Representations of typical hesitant fuzzy rough sets

Abstract

Typical hesitant fuzzy set is a kind of generalization of classical fuzzy set by possessing a membership degree, called as typical hesitant fuzzy element, of finite non-empty subset of the unitary interval. This paper studies the constructive approach to rough set approximation operators in typical hesitant fuzzy background. Firstly, two novel partial orders are introduced to compare two arbitrary typical hesitant fuzzy elements and typical hesitant fuzzy sets, meanwhile, the basic operations, including intersection, union and α-level sets of typical hesitant fuzzy sets are then proposed and their properties are studied in detail. Secondly, typical hesitant fuzzy rough sets are introduced and studied via the new operations of intersection and union. Furthermore, typical hesitant fuzzy rough approximation operators are represented by the rough approximation operators of the α-level set of typical hesitant fuzzy set. Finally, the connections between typical hesitant fuzzy (and crisp, respectively) relations and typical hesitant fuzzy rough (and rough typical hesitant fuzzy, respectively) approximation operators are further established.

Keywords

Typical hesitant fuzzy set partial order typical hesitant fuzzy rough set union intersection

1 Introduction

Since fuzzy set theory was introduced by Zadeh [43], many new approaches and theories treating imprecision and uncertainty have been proposed, such as interval-valued fuzzy sets (IVFSs) [2], intuitionistic fuzzy sets (IFSs) introduced by Atanassov [1], type-2 fuzzy sets [44] and so on. Recently, to deal with hesitant and incongruous problems, Torra and Narukawa [29, 30] introduced the concept of hesitant fuzzy sets (HFSs) which was an extension of classic fuzzy set by arranging a membership degree of a subset of interval [0, 1]. Actually, hesitant fuzzy set is many-valued fuzzy set proposed by Young [42] and Grattan-Guiness [14] and is also a special case of type-2 fuzzy set [44]. HFSs permit the membership degree of an object to be a set of several possible values between 0 and 1, called as hesitant fuzzy element (HFE), which makes HFSs more reasonable and reliable for modeling people’s hesitancy and uncertainty in defining the membership degree of an object. Hesitant fuzzy set theory has gotten much attention in theoretical and applied research [3 , 28] for a few short years. Since HFS is a generalization of interval-valued fuzzy sets that have been researched in detail and most of the discussion are focused on hesitant fuzzy sets with finite non-empty membership degrees, which are called as typical hesitant fuzzy sets (THFSs) and typical hesitant fuzzy elements (THFEs) respectively by Bedregal et al. [4, 5].

The rough set theory, proposed by Pawlak [24, 25], as an efficient tool for the construction of approximations of concept, is an extension of classical set theory for modeling and processing insufficient and incomplete information. As one of the important extensions of rough set theory, fuzzy rough sets have been demonstrated to approximate a fuzzy concept in fuzzy environment and the investigations [7–10 , 48] have shown both rough set theory and fuzzy set theory can be combined into a more flexible framework for the study of imprecise information. The constructive and axiomatic approaches are two mainly methods for the development of rough set theory in fuzzy environment. In constructive approach, some binary relations involving tolerance relations [16, 23] and fuzzy covering [9 , 34] are the primary notions. On the other hand, the frameworks for the study of fuzzy rough and rough fuzzy set theory are investigated by axiomatic approach [7 , 49]. It is very natural to investigate the rough set theory in typical hesitant fuzzy backgrounds. Recently, Yang et al. [41] proposed hesitant fuzzy rough sets by the basic operations: intersection and union [29, 30] and discussed the properties of hesitant fuzzy approximations opeartors. Zhang et al. [45] further defined interval-valued hesitant fuzzy rough sets and explored the constructive and axiomatic approaches. However, the orders and the operators on typical hesitant fuzzy element and typical hesitant fuzzy sets used in these papers are defined in the extended environment where two hesitant fuzzy elements of one object are extended to the same cardinality. This is not reasonable because the losing of information in the transformation is neglected.

Actually, the orders and operators on HFEs (or THFEs) and HFSs (or THFSs) have been discussed in many references. Torra [29], Torra and Narukawa [30] defined an ordered relation by the lower bound and upper bound of the hesitant fuzzy elements. Xia and Xu [39] defined the score function for HFEs to present the superiority of one HFE to another; Liao et al. [17] further proposed the variance function to reflect the ordered relation; Xu and Xia [38] defined the ordered relation under the extended environment. But the losing of information in the extension is neglected. Bedregal et al. [4, 5] proposed a new order by removing the greatest redundant elements of THFEs. Meantime, Bedregal et al. [4, 5] developed Torra’s theory of union and intersection [29, 30] in 2014 via β-function. Although Bedregal’s definition of union and intersection decrease the dimensions of the derived THFEs, they did not satisfy the property of De Morgan’s law. So in this paper, some reasonable novel partial orders and new operators on typical hesitant fuzzy sets will be proposed, furthermore the typical hesitant fuzzy rough sets will be studied based on the new orders andoperators.

The rest of this paper is organized as follows: Section 2 briefly reviews basic concepts of THFSs. Section 3 introduces new definitions, including partial orders, intersection and union for THFSs and THFEs. Section 4 proposes the constructive approach to THFRSs via new operators of union and intersection and presents the relationship between hesitant fuzzy rough approximation operators and the relative crisp rough approximation operators via the α-level set o f THFSs. Section 5 presents a numerical example to show the application of typical hesitant fuzzy rough sets on two universes. Finally, Section 6 concludes the paper.

2 Typical hesitant fuzzy sets

Hesitant fuzzy set is actually a kind of many-valued quantities taking subintervals or subsets of [0, 1] as membership degrees [14 , 42]. Since most of works on HFSs assume explicitly or implicitly that the membership degrees of HFSs, namely the typical hesitant fuzzy elements, are finite and nonempty subsets of [0, 1], Bedregal et al. [4, 5] introduced such kind of hesitant fuzzy sets as typical hesitant fuzzy sets. The basic notations and operators on THFEs and THFSs discussed in [4 , 30] are presented in this section.

Definition 2.1. (typical hesitant fuzzy set) Let U be a nonempty and finite universe of discourse, a typical hesitant fuzzy set on U is a function that when applied to U returns a finite and non-empty subset of [0, 1], which can be represented as follows: $A = {〈 x, h_{A} (x) 〉 | x \in U}$

where h_A (x) is called as typical hesitant fuzzy element (THFE for short) denoting the possible membership degrees of the element x ∈ U to theTHFS A.

The cardinality of h_A (x) is referred to be l (h_A (x)) and $ℕ_{l (h_{A} (x))} = {1, 2, \dots, l (h_{A} (x))}$ . The set of all the finite and non-empty subsets of [0, 1] is denoted as $ℍ$ where $ℍ \subseteq$ epsfboxG :/Tex/IOSPRESS/IFS/0 -2159/IF01 . eps ([0, 1]) ∖ {0} and epsfboxG :/Tex/IOSPRESS/IFS/0 -2159/IF01 . eps ([0, 1]) denotes all the subsets of [0, 1]. For all $h_{i} \in ℍ$ , let $l (h_{i}) = l_{i}, ℕ_{l (h_{i})} = ℕ_{l_{i}}$ for simplicity. The set of all THFSs on U is denoted by THF(U). For all x ∈ U, h_A (x) = {a₁, a₂, …, a_m}, a_i ∈ [1], i = 1, 2, …, m defines a constant THFS, which is denoted by $\hat{a_{1 \dots m}}$ and $a_{1 \dots m} = h_{\hat{a_{1 \dots m}}} = {a_{1}, a_{2}, \dots, a_{m}}$ . Particularly, A =∅ if h_A (x) = {0} for all x in U and A = U if h_A (x) = {1} for all x in U. Obviously, the constant THFS $\hat{a_{1 \dots m}}$ will degenerate into the classical constant fuzzy set for m = 1.

