Research on rough equality and rough inclusion of sets in multi-granulation spaces

Abstract

Rough set theory proposed by professor Pawlak in 1982 is an important tool to solve uncertain problems. In order to make rough sets deal with uncertain problems better, by considering the stringent notions of mathematical equality and inclusion, the definitions of rough equality and rough inclusion were introduced by Pawlak. But the researches of rough equality and rough inclusion are on a certain granularity space. In this paper, some change rules will be given with respect to the relations of rough equality and rough inclusion between two uncertain target sets in multi-granulation spaces, and the similarity degree between two roughly equal sets is proposed to describe the similarity of two uncertain sets in multi-granulation spaces. In order to describe two uncertain sets which are rough equality by an approximation set at the same time, the definitions of optimistic λ-approximation set and pessimistic λ-approximation set will be defined from the points of optimism and pessimism, and some properties of them are discussed in detail.

Keywords

Rough set rough equality rough inclusion similarity multi-granulation spaces approximation set

1 Introduction

More and more experts and scholars focus on the research of uncertain problems with the development of cognitive science [15]. The experts and scholars pay more attention to how to mine the potential knowledge and rules from big data and uncertain information. Since fuzzy set theory [38] was put forward by professor Zadeh in 1965, the research of uncertain problems has made a great breakthrough, and some new theories which can deal with uncertain problems come into being. For example, Pawlak proposed the rough set theory [21, 24] in 1982, it is an important mathematical tool to deal with imprecise, inconsistent, incomplete information and knowledge. Rough set theory used the upper approximation set and lower approximation set to describe the uncertain concept approximately. The quotient space theory proposed by Zhang [41, 42] in 1990, made the most of ”falsity preserving” and ”truth preserving” properties to achieve the composition and decomposition of the extension granule of uncertain concept. The cloud model theory [15, 16] proposed by Li in 1995, by using entropy, excess entropy and expectation, the uncertain concept can be described quantitatively and qualitatively, and the translation between intension and extension of uncertain concept was built by cloud generator. These theories have been widely used in Artificial Intelligence(AI) fields. The rough set theory does not need prior knowledge. And rough set theory has strong complementarity with probability theory, fuzzy set theory, evidence theory and other theories [31]. It has become a very important intelligent information processing technology [5 , 44]. More and more researchers pay attention to rough set theory, and in many fields, a lot of related research achievements have been created by using it, such as machine learning [18, 29], deep learning [10], meteorology [1], medical science [20 , 46], decision analysis [7 , 25] and so on.

In order to make the rough set theory to deal with uncertain problems perfectly, many experts and scholars carried out extended studies on rough set theory by combining with other theories, and some extended models of rough set theory were proposed, such as the probabilistic rough set model [35], the decision-theoretic rough set model [6 , 37], the fuzzy rough set model [4], the rough fuzzy model [4], the game-theoretic rough set model [2 , 12], the variable precision rough set model [47] and so on. Yao and She have studied rough set approximations in multigranulation spaces, and a unified framework was proposed by them to classify and compare existing researches [36]. The extensions of rough set model offer new thoughts to deal with uncertain problems from different views.

In the classical set theory, if two sets are equal, the elements in two sets are identical, and if a set A be included in another set B, then every element in A must be the element of B. The equality and inclusion on classical set are too stringent and they always cannot be applied perfectly in real life situations. In human cognitive activities, people judge whether two sets are equal approximately and whether a set be included in another set approximately by their own knowledge. Even though the approximate description of the relation between two sets conform to human cognition, there are no such situations in the case of mathematics. In order to describe the special situations, the definitions of rough equality and rough inclusion of two uncertain sets were given by Pawlak [21], which are different from equality and inclusion on classical set theory. And some approximate operations and properties were given on rough sets. There is great significance for rough set theory. But the researches of rough equality and rough inclusion were in a given granularity space. However, when dealing with specific problems, people always think problems in different granularity spaces. Therefore, the researches of rough equality and rough inclusion in multi-granulation spaces will be conducted.

Fuzzy set theory, as a mathematical tool has great value that was used in national economy, science and technology and so on [3 , 26]. Based on fuzzy set theory, Zhang et al. [43, 45] proposed λ-approximation set model of rough sets. The basic idea of λ-approximation set model is translating a target set into a fuzzy set at first, then an approximation set of the target set is constructed based on different membership degree for each object in the boundary region by cut sets. Through this method, the target set can be described approximately as far as possible. For two target sets which are rough equality, the upper approximation set and lower approximation set of them are the same. So we consider that given a λ-approximation set to describe them at the same time. People always look at problems from optimistic view and pessimistic view. Refer to this phenomenon, the optimistic λ-approximation set and the pessimistic λ-approximation set are proposed in this paper.

The rest of this paper is organized as follows. Some basic concepts are reviewed briefly in Section 2. In Section 3, the change rules of rough equality relation and rough inclusion relation of two uncertain target sets in multi-granulation spaces are given. In Section 4, the similarity degree of two roughly equal sets is defined in multi-granulation spaces, and some properties are discussed. In Section 5, the definitions of optimistic λ-approximation set and pessimistic λ-approximation set of two uncertain target sets are presented, and some properties are discussed. Finally, the conclusions are drawn in Section 6.

2 Preliminaries

For convenience, many related basic concepts and definitions are introduced briefly at first in this section.

Definition 1. (Decision information system [21, 30]) A decision information system S can be described as S = (U, A, V, f). U is a nonempty finite set, i.e. the domain, A = C ∪ D is an attributes set, where C and D are called condition attributes set and decision attributes set respectively, V = ⋃ _r∈AV_r is the set of attribute values, V_r stands for the value range of attribute r ∈ A. f : U × A → V is an information function, and it specifies the attribute value of every object x in U.□

Definition 2. (Indiscernibility relation [21, 30]) Given a decision information system S = (U, A, V, f), for any subset R ⊆ A, a binary relation(indiscernibility relation) IND (R) can be defined as follows: $IND (R) = {(x, y) | (x, y) \in U^{2}, V_{r \in R} (r (x) = r (y))} .$

Definition 3. (Rough sets [21, 30]) Given a decision information system S = (U, A, V, f), for any subset X ⊆ U and R ⊆ A, the upper approximation set and lower approximation set of X can be defined as follows: $\bar{R} (X) = \cup {Y_{i} | Y_{i} \in U / I N D (R) \land Y_{i} \cap X \neq 0},$ $\underline{R} (X) = \cup {Y_{i} | Y_{i} \in U / I N D (R) \land Y_{i} \subseteq X},$ where U/IND (R) = {X|X ⊆ U ∧ ∀ _{x∈X,y∈X,r∈R} (r (x) = r (y))} is a partition of U induced by indiscernibility relation IND (R), x and y are indiscernible by attribute subset R in S if r (x) = r (y) for every r ∈ R. The upper approximation set and lower approximation set of X can be also defined by the form of set as follows: $\bar{R} (X) = {x | x \in U \land [x]_{R} \cap X \neq \emptyset},$ $\underline{R} (X) = {x | x \in U \land [x]_{R} \subseteq X},$ where [x] _R ∈ U/IND (R), and [x] _R is the equivalence class of x with respect to IND (R).

For any set X ⊆ U, if $\bar{R} (X) = \underline{R} (X)$ , X is called a definable set or crisp set in rough approximation space, and if $\bar{R} (X) \neq \underline{R} (X)$ , X is called rough set.

Definition 4. [21, 30] The set $BN D_{R} (X) = \bar{R} (X) - \underline{R} (X)$ is called the R-boundary region of X, $PO S_{R} (X) = \underline{R} (X)$ is called the R-positive region of X, $NE G_{R} (X) = U - \bar{R} (X)$ is called the R-negative region of X.

Definition 5. (Roughness of rough sets [30]) Given a decision information system S = (U, A, V, f), for an uncertain target set X (X ⊆ U) and nonempty attribute subset R (R ⊆ A), the roughness of X about R can be defined as follows: $β_{R} (X) = 1 - \frac{| \underline{R} (X) |}{| \bar{R} (X) |} = \frac{| BN D_{R} (X) |}{| \bar{R} (X) |} .$

Definition 6. (Fuzzy sets [33, 38]) Let U be a nonempty finite domain, and a mapping on U will be given as follows: $A : U \to [0, 1],$ $x | \to A (x),$ then A is a fuzzy set on U, A (x) is the membership function of A. The fuzzy set of all elements in U denote as F (U), that is F (U) = {A|A : U → [0, 1]}.

Definition 7. (Cut sets [33]) Let A ∈ F (U), λ ∈ [0, 1], and A_λ = {x|x ∈ U, A (x) ≥ λ}, then the λ-Cut set of A will be denoted as A_λ, λ is a threshold value(it also can be called confidence level), the λ-strong cut set of A will be denoted as $\underset{•}{A}_{λ}$ , it can be defined as $\underset{•}{A}_{λ} = {x | x \in U, A (x) > λ}$ .

Definition 8. (Membership function [23]) Let U be a nonempty finite domain, X is a subset of U, for ∀x (x ∈ U), the membership function about x (x ∈ X) can be defined as follows: $μ_{X}^{R} (x) = \frac{| {[x]}_{R} \cap X |}{| {[x]}_{R} |} .$

Obviously, for any element x in U, $0 \leq μ_{X}^{R} (x) \leq 1$ .

In fact, the membership function $μ_{X}^{R} (x)$ is a conditional probability, it reflects the probability that the element x belongs to X. Given a granulation space which induced by R, in order to using the existing knowledge granules to construct a set to describe the uncertain target set X approximately, Zhang [43] proposed the concepts of λ-approximation set, as shown in Definition 9.

Definition 9. (λ-approximation set [43, 45]) Let X be a subset of U, let $R_{λ} (X) = {x \in U | μ_{X}^{R} \geq λ} (0 < λ \leq 1),$ then,R_λ (X) is called λ-approximation set of X.

It is clear that $\underline{R} (X) \subseteq R_{λ} (X) \subseteq \bar{R} (X)$ .

Definition 10. (Similarity degree [40]) Let U be a nonempty finite domain, given two sets X, Y (X ⊆ U, Y ⊆ U). And a mapping can be defined as S : U × U → [0, 1], namely, (X, Y) → S (X, Y). S (X, Y) is the similarity degree of X and Y. If and only if S (X, Y) satisfies the following conditions:

(1) For any X, Y ⊆ U, then 0 ≤ S (X, Y) ≤1;

(2) For any X, Y ⊆ U, then S (X, Y) = S (Y, X);

(3) For any X, Y, Z ⊆ U, and X ⊆ Y ⊆ Z, then S (X, Z) ≤ S (X, Y) and S (X, Z) ≤ S (Y, Z);

(4) For any X, Y ⊆ U, S (X, X) =1 and the necessary and sufficient condition of S (X, Y) =0 is X∩ Y = ∅.

If a formula which satisfies condition(1)-(4), it can be regard as the similarity degree formula of X, Y.

In this paper, the similarity degree formula $S (X, Y) = \frac{| X \cap Y |}{| X \cup Y |}$ will be used [43]. Obviously, $S (X, Y) = \frac{| X \cap Y |}{| X \cup Y |}$ satisfies the four conditions in Definition 10.

3 Rough equality and rough inclusion of rough sets

An uncertain target set can be described by two precise sets(upper approximation set and lower approximation set) in rough sets. Many approximate models can be constructed to describe the uncertain target set in approximate space by this method. However, in the actual research, by using the upper approximation set and lower approximation set of a target set X to solve the problem respectively, both the results of the problem and the accuracies of the solutions have some differences. Therefore, the results may be different if different approximation sets are used.

Given a granularity space, as shown in Fig. 1, $\underline{R} (X) = \underline{R} (Y)$ , if the lower approximation set is used to describe the uncertain target sets X and Y, then the two sets which have differences originally will not be distinguished. Similarly, in Fig. 3, $\bar{R} (X) = \bar{R} (Y)$ , if the upper approximation set is used to describe the uncertain target sets X and Y, then the two sets will not be distinguished too. In Fig. 5, $\underline{R} (X) = \underline{R} (Y)$ and $\bar{R} (X) = \bar{R} (Y)$ , the two sets will also not be distinguished whether the upper approximation set or the lower approximation set is used to describe the uncertain target sets X and Y. For the above three cases, combining with the stringent notions of mathematical equality and inclusion, the definition of rough equality was given by Pawlak, and some properties were also given. However, the properties defined by Pawlak are on a certain granularity space. As we all know, human recognize things always from different granularity spaces. So, we will study rough equality in multi-granulation spaces that can extend the research of approximate equality relation.

Fig.1

X is roughly bottom-equal with Y with respect to R.

Fig.2

X is roughly bottom-equal with Y with respect to P.

Fig.3

X is roughly top-equal with Y with respect to R.

Fig.4

X is roughly top-equal with Y with respect to P.

Fig.5

X is roughly equal with Y with respect to R.

Fig.6

X is roughly equal with Y with respect to P.

Definition 11. [21] Given a decision information system S = (U, A, V, f), let X and Y be two subsets of domain U, and R is a subset of A, which means X ⊆ U, Y ⊆ U and R ⊆ A. If $\underline{R} (X) = \underline{R} (Y)$ , then X and Y will be called roughly bottom-equal with respect to R, denoted by $X \approx_{\underline{R}} Y$ ; if $\bar{R} (X) = \bar{R} (Y)$ , then X and Y will be called roughly top-equal with respect to R, denoted by $X \approx_{\bar{R}} Y$ ; if $\underline{R} (X) = \underline{R} (Y)$ and $\bar{R} (X) = \bar{R} (Y)$ , then X and Y will be called roughly equal with respect to R, denoted by X ≈ _RY.

Obviously, $\approx_{\underline{R}}$ , $\approx_{\bar{R}}$ and ≈_R are equivalence relations on P (U) which induced by IND (R). Next, the formal expression of the three equivalence relations( $\approx_{\underline{R}}$ , $\approx_{\bar{R}}$ and ≈_R) will be given.

Roughly bottom-equal in R: $\approx_{\underline{R}} = {(X, Y) | (X, Y) \in P (U) \times P (U), \underline{R} (X) = \underline{R} (Y)}$ ;

Roughly top-equal in R: $\approx_{\bar{R}} = {(X, Y) | (X, Y) \in P (U) \times P (U), \bar{R} (X) = \bar{R} (Y)}$ ;

Roughly equal in R: $\approx_{R} = {(X, Y) | (X, Y) \in P (U) \times P (U), \underline{R} (X) = \underline{R} (Y) \land \bar{R} (X) = \bar{R} (Y)}$ .

The relations $\approx_{\underline{R}}$ , $\approx_{\bar{R}}$ and ≈_R on P (U) can divide the subsets of U into different equivalence classes respectively. $[X]_{\approx_{\underline{R}}}$ denotes the equivalence class determined by X, and the lower approximation set of all elements in $[X]_{\approx_{\underline{R}}}$ are identical with respect to attribute subset R, obviously, $[X]_{\approx_{\underline{R}}} = {Y \subseteq U | \underline{R} (X) = \underline{R} (Y)}$ . With respect to R, $[X]_{\approx_{\bar{R}}}$ is an equivalence class that expresses the elements in P (U) which have the same upper approximation set with X, and it is clear that $[X]_{\approx_{\bar{R}}} = {Y \subseteq U | \bar{R} (X) = \bar{R} (Y)}$ . With respect to R, [X] _{≈_R} is an equivalence class that expresses the elements in P (U) which have the same lower approximation set and upper approximation set with X, plainly, $[X]_{\approx_{R}} = {Y \subseteq U | \underline{R} (X) = \underline{R} (Y) \land \bar{R} (X) = \bar{R} (Y)}$ .

$P (U) / \approx_{\underline{R}}$ means the family of all equivalence classes of $\approx_{\underline{R}}$ , then we have $P (U) / \approx_{\underline{R}} = {M | M \subseteq P (U) \land \forall_{X \in M, Y \in M} (\underline{R} (X) = \underline{R} (Y))}; P (U) / \approx_{\bar{R}}$ means the family of all equivalence classes of $\approx_{\bar{R}}$ , and it is obvious that $P (U) / \approx_{\bar{R}} = {M | M \subseteq P (U) \land \forall_{X \in M, Y \in M} (\bar{R} (X) = \bar{R} (Y))}; P (U) / \approx_{R}$ means the family of all equivalence classes of ≈_R, and it is also clear that $P (U) / \approx_{R} = {M | M \subseteq P (U) \land \forall_{X \in M, Y \in M} (\underline{R} (X) = \underline{R} (Y) \land \bar{R} (X) = \bar{R} (Y))}$ .

Given a granularity space which induced by the attribute set R, in this granularity space, if X ≈ _RY, the target sets X and Y are indistinguishable. If we want to distinguish X and Y, some new attributes should be given. Then the granularity space maybe become finer because of the new attributes. Next, the rules of rough equality of two target sets in different granularity spaces will be discussed.

Theorem 1. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and P ⊆ R ⊆ A. Then $X \approx_{\underline{R}} Y$ implies $X \approx_{\underline{P}} Y$ .

Proof. Supposing U/IND (R) = {E₁, E₂, ⋯, E_n} and U/IND (P) = {F₁, F₂, ⋯, F_m}, because P ⊆ R ⊆ A, it is clear that ∀_{E_i∈U/IND(R)} (∃ _{F_j∈U/IND(P)} (E_i ⊆ F_j)) and ∀_{F_j∈U/IND(P)} (∃ _{E_i∈U/IND(R)} (E_i ⊆ F_j)). That is to say the equivalence classes in U/IND (P) can be expressed by one or more equivalence classes in U/IND (R).

Because $X \approx_{\underline{R}} Y$ , it can be known that $\underline{R} (X) = \underline{R} (Y)$ .

For any E_i ⊆ X, there must be a F_k (k = 1, 2, ⋯, m) which makes E_i ⊆ F_k, and then the relations between F_k and X have two cases.

F_k ⊆ X, and it is clear that $F_{k} \subseteq \underline{P} (X)$ .

F_k ⊄ X and F_k∩ X ≠ ∅, then $F_{k} \subseteq \bar{P} (X)$ .

So, we have $\underline{P} (X) \subseteq \underline{R} (X)$ . Similarly, let E_i ⊆ Y, there must be a F_k′ (k′ = 1, 2, ⋯, m) which makes E_i ⊆ F_k′, the relations between F_k′ and Y are same as the two relations between F_k and X, then we have $\underline{P} (Y) \subseteq \underline{R} (Y)$ . By means of F_i ∩ F_j = ∅ (i ≠ j) and F_k ∩ F_k′ = E_i, so F_k = F_k′. And then $\underline{P} (X) = \underline{P} (Y)$ . Therefore, $X \approx_{\underline{P}} Y$ .□

Theorem 2. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and P ⊆ R ⊆ A. Then $X \approx_{\bar{R}} Y$ implies $X \approx_{\bar{P}} Y$ .

Proof. Because $X \approx_{\bar{R}} Y$ , we have $\bar{R} (X) = \bar{R} (Y)$ . Let U/IND (R) = {E₁, E₂, ⋯, E_n} and U/IND (P) = {F₁, F₂, ⋯, F_m}. For any $E_{i} \subseteq \bar{R} (X)$ , E_i∩ X ≠ ∅, there must be a F_j (j = 1, 2, ⋯, m) makes E_i ⊆ F_j, $\underset{F_{j} \cap X \neq \emptyset}{\cup} F_{j} = \bar{P} (X)$ can be obtained because of (F_j∪ E_i) ∩ X ≠ ∅. Similarly, for any $E_{i} \subseteq \bar{R} (Y)$ , there must be a F_j′ (j′ = 1, 2, ⋯, m) makes E_i ⊆ F_j′ and $\underset{F_{j^{'}} \cap X \neq \emptyset}{\cup} F_{j^{'}} = \bar{P} (Y)$ . By means of F_i∩ F_j = ∅ and F_j ∩ F_j′ = E_i, we have F_j = F_j′, then $\bar{P} (X) = \bar{P} (Y)$ can be obtained. Therefore, $X \approx_{\bar{P}} Y$ . □

Theorem 3. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and P ⊆ R ⊆ A. Then X ≈ _RY implies X ≈ _PY.

Proof. It is clear that Theorem 3 can be established by Theorem 1 and Theorem 2.□

According to Theorem 3(Theorem 1, Theorem 2), it can be learned that if two uncertain target sets are roughly equal (roughly bottom-equal, roughly top-equal) in a finer granularity space, as shown in Fig. 5(Fig. 1, Fig. 3), they must be roughly equal (roughly bottom-equal, roughly top-equal) in a coarser granularity space, as shown in Fig. 6(Fig. 2, Fig. 4). Conversely, if two uncertain target sets are roughly equal (roughly bottom-equal, roughly top-equal) in a coarser granularity space, it cannot indicate the two uncertain target sets are roughly equal (roughly bottom-equal, roughly top-equal) in a finer granularity space.

Similarly, in classical set theory, it is easy to know whether the inclusion relationship between two sets exists or not. But in rough sets, an uncertain target set is approximately described by its upper approximation set and lower approximation set. So for the inclusion relationship between two uncertain target sets, it can only described by the inclusion relationship of their upper approximation set and lower approximation set. Therefore, in order to describe the inclusion relationship between two sets in rough sets, approximate inclusion relations between two sets were given by Pawlak.

Definition 12. [21] Given a decision information system S = (U, A, V, f), let X and Y be two subsets of domain U, and R is a subset of A, which means X ⊆ U, Y ⊆ U and R ⊆ A, if $\underline{R} (X) \subseteq \underline{R} (Y)$ , then it can be called that X is roughly bottom-included in Y with respect to R, denoted by $X \subseteq_{\underline{R}} Y$ ; if $\bar{R} (X) \subseteq \bar{R} (Y)$ , then it can be called that X is roughly top-included in Y with respect to R, denoted by $X \subseteq_{\bar{R}} Y$ ; if $\underline{R} (X) \subseteq \underline{R} (Y)$ and $\bar{R} (X) \subseteq \bar{R} (Y)$ , then it can be called that X is roughly included in Y with respect to R, denoted by X ⊆ _RY.

As shown in Fig. 11(Fig. 7, Fig. 9), the uncertain target set X is roughly included(roughly bottom-included, roughly top-included) in the uncertain target set Y with respect to attribute set R. Next, the rules of rough inclusion relation of two target sets in different granularity spaces will be discussed.

Fig.7

X is roughly bottom-included in Y with respect to R.

Fig.8

X is roughly bottom-included in Y with respect to P.

Fig.9

X is roughly top-included in Y with respect to R.

Fig.10

X is roughly top-included in Y with respect to P.

Fig.11

X is roughly included in Y with respect to R.

Fig.12

X is roughly included in Y with respect to P.

Theorem 4. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and P ⊆ R ⊆ A. Then $X \subseteq_{\underline{R}} Y$ implies $X \subseteq_{\underline{P}} