Topological structures of interval-valued hesitant fuzzy rough set and its application

Abstract

Interval-valued hesitant fuzzy rough set, defined by Zhang et al. [55], is an extension of hesitant fuzzy rough sets, interval-valued fuzzy rough sets and fuzzy rough sets. For further studying the theories and applications of interval-valued hesitant fuzzy rough sets, in this paper, we mainly investigate the topological structures of interval-valued hesitant fuzzy rough sets. Firstly, the concept of interval-valued hesitant fuzzy topological spaces is introduced by us. Then relationships between interval-valued hesitant fuzzy rough approximation spaces and interval-valued hesitant fuzzy topological spaces are further established. It is proved that the set of all lower approximation sets based on an interval-valued hesitant fuzzy reflexive and transitive approximation space forms an interval-valued hesitant fuzzy topology; and conversely, for an interval-valued hesitant fuzzy rough topological space, there exists an interval-valued hesitant fuzzy reflexive and transitive approximation space such that the topology in the interval-valued hesitant fuzzy rough topological space is just the set of all lower approximation sets in the interval-valued hesitant fuzzy reflexive and transitive approximation space. That is to say, there exists a one-to-one correspondence between the set of all interval-valued hesitant fuzzy reflexive and transitive approximation spaces and the set of all interval-valued hesitant fuzzy rough topological spaces. Finally, a practical application is provided to illustrate the validity of the interval-valued hesitant fuzzy rough set model.

Keywords

Interval-valued hesitant fuzzy set Interval-valued hesitant fuzzy rough set Interval-valued hesitant fuzzy topological space

1 Introduction

Rough set theory, proposed by Pawlak [26, 27], is a mathematical approach to handle imprecision, vagueness and uncertainty in data analysis. It is well known that the equivalence relation is a key notion in Pawlak’s rough set model. However, the equivalence relation is a very stringent condition and limits the application of rough sets in practical problems. Therefore, many researchers have generalized the notion of Pawlak’s rough set by employing non-equivalence binary relations. Recently, rough set approximations have also been developed into fuzzy environment in which the results are called rough fuzzy sets [13 , 44] and fuzzy rough sets [13 , 57]. Moreover, by integrating rough set theory with interval-valued fuzzy sets and intuitionistic fuzzy (IF, for short) sets, many researchers proposed some new rough sets model [6 , 63–66]. Very recently, Yang et al. [52] extended rough set theory into hesitant fuzzy environment, introduced the concept of hesitant fuzzy rough sets and proposed an axiomatic approach to the model. Meanwhile, through hesitant fuzzy relations, Deepak and John [12] also presented a hesitant fuzzy rough set which is different from the one introduced by Yang et al.

The concepts of topology, as a branch of mathematics, exist not only in almost all branches of mathematics but also in many real life applications. For example, topologies are widely used in the research field of machine learning and cybernetics [1 , 24]. Since topological structure is also an important base for knowledge extraction and processing, more and more researchers are devoting to the study of relationships between rough approximation operators and the topological structure of rough sets. In fact, many authors studied topological structures of crisp rough set algebras [3 , 62]. Recently, Qin et al. [28, 29] discussed topological structures of rough sets in the fuzzy environment. One of the main results is that a fuzzy reflexive and transitive relation on a universe of discourse can yield a fuzzy topology on the same universe, and conversely, under some conditions, a fuzzy topology can be associated with a fuzzy reflexive and transitive relation producing the same fuzzy topology. Along the lines of References [28, 29], Zhou and Wu [45, 60] generalized the results to IF rough sets and established relationships between IF rough set approximations and IF topologies. More recently, Zhang [65, 66] constructed interval type-2 rough fuzzy sets and generalized interval type-2 fuzzy rough sets by integrating rough set theory with interval type-2 fuzzy set theory, and also investigated their topological structures.

Hesitant fuzzy (HF, for short) set theory, initiated by Torra [37, 38] as one of the extensions of Zadeh’s fuzzy set [61], permits the membership degree of an element to a set having several possible values. Since the appearance of hesitant fuzzy set, it has attracted more and more scholars’ attention [14 , 48–50]. Another important concept used to cope with imperfect and imprecise information is interval-valued hesitant fuzzy (IVHF, for short) sets originated by Chen et al. [10, 11]. Intuitively, IVHF set is an extension of HF set and has its own merits. In many real decision making problems, due to insufficiency in available information, it is very difficult for decision makers to exactly quantify their opinions by several crisp numbers, but they can be represented by several interval numbers within [0,1]. For example, two decision makers investigate the membership degree of x into A. One wants to assign [0.4,0.6], but the other tends to assign [0.6,0.8]. They cannot persuade with each other, thus the membership degrees of x into A can be represented by {[0.4, 0.6] , [0.6, 0.8]}. Since IVHF sets were introduced, IVHF set theory has been applied in dealing with fuzzy decision making problems [11, 39]. Very recently, similarity, distance and entropy measures for IVHF sets have been investigated by Farhadinia [14]. It should be noted that many of researches about IVHF sets are mainly focusing on IVHF set itself. Although the discussions about fusions of IVHF set theory and other mathematical structures are rarely found in the related literatures, a major step is taken by Zhang et al. in [55]. By means of an IVHF relation, they introduced the concept of IVHF rough sets and proposed an axiomatic approach to the model. However, mathematical structures of the IVHF rough set are not still be investigated, such as lattice structures and topological structures. To develop the application of rough sets in IVHF environment, the topological properties on IVHF rough sets further need to be studied. The purpose of the present paper is mainly to investigate topological structures of IVHF rough sets, and further provides its application in medical diagnosis.

This paper is organized as follows. We review some basic notions related to HF sets, IVHF sets and IVHF rough sets in the next section. Section 3 introduces basic notions and results about IVHF topological spaces. In Section 4, the relationships between IVHF approximation spaces and IVHF topological spaces are established. Section 5 illustrates a practical application of the proposed model. We then conclude the paper with a summary and outlook for further research in Section 6.

2 Preliminaries

2.1 Hesitant fuzzy sets

In the subsection, we recall the concept of hesitant fuzzy sets introduced by Torra [37, 38].

Definition 2.1. [37, 38] Let U be a fixed set, a hesitant fuzzy (HF, for short) set A on U is in terms of a function h_A (x) that when applied to U returns a subset of [0,1], that is, $A = {< x, h_{A} (x) > | x \in U},$ where h_A (x) is a set of some different values in [0,1], representing the possible membership degrees of the element x ∈ U to A.

For convenience, we call h_A (x) a HF element.

Example 2.2. Let U = {x₁, x₂, x₃} be a reference set, h_A (x₁) = {0.7, 0.4, 0.5} , h_A (x₂) = {0.2, 0.4}, and h_A (x₃) = {0.3, 0.1, 0.7, 0.6} the HF elements ofx_i (i = 1, 2, 3) to a set A, respectively. Then A can be considered as a HF set, that is, $\begin{matrix} A = & {< x_{1}, {0.7, 0.4, 0.5} >, < x_{2}, {0.2, 0.4} >, \\ < x_{3}, {0.3, 0.1, 0.7, 0.6} >} . \end{matrix}$

2.2 Lattice and interval-valued hesitant fuzzy sets

First, we recall briefly a special complete lattice on [0, 1] ² with its logical operations proposed by Cornelis et al. [5, 6].

Definition 2.3. [5] Let L^I = {[μ, ν] ∈ [0, 1] × [0, 1] |μ ≤ ν} and denote [μ₁, ν₁] ≤ _{L
^I} [μ₂, ν₂] ⇔ μ₁ ≤ μ₂ and ν₁ ≤ ν₂, ∀ [μ₁, ν₁] , [μ₂, ν₂] ∈ L^I . Then the pair (L^I, ≤ _{L
^I}) is called a complete, bounded lattice.

The operators ∧ and ∨ on (L^I, ≤ _{L
^I}) are defined as follows: $\begin{matrix} [μ_{1}, ν_{1}] \land [μ_{2}, ν_{2}] = [\min {μ_{1}, μ_{2}}, \min {ν_{1}, ν_{2}}], \\ [μ_{1}, ν_{1}] \lor [μ_{2}, ν_{2}] = [\max {μ_{1}, μ_{2}}, \max {ν_{1}, ν_{2}}], \end{matrix}$ for [μ₁, ν₁] , [μ₂, ν₂] ∈ L^I.

Next, we review some basic concepts related to IVHF sets introduced by Chen [10]:

Definition 2.4. [10] Let U be a nonempty and finite universe of discourse, and Int[0,1] be the set of all closed subintervals of [0,1]. An IVHF set $𝔸$ on U is defined as $𝔸 = {< x, h_{𝔸} (x) > | x \in U},$ where $h_{𝔸} : U \to Int [0, 1]$ denotes all possible interval-valued membership degrees of the element x ∈ U to $𝔸$ .

For convenience, we call $h_{𝔸} (x)$ an IVHF element. The set of all IVHF sets on U is denoted by IVHF (U).

Example 2.5. Let U = {x₁, x₂} be a universe set, $h_{𝔸} (x_{1}) = {[0.2, 0.3], [0.4, 0.6], [0.5, 0.6]}$ and $h_{𝔸} (x_{2}) = {[0.3, 0.5], [0.4, 0.7]}$ be the IVHF elements of x_i (i = 1, 2) to a set $𝔸$ , respectively. Then $𝔸$ can be considered as an IVHF set, that is, $\begin{matrix} 𝔸 = { & < x_{1}, {[0.2, 0.3], [0.4, 0.6], [0.5, 0.6]} >, \\ < x_{2}, {[0.3, 0.5], [0.4, 0.7]} >} . \end{matrix}$

Here, we introduce several special IVHF sets as follows [55]: $\forall 𝔸 \in IVHF (U)$ ,

$𝔸$ is referred to as an empty IVHF set if and only if $h_{𝔸} (x) = {[0, 0]}$ for all x ∈ U. In this case, the empty IVHF set is denoted by ∅;

$𝔸$ is referred to as a full IVHF set if and only if $h_{𝔸} (x) = {[1, 1]}$ for all x ∈ U. In this case, the full IVHF set is denoted by $𝕌$ ;

$𝔸$ is referred to as a constant IVHF set if and only if $h_{𝔸} (x) = {[a_{1}^{L}, a_{1}^{U}], [a_{2}^{L}, a_{2}^{U}], \dots,$ $[a_{m}^{L}, a_{m}^{U}]}$ for all x ∈ U, where $[a_{i}^{L}, a_{i}^{U}] \in Int [0, 1], i = 1, \dots, m$ , i.e. $h_{𝔸} (x) \in 2^{Int [0, 1]}$ . In this case, the constant IVHF set is denoted by $\hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]}$ .

Meanwhile, for any y ∈ U, several special IVHF sets [1, 1] _y, its complement [1, 1] _U-y, and [1, 1] _M are defined respectively as follows [55]: for x ∈ U $\begin{matrix} h_{[1, 1]_{y}} (x) = {\begin{matrix} {[1, 1]}, & x = y, \\ {[0, 0]}, & x \neq y . \end{matrix} \\ h_{[1, 1]_{U - {y}}} (x) = {\begin{matrix} {[0, 0]}, & x = y, \\ {[1, 1]}, & x \neq y . \end{matrix} \\ h_{[1, 1]_{M}} (x) = {\begin{matrix} {[1, 1]}, & x \in M, \\ {[0, 0]} . & otherwise . \end{matrix} \end{matrix}$

It should be noted that the number of interval values in different IVHF elements may be different and the interval values are usually out of order. In order to rank the interval values, Xu [47] gave the definition as follows.

Definition 2.6. [47] Let a = [a^L, a^U] , and b = [b^L, b^U] , and let l_a = a^U - a^L and l_b = b^U - b^L; then the degree of possibility of a ≥ b is defined as: $p (a \geq b) = \max {1 - \max (\frac{b^{U} - a^{L}}{l_{a} + l_{b}}, 0), 0}$ (1)

Similarly, the degree of possibility of b ≥ a is defined as: $p (b \geq a) = \max {1 - \max (\frac{a^{U} - b^{L}}{l_{a} + l_{b}}, 0), 0}$ (2)

Equations (1) and (2) are proposed in order to compare two interval values, and to rank all the input arguments. Further details could be found in [47].

Suppose that $l (h_{𝔸} (x))$ stands for the number of interval values in the IVHF element $h_{𝔸} (x)$ . To operate correctly, Chen [10] gave the following assumptions:

(A1) All the elements in each IVHF element $h_{𝔸} (x)$ are arranged in increasing order by Equation (1). Let $h_{𝔸}^{σ (k)} (x)$ stands for the kth largest interval numbers in the IVHF element $h_{𝔸} (x)$ . In this case, $h_{𝔸}^{σ (k)} (x)$ is denoted by $h_{𝔸}^{σ (k)} (x) = [h_{𝔸}^{σ (k) L} (x), h_{𝔸}^{σ (k) U} (x)],$ where $h_{𝔸}^{σ (k) L} (x) = \inf h_{𝔸}^{σ (k)} (x)$ and $h_{𝔸}^{σ (k) U} (x) = \sup h_{𝔸}^{σ (k)} (x)$ respectively represent the lower and upper limits of $h_{𝔸}^{σ (k)} (x)$ .

(A2) If, for two IVHF elements $h_{𝔸} (x), h_{𝔹} (x),$ $l (h_{𝔸} (x)) \neq l (h_{𝔹} (x))$ , then $l = \max {l (h_{𝔸} (x)), l (h_{𝔹} (x))}$ . To have a correct comparison, the two IVHF elements $h_{𝔸} (x)$ and $h_{𝔹} (x)$ should have the same length l. If there are fewer elements in $h_{𝔸} (x)$ than in $h_{𝔹} (x)$ , an extension of $h_{𝔸} (x)$ should be considered optimistically by repeating its maximum element until it has the same length with $h_{𝔹} (x)$ .

For two IVHF sets $𝔸$ and $𝔹$ , Zhang [55] developed some new methods when operating the IVHF elements based on assumptions given by Chen [10], which are slightly different from some operations on IVHF sets that introduced by Chen [10]. The adjusted operational laws are defined as follows.

Definition 2.7. [55] Let U be a nonempty and finite universe of discourse. Suppose that $𝔸$ and $𝔹$ are two IVHF sets, namely, $\forall 𝔸, 𝔹 \in IVHF (U)$ , then, for allx ∈ U

(1) the complement of $𝔸$ , denoted by $𝔸^{c}$ , is given by $\begin{matrix} h_{𝔸^{c}} (x) & = \sim h_{𝔸} (x) = {[1 - h_{𝔸}^{σ (k) U} (x), \\ 1 - h_{𝔸}^{σ (k) L} (x)] | k = 1, 2, \dots, l}, \end{matrix}$ (2) the union of $𝔸$ and $𝔹$ , denoted by $𝔸 ⋓ 𝔹$ , is given by $\begin{matrix} h_{𝔸 ⋓ 𝔹} (x) & = h_{𝔸} (x) ⊻ h_{𝔹} (x) = {[h_{𝔸}^{σ (k) L} (x) \lor h_{𝔹}^{σ (k) L} (x), \\ h_{𝔸}^{σ (k) U} (x) \lor h_{𝔹}^{σ (k) U} (x)] | k = 1, 2, \dots, l}, \end{matrix}$ (3) the intersection of $𝔸$ and $𝔹$ , denoted by $𝔸 ⋒ 𝔹$ , is given by $\begin{matrix} h_{𝔸 ⋒ 𝔹} (x) & = h_{𝔸} (x) ⊼ h_{𝔹} (x) = {[h_{𝔸}^{σ (k) L} (x) \land h_{𝔹}^{σ (k) L} (x), \\ h_{𝔸}^{σ (k) U} (x) \land h_{𝔹}^{σ (k) U} (x)] | k = 1, 2, \dots, l}, \end{matrix}$ (4) the ring sum of $𝔸$ and $𝔹$ , denoted by $𝔸 ⊞ 𝔹$ , is given by $\begin{matrix} h_{𝔸 ⊞ 𝔹} (x) = h_{𝔸} (x) \oplus h_{𝔹} (x) \\ = {[h_{𝔸}^{σ (k) L} (x) + h_{𝔹}^{σ (k) L} (x) - h_{𝔸}^{σ (k) L} (x) h_{𝔹}^{σ (k) L} (x), \\ h_{𝔹}^{σ (k) U} (x) + h_{𝔹}^{σ (k) U} (x) - h_{𝔸}^{σ (k) U} (x) h_{𝔹}^{σ (k) U} (x)] | \\ k = 1, 2, \dots, l}, \end{matrix}$ (5) the ring product of $𝔸$ and $𝔹$ , denoted by $𝔸 ⊠ 𝔹$ , is given by $\begin{matrix} h_{𝔸 ⊠ 𝔹} (x) & = h_{𝔸} (x) \otimes h_{𝔹} (x) = {[h_{𝔸}^{σ (k) L} (x) h_{𝔹}^{σ (k) L} (x), \\ h_{𝔸}^{σ (k) U} (x) h_{𝔹}^{σ (k) U} (x)] | k = 1, 2, \dots, l}, \end{matrix}$ where $l = \max {l (h_{𝔸} (x)), l (h_{𝔹} (x))} .$

Example 2.8. Let $𝔸$ and $𝔹$ be two IVHF sets. Suppose that $h_{𝔸} (x) = {[0.5, 0.6], [0.3, 0.8],$ [0.3, 0.6]} and $h_{𝔹} (x) = {[0.4, 0.5], [0.4, 0.7]}$ are two IVHF elements of x to $𝔸$ and $𝔹$ , respectively. By Equation (1) and assumptions given by Chen [10], $h_{𝔸} (x) = {[0.3, 0.6], [0.3, 0.8],$ [0.5, 0.6]} and $h_{𝔹} (x) = {[0.4, 0.5],$ [0.4, 0.7] , [0.4, 0.7]}. By virtue of Definition 1, we have $\begin{matrix} h_{𝔸 ⋓ 𝔹} (x) & = {[0.3 \lor 0.4, 0.6 \lor 0.5], [0.3 \lor 0.4, 0.8 \lor 0.7], \\ [0.5 \lor 0.4, 0.6 \lor 0.7]} \\ = {[0.4, 0.6], [0.4, 0.8], [0.5, 0.7]}, \end{matrix}$ $\begin{matrix} h_{𝔸 ⋒ 𝔹} (x) & = {[0.3 \land 0.4, 0.6 \land 0.5], [0.3 \land 0.4, 0.8 \land 0.7], \\ [0.5 \land 0.4, 0.6 \land 0.7]} \\ = {[0.3, 0.5], [0.3, 0.7], [0.4, 0.6]}, \end{matrix}$ $\begin{matrix} h_{𝔸 ⊞ 𝔹} (x) & = {[0.3 + 0.4 - 0.3 \cdot 0.4, 0.6 + 0.5 - 0.6 \cdot 0.5], \\ [0.3 + 0.4 - 0.3 \cdot 0.4, 0.8 + 0.7 - 0.8 \cdot 0.7], \\ [0.5 + 0.4 - 0.5 \cdot 0.4, 0.6 + 0.7 - 0.6 \cdot 0.7]} \\ = {[0.58, 0.8], [0.58, 0.94], [0.7, 0.88]}, \end{matrix}$ $\begin{matrix} h_{𝔸 ⊠ 𝔹} (x) & = {[0.3 \cdot 0.4, 0.6 \cdot 0.5], [0.3 \cdot 0.4, 0.8 \cdot 0.7], \\ [0.5 \cdot 0.4, 0.6 \cdot 0.7]} \\ = {[0.12, 0.3], [0.12, 0.56], [0.2, 0.42]} . \end{matrix}$

In [55], in order to compare two IVHF sets, Zhang et al. introduced the concept of the IVHF subset as follows.