For simplicity, the values in a THFE is arranged by the increasing order:

h_A (x) = {σ_{h_A(x)} (1) , σ_{h_A(x)} (2) , … , σ_{h_A(x)} (l)} where σ_{h_A(x)} (1) ≤ σ_{h_A(x)} (2) ≤ … ≤ σ_{h_A(x)} (l).

In the following part of this section, we present the existing orders on the set $ℍ$ . First, we introduce definitions of strict partial order (quasi-order), partial order and pre-order discussed in [26].

Definition 2.2. A binary relation < on a set P is a strict partial order if it satisfies

(spo1) antisymmetry: for all x, y ∈ P, if x < y holds, then y < x does not hold;

(spo2) transitivity: for all x, y, z ∈ P, x < y and y < z imply x < z .

A (non-strict) partial order ≤, on a set P is a binary relation satisfying

(po1) reflexivity: for all x ∈ P, x ≤ x;

(po2) antisymmetry: for all x, y ∈ P, x ≤ y and y ≤ x imply x = y;

(po3) transitivity: for all x, y, z ∈ P, x ≤ y and y ≤ z imply x ≤ z.

Relaxing the conditions by dropping antisymmetry of partial order leads to what is known as a pre-order, namely, pre-order is reflexive and transitive. While a strict partial order is also called as quasi-order which is anti-reflexive and transitive that is different from the definition in [26]. Actually, an anti-reflexive and transitive relation is an antisymmetric and transitive relation, and vice versa. So in the following parts, the strict partial order is called as quasi-order uniformly.

The operators β (h₁, h₂) and γ (h₁, h₂) on $ℍ$ are given in Definition 2.3 because of the different cardinalities of two THFEs.

Definition 2.3. Given $h_{1}, h_{2} \in ℍ$ ,

k = l₁ − l₂ + 1, let $β : ℍ \times ℍ \to ℍ$ and $γ : ℍ \times ℍ \to ℍ$ be functions defined by: $\begin{matrix} β (h_{1}, h_{2}) = {\begin{matrix} h_{1}, if l_{1} \leq l_{2}, \\ {σ_{h_{1}} (i) : i = k, \dots, l_{1}}, else \end{matrix} \\ γ (h_{1}, h_{2}) = {σ_{h_{1}} (i) : i = 1, \dots, \min (l_{1}, l_{2})} \end{matrix}$

β (h₁, h₂) and γ (h₁, h₂) are called as β-normalization and γ-normalization of h₁ with respect to h₂ respectively.

Note 2.1. For any $h_{1}, h_{2} \in ℍ^{(m)} = {h \in ℍ, l (h) = m}$ , with $m \in ℕ$ , we define $h_{1} \leq_{(m)} h_{2} \Leftrightarrow σ_{h_{1}} (i) \leq σ_{h_{2}} (i), \forall i = 1, . . ., m$ Let A = {< x, h_A (x) > , x ∈ U} and B = {< x, h_B (x) > , x ∈ U} be two THFSs on U, the orders defined on $ℍ$ are presented as follows:

the order “≤_T” [29, 30]

A ≤ _TB ⇔ $h_{A}^{-} (x)$ ≤ $h_{B}^{-} (x)$ , and $h_{A}^{+} (x)$ ≤ $h_{B}^{+} (x)$ , for all x ∈ U.

the order ≤_X, Xu and Xia [37] extended two THFEs with same cardinalities and compared the extended THFEs in a simple situation where the partial order is defined as follows:

h_{1} \leq_{X} h_{2} \Leftrightarrow σ_{h_{1}} (i) \leq σ_{h_{2}} (i), i = 1, 2, \dots, l

where h₁ = {σ_{h
₁} (1) , σ_{h
₁} (2) , ⋯ , σ_{h
₁} (l)} and h₂ = {σ_{h
₂} (1) , σ_{h
₂} (2) , ⋯ , σ_{h
₂} (l)} are two THFEs which have been extended with the same cardinality l.

the order ≤_B, Bedregal et al. [4, 5] defined this new order by cutting the preceding and exceeding elements of that THFE which has more numbers of values:

\begin{matrix} h_{1} \leq_{B} h_{2} \Leftrightarrow β (h_{1}, h_{2}) \leq_{(l_{0})} β (h_{2}, h_{1}), \\ where h_{1}, h_{2} \in ℍ, and min (l_{1}, l_{2}) = l_{0} . \end{matrix}

Although the order defined by Bedregal et al. [4, 5] is not a partial order, it presents a good beginning to consider the definition of an order on $ℍ$ in unextended environment.

Now we present the existed definitions of union and intersection for any $ℍ$ and further for THFSs. They were first proposed by Torra and Narukawa [29, 30] in 2009 and 2010. But the dimension of the derived $ℍ$ computed by the union or intersection may increase quickly, which will increase the complexity of the calculations. To overcome the difficulty, Bedregal [4, 5] developed Torra’s theory of union and intersection [29, 30] in2014.

the union ∪ and intersection ∩ defined in [29, 30], for any two THFEs $h_{1}, h_{2} \in ℍ$ ,

$(h_{1} \cup h_{2}) = {h \in (h_{1} \cup h_{2}) | h \geq \max (h_{1}^{-}, h_{2}^{-})}$

$\Leftrightarrow (h_{1} \cup h_{2}) (x) = {max {σ_{h_{1}} (i), σ_{h_{2}} (j)}, i \in ℕ_{l_{1}}, j \in ℕ_{l_{2}}}$ ,

$(h_{1} \cap h_{2}) = {h \in (h_{1} \cup h_{2}), h \leq \min (h_{1}^{+}, h_{2}^{+})}$

$\Leftrightarrow (h_{1} \cap h_{2}) (x) = {min {σ_{h_{1}} (i), σ_{h_{2}} (j)}, i \in ℕ_{l_{1}}, j \in ℕ_{l_{2}}}$ .

the union and intersection defined in [4, 5]

$h_{1} ⊔ h_{2} = {\begin{matrix} h_{1}, l_{1} \leq l_{2}, β (h_{2}, h_{1}) \leq_{l_{1}} h_{1}, \\ {max {σ_{h_{1}} (i), σ_{β (h_{2}, h_{1})} (i)} : i \in ℕ_{l_{1}}} \\ \cup {σ_{h_{2}} (i) : i \in ℕ_{l_{2} - l_{1}}}, \\ l_{1} \leq l_{2}, β (h_{2}, h_{1}) ≰_{l_{1}} h_{1}, \\ h_{2} ⊔ h_{1}, otherwise . \end{matrix}$

$\begin{matrix} h_{1} ⊓ h_{2} = {\begin{matrix} h_{2}, l_{1} \leq l_{2}, β (h_{2}, h_{1}) \leq_{l_{1}} h_{1}, \\ {min {σ_{h_{1}} (i), σ_{β (h_{2}, h_{1})} (i)} : i \in ℕ_{l_{1}}}, \\ l_{1} \leq l_{2}, β (h_{2}, h_{1}) ≰_{l_{1}} h_{1}, \\ h_{2} ⊓ h_{1}, otherwise . \end{matrix} \end{matrix}$

Note 2.3. Although Bedregal’s definition of union and intersection decrease the dimension of the derived $ℍ$ by the union and intersection, they did not satisfy the property of De Morgan’s law.

So in the following subsection, a new partial order is defined between any two THFEs in $ℍ$ and the relative operations such as union and intersection are presented.

3 New partial orders and operations on typical hesitant fuzzy sets

In this section, the basic notations and existing operations of THFSs are introduced. Then, more reasonable partial orders on THFEs and THFSs are introduced and the operations, for example, intersection, union, negation and α-level set of THFEs and THFSs are then discussed in detail.

