (Au → Bu ) , Au → Bu ] . Proof. A → B = {[Cl , Cu ] ∈ I (FL (X)) : [Al , Au ] [Cl , Cu ] [Bl , Bu ]} = {[Cl , Cu ] ∈ I (FL (X)) : Al ∩ Cl ⊆ Bl , Au ∩ Cu ⊆ Bu } = [(∪{Cl ∈ FL (X) : Al ∩ Cl ⊆ Bl }) ∩ (∪{Cu ∈ FL (X) : Au ∩ Cu ⊆ Bu }), ∪{Cu ∈ FL (X) : Au ∩ Cu ⊆ Bu }] = [(Al → Bl ) ∩ (Au → Bu ), Au → Bu ]. It is worth noting that A → B = [Al → Bl , Au → Bu ] is not always true, since ∪{Cl ∈ FL (X) : Al ∩ Cl ⊆ Bl } ⊆ ∪{Cu ∈ FL (X) : Au ∩ Cu ⊆ Bu } is not always true. It can be illustrated by the following example. EXAMPLE 2. Consider Example 1 again, i.e., X = {a, b, c} and P (X). For two interval sets A = [{a}, {a, b, c}] = {{a}, {a, b}, {a, c}, {a, b, c}} and B1 = [{c}, {a, b, c}] = {{c}, {a, c}, {b, c}, {a, b, c}}, International Journal of Intelligent Systems DOI 10.1002/int INTERVAL SETS AND FUZZY INTERVAL SETS 7 We have : {a} ∩ C ⊆ {c} = ∪ {{b}, {c}, {b, c}} {a} → {c} = ∪ Cl ∈ P (X) l = {b, c} and {a, b, c} → {a, b, c} = ∪ Cu ∈ P (X) : {a, b, c} ∩ Cu ⊆ {a, b, c} = {a, b, c}. Thus A → B1 = [{b, c}, {a, b, c}]. If we consider B2 = [{c}, {c}], then {a, b, c} → {c} = ∪ Cu ∈ P (X) : {a, b, c} ∩ Cu ⊆ {c} = ∪{{c}} = {c}. So ({a} → {c}) ⊃ ({a, b, c} → {c}). Thus A → B2 = [{b, c} ∩ {c}, {c}] = [{c}, {c}]. 2.3. Measurement of Interval Sets Inaccuracy of interval sets is caused by the existence of boundary domain. The greater boundary domain of interval sets is, the lower the accuracy of its conceptual description is. To more accurately express it, we introduce the concept of information certainty degree and probability certainty degree. DEFINITION 5. Let A = [Al , Au ] ∈ I (F (X)) and X = {x1 , x2 , · · · , xn }. Then αI (A) = 1, Au = ∅ Au = ∅ |Al | , |Au | is called the information certainty degree of A, where |A| = ni=1 A(xi ) for all A ∈ F (X). 0, Au = ∅ ρI (A) = 1 − αI (A) = |Au −Al | Au = ∅ |Au | is called the information uncertainty degree of A. Consider Example 1 again, we have αI ([[{a}, {a, b, c}]) = 1 2 , ρ ([{a}, {a, b, c}]) = . Information certainty degree reflects “level” to I 3 3 express the concept of approximation of interval set. The smaller αI (A) is (the larger ρI (A) is), the less known information is. It follows from Definition 5 that αI ([∅, A]) = 0 for any nonempty subset A of X. This means that information certainty degree does not reflect true “size’ of boundary domain. To do it, the following concept is introduced: DEFINITION 6. αP (A) = 1 |A| is called the probability certainty degree of A. Magnitude of boundary can be measured by the probability certainty degree. For Example 1, we have αP ([{a}, {a, b, c}]) = 14 , which means that interval set [{a}, {a, b, c}] is a concept represented by the four possible approximations. For a definable interval set A, it is easy to see αP (A) = 12 or 1. Obviously, αP ([Al , Au ]) = 1 iff Al = Au , which shows the interval set expresses an accurate concept. International Journal of Intelligent Systems DOI 10.1002/int 8 HU, WONG, AND YIU 2.4. Interval Sets Based on Partition Let C = {X1 , X2 , . . . , Xn } be a partition of X, i.e. ∀i ∈ {1, 2, . . . , n}, Xi = n Xi = X. ∅, Xi ⊆ X,Xi ∩ Xj = ∅, i = j and i=1 DEFINITION 7. Let C = {X1 , X2 , . . . , Xn } be a partition of X. For A ⊆ X, if A = ∅, or there is Xij ∈ {X1 , X2 , . . . , Xn }, j = 1, 2, . . . , k, such that A = Xi1 ∪ Xi2 ∪ · · · ∪ Xik , then A is called a certain concept based on the partition C . If Al and Au are certain concepts based on the partition C , then interval set A = [Al , Au ] ∈ I (P (X)) is called a C -certain interval set. PROPOSITION 3. Let C = {X1 , X2 , . . . , Xn } be a partition of X. If A = [Al , Au ] and B = [Bl , Bu ] ∈ I (P (X)) are C -certain interval sets, then so are A B, A B, A\B, and ¬A. Proof. In fact we only need to prove that if A and B are certain concepts based on the partition C , then so are A ∩ B,A ∪ B, A − B, and Ac . 1. Let A and B be certain concepts based on the partition C = {X1 , X2 , . . . , Xn }. Without loss of generality, let us assume that A = ∅ and B = ∅. Then there are Xip ∈ {X1 , X2 , . . . , Xn }, p = 1, 2, . . . , m1 (1 ≤ m1 ≤ n) and Xjq ∈ {X1 , X2 , . . . , Xn }, q = 1, 2, . . . , m2 (1 ≤ m2 ≤ n), such that A = Xi1 ∪ Xi2 ∪ · · · ∪ Xim1 and B = Xj1 ∪ Xj2 ∪ · · · ∪ Xjm2 , respectively. Thus A ∩ B = Xi1 ∪ Xi2 ∪ · · · ∪ Xim1 ∩ Xj1 ∪ Xj2 ∪ · · · ∪ Xjm2 = ∅ or Xk1 ∪ Xk2 ∪ · · · ∪ Xkm3 , where Xkr ∈ {Xi1 , Xi2 , . . . , Xim1 }, 1 ≤ r ≤ m3 ≤ m1 , i.e. A ∩ B is a certain concept based on the partition C . 2. For A ∪ B, conclusion is obvious. 3. If A = Xi1 ∪ Xi2 ∪ · · · ∪ Xim , then Ac = ∅ or Ac = Xj1 ∪ Xj2 ∪ · · · ∪ Xjn−m , where Xjr ∈ {X1 , X2 , . . . , Xn } − {Xi1 , Xi2 , . . . , Xim }. 4. Since A − B = A ∩ B c , it is straightforward from (1) and (3). International Journal of Intelligent Systems DOI 10.1002/int 9 INTERVAL SETS AND FUZZY INTERVAL SETS 3. RELATIONSHIP BETWEEN (FUZZY) INTERVAL SETS AND THE SIMILAR CONCEPTS Idea of interval sets seems to be everywhere. What a relationship does interval set have with interval-valued fuzzy sets, shadowed sets, flou sets, intuitionistic fuzzy sets, rough sets, and 3WD? This problem is discussed in the next section. 3.1 Fuzzy Interval Sets and Interval-Valued Fuzzy Sets There are a number of researches on interval-valued fuzzy sets.36–38 An intervalvalued fuzzy set is called the grey set in 39 and ϕ-fuzzy set in 40 . An interval type-2 fuzzy set is an interval-valued fuzzy set.41 An interval-valued type-2 fuzzy set is a fuzzy set with membership of the interval-valued fuzzy set.42 The fuzzy-valued fuzzy set is a fuzzy set with membership of a fuzzy value.43 The following notations are introduced for the ease of exposition: I (2) = [a − , a + ]|0 ≤ a − ≤ a + ≤ 1 , ā = [a, a], [a − , a + ] ≤ [b− , b+ ] ⇔ a − ≤ b− , a + ≤ b+ . If [ai− , ai+ ] ∈ I (2) , i ∈ (any index set), then we define sup[ai− , ai+ ] = [sup ai− , sup ai+ ], inf [ai− , ai+ ] = [inf ai− , inf ai+ ]. i∈ i∈ i∈ i∈ i∈ i∈ DEFINITION 8. 38 A mapping A : X → I (2) is called an interval-valued fuzzy set of X and A(x) is its membership function. FI (2) (X) is a family of all interval-valued fuzzy sets of X. Figure 2 shows interval set characteristic of the interval-valued fuzzy set. For A = [A− , A+ ], B = [B − , B + ] ∈ FI (2) (X), we define union, intersection, and complement pointwise by the formulas (A ∪ B)(x) = [A− (x) ∨ B − (x), A+ (x) ∨ B + (x)], (A ∩ B)(x) = [A− (x) ∧ B − (x), A+ (x) ∧ B + (x)], (Ac )(x) = 1̄ − A(x) = [1 − A+ (x), 1 − A− (x)]. The order relation ⊆ in FI (2) (X) is defined by A ⊆ B if and only if A(x) ≤ B(x), i.e., A− (x) ≤ B − (x) and A+ (x) ≤ B + (x) for all x ∈ X. And ∅, X ∈ FI (2) (X) with membership functions ∅(x) = 0̄ and X(x) = 1̄, ∀x ∈ X, respectively. It follows from Definition 1 that an interval set [Al , Au ] of X is an intervalvalued fuzzy set [χAl , χAu ] of X, where χA is a characteristic function of A. It is easy to see the following proposition that shows the relationship between fuzzy interval sets and interval-valued fuzzy sets through the mapping f : I (F (X)) → FI (2) (X), [Al , Au ] → [χAl , χAu ]. International Journal of Intelligent Systems DOI 10.1002/int 10 HU, WONG, AND YIU Figure 2. Interval-valued fuzzy set A(x) = [A(x), A(x)] PROPOSITION 4. (1) Interval-valued fuzzy sets algebra (FI (2) (X), ∪, ∩, c, ∅, X) is a fuzzy lattice. (2) Fuzzy interval sets algebra (I (F (X)), , , ¬, ∅, X ) is isomorphic to interval-valued fuzzy sets algebra (FI (2) (X), ∪, ∩, c, ∅, X). 3.2 Interval Sets and Shadowed Sets If L = {0, 1, [0, 1]} and its order relation ≤s is defined as 0 ≤s [0, 1] ≤s 1, then (L, ≤s ) is a fuzzy lattice and its algebraic operations are written as ∧s , ∨s , and ¬s . Based on (L, ≤s ), Pedrycz introduced the following concept of shadowed set.44 DEFINITION 9. 44,45 Let X be a universe. Then a set-valued mapping A : X → {0, 1, [0, 1]} is a shadowed set of X. Fs (X) is a family of all shadowed sets of X. It follows from Definition 9 that a shadowed set A of X is an interval-valued fuzzy set [A− , A+ ], where: A− (x) = A(x), A(x) = 0, 1 , A+ (x) = 0, A(x) = [0, 1] A(x), A(x) = 0, 1 . 1, A(x) = [0, 1] PROPOSITION 5. An interval set is a shadowed set; conversely, a shadowed set is an interval set. Proof. Let [Al , Au ] be an interval set of X. Define ⎧ x ∈ Al ⎨ 1, A(x) = [0, 1], x ∈ Au − Al ⎩ 0, otherwise International Journal of Intelligent Systems DOI 10.1002/int INTERVAL SETS AND FUZZY INTERVAL SETS 11 Then A is a shadowed set of X. Conversely, let A be a shadowed set of X. Define Al = {x ∈ X|A(x) = 0}, Au = {x ∈ X|A(x) = 1 or [0, 1]}. Then [Al , Au ] is an interval set of X. In Fs (X), we define the following operations, for all A, B ∈ FS (X), 1. (A ∪s B)(x) = A(x) ∨s B(x), 2. (A ∩s B)(x) = A(x) ∧s B(x), 3. (¬s A)(x) = ¬s (A(x)). The following proposition shows the relationship between interval sets and shadow sets: PROPOSITION 6. (1) A shadowed sets algebra (Fs (X), ∪s , ∩s , ¬s , ∅, X) is a fuzzy lattice. (2) Interval sets algebra (I (P (X)), , , ¬, ∅, X ) is isomorphic to shadowed sets algebra (Fs (X), ∪s , ∩s , ¬s , ∅, X). Proof. Item (1) is straightforward from the definition and operations of shadowed sets. We only prove item (2) in the following. Let f : I (P(X)) → Fs (X) ⎧ x ∈ Al ⎨ 1, f ([Al , Au ])(x) = [0, 1], x ∈ Au − Al ⎩ 0, otherwise. It is clear to see that f is a one to one mapping (injection and surjection) from I (P (X)) to Fs (X). And for A = [Al , Au ], B = [Bl , Bu ] ∈ I (P (X)), we have f ([Al , Au ] [Bl , Bu ])(x) = f ([Al ∪ Bl , Au ∪ Bu ])(x) ⎧ x ∈ Al ∪ Bl ⎨ 1, = [0, 1], x ∈ Au ∪ Bu − Al ∪ Bl ⎩ 0, otherwise. The following discussions are from three cases (see Figure 3): (i) x ∈ Al ∪ Bl Here x ∈ Al or x ∈ Bl . f ([Al , Au ])(x) = 1, f ([Bl , Bu ])(x) ≤s 1 or f ([Al , Au ])(x) ≤s 1, f ([Bl , Bu ])(x) = 1. Thus f ([Al , Au ] [Bl , Bu ])(x) = 1 = f ([Al , Au ])(x) ∨s f ([Bl , Bu ])(x). (ii) x ∈ Au ∪ Bu − Al ∪ Bl International Journal of Intelligent Systems DOI 10.1002/int 12 HU, WONG, AND YIU Figure 3. The relationship diagram of two interval sets f ([Al , Au ])(x) = [0, 1], f ([Bl , Bu ])(x) ≤s [0, 1] or f ([Al , Au ])(x) ≤s [0, 1], f ([Bl , Bu ])(x) = [0, 1]. Thus f ([Al , Au ] [Bl , Bu ])(x) = [0, 1] = f ([Al , Au ])(x) ∨s