A Novel Method of the Generalized Interval-Valued Fuzzy Rough Approximation Operators

Abstract

Rough set theory is a suitable tool for dealing with the imprecision, uncertainty, incompleteness, and vagueness of knowledge. In this paper, new lower and upper approximation operators for generalized fuzzy rough sets are constructed, and their definitions are expanded to the interval-valued environment. Furthermore, the properties of this type of rough sets are analyzed. These operators are shown to be equivalent to the generalized interval fuzzy rough approximation operators introduced by Dubois, which are determined by any interval-valued fuzzy binary relation expressed in a generalized approximation space. Main properties of these operators are discussed under different interval-valued fuzzy binary relations, and the illustrative examples are given to demonstrate the main features of the proposed operators.

1. Introduction

Rough set theory proposed by Pawlak [1] is an extension of set theory for the study of intelligent systems characterized by inexact, uncertain, or insufficient information. The core of the rough set theory and its applications is to define a pair of lower and upper approximation operators, and an equivalence relation is a key and primitive notion in Pawlak’s rough set model [1]. This equivalence relation is the key concept of Pawlak’s rough set model, but also a very strict condition, which may limit the applicability of the rough set model [2, 3]. To solve this problem, several authors have generalized the notion of the approximation operators by using nonequivalent binary relations. The most important research is the amalgamation of fuzzy set theory and rough set theory [2, 4–7] as well as the rough set theory based on generalized binary relations [8–13]. Pawlak first discussed the relation between rough sets and fuzzy sets in [6]. Dubois and Prade [4] proposed the fuzzy rough set theory by amalgamating the fuzzy set theory with the rough set theory. In addition, based on the definition of neighborhood operators, Yao [8–11] studied the rough set theory based on the generalized binary relation, that is, the generalized rough set theory. Recently, Wu et al. [14–19] defined the generalized fuzzy rough set theory based on the study of the fuzzy rough set theory and the generalized rough set theory, and Zhu [12] studied generalized rough sets based on relations.

A rough set model is composed of two parts: the approximation space and the approximated object. Rough set theory comes with a lot of extensions and generalizations. Yao et al. researched the generalized rough sets by considering sets and relations of the approximation space and the approximated object [9, 16]. In Pawlak’s rough set model [6], the relation of approximation spaces is a classical binary equivalence relation and the approximated object is a set. If the equivalence relation is weakened to a general binary relation, the equivalence relation is a special case of the general binary relation. The set theory is generalized to the form of the fuzzy set theory, so that the classical set theory is a special case of the fuzzy set theory. These relationships are outlined in Figure 1.

Figure 1 

Select dependencies in rough set theory.

Most researches on the fuzzy rough set theory focus on point-valued fuzzy sets and point-valued fuzzy binary relations. But the fuzzy notion described by using point values may lose some available information in the real-life information systems sometimes. If the description is done by interval values, it may acquire a better effectiveness than that by using point ones, for example, a self-evolving interval type-2 fuzzy neural network with online structure and parameter learning [20], encoding words into interval type-2 fuzzy sets using an interval approach [21], and corrections to aggregation using the linguistic weighted average and interval type-2 fuzzy sets [22]. Gong et al. [23] proposed a kind of interval-valued rough fuzzy set model based on an equivalent relation and applied the model to acquire rules from the interval-valued fuzzy information systems. It is very significant to apply the interval-valued fuzzy set in researching the rough set theory. Yeung et al. [24] generalized the fuzzy rough sets by means of arbitrary fuzzy relations and presented a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. Wu et al. [25] generalized the concept of fuzzy rough sets to interval type-2 fuzzy environments and proposed a method of attribute reduction within the interval type-2 fuzzy rough set framework. Xue et al. [26] generalized interval-valued fuzzy rough approximation operators. Zhang et al. [27] studied the characterization of generalized interval-valued fuzzy rough sets on two universes of discourse. The positive approximation and converse approximation in interval-valued fuzzy rough sets have been studied in [28]. Zhang and Jiang [29] proposed a note on interval-valued fuzzy rough sets and interval-valued intuitionistic fuzzy sets. Zhang et al. [30] proposed a general frame for intuitionistic fuzzy rough sets. Xu et al. [31] studied an axiomatic approach of interval-valued intuitionistic fuzzy rough sets based on interval-valued intuitionistic fuzzy approximation operators. Zhang and Tian [32] studied interval-valued intuitionistic fuzzy rough sets based on implicators. Wu and Zhou [33] studied intuitionistic fuzzy topologies based on intuitionistic fuzzy reflexive and transitive relations. Zhang et al. [34] proposed a variable-precision-dominance-based rough set approach to interval-valued information systems. Liang and Liu [35] studied three-way decisions with interval-valued decision-theoretic rough sets. Dai et al. [36] proposed an uncertainty measurement for interval-valued decision systems based on extended conditional entropy. Zhang et al. [37] studied multiconfidence rule acquisition and confidence-preserved attribute reduction in interval-valued decision systems. Ma and Hu [38] studied topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets. Hao and Li [39] discussed the relationship between L-fuzzy rough set and L-topology. Zhang et al. [40] studied the union and intersection operations of rough sets based on various approximation spaces. She and He [41–43] studied rough approximation operators on R0-algebras (nilpotent minimum algebras) with an application in formal logic L, the rough consistency measures of logic theories, and approximate reasoning in rough logic and the structure of the multigranulation rough set model as well. Yang et al. [44] studied the combination of interval-valued fuzzy set and soft set. In terms of these researches above, a number of important conclusions are drawn, which exhibit great significance to research the rough fuzzy set theory. However the generalized interval-valued fuzzy rough set theory under the generalized relations needs to be further investigated.

In this paper, we further study the generalized fuzzy rough approximation operators defined in [16]. In particular, from the viewpoint of constructive approach, we reconstruct the lower approximation operator on the premise of the fact that the upper approximation operator is not changed and expand it to interval environments. It is proved that the lower approximation operator is equivalent to the generalized interval Dubois fuzzy rough approximation operator in the approximation space formed by arbitrary binary interval-valued fuzzy relations. Also, properties of the operators are discussed under the different binary interval-valued fuzzy relations.

The rest of the paper is organized as follows. In Section 2, we give some basic notions of interval-valued fuzzy sets and interval-valued fuzzy relations. In Section 3, we study the generalized fuzzy rough approximation operators defined in [16]. In Section 4, from the viewpoint of constructive and interval approach, we reconstruct new lower and upper approximation operators of the generalized interval-valued fuzzy rough sets. In Section 5, we prove some properties of the generalized interval-valued fuzzy rough approximation operators and the presented scheme by the extensive analysis results. In Section 6, we bring forward some conclusions and highlight further work.

2. Basic Concepts of Interval-Valued Fuzzy Sets and Interval-Valued Fuzzy Relations

In this section, we introduce some basic notions and properties related to interval-valued fuzzy sets which will be used in this paper. We first review an interval-valued subset originated by [28]. We first review some basic concepts.

Let 𝐼 be a closed unit interval; that is, 𝐼=[0,1]. [𝐼]={[𝑎⁻,𝑎⁺]∶𝑎⁻≤𝑎⁺,𝑎⁻,𝑎⁺∈𝐼} is the set of all interval-valued subsets of 𝐼. 𝑎=[𝑎⁻,𝑎⁺]∈[𝐼] is an interval value. When 𝑎⁻=𝑎⁺, the interval-valued 𝑎=[𝑎⁻,𝑎⁺] becomes a real number in [𝐼]. In particular, real numbers return intervals of zero length, say 1=[1,1] and 0=[0,0].

Definition 1. Let 𝑎,𝑏∈[𝐼]. 𝑎≤𝑏 if and only if 𝑎⁻≤𝑏⁻, 𝑎⁺≤𝑏⁺; 𝑎=𝑏 if and only if 𝑎⁻=𝑏⁻,𝑎⁺=𝑏⁺; 𝑎<𝑏 if and only if 𝑎≤𝑏 and 𝑎≠𝑏.

Definition 2. Let 𝑎,𝑏∈[𝐼]. 𝑎≰𝑏 indicates that 𝑎 is not less than or equal to 𝑏; 𝑎≮𝑏 indicates that 𝑎 is not less than 𝑏; 𝑎≱𝑏 indicates that 𝑎 is not greater than or equal to 𝑏; 𝑎>𝑏̸ indicates that 𝑎 is not greater than 𝑏.

According to the order relation defined in Definition 1, different elements in [𝐼] may not exhibit order relations, so Definition 2 becomes necessary.

Definition 3. Let 𝑎_𝑖∈[𝐼],𝑏_𝑖∈𝐼,𝑖∈𝐽,𝐽={1,2,⋯,𝑚}; one defines _𝑖∈𝐽𝑏_𝑖=sup𝑏_𝑖∶𝑖∈𝐽,_𝑖∈𝐽𝑏_𝑖=inf𝑏_𝑖∶𝑖∈𝐽_𝑖∈𝐽,𝑎_𝑖=∨𝑎_𝑖∶𝑖∈𝐽=_𝑖∈𝐽𝑎⁻_𝑖,_𝑖∈𝐽𝑎⁺_𝑖,_𝑖∈𝐽𝑎_𝑖=∧𝑎_𝑖∶𝑖∈𝐽=_𝑖∈𝐽𝑎⁻_𝑖,_𝑖∈𝐽𝑎⁺_𝑖,∼𝑎=1−𝑎=1−𝑎⁺,1−𝑎⁻. (1)
Obviously, ([𝐼],≤) is a complete lattice, and the triple ([𝐼],∨,∧) is an algebraic system, which is derived by ([𝐼],≤) with the maximal element [1,1] and the minimum element [0,0].

Definition 4. Let 𝑈 be a finite and nonempty universe of discourse; then a mapping 𝐴∶𝑈→[𝐼] is called an interval-valued fuzzy set on 𝑈. All interval-valued fuzzy sets on 𝑈 are denoted by 𝐹^𝐼(𝑈). In particular, when 𝐴=𝑈,𝐴(𝑥)=[1,1], for all 𝑥∈𝑈, and when 𝐴=∅, 𝐴(𝑥)=[0,0], for all 𝑥∈𝑈.

Similar to fuzzy sets, the operators ⊆,∩,∪, and complement of interval-valued fuzzy sets are defined as follows. For all 𝐴,𝐵∈𝐹^𝐼(𝑈), 𝐴⊆𝐵 means 𝐴(𝑥)≤𝐵(𝑥) and for all 𝑥∈𝑈, (𝐴∩𝐵)(𝑥)=𝐴(𝑥)∧𝐵(𝑥), (𝐴∪𝐵)(𝑥)=𝐴(𝑥)∨𝐵(𝑥), and (∼𝐴)(𝑥)=1−𝐴(𝑥).

Definition 5. Let 𝛼∈[𝐼],𝐴∈𝐹^𝐼(𝑈). 𝛼𝐴 is called numerical product of 𝛼 and 𝐴 and is defined as (𝛼𝐴)(𝑥)=𝛼∧𝐴(𝑥), for all 𝑥∈𝑈.

Definition 6. Let 𝛼∈[𝐼], 𝐴∈𝐹^𝐼(𝑈). 𝐴_𝛼={𝑥∈𝑈∶𝐴(𝑥)≥𝛼} is called 𝛼-cut set of 𝐴 and 𝐴_𝛼={𝑥∈𝑈∶𝐴(𝑥)>𝛼} is called strong 𝛼-cut set of 𝐴.

Theorem 7 (the decomposition theorem of the interval-valued fuzzy sets). Let 𝐴∈𝐹^𝐼(𝑈); then 𝐴=_{𝛼∈[𝐼]}𝛼𝐴_𝛼,𝐴=_{𝛼∈[𝐼]}𝛼𝐴_𝛼. (2)

Proof. For all 𝑥∈𝑈, _{[𝐼]𝛼∈}𝛼𝐴_𝛼(𝑥)=_{[𝐼]𝛼∈}𝛼∧𝐴_𝛼(𝑥)=_{(𝑥)𝛼≤𝐴}𝛼∧𝐴_𝛼(𝑥)∨_{(𝑥)𝛼≰𝐴}𝛼∧𝐴_𝛼(𝑥)=_{(𝑥)𝛼≤𝐴}𝛼∧𝐴_𝛼(𝑥=_{(𝑥)𝛼≤𝐴})𝛼=𝐴(𝑥). (3)
Then 𝐴=∪_{𝛼∈[𝐼]}𝛼𝐴_𝛼.
Similarly, one can show that 𝐴=∪_{𝛼∈[𝐼]}𝛼𝐴_𝛼.

Definition 8. Let 𝑈 and 𝑊 be two finite and nonempty universes of discourse. Then the mapping 𝐼𝑅∶𝑈×𝑊→[𝐼] is called an interval-valued fuzzy relation from 𝑈 to 𝑊, where 𝑈×𝑊={(𝑥,𝑦)∶𝑥∈𝑈,𝑦∈𝑊}. When 𝑈=𝑊, 𝐼𝑅 is called an interval-valued fuzzy relation on 𝑈.

Remark 9. Obviously, an interval-valued fuzzy relation 𝐼𝑅 from 𝑈 to 𝑊 is an interval-valued fuzzy set denoted by 𝐼𝑅∈𝐹^𝐼(𝑈×𝑊). So Definitions 4, 5, and 6 and Theorem 7 are still true in the interval-valued fuzzy relation. For example, 𝐼𝑅∈𝐹^𝐼(𝑈×𝑊) is an interval-valued fuzzy relation. If we see it as an interval-valued fuzzy set, then 𝐼𝑅_𝛼={(𝑥,𝑦)∈𝑈×𝑊∶𝑅(𝑥,𝑦)≥𝛼}.

Definition 10. Let 𝐼𝑅 be an interval-valued fuzzy relation from 𝑈 to 𝑊; then 𝐼𝑅 is said to be serial if and only if for all 𝑥∈𝑈, there exists 𝑦∈𝑊 such that 𝐼𝑅(𝑥,𝑦)=[1,1].

Definition 11. Let 𝐼𝑅 be an interval-valued fuzzy relation on 𝑈; then 𝐼𝑅 is reflexive if and only if 𝐼𝑅(𝑥,𝑥)=[1,1], for all 𝑥∈𝑈; 𝐼𝑅 is symmetric if and only if 𝐼𝑅(𝑥,𝑦)=𝐼𝑅(𝑦,𝑥), for all 𝑥,𝑦∈𝑈; 𝐼𝑅 is transitive if and only if 𝐼𝑅(𝑥,𝑧)≥∨_𝑦∈𝑈(𝐼𝑅(𝑥,𝑦)∧𝐼𝑅(𝑦,𝑧)), for all 𝑥,𝑧∈𝑈; 𝐼𝑅 is Euclidean if and only if 𝐼𝑅(𝑦,𝑧)≥∨_𝑥∈𝑈(𝐼𝑅(𝑥,𝑦)∧𝐼𝑅(𝑥,𝑧)), for all 𝑦,𝑧∈𝑈.

One can prove that the binary relation obtained by calculating 𝛼-cut set or strong 𝛼-cut set to an interval-valued fuzzy relation, for all 𝛼∈[𝐼], still satisfies the corresponding definition of Definition 11 under the classical binary relation; that is, if 𝐼𝑅 is, respectively, reflexive, symmetric, and transitive, then 𝐼𝑅_𝛼(𝐼𝑅_𝑎) is, respectively, reflexive, symmetric, and transitive under the classical binary relation.

3. Generalized Fuzzy Rough Approximation Operators

Definition 12. Let 𝑈 and 𝑊 be two finite universes of discourse. If 𝑅 is an arbitrary binary fuzzy relation from 𝑈 to 𝑊, then the triple (𝑈,𝑊,𝑅) is called a generalized fuzzy approximation space.

Definition 13. Let (𝑈,𝑊,𝑅) be a generalized fuzzy approximation space, for all 𝑥∈𝑈; one defines 𝑅(𝑥)={(𝑦,𝑅(𝑥,𝑦))∶𝑦∈𝑊}.

𝑅(𝑥) is the row of the fuzzy relation which includes 𝑥, and obviously 𝑅_𝛼(𝑥)=(𝑅(𝑥))_𝛼.

Definition 14. Let (𝑈,𝑊,𝑅) be a generalized fuzzy approximation space, for all 𝛼,𝛽∈[0,1], 𝐴∈𝐹(𝑊), 𝑅_𝛼𝐴_𝛽=𝑥∈𝑈∶𝑅_𝛼(𝑥)⊆𝐴_𝛽,𝑅_𝛼𝐴_𝛽=𝑥∈𝑈∶𝑅_𝛼(𝑥)∩𝐴_𝛽≠∅. (4)
𝑅_𝛼(𝐴_𝛽) and 𝑅_𝛼(𝐴_𝛽) are called (𝛼,𝛽) lower and upper approximations of 𝐴 with respect to (𝑈,𝑊,𝑅).

Definition 15. Let (𝑈,𝑊,𝑅) be a generalized fuzzy approximation space, for all 𝐴∈𝐹(𝑊). One defines 𝑅(𝐴)=_𝛼∈𝐼𝛼𝑅_1−𝛼𝐴_𝛼,𝑅(𝐴)=_𝛼∈𝐼𝛼𝑅_𝛼𝐴_𝛼. (5)
The pair (𝑅(𝐴),𝑅(𝐴)) is called the generalized fuzzy rough set of 𝐴 on (𝑈,𝑊,𝑅), and the operators 𝑅(𝐴) and 𝑅(𝐴) are called the generalized fuzzy rough lower and upper approximation operators, respectively.

The dual properties are quite useful in proving the properties of the approximation operators. When one intends to prove two dual properties, it suffices to prove one of them, which simplifies the proof procedure. The properties of the lower and upper approximation operators are characterized as follows.

Theorem 16. Let (𝑈,𝑊,𝑅) be a generalized fuzzy approximation space. Then for all 𝐴∈𝐹(𝑊), 𝑅(𝐴)=∼𝑅(∼𝐴) and 𝑅(𝐴)=∼𝑅(∼𝐴).

Proof. (1) Note that ∩_𝛼∈𝐼(𝛼∪𝑅_1−𝛼(𝐴_𝛼))=∪_𝛼∈𝐼(𝛼∩𝑅_1−𝛼(𝐴_𝛼)). Here 𝛼∈𝐹(𝑈) and 𝛼(𝑥)=𝛼 for all 𝑥∈𝑈.
For all 𝑥∈𝑈, 𝑥∈𝑅_1−𝛼(𝐴_𝛼),thatis,𝑅_1−𝛼(𝐴_𝛼)(𝑥)=1, if and only if 𝑅_1−𝛼(𝑥)⊆𝐴_𝛼, such that for all 𝑦∈𝑊 if 𝑅(𝑥,𝑦)≥1−𝛼, then 𝐴(𝑦)>𝛼; that is, for all 𝑦∈𝑊, 𝑅(𝑥,𝑦)<1−𝛼 or 𝐴(𝑦)>𝛼.
Hence we have ∧_𝑦∈𝑊((1−𝑅(𝑥,𝑦))∨𝐴(𝑦))>𝛼.
For the second case, 𝑥∉𝑅_1−𝛼(𝐴_𝛼); that is, 𝑅_1−𝛼(𝐴_𝛼)(𝑥)=0 if and only if ∧_𝑦∈𝑊((1−𝑅(𝑥,𝑦))∨𝐴(𝑦))≤𝛼. Since ∧_𝑦∈𝑊((1−𝑅(𝑥,𝑦))∨𝐴(𝑦))∈𝐼, there exists 𝛼∈𝐼, so that 𝛼=∧_𝑦∈𝑊((1−𝑅(𝑥,𝑦))∨𝐴(𝑦)).
Hence, _𝛼∈𝐼𝛼∪𝑅_1−𝛼𝐴_𝛼(𝑥)=_𝛼∈𝐼𝛼∨𝑅_1−𝛼𝐴_𝛼𝑥)=_𝑦∈𝑊(1−𝑅𝑥,𝑦∨𝐴𝑦. (6)
Similarly, _𝛼∈𝐼𝛼∩𝑅_1−𝛼𝐴_𝛼(𝑥)=_𝛼∈𝐼𝛼∧𝑅_1−𝛼𝐴_𝛼𝑥)=_𝑦∈𝑊(1−𝑅𝑥,𝑦∨𝐴𝑦. (7)
Therefore, ∩_𝛼∈𝐼(𝛼∪𝑅_1−𝛼(𝐴_𝛼))=∪_𝛼∈𝐼(𝛼∩𝑅_1−𝛼(𝐴_𝛼)).
(2) Now, we prove the validity of the relationship 𝑅(𝐴)=∼𝑅(∼𝐴). In view of Definition 14, from Theorem  3.2(1) of [18], it follows that ∼𝑅(∼𝐴)=∼_𝛼∈𝐼𝛼𝑅_𝛼(∼𝐴)_𝛼=∼_𝛼∈𝐼𝛼𝑅_𝛼∼𝐴_1−𝛼=_𝛼∈𝐼1−𝛼∪∼𝑅_𝛼∼𝐴_1−𝛼=_𝛼∈𝐼1−𝛼∪𝑅_𝛼𝐴_1−𝛼=_𝛼∈𝐼𝛼∪𝑅_1−𝛼𝐴_𝛼=_𝛼∈𝐼𝛼𝑅_1−𝛼𝐴_𝛼=𝑅(𝐴). (8)
Hence, 𝑅(𝐴)=∼𝑅(∼𝐴).
Similarly, 𝑅(𝐴)=∼𝑅(∼𝐴).

Suppose that 𝑦₁=𝛼∨𝑅_1−𝛼(𝐴_𝛼)(𝑥),𝑦₂=𝛼∧𝑅_1−𝛼(𝐴_𝛼)(𝑥) and 𝛽=∧_𝑦∈𝑊((1−𝑅(𝑥,𝑦))∨𝐴(𝑦)); when the variable 𝑥 is a certain value, the variables 𝑦₁ and 𝑦₂ are functions of the variable 𝛼. Refer to Figure 2 for the pertinent detail.

Figure 2 

The coordinate frame of functions 𝑦1 and 𝑦2.

In the proof of Theorem 16, we show that the equation 𝑅(𝐴)=∼𝑅(∼𝐴) holds when the minimum of function 𝑦₁ is equal to the maximum of function 𝑦₂, such that ∩_𝛼∈𝐼(𝛼∪𝑅_1−𝛼(𝐴_𝛼))=∪_𝛼∈𝐼(𝛼𝑅_1−𝛼(𝐴_𝛼)); thus, 𝑅(𝐴)=∼𝑅(∼𝐴) holds. In [16], the lower approximation operator makes function 𝑦₂ equal to zero at the point 𝛽, which makes the maximum of function 𝑦₂ approach 𝛽, but it does not exist. In this paper, the lower and upper approximation operators in Definition 15 have a better duality.

4. Generalized Interval-Valued Fuzzy Rough Sets

Definition 17. Let 𝑈 and 𝑊 be two finite universes of discourse. If 𝐼𝑅 is an arbitrary binary interval-valued fuzzy relation from 𝑈 to 𝑊, then the triple (𝑈,𝑊,𝐼𝑅) is called a generalized interval-valued fuzzy approximation space. In particular, when 𝑈=𝑊, the space is denoted by (𝑈,𝐼𝑅).

Definition 18. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space, for all 𝑥∈𝑈, 𝐼𝑅(𝑥)=𝑦,𝐼𝑅𝑥,𝑦∶𝑦∈𝑊. (9)

Definition 19. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space, 𝐴∈𝐹^𝐼(𝑊), for all 𝑥∈𝑈; one defines 𝑅𝐼𝐹(𝐴)(𝑥)=_𝑦∈𝑊𝑦=∼𝐼𝑅𝑥,𝑦∨𝐴_𝑦∈𝑊1−𝐼𝑅𝑥,𝑦⁺∨𝐴𝑦⁻,_𝑦∈𝑊1−𝐼𝑅𝑥,𝑦⁻∨𝐴𝑦⁺,𝑅𝐼𝐹(𝐴)(𝑥)=_𝑦∈𝑊𝑦=𝐼𝑅𝑥,𝑦∧𝐴_𝑦∈𝑊𝐼𝑅𝑥,𝑦⁻∧𝐴𝑦⁻,_𝑦∈𝑊𝐼𝑅𝑥,𝑦⁺∧𝐴𝑦⁺. (10)
The pair (𝑅𝐼𝐹(𝐴),𝑅𝐼𝐹(𝐴)) is called the generalized interval-valued fuzzy rough set of 𝐴 with respect to the approximation space (𝑈,𝑊,𝐼𝑅). The operators 𝑅𝐼𝐹 and 𝑅𝐼𝐹 are called the generalized interval-valued fuzzy rough lower and upper approximation operators, respectively.

Definition 20. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space, for all 𝛼,𝛽∈[𝐼], 𝐴∈𝐹^𝐼(𝑊); one defines 𝑅𝐼𝐹^′_𝛼𝐴_𝛽=𝑥∈𝑈∶𝐼𝑅_𝛼(𝑥)⊆𝐴_𝛽,𝑅𝐼𝐹^′_𝛼𝐴_𝛽=𝑥∈𝑈∶𝐼𝑅_𝛼(𝑥)∩𝐴_𝛽≠∅. (11)
𝑅𝐼𝐹^′_𝛼(𝐴_𝛽) and 𝑅𝐼𝐹^′_𝛼(𝐴_𝛽) are called the 𝛼,𝛽 lower and upper approximations of 𝐴 with respect to (𝑈,𝑊,𝐼𝑅), respectively.

Definition 21. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space, 𝐴∈𝐹^𝐼(𝑊); one defines 𝑅𝐼𝐹^′(𝐴)=_{[𝐼]𝛼∈}𝛼𝑅𝐼𝐹^′_1−𝛼𝐴_𝛼,𝑅𝐼𝐹^′(𝐴)=_{[𝐼]𝛼∈}𝛼𝑅𝐼𝐹^′_𝛼𝐴_𝛼. (12)
The pair (𝑅𝐼𝐹^′,𝑅𝐼𝐹^′) is called the generalized interval-valued fuzzy rough set of 𝐴 with respect to the approximation space (𝑈,𝑊,𝐼𝑅). The operators 𝑅𝐼𝐹^′ and 𝑅𝐼𝐹^′ are called the generalized interval-valued fuzzy rough lower and upper approximation operators.

Remark 22. The approximation operators introduced in Definition 20 extend the generalized Dubois fuzzy rough approximation operators from numeric value to intervals. The approximation operators defined in Definition 21 provide the same type of generalization. The approximation operators defined in Definition 21 show the inherent relationship between Pawlak’s rough set and interval-valued fuzzy rough sets.

Lemma 23. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space, 𝐴∈𝐹^𝐼(𝑊); then for all 𝛼∈[𝐼],𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)⊆(𝑅𝐼𝐹(𝐴))_𝛼.

Proof. We observe that, for all 𝑥∈𝑈, if 𝑥∈𝑅𝐼𝐹^′_𝛼(𝐴_𝛼), then 𝐼𝑅_𝛼(𝑥)∩𝐴_𝛼≠∅. This means that there exist 𝑦∈𝑊, 𝐼𝑅(𝑥,𝑦)≥𝛼, and 𝐴(𝑦)≥𝛼.
By the interval-valued operations of Definition 3, we have ∨_𝑦∈𝑊(𝐼𝑅(𝑥,𝑦)∧𝐴(𝑦))≥𝛼, so 𝑥∈(𝑅𝐼𝐹(𝐴))_𝛼.
Therefore, 𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)⊆(𝑅𝐼𝐹(𝐴))_𝛼.

Now, we prove that the reverse of Lemma 23 does not hold. Based on the interval-valued operations, which are defined in Definition 3, there exists 𝑦∈𝑊,𝐼𝑅(𝑥,𝑦)∧𝐴(𝑦)≥𝛼; that is, there exists 𝑦∈𝑊, so that 𝐼𝑅(𝑥,𝑦)≥𝛼 and 𝐴(𝑦)≥𝛼 cannot be deduced by ∨_𝑦∈𝑊(𝐼𝑅(𝑥,𝑦)∧𝐴(𝑦))≥𝛼.

Next, we give an example illustrating that the relationship (𝑅𝐼𝐹(𝐴))_𝛼⊆𝑅𝐼𝐹^′_𝛼(𝐴_𝛼) does not hold.

Example 24. Suppose that (𝑈,𝑊,𝐼𝑅) is a generalized interval-valued fuzzy approximation space, 𝑈=𝑥₁,𝑥₂,𝑊=𝑦₁,𝑦₂,𝑦₃[],𝑅𝑥,𝛼=0.2,0.6₁=[0.3,0.5]𝑦₁+[0.1,0.7]𝑦₂+[0.1,0.4]𝑦₃,[]𝐴=0.2,0.5𝑦₁+[0.1,0.9]𝑦₂+[0.2,0.3]𝑦₃, (13)since𝑅𝐼𝐹(𝐴)𝑥₁=_𝑦∈𝑊𝑅𝑥₁𝑦=[]≥[],𝑦∧𝐴0.2,0.70.2,0.6,𝑥₁∈𝑅𝐼𝐹(𝐴)_𝛼. (14)
On the other hand, since 𝐼𝑅_𝛼(𝑥₁)=𝐴_𝛼=∅, we have 𝐼𝑅_𝛼(𝑥₁)∩𝐴_𝛼=∅ and we get 𝑥₁∉𝑅𝐼𝐹^′_𝛼(𝐴_𝛼). This shows that 𝑥₁∈(𝑅𝐼𝐹(𝐴))_𝛼, but 𝑥₁∉𝑅𝐼𝐹^′_𝛼(𝐴_𝛼). So (𝑅𝐼𝐹(𝐴))⊆_𝛼^/𝑅𝐼𝐹^′_𝛼(𝐴_𝛼).

Lemma 25. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space, 𝐴∈𝐹^𝐼(𝑊); then, for all 𝛼∈[𝐼], 𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼)⊆(𝑅𝐼𝐹(𝐴))_𝛼.

Proof. Note that, for all 𝑥∈𝑈, if 𝑥∈𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼) then 𝐼𝑅_1−𝛼(𝑥)⊆𝐴_𝛼. This means that, for all 𝑦∈𝑊, if 𝐼𝑅(𝑥,𝑦)>1−𝛼 then 𝐴(𝑦)≥𝛼; that is, for all 𝑦∈𝑊, 𝐼𝑅(𝑥,𝑦)≤1−𝛼 or 𝐴(𝑦)≥𝛼. By the interval-valued operations as in Definition 3, we have ∧_𝑦∈𝑊(∼𝐼𝑅(𝑥,𝑦)∨𝐴(𝑦))≥𝛼; that is, 𝑅𝐼𝐹(𝐴)(𝑥)≥𝛼, so 𝑥∈(𝑅𝐼𝐹(𝐴))_𝛼.

Now we prove that the reverse of Lemma 25 is not true. Here we use similar reasoning as already used in Lemma 23. ∼𝐼𝑅(𝑥,𝑦)∨𝐴(𝑦)≥𝛼 means that (1−𝐼𝑅(𝑥,𝑦))⁻∨𝐴(𝑦)⁻≥𝛼⁻ and (1−𝐼𝑅(𝑥,𝑦))⁺∨𝐴(𝑦)⁺≥𝛼⁺, which cannot deduce that 1−𝐼𝑅(𝑥,𝑦)≥𝛼 or 𝐴(𝑦)≥𝛼 in Definition 3. Thus for all 𝑦∈𝑊, ∼𝐼𝑅(𝑥,𝑦)∨𝐴(𝑦)≥𝛼, we cannot deduce that 𝐼𝑅(𝑥,𝑦)≤1−𝛼 or 𝐴(𝑦)≥𝛼, and note that, for all 𝑦∈𝑊, ∼𝐼𝑅(𝑥,𝑦)∨𝐴(𝑦)≥𝛼 if and only if ∧_𝑦∈𝑊(∼𝐼𝑅(𝑥,𝑦)∨𝐴(𝑦))≥𝛼. Therefore for all 𝑦∈𝑊, 𝐼𝑅(𝑥,𝑦)≤1−𝛼 or 𝐴(𝑦)≥𝛼 cannot hold.

Next, we show that (𝑅𝐼𝐹(𝐴))_𝛼⊆𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼) does not hold.

Example 26. Suppose that (𝑈,𝑊,𝐼𝑅) is a generalized interval-valued fuzzy approximation space, 𝑈=𝑥₁,𝑥₂,𝑊=𝑦₁,𝑦₂,𝑦₃[],𝑥,𝛼=0.3,0.6𝐼𝑅₁=[0.3,0.8]𝑦₁+[0.4,0.9]𝑦₂+[0.5,0.6]𝑦₃,[]𝐴=0.4,0.5𝑦₁+[0.3,0.4]𝑦₂+[0.4,0.9]𝑦₃, (15) since 𝑅𝐼𝐹(𝐴)𝑥₁=_𝑦∈𝑊∼𝐼𝑅𝑥₁𝑦=[]≥[],𝑦∨𝐴0.3,0.60.3,0.6,𝑥₁∈𝑅𝐼𝐹(𝐴)_𝛼. (16)

On the other hand, since 𝐼𝑅_1−𝛼(𝑥₁)=𝑦₂,𝐴_𝛼=𝑦₃, we have 𝐼𝑅_1−𝛼(𝑥₁)⊆^/𝐴_𝛼; hence 𝑥₁∉𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼). This shows that 𝑥₁∈(𝑅𝐼𝐹(𝐴))_𝛼, but 𝑥₁∉𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼). Therefore (𝑅𝐼𝐹(𝐴))⊆_𝛼^/𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼).

Theorem 27. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space, 𝐴∈𝐹^𝐼(𝑊); then 𝑅𝐼𝐹(𝐴)=𝑅𝐼𝐹^′(𝐴).

Proof. According to Theorem 7, we have 𝑅𝐼𝐹(𝐴)=∪_{𝛼∈[𝐼]}𝛼(𝑅𝐼𝐹(𝐴))_𝛼, and from Lemma 23, we see that 𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)⊆(𝑅𝐼𝐹(A))_𝛼, for all 𝛼∈[𝐼].
Therefore, 𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹(𝐴).
Next we prove that 𝑅𝐼𝐹(𝐴)⊆𝑅𝐼𝐹^′(𝐴).
In fact, for all 𝑥∈𝑈,𝑦∈𝑊, there exists 𝛼=𝐼𝑅(𝑥,𝑦)∧𝐴(𝑦)∈[𝐼], such that 𝑦∈𝐼𝑅_𝛼(𝑥)∩𝐴_𝛼. We observe that 𝑦∈𝐼𝑅_𝛼(𝑥)∩𝐴_𝛼 means that 𝐼𝑅_𝛼(𝑥)∩𝐴_𝛼≠∅, which can deduce that 𝑥∈𝑅𝐼𝐹^′_𝛼(𝐴_𝛼); that is, 𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)(𝑥)=1; hence 𝛼∧𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)(𝑥)=𝛼=𝐼𝑅(𝑥,𝑦)∧𝐴(𝑦).
So, for arbitrary value of 𝑦, 𝑅𝐼𝐹^′(𝐴)(𝑥)=∨_{𝛼∈[𝐼]}(𝛼∧𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)(𝑥))≥∨_𝑦∈𝑊(𝐼𝑅(𝑥,𝑦)∧𝐴(𝑦))=𝑅𝐼𝐹(𝐴)(𝑥), which yields 𝑅𝐼𝐹(𝐴)⊆𝑅𝐼𝐹^′(𝐴).
Therefore, 𝑅𝐼𝐹(𝐴)=𝑅𝐼𝐹^′(𝐴).

Theorem 28. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space, 𝐴∈𝐹^𝐼(𝑊); then 𝑅𝐼𝐹(𝐴)=𝑅𝐼𝐹^′(𝐴).

Proof. In view of Theorem 7, we have 𝑅𝐼𝐹(𝐴)=∪_{𝛼∈[𝐼]}𝛼(𝑅𝐼𝐹(𝐴))_𝛼, and from Lemma 25, 𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼)⊆(𝑅𝐼𝐹(𝐴))_𝛼, for all 𝛼∈[𝐼].
Then 𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹(𝐴).
Now we prove that 𝑅𝐼𝐹(𝐴)⊆𝑅𝐼𝐹^′(𝐴). For all 𝑥∈𝑈, suppose that 𝛼₁=0,_𝑦∈𝑊∼𝐼𝑅𝑥,𝑦∨𝐴𝑦⁺,𝛼₂=_𝑦∈𝑊∼𝐼𝑅𝑥,𝑦∨𝐴𝑦⁻,_𝑦∈𝑊∼𝐼𝑅𝑥,𝑦∨𝐴𝑦⁻. (17)
(1) We verify that 𝑥∈𝑅𝐼𝐹^′_1−𝛼1(𝐴_𝛼1). Let 𝛼₁=0,_𝑦∈𝑊1−𝐼𝑅𝑥,𝑦⁻∨𝐴𝑦⁺,1−𝛼₁=_𝑦∈𝑊𝐼𝑅𝑥,𝑦⁻∧1−𝐴𝑦⁺,1. (18)
Note that, for all 𝑦₀∈𝑊, 𝑦₀∈𝐼𝑅_1−𝛼1(𝑥), such that 𝐼𝑅(𝑥,𝑦₀)>1−𝛼₁, and from (1−𝛼₁)⁺=1, we have 𝐼𝑅(𝑥,𝑦₀)⁻>∨_𝑦∈𝑊(𝐼𝑅(𝑥,𝑦)⁻∧(1−𝐴(𝑦)⁺)).
Further from 𝑦₀∈𝑊, we have 𝐼𝑅(𝑥,𝑦₀)⁻>𝐼𝑅(𝑥,𝑦₀)⁻∧(1−𝐴(𝑦₀)⁺).
Therefore we obtain that 𝐼𝑅(𝑥,𝑦₀)⁻>1−𝐴(𝑦₀)⁺and1−𝐴(𝑦₀)⁺=𝐼𝑅(𝑥,𝑦₀)⁻∧(1−𝐴(𝑦₀)⁺).
Because 𝐼𝑅(𝑥,𝑦₀)⁻∧(1−𝐴(𝑦₀)⁺)≤∨_𝑦∈𝑊(𝐼𝑅(𝑥,𝑦)⁻∧(1−𝐴(𝑦)⁺)), we have 1−𝐴(𝑦₀)⁺≤∨_𝑦∈𝑊(𝐼𝑅(𝑥,𝑦)⁻∧(1−𝐴(𝑦)⁺)), such that 𝐴(𝑦₀)⁺≥∧_𝑦∈𝑊((1−𝐼𝑅(𝑥,𝑦)⁻)∨𝐴(𝑦)⁺), and from 𝛼⁻₁=0, we get 𝐴(𝑦₀)≥𝛼₁; that is, 𝑦₀∈𝐴_𝛼1.
So, for arbitrary value of 𝑦₀, 𝐼𝑅_1−𝛼1(𝑥)⊆𝐴_𝛼1; that is, 𝑥∈𝑅𝐼𝐹^′_1−𝛼1(𝐴_𝛼1).
(2) Similar to the proof shown in (1), we have 𝑥∈𝑅𝐼𝐹^′_1−𝛼2(𝐴_𝛼2). Note that 𝑅𝐼𝐹^′(𝐴𝑥)=)(_{[𝐼]𝛼∈}𝛼∧𝑅𝐼𝐹^′_1−𝛼𝐴_𝛼(𝑥≥𝛼₁)∧𝑅𝐼𝐹^′_1−𝛼1𝐴_𝛼1(𝑥)∨𝛼₂∧𝑅𝐼𝐹^′_1−𝛼2𝐴_𝛼2(𝑥)=𝛼₁∨𝛼₂=_𝑦∈𝑊∼𝐼𝑅𝑥,𝑦∨𝐴𝑦⁻,_𝑦∈𝑊∼𝐼𝑅𝑥,𝑦∨𝐴𝑦⁺=𝑅𝐼𝐹(𝐴)(𝑥). (19)
For any 𝑥, 𝑅𝐼𝐹(𝐴)⊆𝑅𝐼𝐹^′(𝐴).
Therefore, 𝑅𝐼𝐹(𝐴)=𝑅𝐼𝐹^′(𝐴).

Remark 29. According to Theorems 27 and 28, 𝑅𝐼𝐹^′ and 𝑅𝐼𝐹^′ satisfy the property of duality.

Theorem 30. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space, 𝐴∈𝐹^𝐼(𝑊); then ∼𝑅𝐼𝐹(𝐴)=𝑅𝐼𝐹(∼𝐴),∼𝑅𝐼𝐹(𝐴)=𝑅𝐼𝐹(∼𝐴).

Proof. We observe that, for all 𝑥∈𝑈, (𝑥)=𝑅𝐼𝐹∼𝐴)(_𝑦∈𝑊𝐼𝑅𝑥,𝑦∧(∼𝐴)𝑦⁻,_𝑦∈𝑊𝐼𝑅𝑥,𝑦∧(∼𝐴)𝑦⁺=_𝑦∈𝑊𝐼𝑅𝑥,𝑦⁻∧1−𝐴𝑦⁺,_𝑦∈𝑊𝐼𝑅𝑥,𝑦⁺∧1−𝐴𝑦⁻=1−_𝑦∈𝑊1−𝐼𝑅𝑥,𝑦⁺∨𝐴𝑦⁻,_𝑦∈𝑊1−𝐼𝑅𝑥,𝑦⁻∨𝐴𝑦⁺=1−_𝑦∈𝑊1−𝐼𝑅𝑥,𝑦⁻∨𝐴𝑦⁻,_𝑦∈𝑊1−𝐼𝑅𝑥,𝑦⁺∨𝐴𝑦⁺=1−_𝑦∈𝑊1−𝐼𝑅𝑥,𝑦∨A𝑦⁻,_𝑦∈𝑊1−𝐼𝑅𝑥,𝑦∨𝐴𝑦⁺=∼𝑅𝐼𝐹(𝐴)(𝑥). (20)
Hence, ∼𝑅𝐼𝐹(𝐴)=𝑅𝐼𝐹(∼𝐴).
Similarly, ∼𝑅𝐼𝐹(𝐴)=𝑅𝐼𝐹(∼𝐴).

5. Properties of the Approximation Operators

Theorem 31. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space; then the lower approximation operator 𝑅𝐼𝐹^′ and the upper approximation operator 𝑅𝐼𝐹^′ satisfy the following properties.
For all 𝐴,𝐵∈𝐹^𝐼(𝑊),𝑎∈[𝐼],(1)𝑅𝐼𝐹^′(𝐴∪𝑎)=𝑅𝐼𝐹^′(𝐴)∪𝑎,  𝑅𝐼𝐹^′(𝐴∩𝑎)=𝑅𝐼𝐹^′(𝐴)∩𝑎; (2)𝑅𝐼𝐹^′(𝐴∪𝐵)=𝑅𝐼𝐹^′(𝐴)∪𝑅𝐼𝐹^′(𝐵),   𝑅𝐼𝐹^′(𝐴∩𝐵)=𝑅𝐼𝐹^′(𝐴)∩𝑅𝐼𝐹^′(𝐵);(3)𝑖𝑓𝐴⊆𝐵𝑡ℎ𝑒𝑛𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝐵),  𝑎𝑛𝑑𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝐵);(4)𝑅𝐼𝐹^′(𝐴∪𝐵)⊇𝑅𝐼𝐹^′(𝐴)∪𝑅𝐼𝐹^′(𝐵), 𝑅𝐼𝐹^′(𝐴∩𝐵)⊆𝑅𝐼𝐹^′(𝐴)∩𝑅𝐼𝐹^′(𝐵).
Here 𝑎 is a constant interval-valued fuzzy set; that is, 𝑎(𝑥)=𝑎, for all 𝑥∈𝑈 and 𝑥∈𝑊.

Proof. (1) We prove that 𝑅𝐼𝐹^′(𝐴∪𝑎)=𝑅𝐼𝐹^′(𝐴)∪𝑎.
For all 𝑥∈𝑈, let 𝐷₁=𝛼∈[𝐼]∶∀𝑦∈𝐼𝑅_1−𝛼(𝑥)𝑦,𝐷,𝐴≥𝛼or𝑎≥𝛼₂=𝛼∈[𝐼]∶∀𝑦∈𝐼𝑅_1−𝛼(𝑥),𝐴𝑦∨𝑎≥𝛼, (21) where ∀𝑥 is “for all 𝑥” and ∃𝑥 is “there exists 𝑥,” which are the same as follows.
Obviously, 𝐷₁⊆𝐷₂,∨_𝛼∈𝐷1𝛼≤∨_𝛼∈𝐷2𝛼. Set 𝐷₃=𝐷₂−𝐷₁, for all 𝛽∈𝐷₃; two cases appear: 𝛽⁺≤𝑎⁺,𝛽⁻≤_{𝑦∈𝑅1−𝛼(𝑥)}𝐴𝑦⁻ (22) or 𝛽⁻≤𝑎⁻,𝛽⁺≤_{𝑦∈𝑅1−𝛼(𝑥)}𝐴𝑦⁺. (23)
For the first case, suppose 𝑏=[∧_{𝑦∈𝑅1−𝛼(𝑥)}𝐴(𝑦)⁻,∧_{𝑦∈𝑅1−𝛼(𝑥)}𝐴(𝑦)⁺], because 𝑎,𝑏∈𝐷₁; we have 𝛽≤𝑎∨𝑏≤∨_𝛼∈𝐷1𝛼. The proof for the second case is similar.
For arbitrary 𝛽, it is easy to see that ∨_𝛼∈𝐷3𝛼≤∨_𝛼∈𝐷1𝛼, so ∨_𝛼∈𝐷2𝛼=∨_𝛼∈𝐷1𝛼: 𝑅𝐼𝐹^′𝐴∪𝑎(𝑥)=_{[𝐼]𝛼∈}𝛼∧𝑅𝐼𝐹^′_1−𝛼𝐴∪𝑎_𝛼(𝑥)=∨𝛼∈[𝐼]∶𝐼𝑅_1−𝛼(𝑥)⊆𝐴∪𝑎_𝛼=∨𝛼∈[𝐼]∶∀𝑦∈𝐼𝑅_1−𝛼(𝑥)𝑦[𝐼],𝐴∨𝑎≥𝛼=∨𝛼∈∶∀𝑦∈𝐼𝑅_1−𝛼(𝑥)𝑦∨[𝐼],𝐴≥𝛼or𝑎≥𝛼=𝑎∨𝛼∈∶∀𝑦∈𝐼𝑅_1−𝛼(𝑥),𝐴𝑦≥𝛼=𝑎∨𝑅𝐼𝐹^′(𝐴𝑥)=𝑎)(∪𝑅𝐼𝐹^′(𝐴)(𝑥). (24)
Hence, 𝑅𝐼𝐹^′(𝐴∪𝑎)=𝑅𝐼𝐹^′(𝐴)∪𝑎.
Similarly, 𝑅𝐼𝐹^′(𝐴∩𝑎)=𝑅𝐼𝐹^′(𝐴)∩𝑎.
(2) We verify 𝑅𝐼𝐹^′(𝐴∪𝐵)=𝑅𝐼𝐹^′(𝐴)∪𝑅𝐼𝐹^′(𝐵).
For all 𝑥∈𝑈, let 𝐷₄=𝛼∈[𝐼]∶∃𝑦∈𝐼𝑅_𝛼(𝑥)𝑦𝑦;𝐷,𝐴≥𝛼or𝐵≥𝛼₅=𝛼∈[𝐼]∶∃𝑦∈𝐼𝑅_𝛼(𝑥),𝐴𝑦∨𝐵𝑦≥𝛼, (25)
∨_𝛼∈𝐷5𝛼=∨_𝛼∈𝐷4𝛼 holds by using similar arguments as in (1).
We observe that 𝑅𝐼𝐹^′(𝑥)=𝐴∪𝐵)(_{𝛼∈[𝐼]}𝛼∧𝑅𝐼𝐹^′_𝛼(𝐴∪𝐵)_𝛼(𝑥)=∨𝛼∈[𝐼]∶𝐼𝑅_𝛼(𝑥)∩(𝐴∪𝐵)_𝛼[𝐼]≠∅=∨𝛼∈∶∃𝑦∈𝐼𝑅_𝛼(𝑥),()𝑦[𝐼]𝐴∪𝐵≥𝛼=∨𝛼∈∶∃𝑦∈𝐼𝑅_𝛼(𝑥)𝑦𝑦[𝐼],𝐴∨𝐵≥𝛼=∨𝛼∈∶∃𝑦∈𝐼𝑅_𝛼(𝑥)𝑦𝑦=∨[𝐼],𝐴≥𝛼or𝐵≥𝛼𝛼∈∶∃𝑦∈𝐼𝑅_𝛼(𝑥)𝑦∨∨[𝐼],𝐴≥𝛼𝛼∈∶∃𝑦∈𝐼𝑅_𝛼(𝑥)𝑦=,𝐵≥𝛼_{𝛼∈[𝐼]}𝛼∧𝑅𝐼𝐹^′_𝛼(𝐴)_𝛼(𝑥)∨_{𝛼∈[𝐼]}𝛼∧𝑅𝐼𝐹^′_𝛼(𝐵)_𝛼(𝑥)=𝑅𝐼𝐹^′(𝐴)(𝑥)∨𝑅𝐼𝐹^′(𝐵𝑥)=)(𝑅𝐼𝐹^′(𝐴)∪𝑅𝐼𝐹^′(𝐵)(𝑥), (26) which yields that 𝑅𝐼𝐹^′(𝐴∪𝐵)=𝑅𝐼𝐹^′(𝐴)∪𝑅𝐼𝐹^′(𝐵).
Similarly, 𝑅𝐼𝐹^′(𝐴∩𝐵)=𝑅𝐼𝐹^′(𝐴)∩𝑅𝐼𝐹^′(𝐵).
(3) We prove that if 𝐴⊆𝐵 then 𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝐵). If 𝐴⊆𝐵, then 𝐴_𝛼⊆𝐵_𝛼. According to Definition 20 and Theorem  3.2(4) of [18], we have 𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼)⊆𝑅𝐼𝐹^′_1−𝛼(𝐵_𝛼), so ∪_{𝛼∈[𝐼]}𝛼𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼)⊆∪_{𝛼∈[𝐼]}𝛼𝑅𝐼𝐹^′_1−𝛼(𝐵_𝛼); that is, 𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝐵).
Similarly, if 𝐴⊆𝐵, then 𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝐵).
(4) From (3), one immediately obtains (4).