Definition 3.1. For $h_{1}, h_{2} \in ℍ$ , a partial order “≤” between h₁ and h₂ is defined as follows: $h_{1} \leq h_{2} iff {\begin{matrix} h_{1} \leq_{(l_{1})} γ (h_{2}, h_{1}), & if l_{1} \leq l_{2}, \\ β (h_{1}, h_{2}) \leq_{(l_{2})} h_{2}, & else . \end{matrix}$

Then the partial order between any two THFSs A and B∈THF(U) can be defined as follows:

A ⊆ B ⇔ ∀ x ∈ U, h_A (x) ≤ h_B (x).

Theorem 3.1. $(ℍ, \leq)$ is a partially ordered set.

Proof. We need to prove that $(ℍ, \leq)$ satisfies the following properties: reflexivity, antisymmetry and transitivity for all $h_{1}, h_{2}, h_{3} \in ℍ$ .

It is straightforward to prove that ≤ satisfies reflectivity.

The antisymmetry can be proved from the following two situations: first, suppose l₁ < l₂,

h_{1} \leq h_{2} \Leftrightarrow σ_{h_{1}} (i) \leq σ_{h_{2}} (i), i \in ℕ_{l_{1}},

(1) and

σ_{h_{2}} (i) \leq σ_{h_{2}} (i + 1), i \in ℕ_{l_{2} - 1},

(2) on the other hand,

h_{2} \leq h_{1} \Rightarrow σ_{β (h_{2}, h_{1})} (i) \leq σ_{h_{1}} (i), i \in ℕ_{l_{1}}

(3)

Combining (1), (2) and (3), we conclude h₁ = h₂. Similarly, we can show h₁ = h₂, when l₁ > l₂.

So “≤” satisfies antisymmetry.

The following six situations need to be considered when proving the transitivity of “≤ :” l₁ ≤ l₂ ≤ l₃, l₁ ≤ l₃ ≤ l₂, l₂ ≤ l₁ ≤ l₃, l₂ ≤ l₃ ≤ l₁, l₃ ≤ l₁ ≤ l₂, l₃ ≤ l₂ ≤ l₁ .

It is straightforward to prove when l₁ ≤ l₂ ≤ l₃; If l₁ ≤ l₃ ≤ l₂, then $\begin{matrix} h_{1} \leq h_{2} \Rightarrow σ_{h_{1}} (i) \leq σ_{h_{2}} (i), i \in ℕ_{l_{1}} \\ h_{2} \leq h_{3} \Rightarrow σ_{β (h_{2}, h_{3})} (i) \leq σ_{h_{3}} (i), i \in ℕ_{l_{3}} \\ σ_{h_{2}} (i) \leq σ_{h_{2}} (i + 1), i \in ℕ_{l_{2 - 1}} \end{matrix}} \Rightarrow$ $σ_{h_{1}} (i) \leq σ_{h_{3}} (i), i \in ℕ_{l_{1}}$

⇔h₁ ≤ h₃, when l₁ ≤ l₃ ≤ l₂;

Thirdly, if l₂ ≤ l₁ ≤ l₃, then $\begin{matrix} h_{1} \leq h_{2} \Rightarrow σ_{β (h_{1}, h_{2})} (i) \leq σ_{h_{2}} (i), i \in ℕ_{l_{2}} \\ h_{2} \leq h_{3} \Rightarrow σ_{h_{2}} (i) \leq σ_{h_{3}} (i), i \in ℕ_{l_{2}} \\ σ_{h_{1}} (i) \leq σ_{h_{1}} (i + 1), i \in ℕ_{l_{1 - 1}} \end{matrix}} \Rightarrow$ $σ_{h_{1}} (i) \leq σ_{h_{3}} (i), i \in ℕ_{l_{1}}$

The other three situations can be proved by the similar ways. Thus ≤ satisfies transitivity; Then we can conclude $(ℍ, \leq$ ) is a partially ordered set.

Definition 3.2. For $h, h_{1}, h_{2} \in ℍ$ , l₁ = l₁, l₂ = l₂, we define the complement of h, denoted by (h^c), the union (⊻) and intersection (⊼) of h₁ and h₂ as follows:

the complement of h is denoted by h^c such that,

$h^{c} = \sim h = {1 - σ_{h} (l - i + 1) : \forall i \in ℕ_{l (h)}} .$

The intersection of h₁ and h₂ is denoted as follows:

$h_{1} ⊻ h_{2} = {\begin{matrix} h_{2}, if l_{1} \leq l_{2}, β (h_{2}, h_{1}) \leq_{(l_{1})} h_{1}, \\ {σ_{h_{1}} (i) \land σ_{γ (h_{2}, h_{1})} (i), i \in ℕ_{l_{1}},} \\ if l_{1} \leq l_{2}, β (h_{2}, h_{1}) ≰_{(l_{1})} h_{1}, \\ h_{2} ⊻ h_{1}, otherwise . \end{matrix}$

(3) The union of h₁ and h₂ is denoted as follows:

$h_{1} ⊻ h_{2} = {\begin{matrix} h_{2}, if l_{1} \leq l_{2}, h_{1} \leq_{(l_{1})} γ (h_{2}, h_{1}), \\ {σ_{h_{1}} (i) \lor σ_{β (h_{2}, h_{1})} (i), i \in ℕ_{l_{1}}}, \\ if l_{1} \leq l_{2}, h_{1} ≰_{(l_{1})} γ (h_{2}, h_{1}), \\ h_{2} ⊻ h_{1}, otherwise . \end{matrix}$

Theorem 3.2. For $\forall h_{1}, h_{2} \in ℍ$ , ⊼ and ⊻ are defined in Definition 3.2, then the following properties hold:

Idempotent: (1) h₁⊼h₁ = h₁, (2) h₁ ⊻ h₁ = h₁,

Commutativity: (1) h₁⊼h₂ = h₂⊼h₁, (2) h₁ ⊻ h₂ = h₂ ⊻ h₁,

Double negation law: $(h_{1}^{c})^{c} = h_{1}$ ,

De’Morgan law: (1) h₁ ⊼ h₂ = $(h_{1}^{c}$ ⊻ $h_{2}^{c})^{c}$ , (2) h₁ ⊻ h₂ = $(h_{1}^{c}$ ⊼ $h_{2}^{c})^{c}$ ,

h₁ ≤ h₂ ⇔ h₁⊼h₂ = h₁ ⇔ h₁ ⊻ h₂ = h₂,

h₁⊼h₂ ≤ h₁, h₂ ≤ h₁ ⊻ h₂,

Absorption law: (1) h₁⊼ (h₁ ⊻ h₂) = h₁, (2) h₁ ⊻ (h₁⊼h₂) = h₁,

Monotonicity: If $h_{1} \leq h_{2}, \forall h_{3} \in ℍ, h_{1} ⊻ h_{3} \leq h_{2} ⊻ h_{3},$ and h₁ ⊻ h₃ ≤ h₂ ⊻ h₃.

Proof. Properties (1)-(7) can be proved directly based on Definition 3.2, now we will prove (8). For $\forall h_{1}, h_{2}, h_{3} \in ℍ$ , and l₁, l₂, l₃ are the numbers of the values of h₁, h₂ and h₃ respectively. By the definition of ⊼ and ⊻, we only need to prove (8) when l₁ ≤ l₂ ≤ l₃.

(8) For all $h_{1}, h_{2}, h_{3} \in ℍ$ , h₁ ≤ h₂, if β (h₃, h₁) ≤ _(l₁)h₁, we can conclude β (h₃, h₂) ≤ _(l₂)h₂, thus h₁⊼h₃ = h₃ = h₂⊼h₃; If β (h₃, h₁) _{≰ (_l1)}h₁, but β (h₃, h₂) ≤ _(l₂)h₂, we have $h_{1} ⊻ h_{3} = {σ_{h_{1}} (i) \land γ_{(h_{3}, h_{1})} (i), i \in ℕ_{l_{1}}} \leq h_{3} = h_{2} ⊻ h_{3};$ It can be proved straightforwardly when if β (h₃, h₁) _{≰ (_l1)}h₁, β (h₃, h₂) _{≰ (_l2)}h₂ . h₁ ⊻ h₃ ≤ h₂ ⊻ h₃ can be proved by the similar way.