Definition 2.9. [55] Let U be a nonempty and finite universe of discourse. For all $𝔸, 𝔹 \in IVHF (U)$ , $𝔸$ is said to be an IVHF subset of $𝔹$ , if $h_{𝔸} (x) ⪯ h_{𝔹} (x)$ holds for any x ∈ U such that

$\begin{matrix} h_{𝔸} (x) ⪯ h_{𝔹} (x) \Leftrightarrow [h_{𝔸}^{σ (k) L} (x), h_{𝔸}^{σ (k) U} (x)] \\ \leq_{L^{I}} [h_{𝔹}^{σ (k) L} (x), h_{𝔹}^{σ (k) U} (x)] \\ \Leftrightarrow h_{𝔸}^{σ (k) L} (x) \leq h_{𝔹}^{σ (k) L} (x), h_{𝔸}^{σ (k) U} (x) \leq h_{𝔹}^{σ (k) U} (x), \\ k = 1, 2, \dots, l . \end{matrix}$ We denote it by $𝔸 ⊑ 𝔹$ or $𝔹 ⊒ 𝔸$ .

Obviously, the following conclusions hold: $\forall 𝔸, 𝔹,$ $ℂ \in IVHF (U)$ ,

$𝔸 ⊑ 𝔸,$

$𝔸 ⊑ 𝔹, 𝔹 ⊑ ℂ \Rightarrow 𝔸 ⊑ ℂ,$

$𝔸 ⊑ 𝔹, 𝔹 ⊑ 𝔸 \Leftrightarrow 𝔸 = 𝔹 .$

That is, the notation ⊑ is reflexive, transitive and antisymmetric on IVHF(U).

2.3 Interval-valued hesitant fuzzy rough approximation operators

In the subsection, we briefly review some basic results related to IVHF rough approximation operators introduced by Zhang et al. [55].

We first recall the concept of the IVHF relation.

Definition 2.10. [55] Suppose that U is a nonempty and finite universe of discourse. An IVHF relation $ℝ$ on U is an IVHF subset of U × U, namely, $ℝ$ is given by $ℝ = {< (x, y), h_{ℝ} (x, y) > | (x, y) \in U \times U},$ where $h_{ℝ} : U \times U \to Int [0, 1]$ is a set of interval values in Int [0, 1], denoting the possible interval-valued membership degrees of the relationships between x and y.

We denote by IVHFR (U × U) the family of all IVHF relations on U.

Definition 2.11. [55] Let $ℝ \in IVHFR (U \times U)$ .

$ℝ$ is serial, if for any x ∈ U, there exists a y ∈ U such that $h_{ℝ} (x, y) = {[1, 1]}$ ;

$ℝ$ is reflexive, if $h_{ℝ} (x, x) = {[1, 1]}$ for all x ∈ U;

$ℝ$ is symmetric, if for all (x, y) ∈ U × U, $h_{ℝ} (x, y) = h_{ℝ} (y, x)$ ;

$ℝ$ is transitive if, $h_{ℝ} (x, y) ⊼ h_{ℝ} (y, z) ⪯ h_{ℝ} (x, z)$ for all (x, z) ∈ U × U .

Alternatively, $ℝ$ is transitive if the following conditions are satisfied: $\begin{matrix} h_{ℝ}^{σ (k) L} (x, y) \land h_{ℝ}^{σ (k) L} (y, z) \leq h_{ℝ}^{σ (k) L} (x, z), \\ h_{ℝ}^{σ (k) U} (x, y) \land h_{ℝ}^{σ (k) U} (y, z) \leq h_{ℝ}^{σ (k) U} (x, z), \\ k = 1, 2, \dots, l, \end{matrix}$ with $l = \max {l (h_{ℝ} (x, y)), l (h_{ℝ} (y, z)), l (h_{ℝ} (x, z))} .$

In what follows, IVHF rough approximation operators with respect to IVHF approximation space are defined as follows.

Definition 2.12. [55] Let U be a nonempty and finite universe of discourse and $ℝ \in IVHFR (U \times U)$ ; the pair $(U, ℝ)$ is called an IVHF approximation space. For any $𝔸 \in IVHF (U),$ the lower and upper approximations of $𝔸$ with respect to $(U, ℝ)$ , denoted by $\underline{ℝ} (𝔸)$ and $\bar{ℝ} (𝔸)$ , are two IVHF sets and are, respectively, defined as follows: $\underline{ℝ} (𝔸) = {< x, h_{\underline{ℝ} (𝔸)} (x) > | x \in U},$ (3) $\bar{ℝ} (𝔸) = {< x, h_{\bar{ℝ} (𝔸)} (x) > | x \in U},$ (4)where $\begin{matrix} h_{\underline{ℝ} (𝔸)} (x) = ⊼_{y \in U} {h_{ℝ^{c}} (x, y) ⊻ h_{𝔸} (y)}, \\ h_{\bar{ℝ} (𝔸)} (x) = ⊻_{y \in U} {h_{ℝ} (x, y) ⊼ h_{𝔸} (y)} . \end{matrix}$ $\underline{ℝ} (𝔸)$ and $\bar{ℝ} (𝔸)$ are, respectively, called the lower and upper approximations of $𝔸$ with respect to $(U, ℝ)$ . The pair $(\underline{ℝ} (𝔸), \bar{ℝ} (𝔸))$ is called the IVHF rough set of $𝔸$ with respect to $(U, ℝ)$ , and $\underline{ℝ}, \bar{ℝ} : IVHF (U) \to IVHF (U)$ are referred to as lower and upper IVHF rough approximation operators, respectively.

Clearly, the above definition implies equivalences of the following form: $\begin{matrix} h_{\underline{ℝ} (𝔸)} (x) = {[⋀_{y \in U} (h_{ℝ^{c}}^{σ (k) L} (x, y) \lor h_{𝔸}^{σ (k) L} (y)), \\ ⋀_{y \in U} (h_{ℝ^{c}}^{σ (k) U} (x, y) \lor h_{𝔸}^{σ (k) U} (y))] | k = 1, 2, \dots, l}, \end{matrix}$ $\begin{matrix} h_{\bar{ℝ} (𝔸)} (x) = {[⋁_{y \in U} (h_{ℝ}^{σ (k) L} (x, y) \land h_{𝔸}^{σ (k) L} (y)), \\ ⋁_{y \in U} (h_{ℝ}^{σ (k) U} (x, y) \land h_{𝔸}^{σ (k) U} (y))] | k = 1, 2, \dots, l}, \end{matrix}$ where $l = \max {l (h_{ℝ} (x, y)), l (h_{𝔸} (y))} .$

Theorem 2.13. [55] Let $(U, ℝ)$ be an IVHF approximation space. Then the lower and upper IVHF rough approximation operators induced from $(U, ℝ)$ satisfy the following properties: $\forall 𝔸, 𝔹 \in IVHF (U),$

(IVHFL1) $\underline{ℝ} (𝔸^{c}) = (\bar{ℝ} (𝔸))^{c},$

(IVHFU1) $\bar{ℝ} (𝔸^{c}) = (\underline{ℝ} (𝔸))^{c},$

(IVHFL2) $𝔸 ⊑ 𝔹 \Rightarrow \underline{ℝ} (𝔸) ⊑ \underline{ℝ} (𝔹),$

(IVHFU2) $𝔸 ⊑ 𝔹 \Rightarrow \bar{ℝ} (𝔸) ⊑ \bar{ℝ} (𝔹),$

(IVHFL3) $\underline{ℝ} (𝔸 ⋒ 𝔹) = \underline{ℝ} (𝔸) ⋒ \underline{ℝ} (𝔹),$

(IVHFU3) $\bar{ℝ} (𝔸 ⋓ 𝔹) = \bar{ℝ} (𝔸) ⋓ \bar{ℝ} (𝔹),$

(IVHFL4) $\underline{ℝ} (𝔸 ⋓ 𝔹) ⊒ \underline{ℝ} (𝔸) ⋓ \underline{ℝ} (𝔹),$

(IVHFU4) $\bar{ℝ} (𝔸 ⋒ 𝔹) ⊑ \bar{ℝ} (𝔸) ⋒ \bar{ℝ} (𝔹),$

(IVHFL5) $\underline{ℝ} (𝔸 ⋓ \hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]}) = \underline{ℝ} (𝔸) ⋓ \hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]},$

(IVHFU5) $\bar{ℝ} (𝔸 ⋒ \hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]}) = \bar{ℝ} (𝔸) ⋒ \hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]},$

(IVHFL6) $\underline{ℝ} (𝕌) = 𝕌,$

(IVHFU6) $\bar{ℝ} (\emptyset) = \emptyset .$

Proof. It follows directly from Definitions 2.19and 2.9.□

Properties (IVHFL1) and (IVHFU1) show that IVHF rough approximation operators $\bar{ℝ}$ and $\underline{ℝ}$ are dual to each other.

The following Theorem 2.14 shows that an IVHF relation can be represented by the IVHF rough approximation operators.

Theorem 2.14.[55] Let $ℝ \in IVHFR (U \times U)$ ; then ∀x ∈ U, (x, y) ∈ U × U, M ⊆ U,

$h_{\underline{ℝ} ([1, 1]_{M})} (x) = ⊼_{y \notin M} h_{ℝ^{c}} (x, y),$

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

$h_{\underline{ℝ} ([1, 1]_{U - {y}})} (x) = h_{ℝ^{c}} (x, y),$

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

Meanwhile, connections between the properties of several special IVHF relations and the properties of IVHF rough approximation operators are discussed as follows.

Theorem 2.15. [55] Let $ℝ \in IVHFR (U \times U)$ . Suppose that $\underline{ℝ}$ and $\bar{ℝ}$ are the lower and upper IVHF rough approximation operators given in Definition 1; then $ℝ$ is serial iff one of the following properties holds:

$(IVHFL 0) \underline{ℝ} (\emptyset) = \emptyset,$

$(IVHFU 0) \bar{ℝ} (𝕌) = 𝕌 .$

Theorem 2.16. [55] Let (U, R) be an IVHF approximation space. $\underline{R}$ and $\bar{R}$ are the IVHF approximation operators induced from $(U, ℝ)$ ; then $\forall 𝔸 \in IVHF (U),$

$\begin{matrix} (1) ℝ is reflexive & \Leftrightarrow (IVHFLR) \underline{ℝ} (𝔸) ⊑ 𝔸, \\ \Leftrightarrow (IVFHUR) 𝔸 ⊑ \bar{ℝ} (𝔸) . \end{matrix}$ $\begin{matrix} (2) ℝ is symmetric & \Leftrightarrow (IVHFLS) h_{\underline{ℝ} ([1, 1]_{U - {x}})} (y) \\ = h_{\underline{ℝ} ([1, 1]_{U - {y}})} (x), \\ \Leftrightarrow (IVHFUS) h_{\bar{ℝ} ([1, 1]_{x})} (y) \\ = h_{\bar{ℝ} ([1, 1]_{y})} (x) . \end{matrix}$ $\begin{matrix} (3) ℝ is transitive & \Leftrightarrow (IVHFLT) \underline{ℝ} (𝔸) ⊑ \underline{ℝ} (\underline{ℝ} (𝔸)), \\ \Leftrightarrow (IVHFUT) \bar{ℝ} (\bar{ℝ} (𝔸)) ⊑ \bar{ℝ} (𝔸) . \end{matrix}$

The above Theorems 2.15 and 2.16 show that the essential properties of the lower and upper IVHF rough approximation operators can characterize an IVHF relation having special property, such as serializability, reflexivity, symmetry, and transitivity.

Combining (1) and (3) in Theorem 2.16, we can easily obtain the conclusion as follows.

Corollary 2.17. Let $ℝ \in IVHFR (U \times U)$ . If $ℝ$ is reflexive and transitive, and $\underline{ℝ}$ and $\bar{ℝ}$ are the lower and upper IVHF rough approximation operators defined in Definition 2.12, then: $\begin{matrix} (IVHFLRT) \underline{ℝ} (𝔸) = \underline{ℝ} (\underline{ℝ} (𝔸)), \forall 𝔸 \in IVHF (U), \\ (IVHFURT) \bar{ℝ} (\bar{ℝ} (𝔸)) = \bar{ℝ} (𝔸), \forall 𝔸 \in IVHF (U) . \end{matrix}$

3 Interval-valued hesitant fuzzy topological spaces

In this section, we introduce basic concepts related to IVHF topological spaces.

Definition 3.1. An IVHF topology on a nonempty set U is a family τ of IVHF sets on U that satisfies the following conditions:

$\emptyset, 𝕌 \in τ,$

$𝔸 ⋒ 𝔹 \in τ$ for any $𝔸, 𝔹 \in τ$

$⋓_{i \in I} 𝔸_{i} \in τ$ for any $𝔸_{i} \in τ, i \in I .$

The pair (U, τ) is called an IVHF topological space and each IVHF set $𝔸$ in τ is referred to as an IVHF open set in (U, τ). The complement of an IVHF open set in the IVHF topological space (U, τ) is called an IVHF closed set in (U, τ) .

Example 3.2. Let U = {x₁, x₂}. Suppose that $𝔸, 𝔹, ℂ$ and $𝔻$ are four IVHF sets on U defined as follows: $\begin{matrix} 𝔸 = { & < x_{1}, {[0.4, 0.5], [0.6, 0.7]} >, \\ < x_{2}, {[0.4, 0.6]} >}, \end{matrix}$ $\begin{matrix} 𝔹 = { & < x_{1}, {[0.6, 0.7], [0.8, 0.9]} >, \\ < x_{2}, {[0.3, 0.4], [0.6, 0.8]} >}, \end{matrix}$ $\begin{matrix} ℂ = { & < x_{1}, {[0.4, 0.5], [0.6, 0.7]} >, \\ < x_{2}, {[0.3, 0.4], [0.4, 0.6]} >}, \end{matrix}$ $\begin{matrix} 𝔻 = { & < x_{1}, {[0.6, 0.7], [0.8, 0.9]} >, \\ < x_{2}, {[0.4, 0.6], [0.6, 0.8]} >} . \end{matrix}$

Then, the family $τ = {\emptyset, 𝕌, 𝔸, 𝔹, ℂ, 𝔻}$ of IVHF sets on U is an IVHF topology on U.

Now we define IVHF interior and closure operators in an IVHF topological space.

Definition 3.3. Let (U, τ) be an IVHF topological space. For any $𝔸 \in IVHF (U)$ , the IVHF interior and IVHF closure of $𝔸$ are, respectively, defined as follows: $\begin{matrix} int (𝔸) = ⋓ {𝔾 | 𝔾 \in τ and 𝔾 ⊑ 𝔸}, \\ cl (𝔸) = ⋒ {𝕂 | 𝕂^{c} \in τ and 𝔸 ⊑ 𝕂}, \end{matrix}$ where int and cl : IVHF (U) ⟶ IVHF (U) are, respectively, called the IVHF interior operator and the IVHF closure operator of τ.

Example 3.4. Reconsidering Example 3.2, we know that (U, τ) is an IVHF topological space. Let $𝔼$ be another IVHF set on U which is defined as follows: $\begin{matrix} 𝔼 = { & < x_{1}, {[0.7, 0.8], [0.8, 0.9]} >, \\ < x_{2}, {[0.5, 0.7], [0.6, 0.8]} >} . \end{matrix}$ By Definition 3.3, then $\begin{matrix} int (𝔼) = 𝔻 = { & < x_{1}, {[0.6, 0.7], [0.8, 0.9]} >, \\ < x_{2}, {[0.4, 0.6], [0.6, 0.8]} >}, \end{matrix}$ and $cl (𝔼) = 𝕌 .$

In the following, some properties of the IVHF interior operator and the IVHF closure operator of τ are discussed.

Theorem 3.5. Let (U, τ) be an IVHF topological space. For any $𝔸 \in IVHF (U)$ , then

$𝔸$ is an IVHF open set in (U, τ) iff $int (𝔸) = 𝔸$ ,

$𝔸$ is an IVHF closed set in (U, τ) iff $cl (𝔸) = 𝔸$ .

Proof. It is straightforward from Definition 3.3.□

From Example 3.2, we can also easily verify that Theorem 3.5 holds.

Theorem 3.6. Let (U, τ) be an IVHF topological space and $𝔸, 𝔹 \in IVHF (U)$ . Then, the following properties hold:

(Int0) $(int (𝔸))^{c} = cl (𝔸^{c})$ ,

(Cl0) $(cl (𝔸))^{c} = int (𝔸^{c})$ ;

(Int1) $int (𝕌) = 𝕌, int (\emptyset) = \emptyset$ ,

(Cl1) $cl (𝕌) = 𝕌, cl (\emptyset) = \emptyset$ ;

(Int2) $int (𝔸) ⊑ 𝔸$ ,

(Cl2) $𝔸 ⊑ cl (𝔸)$ ;

(Int3) $int (int (𝔸)) = int (𝔸)$ ,

(Cl3) $cl (cl (𝔸)) = cl (𝔸)$ ;

(Int4) $int (𝔸 ⋒ 𝔹) = int (𝔸) ⋒ int (𝔹)$ ,

(Cl4) $cl (𝔸 ⋓ 𝔹) = cl (𝔸) ⋓ cl (𝔹)$ .

Proof. It can easily be verified from Definition 3.3 and Theorem 3.5.□

Properties (Int0) and (Cl0) state that the IVHF interior operator and the IVHF closure operator of τ are dual to each other. Moreover, it is easy to observe that properties (Int4) and (Cl4) imply, respectively, the following properties (Int4) ′ and (Cl4) ′: $\begin{matrix} (Int 4)^{'} 𝔸 ⊑ 𝔹 \Rightarrow int (𝔸) ⊑ int (𝔹), \\ (Cl 4)^{'} 𝔸 ⊑ 𝔹 \Rightarrow cl (𝔸) ⊑ cl (𝔹) . \end{matrix}$

The following theorem shows that an IVHF operator satisfying properties (Int1)-(Int4) (respectively, properties (Cl1)-(Cl4)) is the IVHF interior operator (respectively, the IVHF closure operator) of certain IF topology.

Theorem 3.7. (1) If an IVHF operator int : IVHF (U) ⟶IVHF (U) satisfies properties (Int1)-(Int4), then there exists an IVHF topology τ_int on U such that int_{τ
_int} = int .

(2) If an IVHF operator cl : IVHF (U) ⟶ IVHF (U) satisfies properties (Cl1)-(Cl4), then there exists an IVHF topology τ_cl on U such that cl_{τ
_cl} = cl .

Proof. (1) Define $τ_{int} = {𝔸 \in IVHF (U) | int (𝔸) = 𝔸} .$ Then we can prove that τ_int is an IVHF topology on U.