Remark 32. From Theorem 31 (1), one can see that 𝑅𝐼𝐹^′(𝑊)=𝑈,𝑅𝐼𝐹^′(∅)=∅.

Theorem 33. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: (1)𝐼𝑅 is serial;(2)𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝐴), for all 𝐴⊆𝐹^𝐼(𝑊);(3)𝑅𝐼𝐹^′(𝑊)=𝑈;(4)𝑅𝐼𝐹^′(∅)=∅.

Theorem 34. Let (𝑈,𝐼𝑅) be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: (1)𝐼𝑅 is reflexive;(2)𝑅𝐼𝐹^′(𝐴)⊆𝐴, for all 𝐴⊆𝐹^𝐼(𝑊);(3)𝐴⊆𝑅𝐼𝐹^′(𝐴), for all 𝐴⊆𝐹^𝐼(𝑊).

Lemma 35. Let (𝑈,𝑊,𝐼𝑅) be a generalized interval-valued fuzzy approximation space; then the following properties hold: (1)𝑅𝐼𝐹^′(1_{𝑦})(𝑥)=𝐼𝑅(𝑥,𝑦), (𝑥,𝑦)∈𝑈×𝑊;(2)𝑅𝐼𝐹^′(1_{𝑊⧵{𝑦}})(𝑥)=1−𝐼𝑅(𝑥,𝑦),(𝑥,𝑦)∈𝑈×𝑊.
Here 1_𝐴 is an interval-valued fuzzy set which gets interval value [1,1] in the set 𝐴 and interval value [0,0] in the set ∼𝐴, respectively.

Remark 36. The proofs of Theorems 33 and 34 as well as Lemma 35 are similar to Theorems 3.8, 3.9, and  3.7 in [16], respectively; it suffices to change point values to interval values in the proof.

Lemma 37. Let (𝑈,𝐼𝑅) be a generalized interval-valued fuzzy approximation space and 𝐴 is an interval-valued fuzzy set on 𝑈; then ∀𝛼∈[𝐼], 𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)⊆(𝑅𝐼𝐹^′(𝐴))_𝛼, and 𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼)⊆(𝑅𝐼𝐹^′(𝐴))_𝛼.

Proof. Clearly, 𝑅𝐼𝐹^′_𝛼𝐴_𝛼=𝛼𝑅𝐼𝐹^′_𝛼𝐴_𝛼𝛼⊆𝛼𝑅𝐼𝐹^′_𝛼𝐴_𝛼∪_{[𝐼]𝛽∈−𝛼}𝛽𝑅𝐼𝐹^′_𝛽𝐴_𝛽𝛼=_{[𝐼]𝛽∈}𝛽𝑅𝐼𝐹^′_𝛽𝐴_𝛽𝛼=𝑅𝐼𝐹^′(𝐴)_𝛼. (27)
Hence, 𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)⊆(𝑅𝐼𝐹^′(𝐴))_𝛼.
Similarly, 𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼)⊆(𝑅𝐼𝐹^′(𝐴))_𝛼.

Remark 38. 𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)=(𝑅𝐼𝐹^′(𝐴))_𝛼 and 𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼)=(𝑅𝐼𝐹^′(𝐴))_𝛼 hold for the fuzzy rough set in Lemma 37, but these are not true for the interval-valued fuzzy rough set. The reason is that the two interval values cannot always be comparable. Next, we give two examples to visualize this effect.

Example 39. Suppose that (𝑈,𝐼𝑅) is a generalized interval-valued fuzzy approximation space, where 𝑈={𝑥₁,𝑥₂}, 𝐼𝑅(𝑥₁)=[0.5,0.5]/𝑥₁+[0.3,0.7]/𝑥₂,𝐴=[0.9,1]/𝑥₁+[0.7,0.9]/𝑥₂,𝛼=[0.4,0.6], 𝛽₁=[0.5,0.5], and 𝛽₂=[0.3,0.7]. Since 𝐼𝑅_𝛼(𝑥₁)=∅,𝐴_𝛼={𝑥₁,𝑥₂}, we have 𝐼𝑅_𝛼(𝑥₁)∩𝐴_𝛼=∅. Hence 𝑥₁∉𝑅𝐼𝐹^′_𝛼(𝐴_𝛼).
On the other hand, since 𝐼𝑅_𝛽1(𝑥₁)={𝑥₁},𝐴_𝛽1={𝑥₁,𝑥₂},𝐼𝑅_𝛽2(𝑥₁)={𝑥₂},𝐴_𝛽2={𝑥₁,𝑥₂}, we see that 𝑥₁∈𝑅𝐼𝐹^′_𝛽1(𝐴_𝛽1) and 𝑥₁∈𝑅𝐼𝐹^′_𝛽2(𝐴_𝛽2)get 𝛽₁∧𝑅𝐼𝐹^′_𝛽1(𝐴_𝛽1)(𝑥₁)=𝛽₁ and 𝛽₂∧𝑅𝐼𝐹^′_𝛽2(𝐴_𝛽2)(𝑥₁)=𝛽₂.
Note that ∨_{𝛽∈[𝐼]}(𝛽∧𝑅𝐼𝐹^′_𝛽(𝐴_𝛽)(𝑥₁))≥𝛽₁∨𝛽₂>𝛼; then 𝑅𝐼𝐹^′(𝐴)(𝑥₁)>𝛼. Hence, 𝑥₁∈(𝑅𝐼𝐹^′(𝐴))_𝛼.

Example 40. Suppose that (𝑈,𝐼𝑅) is a generalized interval-valued fuzzy approximation space, where 𝑈={𝑥₁,𝑥₂,𝑥₃}, 𝐼𝑅(𝑥₁)=[0.3,0.7]/𝑥₁+[0.1,0.8]/𝑥₂+[0.5,0.5]/𝑥₃,𝐴=[0.5,0.7]/𝑥₁+[0.6,1]/𝑥₂+[0.7,1]/𝑥₃,𝛼=[0.4,0.8],𝛽₁=[0.2,1],𝛽₂=[0.5,0.6].
Since 𝐼𝑅_1−𝛼(𝑥₁)={𝑥₁},𝐴_𝛼={𝑥₂,𝑥₃}, we have 𝐼𝑅_1−𝛼(𝑥₁)⊆^/𝐴_𝛼; hence, 𝑥₁∉𝑅𝐼𝐹^′_1−𝛼(𝐴_𝛼).
On the other hand, since 𝐼𝑅_1−𝛽1(𝑥₁)={𝑥₂},𝐴_𝛽1={𝑥₂,𝑥₃},𝐼𝑅_1−𝛽2(𝑥₁)={𝑥₃},𝐴_𝛽2={𝑥₂,𝑥₃}, we conclude that 𝑥₁∈𝑅𝐼𝐹^′_1−𝛽1(𝐴_𝛽1)and𝑥₁∈𝑅𝐼𝐹^′_1−𝛽2(𝐴_𝛽2) get 𝛽₁∧𝑅𝐼𝐹^′_1−𝛽1(𝐴_𝛽1)(𝑥₁)=𝛽₁ and 𝛽₂∧𝑅𝐼𝐹^′_1−𝛽2(𝐴_𝛽2)(𝑥₁)=𝛽₂.
Note that ∨_{𝛽∈[𝐼]}(𝛽∧𝑅𝐼𝐹^′_1−𝛽(𝐴_𝛽)(𝑥₁))≥𝛽₁∨𝛽₂>𝛼; then 𝑅𝐼𝐹^′(𝐴)(𝑥₁)>𝛼.
Therefore, 𝑥₁∈(𝑅𝐼𝐹^′(𝐴))_𝛼.

Theorem 41. Let (𝑈,𝐼𝑅) be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: (1)𝐼𝑅 is transitive;(2)𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)), for all 𝐴∈𝐹^𝐼(𝑈);(3)𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)), for all 𝐴∈𝐹^𝐼(𝑈).

Proof. (1)⇒(2) For all 𝐴∈𝐹^𝐼(𝑈), from Definition 20, Lemma 37, and Theorem 3.6 of [18], we have 𝑅𝐼𝐹^′𝑅𝐼𝐹^′(𝐴)=_{[𝐼]𝛼∈}𝛼𝑅𝐼𝐹^′_1−𝛼𝑅𝐼𝐹^′(𝐴)_𝛼⊇_{[𝐼]𝛼∈}𝛼𝑅𝐼𝐹^′_1−𝛼𝑅𝐼𝐹^′_1−𝛼𝐴_𝛼⊇_{[𝐼]𝛼∈}𝛼𝑅𝐼𝐹^′_1−𝛼𝐴_𝛼=𝑅𝐼𝐹^′(𝐴). (28)
Hence 𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)).
(3)⇒(1) For all 𝑥,𝑦,𝑧∈𝑈, let 𝐷₆=𝛼∈[𝐼]∶∃𝑢∈𝑈,𝐼𝑅(𝑥,𝑢)≥𝛼,𝑅𝐼𝐹^′1_{𝑧}𝑢)(),𝐷(=𝐼𝑅𝑢,𝑧≥𝛼₇𝑦=𝐼𝑅𝑥,𝑦∧𝐼𝑅𝑦,𝑧,∀𝑦∈𝑈. (29)
For all 𝑦∈𝑈, suppose that 𝛼=𝐷₇(𝑦)=𝐼𝑅(𝑥,𝑦)∧𝐼𝑅(𝑦,𝑧); then 𝐼𝑅(𝑥,𝑦)≥𝛼,𝐼𝑅(𝑦,𝑧)≥𝛼; hence 𝛼∈𝐷₆ and by the arbitrary 𝑦, ∨_𝑦⊆𝑈𝐷₇(𝑦)≤∨_𝛼∈𝐷6𝛼.
For all 𝛼∈𝐷₆, there exists 𝑦∈𝑈, such that 𝛼≤𝐼𝑅(𝑥,𝑦)∧𝐼𝑅(𝑦,𝑧). For arbitrary 𝛼, ∨_𝑦⊆𝑈𝐷₇(𝑦)≥∨_𝛼∈𝐷6𝛼, so ∨_𝑦⊆𝑈𝐷₇(𝑦)=∨_𝛼∈𝐷6𝛼. We observe that 𝑅𝐼𝐹^′𝑅𝐼𝐹^′1_{𝑧}(𝑥)=∨𝛼∈[𝐼]∶𝑥∈𝑅𝐼𝐹^′_𝛼𝑅𝐼𝐹^′1_{𝑧}𝛼=∨𝛼∈[𝐼]∶𝐼𝑅_𝛼(𝑥)∩𝑅𝐼𝐹^′1_{𝑧}𝛼[𝐼]()≠∅=∨𝛼∈∶∃𝑢∈𝑈,𝐼𝑅𝑥,𝑢≥𝛼,𝑅𝐼𝐹^′1_{𝑧}(𝑢)()==𝐼𝑅𝑢,𝑧≥𝛼_𝑦∈𝑈𝐼𝑅𝑥,𝑦∧𝐼𝑅𝑦,𝑧. (30)
Hence, by Lemma 35 (1), we have 𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(1_{𝑧}))(𝑥)≤𝑅𝐼𝐹^′(1_{𝑧})(𝑥)=𝐼𝑅(𝑥,𝑧); then 𝐼𝑅(𝑥,𝑧)≥∨_𝑦∈𝑈(𝐼𝑅(𝑥,𝑦)∧𝐼𝑅(𝑦,𝑧)). Therefore, 𝐼𝑅 is transitive.
(2)⇔(3) This conclusion follows immediately from the duality.

Remark 42. In [18], if 𝐼𝑅 is symmetric, then the approximation operators satisfy 𝐴⊆𝑅𝑅(𝐴) and 𝑅𝑅(𝐴)⊆𝐴 for all 𝐴⊆𝑈; if 𝐼𝑅 is Euclidean, then the approximation operators satisfy 𝑅(𝐴)⊆𝑅𝑅(𝐴) and 𝑅𝑅(𝐴)⊆𝑅(A) for all 𝐴⊆𝑈. These properties do not hold in the interval-valued fuzzy rough sets. Next, we give a counterexample to show it.

Example 43. Suppose that (𝑈,𝐼𝑅) is a generalized interval-valued fuzzy approximation space, 𝑈={𝑥₁,𝑥₂,𝑥₃}, and [][][],[]𝐼𝑅=0.9,1][0.6,0.6][0.1,0.60.6,0.6][0.8,0.9][0.1,0.90.1,0.6][0.1,0.9][0.1,1𝐴=0.3,0.4𝑥₁+[0.4,0.6]𝑥₂+[0.2,0.8]𝑥₃,𝑅𝐼𝐹^′(𝐴)=[0.4,0.6]𝑥₁+[0.4,0.8]𝑥₂+[0.1,0.8]𝑥₃,𝑅𝐼𝐹^′𝑅𝐼𝐹^′(𝐴=[])0.4,0.6𝑥₁+[0.1,0.6]𝑥₂+[0.1,0.9]𝑥₃. (31)
From Definition 11, 𝐼𝑅 is symmetric and Euclidean, but 𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)) and 𝐴⊆𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)) do not hold. According to the duality, 𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴))⊆𝑅𝐼𝐹^′(𝐴) and 𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴))⊆𝐴 are not true.

Theorem 44. Let (𝑈,𝐼𝑅) be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: (1)𝐼𝑅(𝑥,𝑧)≤∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑧)),𝑥,𝑧∈𝑈;(2)𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴))⊆𝑅𝐼𝐹^′(𝐴),𝐴∈𝐹^𝐼(𝑈);(3)𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)),𝐴∈𝐹^𝐼(𝑈).

Proof. (2)⇒(1) For all 𝑥,𝑧∈𝑈, we have 𝑅𝐼𝐹^′𝑅𝐼𝐹^′1_{𝑈⧵{𝑧}}(𝑥)=∨𝛼∈[𝐼]∶𝑥∈𝑅𝐼𝐹^′_𝛼𝑅𝐼𝐹^′1_{𝑈⧵{𝑧}𝛼}=∨𝛼∈[𝐼]∶𝐼𝑅_𝛼(𝑥)∩𝑅𝐼𝐹^′1_{𝑈⧵{𝑧}𝛼}[𝐼]≠∅=∨𝛼∈∶∃𝑦∈𝑈,𝐼𝑅𝑥,𝑦≥𝛼,𝑅𝐼𝐹^′1_{𝑈⧵{𝑧}}𝑦[𝐼][𝐼]∧=≥𝛼=∨𝛼∈∶∃𝑦∈𝑈,𝐼𝑅𝑥,𝑦≥𝛼,1−𝐼𝑅𝑦,𝑧≥𝛼=∨𝛼∈∶∃𝑦∈𝑈,𝐼𝑅𝑥,𝑦1−𝐼𝑅𝑦,𝑧≥𝛼_𝑦∈𝑈𝐼𝑅𝑥,𝑦∧1−𝐼𝑅𝑦,𝑧. (32)
By Lemma 35 (2), we get 𝑅𝐼𝐹^′(1_{𝑈⧵{𝑧}})(𝑥)=1−𝐼𝑅(𝑥,𝑧).
Hence, ∨_𝑦∈𝑈(𝐼𝑅(𝑥,𝑦)∧(1−𝐼𝑅(𝑦,𝑧)))≤1−𝐼𝑅(𝑥,𝑧).
At the same time, we have 𝐼𝑅(𝑥,𝑧)≤1−_𝑦∈𝑈∧=[]−𝐼𝑅𝑥,𝑦1−𝐼𝑅𝑦,𝑧1,1_𝑦∈𝑈𝐼𝑅𝑥,𝑦⁻∧1−𝐼𝑅𝑦,𝑧⁺,_𝑦∈𝑈𝐼𝑅𝑥,𝑦⁺∧1−𝐼𝑅𝑦,𝑧⁻=1−_𝑦∈𝑈𝐼𝑅𝑥,𝑦⁺∧1−𝐼𝑅𝑦,𝑧⁻,1−_𝑦∈𝑈𝐼𝑅𝑥,𝑦⁻∧1−𝐼𝑅𝑦,𝑧⁺=_𝑦∈𝑈1−𝐼𝑅𝑥,𝑦⁺∨𝐼𝑅𝑦,𝑧⁻,_𝑦∈𝑈1−𝐼𝑅𝑥,𝑦⁻∨𝐼𝑅𝑦,𝑧⁺=_𝑦∈𝑈1−𝐼𝑅𝑥,𝑦∨𝐼𝑅𝑦,𝑧. (33)
Therefore (1) has been proven.
(1)⇒(3) First we prove that 𝑅𝐼𝐹^′_𝛼𝐴_𝛼⊆𝑅𝐼𝐹^′_1−𝛼𝑅𝐼𝐹^′_𝛼𝐴_𝛼. (34)
Note that, for all 𝑥∈𝑈, if 𝑥∈𝑅𝐼𝐹^′_𝛼(𝐴_𝛼), then 𝐼𝑅_𝛼(𝑥)∩𝐴_𝛼≠∅; that is, there exists 𝑧∈𝐼𝑅_𝛼(𝑥)∩𝐴_𝛼. So we have 𝐼𝑅(𝑥,𝑧)≥𝛼 and 𝐴(𝑧)≥𝛼.
Since 𝛼≤𝐼𝑅(𝑥,𝑧)≤_𝑦∈𝑈=1−𝐼𝑅𝑥,𝑦∨𝐼𝑅𝑦,𝑧_𝑦∈𝑈1−𝐼𝑅𝑥,𝑦⁺∨𝐼𝑅𝑦,𝑧⁻,_𝑦∈𝑈1−𝐼𝑅𝑥,𝑦⁻∨𝐼𝑅𝑦,𝑧⁺, (35)
we have _𝑦∈𝑈1−𝐼𝑅𝑥,𝑦⁺∨𝐼𝑅𝑦,𝑧⁻≥𝛼⁻,_𝑦∈𝑈1−𝐼𝑅𝑥,𝑦⁻∨𝐼𝑅𝑦,𝑧⁺≥𝛼⁺. (36)
Hence, for all 𝑦∈𝑈, 𝐼𝑅(𝑥,𝑦)⁺≤1−𝛼⁻ or 𝐼𝑅(𝑦,𝑧)⁻≥𝛼⁻and 𝐼𝑅(𝑥,𝑦)⁻≤1−𝛼⁺ or 𝐼𝑅(𝑦,𝑧)⁺≥𝛼⁺ imply that if 𝐼𝑅(𝑥,𝑦)⁺>1−𝛼⁻, then 𝐼𝑅(𝑦,𝑧)⁻≥𝛼⁻, and if 𝐼𝑅(𝑥,𝑦)⁻>1−𝛼⁺, then 𝐼𝑅(𝑦,𝑧)⁺≥𝛼⁺. It follows that 𝐼𝑅(𝑥,𝑦)⁺>1−𝛼⁻ and 𝐼𝑅(𝑥,𝑦)⁻>1−𝛼⁺ imply that 𝐼𝑅(𝑦,𝑧)⁻≥𝛼⁻ and 𝐼𝑅(𝑦,𝑧)⁺≥𝛼⁺; therefore, 𝐼𝑅(𝑥,𝑦)>1−𝛼 implies that 𝐼𝑅(𝑦,𝑧)≥𝛼, because 𝐼𝑅(𝑥,𝑦)>1−𝛼 and 𝐼𝑅(𝑦,𝑧)≥𝛼 are equivalent to 𝑦∈𝐼𝑅_1−𝛼(𝑥) and 𝑧∈𝐼𝑅_𝛼(𝑦), respectively. If 𝑦∈𝐼𝑅_1−𝛼(𝑥), then 𝑧∈𝐼𝑅_𝛼(𝑦), and since 𝐴(𝑧)≥𝛼, 𝑦∈𝐼𝑅_1−𝛼(𝑥) implies that 𝑧∈𝐼𝑅_𝛼(𝑦)∩𝐴_𝛼; that is, 𝐼𝑅_𝛼(𝑦)∩𝐴_𝛼≠∅. So 𝑦∈𝐼𝑅_1−𝛼(𝑥) implies that 𝑦∈𝑅𝐼𝐹^′_𝛼(𝐴_𝛼).
For arbitrary 𝑦, it follows that 𝐼𝑅_1−𝛼(𝑥)⊆𝑅𝐼𝐹^′_𝛼(𝐴_𝛼). Hence, 𝑥∈𝑅𝐼𝐹^′_1−𝛼(𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)). Then, for arbitrary 𝑥, we obtain 𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)⊆𝑅𝐼𝐹^′_1−𝛼(𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)).
By Lemma 37, it follows that 𝑅𝐼𝐹^′𝑅𝐼𝐹^′(𝐴)=_{[𝐼]𝛼∈}𝛼𝑅𝐼𝐹^′_1−𝛼𝑅𝐼𝐹^′(𝐴)_𝛼⊇_{[𝐼]𝛼∈}𝛼𝑅𝐼𝐹^′_1−𝛼𝑅𝐼𝐹^′_𝛼𝐴_𝛼⊇_{[𝐼]𝛼∈}𝛼𝑅𝐼𝐹^′_𝛼𝐴_𝛼=𝑅𝐼𝐹^′(𝐴). (37)
Therefore 𝑅𝐼𝐹^′(𝐴)⊆𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)).
(2)⇔(3) This conclusion follows immediately from the duality.