Note 3.1. It should be noted that the distributivity laws do not hold for operators ⊼ and ⊻ on $ℍ$ . Let h₁ = {0.2, 0.7} , h₂ = {0.3, 0.5, 0.7} , h₃ = {0.1, 0.4, 0.6, 0.8}, then we have

h₁⊼ (h₂ ⊻ h₃) = {0.2, 0.6} ≠ (h₁⊼h₂) ⊻ (h₁⊼h₃) = {0.2, 0.5},

h₁ ⊻ (h₂⊼h₃) = {0.4, 0.7} ≠ (h₁ ⊻ h₂) ⊼ (h₁ ⊻ h₂) = {0.5, 0.7}.

But the following inequalities hold for all h₁, h₂, h₃ ∈ $ℍ$ ,

(h₁⊼h₂) ⊻ (h₁⊼h₃) ≤ h₁⊼ (h₂ ⊻ h₃) ,

h₁ ⊻ (h₂⊼h₃) ≤ (h₁ ⊻ h₂) ⊼ (h₁ ⊻ h₂).

Definition 3.3. For all A, B∈ THF(U), then

the complement of A is denoted by A^c such that ∀x ∈ U,

h_A
^c (x) = ∼ h_A (x) = {1 - σ_{h_A(x)} (i), i ∈ $ℕ_{l (h_{A (x)})}}$ ;

the intersection of A and B is denoted by A⋒B such that ∀ x ∈ U,

h_A⋒B (x) = h_A(x)⊼h_B(x);

the union of A and B is denoted by A⋓B such that ∀ x ∈ U,

h_A⋓B (x) = h_A(x) ⊻ h_B(x).

At last, we first introduce the α-level sets, strong α-level sets for THFS as follows:

Definition 3.4. For any A∈ THF(U), $α_{1 \dots m} \in ℍ$ , the α_1⋯m-level sets A_{α
_1⋯m}, strong α_1⋯m-level sets A_{α_1⋯m+} can be defined as follows:

A_{α
_1⋯m} = {x ∈ U : h_A (x) ≥ α_1⋯m};

A_{α_1⋯m+} = {x ∈ U : h_A (x) > α_1⋯m}.

Note 3.2. For ∀A∈ THF(U), $α_{1 \dots m} \in ℍ$ , we have:

h_A (x) ≥ α_1⋯m ⇔ α_1⋯m ≤ h_A (x), for x ∈ U,

h_A (x) > α_1⋯m ⇔ α_1⋯m < h_A (x), for x ∈ U.

In this section, on the basis of the preliminary notations of THFEs and THFSs, the more reasonable partial orders on THFEs and THFSs are proposed and the basic operations are further defined. In the following section, typical fuzzy rough sets are defined and studied based on the aforementioned fundamental operations.

4 Typical hesitant fuzzy rough sets

In this section, typical hesitant fuzzy rough set will be introduced to approximate a typical hesitant fuzzy target through a typical hesitant fuzzy relation.

4.1 Definitions of typical hesitant fuzzy approximation operators

Definition 4.1. Let U and W be two universes of discourse, a typical hesitant fuzzy relation from U to W is the typical hesitant fuzzy subset such that R ∈ THF (U × W) where $R = {〈 (x, y), h_{R} (x, y) 〉 : (x, y) \in U \times W}, \forall (x, y) \in U \times W, h_{R} (x, y) \in ℍ$ , which is used to denote the possible membership degrees of the relationship between x and y.

Definition 4.2. Let R∈ THF(U× W),

R is serial, if for each x ∈ U, there exists y ∈ W such that h_R (x, y) = {1},

R is reflexive, if h_R (x, x) = {1} for all x ∈ U, when U = W,

R is symmetric, if h_R (x, y) = h_R (y, x) for all x, y ∈ U, when U = W,

R is transitive, if ⊻_y∈U (h_R (x, y) ⊼h_R (y, z)) ≤ h_R (x, z) for all x, z ∈ U, when U = W.

where ≤ be a partial order defined in Definition 3.1.

Theorem 4.1. Let R be a binary typical hesitant fuzzy relationship between the discourse of universe U and W, R_{α
_1⋯m} is the α_1⋯m-level set of R, where $α_{1 \dots m} \in ℍ$ , then:

R is serial ⇔ R_{α
_1⋯m} is serial,

R is reflexive ⇔ R_{α
_1⋯m} is reflexive,

R is symmetric ⇔ R_{α
_1⋯m} is symmetric,

R is transitive ⇔ R_{α
_1⋯m} is transitive.

Proof. It can be proved straightforwardly from Definition 4.2.

Definition 4.3. Let R be a typical hesitant fuzzy relation from U to W. The tripe (U, W, R) is called a generalized typical hesitant fuzzy approximation space. For any set A∈ THF(W), the lower and upper approximations of A, $\underline{R} (A)$ and $\bar{R} (A)$ , with respect to the approximation space (U, W, R) are THFs of U whose membership functions, for each x ∈ U, are defined respectively by $h_{\underline{R} (A)} (x) = ⊻_{y \in W} [h_{R}^{c} (x, y) ⊻ h_{A} (y)], x \in U$ (4) $h_{\bar{R} (A)} (x) = ⊻_{y \in W} [h_{R} (x, y) ⊻ h_{A} (y)], x \in U$

The pair $(\underline{R} (A)$ , $\bar{R} (A))$ is referred to as a generalized THFRS of A in terms of the typical hesitant fuzzy relation R. $\underline{R}$ and $\bar{R}$ THF(W)→ THF(U) are referred to as lower and upper generalized typical hesitant fuzzy rough approximation operators respectively.

Note 4.1. If R degenerates to a crisp relation from U to W, the typical hesitant fuzzy rough operators are degenerated to rough typical hesitant fuzzy approximation operators. Under such circumstance, the pair $(\underline{R} (A), \bar{R} (A))$ is referred to as a generalized rough typical hesitant fuzzy set and they satisfy the followings:

$\begin{matrix} h_{\underline{R} (A)} (x) = ⊻_{y \in W} h_{A} (y), x \in U \\ h_{\bar{R} (A)} (x) = ⊻_{y \in W} h_{A} (y), x \in U \end{matrix}$ (5)

For simplicity, we call both typical hesitant fuzzy rough approximation operators and rough typical hesitant fuzzy approximation operators the typical hesitant fuzzy approximation operators.

4.2 Representations of typical hesitant fuzzy approximation operators

Definition 4.4. A set-valued mapping $N : ℍ \to P (U)$ is said to be nested if for all $α, α^{'} \in ℍ$ ,

$α_{1 \dots m} \leq α_{1 \dots m}^{'} \Rightarrow N (α_{1 \dots m}^{'}) \subseteq N (α_{1 \dots m})$ .

The class of all $P (U)$ - valued nested mappings on $ℍ$ will be denoted by $N (U)$ .

Obviously, The following theorem holds from the above definition.

Definition 4.5. Let $N \in N (U)$ . A function $f : N (U) \to$ THF(U) is defined by:

h_A (x) = h_f(N) (x)

$= ⊻_{α_{1 \dots m} \in ℍ} (α ⊻ h_{N (α_{1 \dots m})} (x)), x \in U,$

In the function, h_{N(α_1⋯m)} (x) is the characteristic function of N (α_1⋯m). f is a surjective homomorphism, and the following properties hold:

A_{α_1⋯m+} ⊆ N (α_1⋯m) ⊆ A_{α
_1⋯m},

A_{α
_1⋯m} = ∩ _{λ_1⋯n<α_1⋯m}N (λ_1⋯n) ,

A_{α_1⋯m+} = ∪ _{λ_1⋯n>α_1⋯m}N (λ_1⋯n) ,

$A = ⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ A_{α_{1 \dots m} +})$

$= ⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ A_{α_{1 \dots m}})$ .