f ([Bl , Bu ])(x). (iii) x ∈ / Au ∪ Bu (f ([Al , Au ]) ∪s f ([Bl , Bu ]))(x) = 0 = f ([Al , Au ])(x) = f ([Bl , Bu ])(x), i.e., f ([Al , Au ] [Bl , Bu ]) = f ([Al , Au ]) ∪s f ([Bl , Bu ]). It can be proved that f ([Al , Au ] [Bl , Bu ]) = f ([Al , Au ]) ∩s f ([Bl , Bu ]) in a similar way. ⎧ x ∈ (Au )c ⎨ 1, f (¬ [Al , Au ])(x) = f ([(Au )c , (Al )c ])(x) = [0, 1], x ∈ (Al )c − (Au )c ⎩ 0, otherwise ⎧ x ∈ (Au )c ⎨ ¬s 0, = ¬s [0, 1], x ∈ Au − Al ⎩ ¬s 1, x ∈ Al = ¬s (f ([Al , Au ]))(x). This completes the proof that f is an isomorphic mapping from I (P (X)) to Fs (X). Cattanco and Ciucci 46 adopted a membership value 0.5 instead of [0, 1], which is equivalent to Pedrycz’s method. International Journal of Intelligent Systems DOI 10.1002/int INTERVAL SETS AND FUZZY INTERVAL SETS 13 Figure 4. The tree structure of English word “act” 3.3 Interval Sets and Flou Sets The flou set was introduced by Gentilhomme in 1968 47 through considering the natural language words. For example, starting from English word “act,” one can form other words by adding prefix such as “in,” “un,” “re,” “dis,” etc. and/or by adding suffix such as “ive,” “ivity,” “ion,” “ionability,” “able,” etc. That is to say, we can consider the following tree structure, as shown in Figure 4. Some combination from the tree produces some common words such as inactive, action, activate, actable etc. Some combinations are clearly less common words such as inactable. Some combinations seem to be acceptable, but no one can find them in the dictionary. Through this consideration from Gentilhomme, a universe is divided into three categories: the first class is the central elements, i.e. these elements must satisfy certain properties; the second is the surrounding elements, i.e., suspicious elements; the third is nonelements, i.e. these elements do not satisfy the given property. It formalizes these ideas as follows: DEFINITION 10. 47 Let X be a universe. Then the pair (E, F ) is called a flou set of X, where E, F ⊆ X, E ⊆ F . E is a certain domain, F is a maximum domain, and F − E is called a flou area. FL (X) is a family of all flou sets of X, i.e. FL (X) = (E, F ) : E, F ∈ P (X), E ⊆ F . Obviously, crisp set A of X is a special flou set, written as (A, A). The flou set (E, F )may be interpreted as E is a set of “core” elements of A; F − E is a set of “periphery” elements of A; elements of E are more likely to belong to A than elements of F − E. International Journal of Intelligent Systems DOI 10.1002/int 14 HU, WONG, AND YIU EXAMPLE 3. (1) Let X = [0, 100] be a universe of age. Old age can be represented by a flou set. For example, ‘the interval set ([60, 100], [50,100]) indicates, every people, not less than 60 years of age, must be aged, every one below 50 years of age is certainly not old people and there is a question whether they are old or not for people having the ages from 50 to 60. (2) Let R be an equivalence relation on finite universe, A be a subset of X, R(A) and R(A) be Pawlak’s lower and upper approximation of A, respectively. Then the pair (R(A), R(A)) is a flou set. Elements of R(A) must belong to A (called a positive domain in rough sets theory), elements of (R(A))c do not belong to A certainly (called a negative domain in rough sets theory), and it is doubtful whether elements of R(A) − R(A) belong to A (called boundary domain in rough sets theory). DEFINITION 11. Let A = (E, F ), B = (E , F ) ∈ FL (X). Then we define 1. 