Theorem 45. Let (𝑈,𝐼𝑅) be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: (1)∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑥))=1,𝑥∈𝑈; (2)𝐼𝑅(𝑥,𝑦)=0𝑜𝑟𝐼𝑅(𝑦,𝑥)=1,𝑥,𝑦∈𝑈; (3)𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴))⊆𝐴,𝐴∈𝐹^𝐼(𝑈);(4)𝐴⊆𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)),𝐴∈𝐹^𝐼(𝑈).

Proof. (1)⇔(2) We observe that, for all 𝑥∈𝑈, ∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑥))=1 if and only if ((1−𝐼𝑅(𝑥,𝑦)⁺)∨𝐼𝑅(𝑦,𝑥)⁻)=1 and ((1−𝐼𝑅(𝑥,𝑦)⁻)∨𝐼𝑅(𝑦,𝑥)⁺)=1; namely, for all 𝑥∈𝑈, 𝐼𝑅(𝑥,𝑦)⁺=0or𝐼𝑅(𝑦,𝑥)⁻=1 and 𝐼𝑅(𝑥,𝑦)⁻=0 or 𝐼𝑅(𝑦,𝑥)⁺=1.
On the one hand, if 𝐼𝑅(𝑥,𝑦)⁺=0, then 𝐼𝑅(𝑥,𝑦)⁻=0; we have 𝐼𝑅(𝑥,𝑦)=0. If 𝐼𝑅(𝑥,𝑦)⁺≠0, then 𝐼𝑅(𝑦,𝑥)⁻=1 and 𝐼𝑅(𝑦,𝑥)⁺=1; we have 𝐼𝑅(𝑦,𝑥)=1.
Hence, ∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑥))=1 if and only if for all 𝑦∈𝑈, 𝐼𝑅(𝑥,𝑦)=0 or 𝐼𝑅(𝑦,𝑥)=1.
(2)⇒(4) We first prove that 𝐴_𝛼⊆𝑅𝐼𝐹^′_1−𝛼(𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)).
Suppose that 𝑥∈𝐴_𝛼, for all 𝑦∈𝑈, 𝐼𝑅(𝑥,𝑦)=0 or 𝐼𝑅(𝑦,𝑥)=1 if and only if 𝐼𝑅(𝑥,𝑦)≠0 deduces 𝐼𝑅(𝑦,𝑥)=1. It follows that if 𝐼𝑅(𝑥,𝑦)>1−𝛼, then 𝐼𝑅(𝑦,𝑥)≥𝛼; that is, 𝑥∈𝐼𝑅_𝛼(𝑦). Further since 𝑥∈𝐴_𝛼, 𝐼𝑅(𝑥,𝑦)>1−𝛼 implies that 𝑥∈𝐼𝑅_𝛼(𝑦)∩𝐴_𝛼. So 𝐼𝑅_𝛼(𝑦)∩𝐴_𝛼≠∅. Note that 𝐼𝑅(𝑥,𝑦)>1−𝛼 and 𝐼𝑅_𝛼(𝑦)∩𝐴_𝛼≠∅ are equivalent to 𝑦∈𝐼𝑅_1−𝛼(𝑥) and 𝑦∈𝑅𝐼𝐹^′_𝛼(𝐴_𝛼), respectively; we have if 𝐼𝑅(𝑥,𝑦)>1−𝛼, then 𝑥∈𝐼𝑅_𝛼(𝑦)∩𝐴_𝛼. It shows that if 𝑦∈𝐼𝑅_1−𝛼(𝑥), then 𝑦∈𝑅𝐼𝐹^′_𝛼(𝐴_𝛼). By the arbitrary 𝑦, 𝐼𝑅_1−𝛼(𝑥)⊆𝑅𝐼𝐹^′_𝛼(𝐴_𝛼) holds; namely, 𝑥∈𝑅𝐼𝐹^′_1−𝛼(𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)). For any 𝑥, 𝐴_𝛼∈𝑅𝐼𝐹^′_1−𝛼(𝑅𝐼𝐹^′_𝛼(𝐴_𝛼)) holds.
On the other hand, in view of Lemma 37, we have 𝑅𝐼𝐹^′𝑅𝐼𝐹^′(𝐴)=_[𝛼∈I]𝛼𝑅𝐼𝐹^′_1−𝛼𝑅𝐼𝐹^′(𝐴)_𝛼⊇_{[𝐼]𝛼∈}𝛼𝑅𝐼𝐹^′_1−𝛼𝑅𝐼𝐹^′_𝛼𝐴_𝛼⊇_{𝛼∈[𝐼]}𝛼𝐴_𝛼=𝐴. (38)
(3)⇒(1) For all 𝑥,𝑧∈𝑈, from the proof of “(2)⇒(1)” in Theorem 44, we know that ∨_𝑦∈𝑈(𝐼𝑅(𝑥,𝑦)∧(1−𝐼𝑅(𝑦,𝑧)))=𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(1_𝑈/{𝑥}))(𝑧).
Furthermore, since 𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(1_𝑈/{𝑥}))(𝑧)≤1_𝑈/{𝑥}(𝑧), it follows that ∨_𝑦∈𝑈(𝐼𝑅(𝑥,𝑦)∧(1−𝐼𝑅(𝑦,𝑧)))≤1_𝑈/{𝑥}(𝑧); namely, ∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑧))≥1−1_𝑈/{𝑥}(𝑧).
When 𝑧=𝑥, we have ∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑥))≥1. Because the value of ∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑥)) is restricted in [𝐼], we have ∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑥))=1.
(3)⇔(4) This conclusion follows immediately from the duality.

Theorem 46. Let (𝑈,𝐼𝑅) be a generalized interval-valued fuzzy approximation space.(1)If 𝐼𝑅 is reflexive and transitive, then 𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)) and 𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)),   for all 𝐴∈𝐹^𝐼(𝑈).(2)If 𝐼𝑅 is reflexive and 𝐼𝑅(𝑥,𝑧)≤∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑧)), for all 𝑥,𝑧∈𝑈, then 𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴))and 𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′(𝑅𝐼𝐹^′(𝐴)), for all 𝐴∈𝐹^𝐼(𝑈).

Proof. Theorem 46 is proved easily by Theorems 34, 41, and 44.

According to duality and Theorem 46, one can obtain the next corollary.

Corollary 47. Suppose that (𝑈,𝐼𝑅) is a generalized interval-valued fuzzy approximation space. (1)If 𝐼𝑅 is reflexive and transitive, then ∼𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′∼𝑅I𝐹^′(𝐴)=𝑅𝐼𝐹^′𝑅𝐼𝐹^′(∼𝐴),∀𝐴∈𝐹^𝐼(𝑈),∼𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′∼𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′𝑅𝐼𝐹^′(∼𝐴),∀𝐴∈𝐹^𝐼(𝑈). (39)(2)If 𝐼𝑅 is reflexive and 𝐼𝑅(𝑥,𝑧)≤∧_𝑦∈𝑈((1−𝐼𝑅(𝑥,𝑦))∨𝐼𝑅(𝑦,𝑧)), for all 𝑥,𝑧∈𝑈, then ∼𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′∼𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′𝑅𝐼𝐹^′(∼𝐴),∀𝐴∈𝐹^𝐼(𝑈),∼𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′∼𝑅𝐼𝐹^′(𝐴)=𝑅𝐼𝐹^′𝑅𝐼𝐹^′(∼𝐴),∀𝐴∈𝐹^𝐼(𝑈). (40)

6. Conclusion and Future Work

In this paper, we proposed two types of the generalized interval-valued fuzzy approximation operators by integrating the generalized rough set theory and interval-valued fuzzy sets as well as fuzzy relations. The equivalence of these two types of the generalized interval-valued fuzzy approximation operators has been examined. Furthermore, we also demonstrated the duality of the lower and upper generalized interval-valued fuzzy approximation operators and discussed the properties of the generalized interval-valued fuzzy approximation operators under different interval-valued fuzzy relations.

In this paper, one can prove that the binary relation obtained by calculating 𝛼-cut set or strong 𝛼-cut set to an interval-valued fuzzy relation, for all 𝛼∈[𝐼], still satisfies the corresponding definition of Definition 11 under the classical binary relation; that is, if IR is reflexive, symmetric, and transitive, respectively, then 𝐼𝑅(𝐼𝑅𝑎) is reflexive, symmetric, and transitive, respectively, under the classical binary relation. Thus, if 𝐼𝑅 can satisfy the above functions, this technology can be applied in reasoning, learning, and decision-making. In Sections 4 and 5, the definitions and theorems provide some theoretical bases for reasoning, learning, and decision-making.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 61273018, the Foundation and Advanced Technology Research Program of Henan Province of China under Grant no. 132300410174, and the Key Scientific and Technological Project of Education Department of Henan Province of China under Grant no. 14A520082.