Definition 4.6. Let (U, W, R) be a typical hesitant fuzzy approximation space, (U, W, R_{α
_1⋯m}) and (U, W, R_{α_1⋯m+}) are two crisp approximation spaces induced by the crisp relations R_{α
_1⋯m} and R_{α_1⋯m+}, ∀X ∈ $P (W)$ . The upper and lower approximations of X with respect to (U, W, R_{α
_1⋯m}) and (U, W, R_{α_1⋯m+}) are defined respectively as

$\bar{R_{α_{1 \dots m}}} (X) = {x \in U : r_{α_{1 \dots m}} (x) \cap X \neq \emptyset}, \underline{R_{α_{1 \dots m}}} (X) = {x \in U : r_{α_{1 \dots m}} (x) \subseteq X},$

$\bar{R_{α_{1 \dots m} +}} (X) = {x \in U : r_{α_{1 \dots m} +} (x) \cap X \neq \emptyset},$ $\underline{R_{α_{1 \dots m} +}} (X) = {x \in U : r_{α_{1 \dots m} +} (x) \subseteq X}$ .

Where r_{α
_1⋯m} (x) = {y ∈ W : (x, y) ∈ R_{α
_1⋯m}} and r_{α_1⋯m+} (x) = {y ∈ W : (x, y) ∈ R_{α_1⋯m+}}.

Crearly, ∀X ∈ $P (W)$ and ${0} \leq α_{1 \dots m} \leq α_{1 \dots m}^{'} \leq {1},$

$\underline{R_{α_{1 \dots m}}} (X) \subseteq \underline{R_{α_{1 \dots m}^{'}}} (X), \underline{R_{α_{1 \dots m} +}} (X) \subseteq \underline{R_{α_{1 \dots m}^{'} +}} (X),$

$\bar{R_{α_{1 \dots m}^{'}}} (X) \subseteq \bar{R_{α_{1 \dots m}}} (X), \bar{R_{α_{1 \dots m}^{'} +}} (X) \subseteq \bar{R_{α_{1 \dots m} +}} (X) .$

Definition 4.7. ∀A∈ THF(W) and ${0} \leq α_{1 \dots m}^{'} \leq {1},$ the upper and lower approximations of $A_{α_{1 \dots m}^{'}}$ and $A_{α_{1 \dots m}^{'} +}$ with respect to (U, W, R_{α
_1⋯m}) and (U, W, R_{α_1⋯m+}) are defined respectively as

$\underline{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'}}) = {x \in U : r_{α_{1 \dots m}} (x) \subseteq A_{α_{1 \dots m}^{'}}},$

$\bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'}}) = {x \in U : r_{α_{1 \dots m}} (x) \cap A_{α_{1 \dots m}^{'}} \neq \emptyset},$

$\underline{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'} +}) = {x \in U : r_{α_{1 \dots m}} (x) \subseteq A_{α_{1 \dots m}^{'} +}},$

$\bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'} +}) = {x \in U : r_{α_{1 \dots m}} (x) \cap A_{α_{1 \dots m}^{'} +} \neq \emptyset},$

$\underline{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'}}) = {x \in U : r_{α_{1 \dots m} +} (x) \subseteq A_{α_{1 \dots m}^{'}}},$

$\bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'}}) = {x \in U : r_{α_{1 \dots m} +} (x) \cap A_{α_{1 \dots m}^{'}} \neq \emptyset},$

$\underline{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'} +}) = {x \in U : r_{α_{1 \dots m} +} (x) \subseteq A_{α_{1 \dots m}^{'} +}},$

$\bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'} +}) = {x \in U : r_{α_{1 \dots m} +} (x) \cap A_{α_{1 \dots m}^{'} +} \neq \emptyset} .$

Theorem 4.2. Let (U, W, R) be a typical hesitant fuzzy approximation space, (U, W, R_{α
_1⋯m}) and (U, W, R_{α_1⋯m+}) are two crisp approximation spaces induced by the crisp relations R_{α
_1⋯m} and R_{α_1⋯m+}, ∀A∈THF(W), the approximate operators with respect to $A_{α_{1 \dots m}^{'}}$ and $A_{α_{1 \dots m}^{'} +}$ satisfy the following properties:

$\underline{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'} +}) \subseteq \underline{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'} +}) \subseteq \underline{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'}}),$

$\underline{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'} +}) \subseteq \underline{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'}}) \subseteq \underline{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'}}),$

$\bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'} +}) \subseteq \bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'}}) \subseteq \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'}}),$

$\bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}^{'} +}) \subseteq \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'} +}) \subseteq \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}^{'}}) .$

$\forall α_{1 \dots m} \in ℍ$ , it can be observed with Definition 4.4 that all of the following classes are $P (U)$ valued nested mapping on THF(U): ${\underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m}})}$ , ${\underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m} +})}$ , ${\underline{R_{α_{1 \dots m}^{c} +}} (A_{α_{1 \dots m}})}$ , ${\underline{R_{α_{1 \dots m}^{c} +}}$ (A_{α_1⋯m+})}, ${\bar{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m}})}$ , ${\bar{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m} +})}$ , ${\bar{R_{α_{1 \dots m}^{c} +}} (A_{α_{1 \dots m}})}$ , ${\bar{R_{α_{1 \dots m}^{c} +}} (A_{α_{1 \dots m} +})}$ . The following theorem shows that the typical hesitant fuzzy rough approximation operators can be represented by the relative characteristic functions.

Theorem 4.3. Let (U, W, R) be a typical hesitant fuzzy approximation space and A∈ THF(W), then

$\begin{matrix} \bar{R} (A) = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}})] \\ = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m} +})] \\ = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}})] \\ = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m} +})] \end{matrix}$ ,

$\begin{matrix} \underline{R} (A) = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m}})] \\ = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m} +})] \\ = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R_{(α_{1 \dots m}^{c}) +}} (A_{α_{1 \dots m}})] \\ = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R_{(α_{1 \dots m}^{c}) +}} (A_{α_{1 \dots m} +})] \end{matrix}$ ,

$\begin{matrix} [\bar{R} (A)]_{α_{1 \dots m} +} \subseteq \bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m} +}) \subseteq \bar{R_{α_{1 \dots m} +}} \\ (A_{α_{1 \dots m}}) \subseteq \bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}}) \subseteq [\bar{R} (A)]_{α_{1 \dots m}} \end{matrix}$ ,

$\begin{matrix} [\bar{R} (A)]_{α_{1 \dots m} +} \subseteq \bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m} +}) \subseteq \bar{R_{α_{1 \dots m}}} \\ (A_{α_{1 \dots m} +}) \subseteq \bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}}) \subseteq [\bar{R} (A)]_{α_{1 \dots m}} \end{matrix}$ ,

$\begin{matrix} [\underline{R} (A)]_{α_{1 \dots m} +} \subseteq \underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m} +}) \subseteq \underline{R_{(α_{1 \dots m}^{c}) +}} \\ (A_{α_{1 \dots m} +}) \subseteq \underline{R_{(α_{1 \dots m}^{c}) +}} (A_{α_{1 \dots m}}) \subseteq [\underline{R} (A)]_{α_{1 \dots m}} \end{matrix}$ ,

$\begin{matrix} [\underline{R} (A)]_{α_{1 \dots m} +} \subseteq \underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m} +}) \subseteq \underline{R_{α_{1 \dots m}^{c}}} \\ (A_{α_{1 \dots m}}) \subseteq \underline{R_{(α_{1 \dots m}^{c}) +}} (A_{α_{1 \dots m}}) \subseteq [\underline{R} (A)]_{α_{1 \dots m}} \end{matrix}$ .