2. 3. 4. (E, F ) ⊆ (E , F ) ⇔ E ⊆ E , F ⊆ F . (E, F ) ∪ (E , F ) = (E ∪ E , F ∪ F ). (E, F ) ∩ (E , F ) = (E ∩ E , F ∩ F ). (E, F )c = (F c , E c ). The following are immediate consequences of Definition 11. PROPOSITION 7. (1) Flou sets algebra (FL (X), ∪, ∩, c, (∅, ∅), (X, X)) is a fuzzy lattice. (2) Interval sets algebra (I (P (X)), , , ¬, ∅, X ) is isomorphic to shadowed sets algebra (FL (X), ∪, ∩, c, (∅, ∅), (X, X)). The interval set and flou set are two equivalent concepts from the mathematical point of view. However, the interval set stresses not only results of 3WD, i.e., acceptance, rejection, and nonpromise, and also emphasizes the extension of concept described by the sets between the given two sets. But the flou set emphasizes only a result of 3WD. The Flou set can be generalized to an n-flou set. DEFINITION 12. Let X be a universe. Then an n-flou set of X is an n-tuple (E1 , E2 , . . . , En ), where E1 ⊆ E2 ⊆ · · · ⊆ En ⊆ X. Fn (X) is a family of all nflou sets of X. DEFINITION 13. Let (E1 , E2 , . . . , En ), (F1 , F2 , . . . , Fn ) ∈ FL n (X). Then 1. 2. 3. 4. (E1 , E2 , . . . , En ) ⊆ (F1 , F2 , . . . , Fn ) ⇔ (∀i ∈ {1, 2, · · · , n})(Ei ⊆ Fi ). (E1 , E2 , . . . , En ) ∪ (F1 , F2 , . . . , Fn ) = (E1 ∪ F1 , E2 ∪ F2 , . . . , En ∪ Fn ). (E1 , E2 , . . . , En ) ∩ (F1 , F2 , . . . , Fn ) = (E1 ∩ F1 , E2 ∩ F2 , . . . , En ∩ Fn ). c , . . . , E1c ). (E1 , E2 , . . . , En )c = (Enc , En−1 It is trivial to show the following proposition: PROPOSITION 8. (FL n (X), ∪, ∩, c) is a fuzzy lattice. Negoita and Ralescu48 introduced the concept of L-flou set and proved the equivalency between the L-fuzzy set and L-flou set. International Journal of Intelligent Systems DOI 10.1002/int INTERVAL SETS AND FUZZY INTERVAL SETS 15 DEFINITION 14. 48 Let X be a universe. Then an L-flou set of X is a mapping from L to P (X), i.e. subset of X, which satisfies the following conditions: 1. E0 = ∅; 2. Esup αi = Eαi , ∀(αi )i∈I , αi ∈ L. i∈I FL L (X) is a family of all L-flou sets of X. It is clear to see that an n-flou set is a special L-flou set, where L = {0, 1, 2, . . . , n} with an nature order, and E0 = ∅, (Eα )α∈{1,2,...,n} ∈ FL n (X). 3.4 Fuzzy Interval Sets and Intuitionistic Fuzzy Sets The concept of intuitionistic fuzzy sets was introduced by Atanassov,49,50 as a generalization of fuzzy sets which were developed by Zadeh. DEFINITION 15. 49 The object A = (μA , νA ) is called an intuitionistic fuzzy set on X, where μA , vA ∈ F (X), μA (x) i s a membership degree of x to A, and νA (x) a nonmembership degree of x to A with μA (x) + νA (x) ≤ 1. Union, intersection, complement, and order relation of intuitionistic fuzzy sets are defined as follows: DEFINITION 16. Let A = (μA , νA ) and B = (μB , νB ) be two intuitionistic fuzzy sets of X. Then the following statements hold: 1. 2. 3. 4. 5. 6. A ∪ B = {(x, max{μA (x), μB (x)}, min{νA (x), νB (x)})|x ∈ X}. A ∩ B = {(x, min{μA (x), μB (x)}, max{νA (x), νB (x)})|x ∈ X}. Ac = {(x, νA (x), μA (x))|x ∈ X}. A ⊆ B ⇔ μA ⊆ μB and νA ⊇ νB . A = B ⇔ A ⊆ B and B ⊆ A. A≺B ⇔ μA ⊆ μB and νA ⊆ νB . The following proposition shows the relationship between fuzzy interval sets and intuitionistic fuzzy sets: PROPOSITION 9. (1) Intuitionistic fuzzy sets algebra (I F (X), ∪, ∩, c, ∅, X) is a fuzzy lattice. (2) Fuzzy interval sets algebra(I (F (X)), , , ¬, ∅, X ) is isomorphic to intuitionistic fuzzy sets algebra (I F (X), ∪, ∩, c, ∅, X). Proof. Let f : I (F (X)) → I F (X), [Al , Au ] → (Al , (Au )c ). International Journal of Intelligent Systems DOI 10.1002/int 16 HU, WONG, AND YIU Obviously (∀x ∈ X)(Al (x) ≤ Au (x) ⇔ Al (x) + (Au )c (x) ≤ 1) and f is a bijection (injection and surjection). f ([Al , Au ] [Bl , Bu ]) = f ([Al ∪ Bl , Au ∪ Bu ]) = (Al ∪ Bl , (Au ∪ Bu )c ) = (Al ∪ Bl , (Au )c ∩ (Bu )c ) = (Al , (Au )c ) ∪ (Bl , (Bu )c ) = f ([Al , Au ]) ∪ f ([Bl , Bu ]) f (¬[Al , Au ]) = f ([Acu , Acl ]) = (Acu , Al ) = (Al , Acu )c = (f ([Al , Au ]))c . Gau and Buehrer51 introduced vague sets. Bustince and Burillo52 proved that vague sets are intuitionistic fuzzy sets soon after. Atanassov and Stoeva discussed interval-valued intuitionistic fuzzy sets53 and intuitionistic L-fuzzy sets.54 3.5 Interval Sets and Rough Sets Let R be an equivalence relation on a finite universe, A be a subset of X, R(A) and R(A) be Pawlak’s lower and upper approximation of A, respectively. Then [R(A), R(A)] is an interval set. The lower approximation and upper approximation are extended to all kinds of rough sets, which interval set features are listed in Table I. There are some differences in operations of rough sets and classic sets. In the classic set, if the two sets have the same elements, then these two sets are equal. In rough sets theory, we need another concept on equality of sets, i.e. proximately (or roughly) equality. Two sets are not equal in the classic sets, but there may be approximately equal in rough sets. Whether two sets are approximately equal or not, it is based on our judgments on knowledge we have obtained. In the following, we introduce the concepts on approximately equality and inclusion of sets. DEFINITION 17. Let (X, R) be a Pawlak’s approximation space and A, B ⊆ X. 1. If RA = RB (respectively, RA = RB), then A and B are called R-lower equality (respectively, R-upper equality), written as A =R B (resp. A =R̄ B). 2. If A =R B and A =R̄ B, then A and B are called R-equality, written as A =R B. 3. If RA ⊆ RB (respectively, RA ⊆ RB), then A is said to be R-lower contained in B (respectively, R-upper contained in B), written as A ⊆R B(respectively, A ⊆R B). 4. If A ⊆R B and A ⊆R̄ B, then A is called R-contained in B, written as A ⊆R B. Obviously, for any equivalence relation of X, =R , =R̄ and =R are equivalence relations on P (X) and ⊆R ,⊆R and ⊆R are preorder relations on P (X). International Journal of Intelligent Systems DOI 10.1002/int Fuzzy relation / fuzzy set Fuzzy relation / fuzzy set Fuzzy relation /fuzzy set of Y IVF relation / IVF set of X Equivalence relation / IVF set of X Fuzzy relation / fuzzy set General relation / fuzzy set X×Y X×Y X×Y X×X X×X X×Y X×Y (⊥, T )- generalized fuzzy RS5 (θT , σ⊥ )- generalized fuzzy RS6 (I, T )- generalized fuzzy RS8 Interval-valued rough fuzzy set9 Generalized fuzzy RS3,4 Generalized rough fuzzy set3 IVF IVF relation / IVF set X×Y (⊥, T )- generalized IVF RS10–12 RS9 Relation (R)/objet (A) Universe RS’s family y∈Y T y∈Y y∈Y y∈X International Journal of Intelligent Systems y∈X y∈X DOI 10.1002/int R(A)(x) = ∧{A(y)|y ∈ F (x)} R̄(A)(x) = ∨{A(y)|y ∈ F (x)} F (x) = {y ∈ Y |(x, y) ∈ R}, x ∈ X y∈X R(A)(x) = sup (A(y) ∧ R(x, y)) R(A)(x) = inf (A(y) ∨ (1 − R(x, y))) R(A)(x) = ∈ [x]R }, ∧{A+ (y)|y ∈ [x]R }] − R̄(A)(x) = [∨{A (y)|y ∈ [x]R }, ∨{A+ (y)|y ∈ [x]R }] [∧{A− (y)|y y∈X R(A)(x) = [ ∨ {A− (y) ∧ R − (x, y)}, ∨ {A+ (y) ∧ R + (x, y)}] y∈X (Continued) R(A)(x) = [ ∧ {A− (y) ∨ (1 − R + (x, y))}, ∧ {A+ (y) ∨ (1 − R − (x, y))}] y∈Y R T (A)(x) = sup T (R(x, y), A(y)) R I (A)(x) = inf I (R(x, y), A(y)) y∈Y R σ⊥ (A)(x) = sup σ⊥ (N (R(x, y)), A(y)) R θ (A)(x) = inf θT (R(x, y), A(y)) y∈Y R T (A)(x) = sup T (R(x, y), A(y)) R ⊥ (A)(x) = inf ⊥(N (R(x, y)), A(y)) R ⊥ (A) = [R + ⊥1 (A− ), R − ⊥2 (A+ )] R T (A) = [R − T1 (A− ), R + T2 (A+ )] Lower and upper approximation Table I. Interval set features of all kinds of rough sets INTERVAL SETS AND FUZZY INTERVAL SETS 17 Fuzzy relation / fuzzy set Fuzzy relation / Cantor set Equivalence relation / fuzzy set Equivalence relation / Cantor set X×X X×X X×X X×X Fuzzy RS2 RS2 International Journal of Intelligent Systems IVF stands for interval-valued fuzzy, and RS stands for rough set. Pawlak RS1 Rough fuzzy set2 General relation / Cantor set X×Y Generalized RS14 Fuzzy Relation (R)/objet (A) Universe RS’s family y∈X y ∈A / R(A) = {x ∈ X : [x]R ⊆ A} = ∪{[x]R ∈ X/R : [x]R ⊆ A}, R(A) = {x ∈ X : [x]R ∩ A = ∅} = ∪{[x]R ∈ X/R : [x]R ∩ A = ∅}. R(A)(x) = inf{A(y)|y ∈ [x]R } R(A)(x) = sup{A(y)|y ∈ [x]R } y∈A R(A)(x) = ∨ {R(x, y)} R(A)(x) = ∧ {1 − R(x, y)} y∈X R(A)(x) = sup (A(y) ∧ R(x, y)) R(A)(x) = inf (A(y) ∨ (1 − R(x, y))) R(A) = {x ∈ X|F (x) ⊆ A} R̄(A) = {x ∈ X|F (x) ∩ A = ∅} F (x) = {y ∈ Y |(x, y) ∈ R}, x ∈ X Lower and upper approximation Table I. Continued 18 HU, WONG, AND YIU DOI 10.1002/int INTERVAL SETS AND FUZZY INTERVAL SETS 19 PROPOSITION 10. ∀A ∈ P (X), interval set [R(A), R(A)] possess the following properties: 1. [R(A), R(A)] ⊇ [A]=R = {B ∈ P (X)|B =R A}; 2. [R(A), R(A)] = {B ∈ P (X)|A ⊆R B ⊆R A}. Proof. It is easy to show (1) and we only prove (2). ∀B ∈ [R(A), R(A)], R(A) ⊆ B ⊆ R(A) ⇒ R R(A) ⊆ R (B) , R (B) ⊆ R R(A) ⇒ R(A) ⊆ R (B) , R (B) ⊆ R(A) ⇒ A ⊆R B ⊆R A. Conversely, A ⊆R B ⊆R A implies R(A) ⊆ R(B) ⊆ B ⊆ R(B) ⊆ R(A). Figure 5 shows the relationship between interval sets constituted by rough sets and R-equality (R-contained). From properties of rough sets, we have the following proposition: PROPOSITION 11. Let (X, R) be a Pawlak’s approximation space and A, B ⊆ X. Then the following holds: 1. 2. 3. 4. 5. [R(A), R̄(A)] [R(B), R̄(B)] [R(A ∪ B), R̄(A ∪ B)]; [R(A), R̄(A)] [R(B), R̄(B)] [R(A ∩ B), R̄(A ∩ B)]; ¬[R(A), R̄(A)] = [R(Ac ), R̄(Ac )]; [R(∅), R̄(∅)] = ∅, [R(X), R̄(X)] = X ; A ⊆ B implies [R(A), R̄(A)] [R(B), R̄(B)]. Proof. We prove only (1), and others are similar in their proof. [R (A), R̄(A)] [R(B), R̄(B)] = [R(A) ∪ R(B), R̄(A) ∪ R̄(B)] − = [R(A) ∪ R(B), R̄(A ∪ B)] [R(A ∪ B), R̄(A ∪ B)]. From the discussion above, for any rough set (R(A), R̄(A)), it forms an interval set [R(A), R̄(A)]. It is nature to ask if there is an equivalence relation R over X and a subset A of X such that [Al , Au ] = [R(A), R̄(A)] for any interval set [Al , Au ] over X. The following proposition is an answer. PROPOSITION 12. Let [Al , Au ] be an interval set over X with Al = Au and R = (x, y) : x, y ∈ Al or x, y ∈ Au \Al or x, y ∈ Acu Then [Al , Au ] = [R(A), R̄(A)] if and only if A ∈ [Al , Au ] and A = Al ,A = Au . Proof. Suppose that A ∈ [Al , Au ] and A = Al ,A = Au . Then R(A) = {x ∈ X : [x]R ⊆ A} = Al becauseA ∈ [Al , Au ] and A = Au . R̄(A) = {x ∈ X : [x]R ∩ A = ∅} = Au because A ∈ [Al , Au ] and A = Al . International Journal of Intelligent Systems DOI 10.1002/int Figure 5. Equivalence classes determined by interval sets constituted by rough sets 20 HU, WONG, AND YIU International Journal of Intelligent Systems DOI 10.1002/int INTERVAL SETS AND FUZZY INTERVAL SETS 21 Conversely, let [Al , Au ] = [R(A), R̄(A)], i.e., R(A) = {x ∈ X : [x]R ⊆ A} = Al and R̄(A) = {x ∈ X : [x]R ∩ A = ∅} = Au . i. If Al ⊆ A, then R(A) = {x ∈ X : [x]R ⊆ A} = Al . This is a contradiction. ii. If A ⊆ Au ,R̄(A) = {x ∈ X : [x]R ∩ A = ∅} = Au . This is a contradiction. Thus, A ∈ [Al , Au ] and A = Al , A = Au . 3.6. Interval Sets and Three-Way Decisions Another description of the lower and supper approximation of rough sets is 3WD proposed by Yao.55–57 For theoretical research of 3WD, Hu established threeway decision space such that the researches on 3WD are unified to a theoretical framework.58–62 In the following, we discuss the relationship between 3WD and interval sets. Semantically speaking, an interval set describes a concept partly known. Despite the extension of the concept is a subset of X, it is difficult to precisely present the subset owing to the incompleteness of information. One possible approach is to describe the concept by a lower bound Al and an upper bound Au . For any subset A of X, if Al ⊆ A ⊆ Au , then, as it is, A is a real extension of the concept, as shown in Figure 6. 3WD of interval set A = [Al , Au ] = {A ⊆ X|Al ⊆ A ⊆ Au } are 1. Acceptance: Al ; 2. Rejection: (Au )c ; 3. Uncertain: Au − Al . For [Al , Au ] ∈ I (P (X)) (resp. [Al , Au ] ∈ I (F (X)), [Al , Au ] ∈ I (FL (X))), Al and Au are called the lower and upper bound of (resp. fuzzy, L-fuzzy) interval Figure 6. Interval set description of the concept International Journal of Intelligent Systems DOI 10.1002/int HU, WONG, AND YIU Figure 7. The relationship between interval sets and their likeness 22 International Journal of Intelligent Systems DOI 10.1002/int INTERVAL SETS AND FUZZY INTERVAL SETS 23 set [Al , Au ], respectively. Al , (Au )N and Au − Al are called acceptance region, rejection region, and uncertain region of [Al , Au ], respectively, which be denoted by ACP([Al , Au ]), REJ([Al , Au ]), and UNC([Al , Au ]). EXAMPLE 4. Consider the course evaluation for students, we use an interval set ([60, 100], [50, 100]) = {[x, 100]|50 ≤ x ≤ 60} The acceptance region is [60,100], i.e., if course examination grade of a student is not less than 60, he/she passes. The rejection region is [0, 50), i.e., if course examination grade of a student is less than 50, he/she does not pass. The uncertain region is [50, 60), i.e., if course examination grade of a student is not less than 50 and less than 60, it is not sure whether he/she passes the examination and further evaluation is needed. 4. CONCLUSIONS This paper discusses the relationship between the interval sets and sets with similar concepts. Integrating the above results, we draw the following relations, as shown in Figure 7. As has been said that the concepts of interval sets are mathematically equivalent to shadowed sets and flou sets, respectively, and the concepts of fuzzy interval sets are mathematically equivalent to interval-valued fuzzy sets and intuitionistic fuzzy sets, respectively. These concepts can be discussed uniformly on the interval sets. 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