(T₁) By (Int1), $\emptyset, 𝕌 \in τ_{int} .$

(T₂) For any $𝔸, 𝔹 \in τ_{int},$ then $int (𝔸) = 𝔸$ and $int (𝔹) = 𝔹 .$ Thus, by (Int4) we have $int (𝔸 ⋒ 𝔹) = int (𝔸) ⋒ int (𝔹) = 𝔸 ⋒ 𝔹,$ which implies that $𝔸 ⋒ 𝔹 \in τ_{int} .$

(T₃) Suppose that $𝔸_{i} \in τ_{int};$ then $int (𝔸_{i}) = 𝔸_{i},$ for all i ∈ I . By virtue of (Int2), we have $int (⋓_{i \in I} 𝔸_{i}) ⊑ ⋓_{i \in I} 𝔸_{i} .$

On the other hand, noting that $⋓_{i \in I} int (𝔸_{i}) ⊒ int (𝔸_{i})$ , by (Int4) ′ and (Int3), we have $int (⋓_{i \in I} int (𝔸_{i}))$ $⊒ int (int (𝔸_{i})) = int (𝔸_{i}), \forall i \in I .$ Thus, $int (⋓_{i \in I} int (𝔸_{i}))$ $⊒ ⋓_{i \in I} int (𝔸_{i})$ . Moreover, from the assumption, then $int (⋓_{i \in I} 𝔸_{i}) ⊒ ⋓_{i \in I} 𝔸_{i} .$ From the above discussions, we can conclude that $int (⋓_{i \in I} 𝔸_{i}) = ⋓_{i \in I} 𝔸_{i},$ which implies that $⋓_{i \in I} 𝔸_{i} \in τ_{int}$ . So we have proved that τ_int is an IVHF topology on U. Obviously, int_{τ
_int} = int .

(2) By defining $τ_{cl} = {𝔸 \in IVHF (U) | cl (𝔸^{c}) = 𝔸^{c}},$ it is similar to the proof of (1).□

Theorem 3.8.(1) Let int : IVHF (U) ⟶ IVHF (U) be an IVHF operator satisfying properties (Int1)-(Int4). Define $τ_{int}^{'} = {int (𝔸) | 𝔸 \in IVHF (U)},$ then $τ_{int}^{'} = τ_{int} .$

(2) Let cl : IVHF (U) ⟶ IVHF (U) be an IVHF operator satisfying properties (Cl1)-(Cl4). Define $τ_{cl}^{'} = {(cl (𝔸))^{c} | 𝔸 \in IVHF (U)},$ then $τ_{cl}^{'} = τ_{cl} .$

Proof. (1) Obviously, $τ_{int} = {𝔸 \in IVHF (U) | int (𝔸)$ $= 𝔸} \subseteq τ_{int}^{'} .$ On the other hand, for any $𝔸 \in IVHF (U)$ , by (Int3) we have $int (int (𝔸)) = int (𝔸)$ , from which we know that $int (𝔸) \in τ_{int} .$ Hence, $τ_{int}^{'} \subseteq τ_{int} .$ From the above discussions, we can conclude that $τ_{int}^{'} = τ_{int} .$

(2) It is similar to the proof of (1).□

The above Theorem 3.8 states that two IVHF topological spaces induced an IVHF operator satisfying properties (Int1)-(Int4) are equal and two IVHF topological spaces induced an IVHF operator satisfying properties (Cl1)-(Cl4) are also equal. In the following we are to prove that the four IVHF topological spaces are equal to each other if (Int0) and (Cl0) hold.

Theorem 3.9. Let int : IVHF (U) ⟶ IVHF (U) be an IVHF operator satisfying properties (Int1)-(Int4) and cl : IVHF (U) ⟶ IVHF (U) be an IVHF operator satisfying properties (Cl1)-(Cl4). If (Int0) and (Cl0) hold, then $τ_{int}^{'} = τ_{int} = τ_{cl}^{'} = τ_{cl} .$

Proof. According to Theorem 3.8, we are to prove that $τ_{int}^{'} = τ_{cl}^{'} .$

In fact, by (Int0) and (Cl0), we have $\begin{matrix} τ_{int}^{'} & = {int (𝔸) | 𝔸 \in IVHF (U)} \\ = {(cl (𝔸^{c}))^{c} | 𝔸 \in IVHF (U)} \\ = {(cl (𝔸))^{c} | 𝔸^{c} \in IVHF (U)} \\ = {(cl (𝔸))^{c} | 𝔸 \in IVHF (U)} = τ_{cl}^{'} . \end{matrix}$ □

4 Relationships between IVHF approximation spaces andIVHF topological spaces

In this section, we generalize the IVHF rough set theory in the framework of IVHF topological spaces and discuss the relationships between IVHF rough approximation spaces and IVHF topological spaces.

4.1 From IVHF approximation spaces to IVHF topological spaces

In this subsection, we assume that U is a nonempty and finite universe of discourse, $ℝ$ is an IVHF relation on U, and $\underline{ℝ}$ and $\bar{ℝ}$ are two IVHF rough approximation operators in Definition 2.12. Define $τ_{ℝ} = {𝔸 \in IVHF (U) | \underline{ℝ} (𝔸) = 𝔸} .$ (5)

Proposition 4.1. Let I be an index set, and $𝔸_{i} \in IVHF (U)$ for all i ∈ I. If $ℝ$ is an IVHF reflexive and transitive relation on U, then $\underline{ℝ} (⋓_{i \in I} \underline{ℝ} (𝔸_{i})) = ⋓_{i \in I} \underline{ℝ} (𝔸_{i}) .$

Proof. On the one hand, by the reflexivity of $ℝ$ and Theorem 2.16, we have $\underline{ℝ} (⋓_{i \in I} \underline{ℝ} (𝔸_{i})) ⊑ ⋓_{i \in I} \underline{ℝ} (𝔸_{i}) .$ On the other hand, since $⋓_{i \in I} \underline{ℝ} (𝔸_{i}) ⊒ \underline{ℝ} (𝔸_{i})$ , in terms of (IVHFL2) in Theorem 2.13, we have $\underline{ℝ} (⋓_{i \in I} \underline{ℝ} (𝔸_{i})) ⊒ \underline{ℝ} (\underline{ℝ} (𝔸_{i})) .$ Noting that $ℝ$ is an IVHF reflexive and transitive relation on U, then by Corollary 2.17 we can conclude that $\underline{ℝ} (⋓_{i \in I} \underline{ℝ} (𝔸_{i})) ⊒ \underline{ℝ} (𝔸_{i}) .$ Thus $\underline{ℝ} (⋓_{i \in I} \underline{ℝ} (𝔸_{i})) ⊒ ⋓_{i \in I} \underline{ℝ} (𝔸_{i}) .$ Consequently, $\underline{ℝ} (⋓_{i \in I} \underline{ℝ} (𝔸_{i})) = ⋓_{i \in I} \underline{ℝ} (𝔸_{i}) .$ □

The following Theorem 4.2 shows that an IVHF reflexive and transitive relation on U can induce an IVHF topology on U.

Theorem 4.2. If $ℝ$ is an IVHF reflexive and transitive relation on U, then $τ_{ℝ}$ is an IVHF topology on U.

Proof. (T₁) Since an IVHF reflexive relation must be serial, according to Theorem 2.15 we have $\underline{ℝ} (\emptyset) = \emptyset$ . Meanwhile, by Theorem 2.13, then $\underline{ℝ} (𝕌) = 𝕌 .$ Consequently, $\emptyset, 𝕌 \in τ_{ℝ} .$

(T₂) For any $𝔸, 𝔹 \in τ_{ℝ},$ then $\underline{ℝ} (𝔸) = 𝔸$ and $\underline{ℝ} (𝔹) = 𝔹 .$ Thus, in terms of Theorem 2.13 we have $\underline{ℝ} (𝔸 ⋒ 𝔹) = \underline{ℝ} (𝔸) ⋒ \underline{ℝ} (𝔹) = 𝔸 ⋒ 𝔹,$ which implies that $𝔸 ⋒ 𝔹 \in τ_{ℝ} .$

(T₃) Suppose that $𝔸_{i} \in τ_{ℝ};$ then $\underline{ℝ} (𝔸_{i}) = 𝔸_{i},$ for all i ∈ I . Since $ℝ$ is reflexive and transitive, by virtue of Proposition 4.1, we have $\underline{ℝ} (⋓_{i \in I} \underline{ℝ} (𝔸_{i})) = ⋓_{i \in I} \underline{ℝ} (𝔸_{i}) .$ Moreover, from the assumption, then we can conclude that $\underline{ℝ} (⋓_{i \in I} 𝔸_{i}) = ⋓_{i \in I} 𝔸_{i},$ which implies that $⋓_{i \in I} 𝔸_{i} \in τ_{ℝ}$ . So we have proved that $τ_{ℝ}$ is an IVHF topology on U.□

Example 4.3. Let $(U, ℝ)$ be an IVHF approximation space, where U = {x₁, x₂} and $ℝ$ is defined by the matrix as follows: $ℝ = \begin{matrix} x_{1} & x_{2} \\ \begin{matrix} x_{1} \\ x_{2} \end{matrix} ( & \begin{matrix} {[0.2, 0.7]} \\ {[0.3, 0.7], [0.6, 0.9]} \end{matrix} & \begin{matrix} {[0.4, 0.6], [0.7, 0.8]} \\ {[1, 1]} \end{matrix}) \end{matrix}$ It should be noted that $ℝ$ is not reflexive. By Definition 2.12, we have $\begin{matrix} h_{\underline{ℝ} (\emptyset)} (x_{1}) & = ⊼_{y \in U} {h_{ℝ^{c}} (x_{1}, y) ⊻ h_{\emptyset} (y)} \\ = {[0.3, 0.8]} ⊼ {[0.2, 0.3], [0.4, 0.6]} \\ = {[0.2, 0.3], [0.3, 0.6]}, \end{matrix}$ $\begin{matrix} h_{\underline{ℝ} (\emptyset)} (x_{2}) & = ⊼_{y \in U} {h_{ℝ^{c}} (x_{2}, y) ⊻ h_{\emptyset} (y)} \\ = {[0.1, 0.4], [0.3, 0.7]} ⊼ {[0, 0]} = {[0, 0]} . \end{matrix}$ Thus, $\underline{ℝ} (\emptyset) = {< x_{1}, {[0.2, 0.3], [0.3, 0.6]} >, < x_{2},$ {[0, 0]} >} ≠ ∅ . Hence, $\emptyset \notin τ_{ℝ},$ which implies that $τ_{ℝ}$ does not form an IVHF topology.

From the above example, we can note that if an IVHF relation $ℝ$ is not reflexive, then $τ_{ℝ}$ defined by Equation (5) may not be an IVHF topology. Denote $τ_{ℝ}^{'} = {\underline{ℝ} (𝔸) | 𝔸 \in IVHF (U)} .$

In what follows, we are to prove that $τ_{ℝ}^{'} = τ_{ℝ},$ that is, $τ_{ℝ}^{'}$ and $τ_{ℝ}$ are the same IVHF topology on U.

Theorem 4.4. If $ℝ$ is an IVHF reflexive and transitive relation on U, then $τ_{ℝ}^{'}$ is an IVHF topology on U.

Proof. Obviously, $τ_{ℝ} \subseteq τ_{ℝ}^{'} .$ On the other hand, since $ℝ$ is an IVHF reflexive and transitive relation on U, by Corollary 2.17, we have $\underline{ℝ} (\underline{ℝ} (𝔸)) = \underline{ℝ} (𝔸),$ which implies that $\underline{ℝ} (𝔸) \in τ_{ℝ}$ . So it follows that $τ_{ℝ}^{'} \subseteq τ_{ℝ} .$ Consequently, $τ_{ℝ}^{'} = τ_{ℝ} .$ Hence, by Theorem 4.2, we conclude that $τ_{ℝ}^{'}$ is an IVHF topology on U.□

Theorem 4.4 states that an IVHF reflexive and transitive approximation space can generate an IVHF topological space such that the family of all lower approximations of IVHF sets with respect to the IVHF approximation space forms the IVHF topology. Theorem 4.5 below shows that the lower and upper IVHF rough approximation operators are respectively the interior and closure operators of the IVHF topological space.

Theorem 4.5. Let $(U, τ_{ℝ})$ be an IVHF topological space induced from an IVHF reflexive and transitive approximation space $(U, ℝ))$ , i.e. $τ_{ℝ} = {\underline{ℝ} (𝔸) | 𝔸 \in IVHF (U)} .$ Then, ∀A ∈ IVHF (U), $\begin{matrix} (1) \underline{ℝ} (𝔸) & = {int}_{τ_{ℝ}} (A) \\ = ⋓ {\underline{ℝ} (𝔹) | \underline{ℝ} (𝔹) ⊑ 𝔸, 𝔹 \in IVHF (U)}, \end{matrix}$ $\begin{matrix} (2) \bar{ℝ} (𝔸) & = {cl}_{τ_{ℝ}} (A) \\ = ⋒ {(\underline{ℝ} (𝔹))^{c} | (\underline{ℝ} (𝔹))^{c} ⊒ 𝔸, 𝔹 \in IVHF (U)} \\ = ⋒ {\bar{ℝ} (𝔹) | \bar{ℝ} (𝔹) ⊒ 𝔸, 𝔹 \in IVHF (U)} . \end{matrix}$

Proof. (1) Since $ℝ$ is an IVHF reflexive, by Theorem 2.16, we obtain $\underline{ℝ} (𝔸) ⊑ 𝔸 .$ Then $\underline{ℝ} (𝔸) ⊑ ⋓ {\underline{ℝ} (𝔹) | \underline{ℝ} (𝔹) ⊑ 𝔸, 𝔹 \in IVHF (U)} .$ On the other hand, we have $⋓ {\underline{ℝ} (𝔹) | \underline{ℝ} (𝔹) ⊑ 𝔸, 𝔹 \in IVHF (U)} ⊑ 𝔸$ , from which it follows that $\underline{ℝ} (⋓ {\underline{ℝ} (𝔹) | \underline{ℝ} (𝔹) ⊑ 𝔸, 𝔹 \in IVHF (U)}) ⊑ \underline{ℝ} (𝔸) .$ Consequently, by Proposition 4.1, then $⋓ {\underline{ℝ} (𝔹) | \underline{ℝ} (𝔹) ⊑ 𝔸, 𝔹 \in IVHF (U)}$ $⊑ \underline{ℝ} (𝔸) .$ From the above discussions, we conclude that $⋓ {\underline{ℝ} (𝔹) | \underline{ℝ} (𝔹) ⊑ 𝔸, 𝔹 \in IVHF (U)} = \underline{ℝ} (𝔸) .$

(2) It follows immediately from the duality of $\underline{ℝ}$ and $\bar{ℝ}$ and (1).□

Theorem 4.6 below shows that an IVHF reflexive and transitive relation can also be represented by its producing IVHF topology.

Theorem 4.6. Let $(U, ℝ)$ be an IVHF reflexive and transitive approximation space and $(U, τ_{ℝ})$ be the IVHF topological space induced by $(U, ℝ)$ . Then $h_{ℝ} (x, y) = ⊼_{B \in (y)_{τ_{ℝ}}} h_{𝔹} (x),$ where $(y)_{τ_{ℝ}} = {𝔹 \in IVHF (U) | 𝔹^{c} \in τ_{ℝ}, h_{𝔹} (y) = {[1, 1]}} .$

Proof. For any x, y ∈ U, by Theorem 4.5, then $\bar{ℝ} ([1, 1]_{y}) = {cl}_{τ_{ℝ}} ([1, 1]_{y})$ . Moreover, from Theorem 2.14, we have $h_{ℝ} (x, y) = h_{\bar{ℝ} ([1, 1]_{y})} (x) .$ On the other hand, noting that ${cl}_{τ_{ℝ}} ([1, 1]_{y}) = ⋒ {𝔹 \in IVHF (U) | 𝔹^{c} \in τ_{ℝ} and [1, 1]_{y}$ $⊑ 𝔹},$ for all x ∈ U we have $\begin{matrix} h_{{cl}_{τ_{ℝ}} ([1, 1]_{y})} (x) & = ⊼ {h_{𝔹} (x) | 𝔹^{c} \in τ_{ℝ}, h_{[1, 1]_{y}} (x) ⪯ h_{𝔹} (x)} \\ = ⊼ {h_{𝔹} (x) | 𝔹^{c} \in τ_{ℝ}, h_{𝔹} (y) = {[1, 1]}} \\ = ⊼_{B \in (y)_{τ_{ℝ}}} h_{𝔹} (x) . \end{matrix}$ From the above discussions, we conclude that $h_{ℝ}$ $(x, y) = ⊼_{B \in (y)_{τ_{ℝ}}} h_{𝔹} (x) .$ □

4.2 From IVHF topological spaces to IVHF approximation spaces

As can be seen from Subsection 4.1, an IVHF reflexive and transitive approximation space yields an IVHF topological space. In this subsection, we consider the reverse problem, that is, under which conditions can an IVHF topological space be associated with an IVHF approximation space which produces the same IVHF topological space? The following Theorem 4.7 answers the question.

Theorem 4.7. Let (U, τ) be an IVHF topological space and int, cl : IVHF (U) ⟶ IVHF (U) be its IVHF interior operator and IVHF closure operator, respectively. Then there exists an IVHF reflexive and transitive relation $ℝ_{τ}$ on U such that $\underline{ℝ_{τ}} (𝔸) = int (𝔸)$ , and $\bar{ℝ_{τ}} (𝔸) = cl (𝔸)$ for all $𝔸 \in IVHF (U)$ iff int satisfies axioms (I2) and (I3), or equivalently, cl satisfies axioms (C2) and (C3): $\forall 𝔸, 𝔹 \in IVHF (U), \hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]} \in 2^{Int [0, 1]},$

(I2) $int (𝔸 ⋓ \hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]})$

$= int (𝔸) ⋓ \hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]},$

(I3) $int (𝔸 ⋒ 𝔹) = int (𝔸) ⋒ int (𝔹),$

(C2) $cl (𝔸 ⋒ \hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]})$

$= cl (𝔸) ⋒ \hat{[a_{1, \dots, m}^{L}, a_{1, \dots, m}^{U}]},$

(C3) $cl (𝔸 ⋓ 𝔹) = cl (𝔸) ⋓ cl (𝔹) .$

Proof. “ ⇒ " Suppose that there exists an IVHF reflexive and transitive relation $ℝ_{τ}$ on U such that $\underline{ℝ_{τ}} (𝔸) = int (𝔸)$ , and $\bar{ℝ_{τ}} (𝔸) = cl (𝔸)$ for all $𝔸 \in IVHF (U)$ , then by Theorem 2.13, it can be easily observed that conditions (I2), (I3), (C2), and (C3) hold.

“ ⟸ " Assume that the operator cl satisfies axioms (C2) and (C3). Then we can define an IVHF relation $ℝ_{τ} = {< (x, y), h_{ℝ_{τ}} (x, y) > | (x, y) \in U \times U}$ on U by cl as follows: $h_{ℝ_{τ}} (x, y) = h_{cl ([1, 1]_{y})} (x), (x, y) \in U \times U .$ (6)

Moreover, we can prove that for any $𝔸 \in IVHF (U),$ $𝔸 = ⋓_{y \in U} ([1, 1]_{y} ⋒ \hat{h_{𝔸} (y)}) .$

In fact, for all x ∈ U, then $\begin{matrix} h_{⋓_{y \in U} ([1, 1]_{y} ⋒ \hat{h_{𝔸} (y)})} (x) & = ⊻_{y \in U} h_{([1, 1]_{y} ⋒ \hat{h_{𝔸} (y)})} (x) \\ = ⊻_{y \in U} (h_{[1, 1]_{y}} (x) ⊼ h_{\hat{h_{𝔸} (y)}} (x)) \\ = ({[1, 1]} ⊼ h_{𝔸} (x)) ⊻ {[0, 0]} \\ = h_{𝔸} (x), \end{matrix}$ which implies that $𝔸 = ⋓_{y \in U} ([1, 1]_{y} ⋒ \hat{h_{𝔸} (y)})$ .

For any x ∈ U, by Equation (4), (C2) and (C3), we have $\begin{matrix} h_{\bar{ℝ_{τ}} (𝔸)} (x) & = ⊻_{y \in U} {h_{ℝ_{τ}} (x, y) ⊼ h_{𝔸} (y)} \\ = ⊻_{y \in U} {h_{cl ([1, 1]_{y})} (x) ⊼ h_{\hat{h_{𝔸} (y)}} (x)} \\ = ⊻_{y \in U} {h_{cl ([1, 1]_{y}) ⋒ \hat{h_{𝔸} (y)}} (x)} \\ = ⊻_{y \in U} {h_{cl ([1, 1]_{y} ⋒ \hat{h_{𝔸} (y)})} (x)} \\ = h_{⋓_{y \in U} cl ([1, 1]_{y} ⋒ \hat{h_{𝔸} (y)})} (x) \\ = h_{cl (⋓_{y \in U} ([1, 1]_{y} ⋒ \hat{h_{𝔸} (y)}))} (x) \\ = h_{cl (A)} (x) . \end{matrix}$

Thus, $cl (𝔸) = \bar{ℝ_{τ}} (𝔸)$ . Noticing that cl and int are dual to each other, from $cl (𝔸) = \bar{ℝ_{τ}} (𝔸)$ we immediately conclude that $int (𝔸) = \underline{ℝ_{τ}} (𝔸) .$ On the other hand, in terms of (Int2) of Theorem 3.6, we have $\underline{ℝ_{τ}} (𝔸) ⊑ 𝔸$ . Then, by Theorem 2.16, we conclude that $ℝ_{τ}$ is reflexive. Moreover, by (Int4) ′ of Theorem 3.6 again, we have $\underline{ℝ_{τ}} (\underline{ℝ_{τ}} (𝔸)) ⊑ \underline{ℝ_{τ}} (𝔸) .$ Meanwhile, by (Int3) of Theorem 3.6, then $\underline{ℝ_{τ}} (\underline{ℝ_{τ}} (𝔸)) = \underline{ℝ_{τ}} (𝔸) .$ Consequently, we must have $\underline{ℝ_{τ}} (𝔸) ⊑ \underline{ℝ_{τ}} (\underline{ℝ_{τ}} (𝔸)) .$ Therefore, by Theorem 2.16, we conclude that $ℝ_{τ}$ is transitive. So the IVHF relation $ℝ_{τ}$ is reflexive and transitive.□

The above Theorem 4.7 gives the sufficient and necessary conditions that an IVHF interior (closure, respectively) operator derived from an IVHF topological space can associate with an IVHF reflexive and transitive relation such that the induced lower (upper, respectively) IVHF rough approximation operator is just the IVHF interior (closure, respectively) operator.

Definition 4.8. Let (U, τ) be an IVHF topological space and int and cl : IVHF (U) ⟶ IVHF (U) be the induced IVHF interior operator and IVHF closure operator, respectively. If int satisfies the conditions (I2) and (I3), or equivalently, cl satisfies the conditions (C2) and (C3), then we call (U, τ) an IVHF rough topological space.

Let $R$ be the set of all IVHF reflexive and transitive relations on U and $T$ be the set of all IVHF rough topological spaces.

Theorem 4.9. (1) If $ℝ \in R$ , $τ_{ℝ}$ is defined by Equation 5 and $ℝ_{τ_{ℝ}}$ is defined by Equation 6, then $ℝ_{τ_{ℝ}} = ℝ .$

(2) If $τ \in T$ , $ℝ_{τ}$ is defined by Equation 6 and $τ_{ℝ_{τ}}$ is defined by Equation 5, then $τ_{ℝ_{τ}} = τ .$

Proof. (1) Since $ℝ$ is an IVHF reflexive and transitive relation on U, by Theorem 4.5, we obtain $\underline{ℝ} = {int}_{τ_{ℝ}}$ and $\bar{ℝ} = {cl}_{τ_{ℝ}} .$ According to Equation 6 and Theorem 2.14, for any x, y ∈ U we have $h_{ℝ_{τ_{ℝ}}} (x, y) = h_{{cl}_{τ_{ℝ}} ([1, 1]_{y})} (x) = h_{\bar{ℝ} ([1, 1]_{y})} (x) = h_{ℝ} (x, y) .$

Thus, $ℝ_{τ_{ℝ}} = ℝ .$

(2) By Equation 5 and Theorem 4.7, we have $τ_{ℝ_{τ}} = {𝔸 \in IVHF (U) | \underline{ℝ_{τ}} (𝔸) = 𝔸} = {𝔸 \in IVHF (U) | int (𝔸)$ $= 𝔸} = τ .$ □

Theorem 4.10. There exists a one-to-one correspondence between $R$ and $T$ .

Proof. Define a mapping $f : R ⟶ T$ as follows: $f (ℝ) = τ_{ℝ}, ℝ \in R .$

On the other hand, define a mapping $g : T ⟶ R$ as follows: $g (τ) = ℝ_{τ}, τ \in T .$ Then, by Theorem 4.9, it is easy to verify that both f and g are one-to-one correspondences between $R$ and $T$ .□

The above Theorem 4.10 shows that there exists a one-to-one correspondence between the set of all IVHF reflexive and transitive approximation spaces and the set of all IVHF rough topological spaces such that the lower and upper IVHF rough approximation operators are respectively the IVHF interior and closure operators.

5 Application of the IVHF rough set model in medical diagnosis

Firstly, we generalize the IVHF relation on one same universe in Section 3 and give the IVHF relation on two universes.

Definition 5.1. Let U, V be two nonempty and finite universes. An IVHF subset $ℝ$ of the universe U × V is called an IVHF relation from U to V, namely, $ℝ$ is given by $ℝ = {< (x, y), h_{ℝ} (x, y) > | (x, y) \in U \times V},$ where $h_{ℝ} : U \times V \to Int [0, 1]$ is a set of interval values in Int [0, 1].

It is noted that if U = V, then $ℝ$ degenerates to an IVHF fuzzy relation on U given in Definition 2.10.

In generally, for any x ∈ U, y ∈ V, $h_{ℝ} (x, y)$ denotes the possible interval membership degrees of the relationships between x and y. We denote byIVHFR (U × V) the family of all IVHF relations on U × V.

Definition 5.2. Let U, V be two nonempty and finite universes and $ℝ$ be an IVHF fuzzy relation from U to V. For any [α^L, α^U] ∈ Int [0, 1], we define a binary crisp relation between the universe U and V as follows: $\begin{matrix} ℝ_{{[α^{L}, α^{U}]}} & = {(x, y) \in U \times V | h_{ℝ} (x, y) ⪰ {[α^{L}, α^{U}]}} \\ = {(x, y) \in U \times V | h_{ℝ}^{σ (k) L} (x, y) \geq α^{L}, \\ h_{ℝ}^{σ (k) U} (x, y) \geq α^{U}}, k = 1, 2, \dots, l, \end{matrix}$ which is called the [α^L, α^U]-level cut set of $ℝ$ .

Furthermore, we define the successor neighborhood of x ∈ U with respect to $ℝ_{{[α^{L}, α^{U}]}}$ as follows: $\begin{matrix} ℝ_{{[α^{L}, α^{U}]}} (x) & = {y \in V | h_{ℝ} (x, y) ⪰ {[α^{L}, α^{U}]}} \\ = {y \in V | h_{ℝ}^{σ (k) L} (x, y) \geq α^{L}, \\ h_{ℝ}^{σ (k) U} (x, y) \geq α^{U}}, k = 1, 2, \dots, l . \end{matrix}$

According to the above definition, if $(x, y) \in ℝ_{{[α^{L}, α^{U}]}}$ , it indicates that the membership degree of the relationships of x and y with respect to the IVHF fuzzy relation $ℝ$ is not less than [α^L, α^U].

In what follows we give an IVHF rough set model based on a binary crisp relation $ℝ_{{[α^{L}, α^{U}]}}$ induced by an IVHF fuzzy relation $ℝ$ from U to V.

Definition 5.3. Let U and V be two nonempty and finite universes of discourse. Suppose that $ℝ_{{[α^{L}, α^{U}]}}$ is an arbitrary crisp relation given in Definition 5.2 from U to V. For any set A ⊆ V, a pair of lower and upper approximations, $\underline{ℝ_{{[α^{L}, α^{U}]}}} (A)$ and $\bar{ℝ_{{[α^{L}, α^{U}]}}} (A)$ , are defined by $\begin{matrix} \underline{ℝ_{{[α^{L}, α^{U}]}}} (A) = {x \in U | ℝ_{{[α^{L}, α^{U}]}} (x) \subseteq A}, \\ \bar{ℝ_{{[α^{L}, α^{U}]}}} (A) = {x \in U | ℝ_{{[α^{L}, α^{U}]}} (x) \cap A \neq \emptyset} . \end{matrix}$

Furthermore, we also define the positive region ${pos}_{ℝ_{{[α^{L}, α^{U}]}}} (A)$ , negative region ${neg}_{ℝ_{{[α^{L}, α^{U}]}}} (A)$ and boundary region ${bn}_{ℝ_{{[α^{L}, α^{U}]}}} (A)$ of A about $ℝ_{{[α^{L}, α^{U}]}}$ as follows, respectively: $\begin{matrix} {pos}_{ℝ_{{[α^{L}, α^{U}]}}} (A) = \underline{ℝ_{{[α^{L}, α^{U}]}}} (A), \\ {neg}_{ℝ_{{[α^{L}, α^{U}]}}} (A) = U - \bar{ℝ_{{[α^{L}, α^{U}]}}} (A), \\ {bn}_{ℝ_{{[α^{L}, α^{U}]}}} (A) = \bar{ℝ_{{[α^{L}, α^{U}]}}} (A) - \underline{ℝ_{{[α^{L}, α^{U}]}}} (A) . \end{matrix}$

Remark 5.4. In [33], Sun and Ma introduced a fuzzy rough set model on two different universes and applied it to clinical diagnosis problems. However, the method is imperfect whose disadvantages can be viewed in the two aspects. On the one hand, it is unreasonable to invite only a doctor to evaluate the membership degrees for every sufferer with respect to the symptoms, which may increase the risk of misdiagnosis. So the number of doctors need to be added in order to make diagnostic results more objective. On the other hand, due to the shortage of the doctors’ available information and experience, it is very difficult for doctors to quantify their opinions by several crisp numbers. Instead, it seems to be more reasonable that they can be represented by several interval numbers within [0,1]. In order to overcome the two defects, we extend the fuzzy compatible relation in [33] to IVHF environment, introduce the IVHF rough set model based on a binary crisp relation induced by an IVHF fuzzy relation $ℝ$ from U to V, and apply the model to medical diagnostic problems. It turns out that IVHF rough set model based on a binary crisp relation works better than the method proposed by Sun and Ma [33], and can overcome the two disadvantages that we pointed out.

Suppose that the universe U = {x₁, x₂, ⋯ , x_m} denotes the set of m patients, and the universe V = {y₁, y₂, ⋯ , y_n} denotes the set of n symptoms. Let $ℝ \in IVHFR (U \times V)$ be an IVHF relation from U to V. For any (x_i, y_j) ∈ U × V, $h_{ℝ} (x_{i}, y_{j})$ represents interval membership degree of the relationships of the patient x_i and the symptom y_j, which are evaluated by several doctors. In clinical practice, a patient can see different doctors and may get different diagnoses. To decrease the risk of misdiagnosis, we should carefully consider all the doctors’ comments. In this case, the threshold [α^L, α^U] is given in advance by several doctors, and represents the doctors’ the lowest requirements on the membership degree. For any set A ⊆ V, A denotes a certain disease and has the basic symptoms y_j ∈ A. As for a specific patient x_i ∈ U, if $x_{i} \in \underline{ℝ_{{[α^{L}, α^{U}]}}} (A)$ , then we are sure that the patient x_i is suffering from the disease A and must receive treatment immediately. If $x_{i} \in \bar{ℝ_{{[α^{L}, α^{U}]}}} (A) - \underline{ℝ_{{[α^{L}, α^{U}]}}} (A)$ , i.e. $x_{i} \in {bn}_{ℝ_{{[α^{L}, α^{U}]}}} (A)$ , then we cannot currently sure whether the patient x_i is suffering from the disease A or not. So the doctors will carry out a second choice to determine whether the patient x_i is suffering from the disease A or not. If $x_{i} \in U - \bar{ℝ_{{[α^{L}, α^{U}]}}} (A)$ , then we conclude that the patient x_i is not suffering from the disease A and does not need to receive treatment at all.

From the above analysis, it is clear that our method can help several doctors know the disease better such that the diagnosis results are more scientific and more comprehensive. After all, it also can decrease the risk of misdiagnosis in clinical practice.

To illustrate our method, let us consider the following example.

Example 5.5. Let U = {x₁, x₂, x₃, x₄, x₅} be a set of five patients, V = {y₁, y₂, y₃, y₄} be a set of four symptoms, and $ℝ \in IVHFR (U \times V)$ be an IVHF relation from U to V (see Table 1). For any (x_i, y_j) ∈ U × V, $h_{ℝ} (x_{i}, y_{j})$ represents different interval membership degree of the patient x_i with the symptom y_j, which are evaluated by different doctors. For example, $h_{ℝ} (x_{1}, y_{1}) = {[0.3, 0.4], [0.5, 0.6]}$ indicates that one doctor thinks that the membership degree of the relationships of the patient x₁ and the symptom y₁ is [0.3,0.4], and another doctor thinks that the membership degree of the relationships of the patient x₁ and the symptom y₁ is [0.5,0.6]. Let A = {y₁, y₃, y₄} ⊆ V denotes a certain disease showing three basic symptoms clinically.

First, taking [α^L, α^U] = [0.4, 0.5], by Definition 5.2 we can obtain the binary crisp relation $ℝ_{{[0.4, 0.5]}}$ (see Table 2).

According to Table 2, we can obtain the results as follows: $\begin{matrix} ℝ_{{[0.4, 0.5]}} (x_{1}) = {y_{3}, y_{4}}, \\ ℝ_{{[0.4, 0.5]}} (x_{2}) = {y_{1}, y_{2}, y_{3}, y_{4}}, \\ ℝ_{{[0.4, 0.5]}} (x_{3}) = {y_{3}}, \\ ℝ_{{[0.4, 0.5]}} (x_{4}) = {y_{1}, y_{2}}, ℝ_{{[0.4, 0.5]}} (x_{5}) = {y_{1}, y_{3}} . \end{matrix}$

Then, by virtue of Definition 5.3, we can calculate the lower approximation, the upper approximation and the boundary region of A as follows, respectively.

$\underline{ℝ_{{[0.4, 0.5]}}} (A) = {x_{1}, x_{3}, x_{5}},$

$\bar{ℝ_{{[0.4, 0.5]}}} (A) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}},$

${bn}_{ℝ_{{[0.4, 0.5]}}} (A) = \bar{ℝ_{{[0.4, 0.5]}}} (A) - \underline{ℝ_{{[0.4, 0.5]}}} (A) = {x_{2}, x_{4}}$ , ${neg}_{ℝ_{{[0.4, 0.5]}}} (A) = \emptyset .$

Then, in terms of the above analysis, we can obtain the following conclusions:

Patients x₁, x₃ and x₅ are suffering from the disease A and must receive treatment immediately.

According to the current symptoms, we cannot determine whether patients x₂ and x₄ are suffering from the disease A or not. Whether or not they are suffering from the disease A will be carried out a second choice.

None of the patients are healthy, and all the patients need to receive treatment.

For the second time, taking [α^L, α^U] = [0.5, 0.6], by Definition 5.2 we can obtain the binary crisp relation $ℝ_{{[0.5, 0.6]}}$ (see Table 3).

By Table 3, we can obtain the results as follows: $\begin{matrix} ℝ_{{[0.5, 0.6]}} (x_{1}) = {y_{3}, y_{4}}, ℝ_{{[0.5, 0.6]}} (x_{2}) = {y_{2}, y_{3}}, \\ ℝ_{{[0.5, 0.6]}} (x_{3}) = \emptyset, ℝ_{{[0.5, 0.6]}} (x_{4}) = {y_{2}}, \\ ℝ_{{[0.5, 0.6]}} (x_{5}) = {y_{1}} . \end{matrix}$

Then, according to Definition 5.3, we can calculate the lower approximation, the upper approximation and the boundary region of A as follows, respectively. $\begin{matrix} \underline{ℝ_{{[0.5, 0.6]}}} (A) & = {x_{1}, x_{3}, x_{5}}, \\ \bar{ℝ_{{[0.5, 0.6]}}} (A) & = {x_{1}, x_{2}, x_{3}, x_{5}}, \\ {bn}_{ℝ_{{[0.5, 0.6]}}} (A) & = \bar{ℝ_{{[0.5, 0.6]}}} (A) - \underline{ℝ_{{[0.5, 0.6]}}} (A) \\ = {x_{2}}, {neg}_{ℝ_{{[0.5, 0.6]}}} (A) = {x_{4}} . \end{matrix}$

In terms of the above analysis, we can obtain the following conclusions:

Patients x₁, x₃ and x₅ are suffering from the disease A and must receive treatment immediately.

According to the current symptoms, we cannot determine whether patients x₂ is suffering from the disease A or not. Whether or not he or she is suffering from the disease A will be carried out a second choice.

Patient x₄ is not suffering from the disease A at all and does not need to receive treatment at all.

From the above analysis, we know that when the doctors take the threshold value [α^L, α^U] = [0.4, 0.5], patients x₁, x₃ and x₅ are suffering from the disease A and none of the patients are not suffering from the disease A. Furthermore, when the doctors take the threshold value [α^L, α^U] = [0.5, 0.6], we can note that patients x₁, x₃ and x₅ are still suffering from the disease A, but patient x₄ is not suffering from the disease A at all.

Comparing with the decision-making method in [33], we can see that the available information (see Table 1) in the above IVHF rough sets is more comprehensive and objective than fuzzy rough sets proposed by Sun and Ma [33], which helps the doctors make a more comprehensive and objective diagnosis. On the other hand, due to the involvement of several doctors, it is evident that our adjustable method can also help the doctors make a scientific, comprehensive and precise decision such that the risk of misdiagnosis will be decreased in clinical practice. All in all, since the available information is more comprehensive and the opinions of several doctors are considered, the greatest advantage of our adjustable method is to decrease the risk of misdiagnosis in clinical practice.

6 Conclusion

In this study, we have explored the topological structures of IVHF rough sets. On the one hand, we have proved that a pair of dual IVHF rough approximation operators can induce an IVHF topological space if and only if the IVHF relation is reflexive and transitive. On the other hand, we have investigated that an IVHF rough topological space can be associated with an IVHF reflexive and transitive rough approximation space such that the induced lower (upper, respectively) IVHF rough approximation operator is just the IVHF interior (closure, respectively) operator of the given topology. Finally, applications of IVHF rough sets in medical diagnosis problem have been shown.

In the future, we can use the IVHF rough set model to address the application to knowledge discovery and reduction. Moreover, it is important and interesting to further investigate uncertain measures of IVHF rough sets with application to data analysis.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 71261022).

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