Proof. (1) ∀A∈ THF(W) and x ∈ U, it should be noted for all $α_{1 \dots m} \in ℍ$ ,

$⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}})] (x)$

$= ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}}) (x)]$

$= sup {α_{1 \dots m} \in ℍ : \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}}) (x) = {1}}$

$= sup {α_{1 \dots m} \in ℍ : x \in \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}})}$

$= sup {α_{1 \dots m} \in ℍ : r_{α_{1 \dots m}} (x) \cap A_{α_{1 \dots m}} \neq \emptyset}$

$= sup {α_{1 \dots m} \in ℍ : \exists y \in W, h_{R} (x, y) \geq α_{1 \dots m},$

h_A (y) ≥ α_1⋯m}

$= sup {α_{1 \dots m} \in ℍ : \exists y \in W, h_{R} (x, y) ⊻ h_{A} (y) \geq$

α_1⋯m}

$= sup {α_{1 \dots m} \in ℍ : ⊻_{y \in W} [h_{R} (x, y) ⊻ h_{A} (y)] \geq$

α_1⋯m}

$= ⊻_{y \in W} [h_{R (x, y)} ⊻ h_{A} (y)] = \bar{R} (A) (x)$ .

i.e., $⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}})] = \bar{R} (A) .$

Likewise,

$\bar{R} (A) = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m} +})]$

$= ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m}})]$

$= ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R_{α_{1 \dots m} +}} (A_{α_{1 \dots m} +})] .$

∀x ∈ U,

$⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m}})] (x)$

$= ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ (\underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m}}) (x))]$

$= sup {α_{1 \dots m} \in ℍ : \underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m}}) (x) = {1}}$

$= sup {α_{1 \dots m} \in ℍ : r_{α_{1 \dots m}^{c}} (x) \subseteq A_{α_{1 \dots m}}}$

$= sup {α_{1 \dots m} \in ℍ : \forall y \in W [y \in r_{α_{1 \dots m}^{c}} (x) \Rightarrow y \in$

A_{α
_1⋯m}]}

$= sup {α_{1 \dots m} \in ℍ : \forall y \in W [h_{R} (x, y) \geq α_{1 \dots m}^{c} \Rightarrow$

h_A (y) ≥ α_1⋯m]}

$= sup {α_{1 \dots m} \in ℍ : \forall y \in W [h_{R^{c}} (x, y) \leq α_{1 \dots m} \Rightarrow$

h_A (y) ≥ α_1⋯m]}

$= sup {α_{1 \dots m} \in ℍ : \forall y \in W [(h_{R^{c}} (x, y)) ⊻ h_{A} (y)) \geq$

α_1⋯m]}

$= sup {α_{1 \dots m} \in ℍ : ⊻_{y \in W} [(h_{R^{c}} (x, y)) ⊻ h_{A} (y))] \geq$

α_1⋯m}

= ⊻ _y∈W [h_{R
^c} (x, y) ⊻ h_A (y)],

That is,

$\underline{R} (A) = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m}})]$ .

Similarly, it can be observed that

$\underline{R} (A) = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R_{α_{1 \dots m}^{c}}} (A_{α_{1 \dots m} +})]$

$= ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R_{(α_{1 \dots m}^{c}) +}} (A_{α_{1 \dots m}})]$

$= ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R_{(α_{1 \dots m}^{c}) +}} (A_{α_{1 \dots m} +})]$ .

Combining formulas of (4) and (5) in Definition 4.3 and Note 4.1 and Definition 4.5, (3)-(6) can be proved by the similar way.

If the typical hesitant fuzzy relation in Theorem 4.2 degenerate to a crisp relation, then the representation theorem for rough typical hesitant fuzzy approximation operators can hold.

Theorem 4.4. Let (U, W, R) be a generalized crisp approximation space and A∈ THF(W), then

$\bar{R} (A) = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R} (A_{α_{1 \dots m}})] = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \bar{R} (A_{α_{1 \dots m} +})]$

$\underline{R} (A) = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R} (A_{α_{1 \dots m}})] = ⊻_{α_{1 \dots m} \in ℍ} [α_{1 \dots m} ⊻ \underline{R} (A_{α_{1 \dots m} +})]$

$[\bar{R} (A)]_{α_{1 \dots m} +} \subseteq \bar{R} (A_{α_{1 \dots m} +}) \subseteq \bar{R} (A_{α_{1 \dots m}}) \subseteq [\bar{R} (A)]_{α_{1 \dots m}}$ ,

$[\underline{R} (A)]_{α_{1 \dots m} +} \subseteq \underline{R} (A_{α_{1 \dots m} +}) \subseteq \underline{R} (A_{α_{1 \dots m}}) \subseteq [\underline{R} (A)]_{α_{1 \dots m}}$ .

4.3 Properties of typical hesitant fuzzy approximation operators

Theorem 4.5. For all $A, B, a = \hat{a_{1 \dots m}} \in$ THF(W), the lower and upper typical hesitant fuzzy rough (rough typical hesitant, respectively) approximation operators, $\underline{R}$ and $\bar{R}$ defined by Eq.(3) and Eq.(4) respectively, satisfy the properties:

$\underline{R} (A) = \sim \bar{R} (\sim A)$ ,

$\bar{R} (A) = \sim \underline{R} (\sim A)$ ,

$\underline{R} (A ⋓ \hat{a_{1 \dots m}}) \supseteq \underline{R} (A) ⋓ \hat{a_{1 \dots m}}$ ,

$\bar{R} (A ⋒ \hat{a_{1 \dots m}}) \subseteq \bar{R} (A) ⋒ \hat{a_{1 \dots m}},$

$A \subseteq B \Rightarrow \underline{R} (A) \subseteq \underline{R} (B),$

$A \subseteq B \Rightarrow \bar{R} (A) \subseteq \bar{R} (B),$

$\underline{R} (A ⋓ B) \supseteq \underline{R} (A) ⋓ \underline{R} (B)$ ,

$\bar{R} (A ⋒ B) \subseteq \bar{R} (A) ⋒ \bar{R} (B),$

$\underline{R} (A ⋒ B) \subseteq \underline{R} (A) ⋒ \underline{R} (B)$ ,

$\bar{R} (A ⋓ B) \supseteq \bar{R} (A) ⋓ \bar{R} (B) .$

Proof. (L1) and (U1) can be obtained straightforwardly from Definition 4.3.

∀x ∈ U,

$h_{\underline{R} (A ⋓ \hat{a_{1 \dots m}})} (x)$

$= ⊻_{y \in W} [h_{R}^{c} (x, y) ⊻ (h_{A} (y) ⊻ a_{1 \dots m})]$

$= ⊻_{y \in W} [h_{R}^{c} (x, y) ⊻ h_{A} (y)) ⊻ a_{1 \dots m}]$

$\geq ⊻_{y \in W} [h_{R}^{c} (x, y) ⊻ h_{A} (y) ⊻ a_{1 \dots m}]$

$= h_{\underline{R} (A)} (x) ⊻ a_{1 \dots m}$

Thus

$\underline{R} (A ⋓ \hat{a_{1 \dots m}}) \supseteq \underline{R} (A) ⋓ \hat{a_{1 \dots m}}$ ,

Combining with Theorem 4.2, ∀x ∈ U,

A ⊆ B ⇔ h_A (x) ≤ h_B (x)

$\Rightarrow h_{R}^{c} (x, y) ⊻ h_{A} (x) \leq h_{R}^{c} (x, y) ⊻ h_{B} (x)$

$\Rightarrow ⊻_{y \in W} (h_{R}^{c} (x, y) ⊻ h_{A} (x)) \leq ⊻_{y \in W} (h_{R}^{c} (x, y) ⊻ h_{B} (x))$

$\Leftrightarrow h_{\underline{R} (A)} (x) \leq h_{\underline{R} (B)} (x)$ ,

thus,

$\underline{R} (A) \subseteq \underline{R} (B),$

can be proved similar to (L3),

∀ x ∈ U, by Note 4.1, we know

$h_{\underline{R} (A ⋒ B)} (x) = ⊻_{y \in W} (h_{R}^{c} (x, y) ⊻ h_{(A ⋒ B)} (x))$

$\leq ⊻_{y \in W} (((h_{R}^{c} (x, y) ⊻ h_{A} (x)) ⊻ (h_{R}^{c} (x, y) ⊻ h_{B} (x)))$

$= h_{\underline{R} (A)} (x) ⊻ h_{\underline{R} (B)} (x)$ .

i.e.,

$\underline{R} (A ⋒ B) \subseteq \underline{R} (A) ⋒ \underline{R} (B)$ .

Similarly, (U2)-(U5) can be checked.

Proposition 4.1. Let R be an arbitrary typical hesitant fuzzy relationship from U to W, for all (x, y)∈ U × W, X ∈ THF(W), we have

$h_{\bar{R} ({1}_{{y}})} (x) = h_{R} (x, y),$

$h_{\underline{R} ({1}_{W \ {y}})} (x) = 1 - h_{R} (x, y),$

$h_{\bar{R} ({1}_{X})} (x) = ⊻_{y \in X} {h_{R} (x, y) : y \in X},$

$h_{\underline{R} ({1}_{X})} (x) = ⊻_{y \notin X} {1 - h_{R} (x, y)}$ .

Where {1}_{y} denotes the typical hesitant fuzzy singleton with typical hesitant fuzzy value {1} at y and {0} elsewhere, {1} _X denotes the characteristic function of X.

Proof. (1) ∀ (x, y) ∈ U × W,

$h_{\bar{R} ({1}_{y})} (x) = ⊻_{z \in W} [h_{R} (x, z) ⊻ h_{{1}_{y}} (z)]$

= (h_R (x, y) ⊼ {1}) ⊻ (⊻ _z∈W∖{y} (h_R (x, z) ⊼ {0}))

= h_R (x, y)

∀ (x, y) ∈ U × W,

$h_{\underline{R} ({1}_{W \ {y}})} (x)$

= _⊼z ∈W [(1 - h_R (x, z)) ⊻ h_{({1}_W\{y})} (z)]

= {_{⊼z ∈W ∖{y}} ((1 - h_R (x, z)) ⊻ {1})} ⊼ {(1-

h_R (x, y)) ⊻ {0}}

= {1} ⊼ (1 - h_R (x, y))

= 1 - h_R (x, y)

(3) and (4) are straightforward from Equation (4) in Definition 4.3.

Theorem 4.6. If R is an arbitrary typical hesitant fuzzy relation from U to W, then

R is serial $\Leftrightarrow \bar{R} (W) = U$ ,

$\Rightarrow \underline{R} (A) \subseteq \bar{R} (A), \forall A \in$ THF(W).

R is reflexive, U = W

$\Leftrightarrow \underline{R} (A) \subseteq A, \forall A \in$ THF(U),

$\Leftrightarrow A \subseteq \bar{R} (A), \forall A \in$ THF(U).

R is symmetric, U = W

$\Leftrightarrow h_{\underline{R}} (1_{U - {x}}) (y) = h_{\underline{R}} (1_{U - {y}}) (x),$

∀ (x, y) ∈ U × U,

$\Leftrightarrow h_{\bar{R}} ({1}_{{x}}) (y) = h_{\bar{R}} ({1}_{{y}}) (x),$

∀ (x, y) ∈ U × U.

R is transitive

$\Leftrightarrow \underline{R} (A) \subseteq \underline{R} (\underline{R} (A))$ ,

$\Leftrightarrow \bar{R} (\bar{R} (A)) \subseteq \bar{R} (A)$ .

Proof. (1) From Proposition 4.1, that for any x ∈ U,

$\bar{R} (W) (x) = ⋁_{y \in W} R (x, y)$ .

Then

R is serial ⇔ ∃ y ∈ W, suchthat h_R (x, y) = {1}

⇔ ⋁ _y∈Wh_R (x, y) = {1}

$\Leftrightarrow \bar{R} (W) = U .$

If R is serial, then ∀x ∈ U, ∃ z ∈ W, such that $h_{R} (x, z) = {1}$ , it follows that h_∼R(x, z) = {0},

$h_{\underline{R} (A)} (x) = ⊻_{y \in W} {h_{R}^{c} (x, y) ⊻ h_{A} (y)}$

$= {h_{R}^{c} (x, z) ⊻ h_{A} (z)} ⊻ (⊻_{y \neq z} {h_{R}^{c} (x, y) ⊻ h_{A} (y))$

$= h_{A} (z) ⊻ (⊻_{y \neq z} {h_{R}^{c} (x, y) ⊻ h_{A} (y))$ .

Moreover,

$h_{\bar{R} (A)} (x) = ⊻_{y \in W} {h_{R} (x, y) ⊻ h_{A} (y)}$

= {h_R (x, z) ⊼h_A (z)} ⊻ (⊻ _y≠z {h_R (x, y) ⊼h_A (y))

= h_A (z)} ⊻ (⊻ _y≠z {h_R (x, y) ⊼h_A (y))

By Theorem 4.5, we know that $h_{\underline{R} (A)} (x) \leq h_{A} (z) \leq h_{\bar{R} (A)} (x)$ .

i.e.,

$\underline{R} (A) \subseteq \bar{R} (A), \forall A \in$ THF(W).

(2) (1) and (2) are equivalent because of (L1) and (U1). We only need to prove that the reflexivity of R is equivalent to (2).

⇒: if R is reflexive, for ∀ A∈ THF(U), and x ∈ U, let h_A (x) = α_1⋯m, clearly, x ∈ A_{α
_1⋯m}. Since R is reflexive, R_{α
_1⋯m} is an ordinary reflexive binary relation as well. Hence x ∈ R_{α
_1⋯m} (x), and furthermore, x ∈ R_{α
_1⋯m} ∩ A_{α
_1⋯m}, i.e., $x \in \bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}})$ . Since, $\bar{R_{α_{1 \dots m}}} (A_{α_{1 \dots m}}) \subseteq (\bar{R} (A))_{α_{1 \dots m}}$ , we have that $x \in (\bar{R} (A))_{α_{1 \dots m}}$ , that is $h_{\bar{R} (A)} (x) \geq α_{1 \dots m} = h_{A} (x)$ which implies (2).

⇐ : Assume $A \subseteq \bar{R} (A), \forall A \in$ THF(U). Let A = {1} _x, for all x ∈ U. From Proposition 4.1 and the assumption, we conclude

${1} = {1}_{x} (x) \leq \bar{R} ({1}_{x}) (x) = R (x, x)$ .

It follows R is reflexive.

It is straightforward from Proposition 4.1.

We only need to prove (b) because of the dual operators of (L1) and (U1).

⇒: Assume R is transitive, then by Definition 4.3 and Theorem 4.3, we have ∀A∈ THF(U), $α_{1 \dots m} \in ℍ$ ,

$(\bar{R} (A))_{α_{1 \dots m} +} \subseteq {\bar{R}}_{α_{1 \dots m}} (A_{α_{1 \dots m} +})$

$\subseteq (\bar{R} (A))_{α_{1 \dots m}}$ ,

combining with Theorem 4.2 and Theorem 4.4,

$⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ (\bar{R} (A))_{α_{1 \dots m} +})$

$\subseteq ⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ {\bar{R}}_{α_{1 \dots m}} (A_{α_{1 \dots m} +}))$

$\subseteq ⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ (\bar{R} (A))_{α_{1 \dots m}})$ ,

which implies

$⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ {\bar{R}}_{α_{1 \dots m}} (A_{α_{1 \dots m} +})) = \bar{R} (A) .$

With the assumption, we get

$\bar{R} (\bar{R} (A))$

$= ⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ {\bar{R}}_{α_{1 \dots m}} ((\bar{R} (A))_{α_{1 \dots m} +}))$

$\subseteq ⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ {\bar{R}}_{α_{1 \dots m}} ({\bar{R}}_{α_{1 \dots m}} (A_{α_{1 \dots m} +})))$

$\subseteq ⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ {\bar{R}}_{α_{1 \dots m}} ({\bar{R}}_{α_{1 \dots m}} (A_{α_{1 \dots m}})))$

$\subseteq ⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ {\bar{R}}_{α_{1 \dots m}} (A_{α_{1 \dots m} +}))$

$= \bar{R} (A)$ .

Thus

$\bar{R} (\bar{R} (A)) \subseteq \bar{R} (A)$ .

⇐: Let x, y, z ∈ U and $α = \hat{α_{1 \dots m}}, λ = \hat{λ_{1 \dots n}} \in$ THF(U) such that h_R(x, y) ≥ λ_1⋯n, and h_R(y, z) ≥ λ_1⋯n. Then on the one hand,

$h_{\bar{R} (\bar{R} ({1}_{z}))} (x) \leq h_{\bar{R} ({1}_{z})} (x) = h_{R (x, z)}$ ,

and on the other hand,

$\bar{R} (\bar{R} ({1}_{z})) (x)$

$= sup {α_{1 \dots m} : x \in {\bar{R}}_{α_{1 \dots m}} (\bar{R} ({1}_{z}))_{α_{1 \dots m}}}$

$= sup {α_{1 \dots m} : R_{α_{1 \dots m}} \cap (\bar{R} ({1}_{z}))_{α_{1 \dots m}} \neq \emptyset}$

= sup {α_1⋯m : ∃ u ∈ U [h_R (x, u) ≥ α_1⋯m,

$(h_{\bar{R} ({1}_{z}))} (u) = h_{R} (u, z) \geq α_{1 \dots m}]}$

≥ min {h_R (x, y) , h_R (y, z)} ≥ λ_1⋯n,

thus h_R (x, z) ≥ λ_1⋯n, it follows that R_{α
_1⋯m} is transitive. We can conclude that R is transitive by Theorem 4.1.

Corollary 4.1. If R is a reflexive and transitive typical hesitant fuzzy relation on U, ∀ A ∈ THF(U), the following properties hold:

$\underline{R} (A) = \underline{R} (\underline{R} (A))$ ,

$\bar{R} (A) = \bar{R} (\bar{R} (A)) .$

Proof. It can be straightforward from Theorem 4.6.

Theorem 4.7. If R is a reflexive relation on U, then the following properties hold:

$\underline{R} (\hat{a_{1 \dots m}})$ = $\bar{R} (\hat{a_{1 \dots m}})$ = $\hat{a_{1 \dots m}}$ , ∀ $\hat{a_{1 \dots m}}$ ∈ THF (U),

$inf {h_{\underline{R} (A)} (x) : x \in U} = inf {h_{A} (x) : x \in U}, A \in$ THF(U),

$sup {h_{\bar{R} (A)} (x)$ : x ∈ U} = sup {h_A (x) : x ∈ U}, A∈ THF(U).

Proof. We only need to prove $\bar{R} (\hat{a_{1 \dots n}}) = \hat{a_{1 \dots n}}$ because of the duality of $\bar{R}$ and $\underline{R}$ .

$\forall α_{1 \dots m} \in ℍ, \hat{a_{1 \dots n}} \in$ THF(U),

$(\hat{a_{1 \dots n}})_{α_{1 \dots m}} \neq \emptyset \Leftrightarrow a_{1 \dots n} \geq α_{1 \dots m} \Leftrightarrow (\hat{a_{1 \dots n}})_{α_{1 \dots m}} = U$ .

Since R is reflexive⇔R_{α
_1⋯m} is reflexive, $\forall α_{1 \dots m} \in ℍ$ . Then ∀x ∈ U, x ∈ R_{α
_1⋯m} (x), i.e., $\forall α_{1 \dots m} \in ℍ, R_{α_{1 \dots m}} \neq \emptyset$ , then we conclude

$\bar{R} (\hat{a_{1 \dots n}}) (x)$

$= ⊻_{α_{1 \dots m} \in ℍ} (α_{1 \dots m} ⊻ {\bar{R}}_{α_{1 \dots m}} (\hat{a_{1 \dots n}})_{α_{1 \dots m}}) (x)$

$= sup {α_{1 \dots m} \in ℍ : x \in {\bar{R}}_{α_{1 \dots m}} (\hat{a_{1 \dots n}})_{α_{1 \dots m}}}$

$= sup {α_{1 \dots m} \in ℍ : R_{α_{1 \dots m}} (x) \cap (\hat{a_{1 \dots n}})_{α_{1 \dots m}} \neq \emptyset}$

= sup {α_1⋯m ≤ a_1⋯n : R_{α
_1⋯m} (x) ≠ ∅} = a_1⋯n,

which implies $\bar{R} (\hat{a_{1 \dots n}}) = \hat{a_{1 \dots n}}$ .

Let inf {h_A (x) : x ∈ U} = a_1⋯m, obviously, $\hat{a_{1 \dots m}} \subseteq A$ . Since R is reflexive, combining with (1) and Theorem 4.4, we have

$\hat{a_{1 \dots m}} = \underline{R} (\hat{a_{1 \dots m}}) \subseteq \underline{R} (A) \subseteq A .$

Hence

$a_{1 \dots m} \leq inf {h_{\underline{R} (A)} (x) : x \in U} \leq inf {h_{A} (x) : x \in U} = a_{1 \dots m}$ .

such that (2) holds.

(3) It is straightforward from (2).

5 A numerical example over two universes

In this section, a simple numerical example is presented to illuminate the application of THFRSs over two universes, which have been defined in Section 3, in clinical diagnoses.

5.1 Background statement

Fuzzy rough set model on two different universes has been applied effectively to clinical diagnoses systems [28, 32]. But in some practical situations, typical hesitant fuzzy clinical diagnosis systems may be obtained because of the inconsistency and incompleteness. So a typical hesitant fuzzy medical diagnosis system is presented as follows.

Let U be a finite non-empty set of patients andW be a finite non-empty set of symptoms. It issometimes difficult for the expert to diagnose the disease and in some circumstances the symptoms are given by different experts. Let x (x ∈ U) be one patient and $h (x, y) \in ℍ$ denotes the membership degree of object x with the symptom y, y ∈ W. Especially, h (x, y) =∅ means the patient x has no symptom y, x ∈ U, y ∈ W.

5.2 Numerical test

In what follows, we present the numerical example of a medical diagnosis.

Let U = {x₁, x₂, x₃, x₄, x₅} be finite universe of five diseases, W = {y₁, y₂, y₃, y₄} be a finite universe of symptoms and R be a typical hesitant fuzzy decision matrix which is defined from universe U to universe W which is shown in Table 1. X = {〈y₁, {0.991} , 〉, 〈y₂, {0.995} 〉, 〈y₃, ∅ 〉, 〈y₄, {0.998} 〉} is the membership function of a possible patient about the symptoms W = {y₁, y₂, y₃, y₄}.

We in what follows use the model of THFRS to check out which patient may have disease X.

$\underline{R_{0.991}} (X_{0.991}) = \underline{R_{0.995}} (X_{0.995}) = {x_{1}, x_{2}, x_{4}}$ , $\underline{R_{1}} (X_{1}) = {x_{1}, x_{4}}, \underline{R_{0.998}} (X_{0.998}) = {x_{1}, x_{4}}$ .

We conclude that x₁ and x₄ have disease X.

6 Conclusion

In this paper, we have studied the generalized rough set approximation operators in typical hesitant fuzzy environment. The existing preliminary notions and orders of THFES and THFSs have been reviewed, then the reasonable new partial orders and operations for THFSs and THFEs have been proposed and their properties have been discussed in detail. Furthermore, typical hesitant fuzzy rough sets have been constructed by the new operators of union and intersection defined in this paper. The paper proposes a general framework for the study of hesitant fuzzy rough sets. Attribute reduction is important to rough sets and we will study it in the future.

Footnotes

Acknowledgments

This work was supported by grants from the National Natural Science Foundation of China (Nos. 61005042), the Natural Science Foundation of Shaanxi Province (Nos. 2014JQ8348) and theFundamental Research Funds for the CentralUniversities.

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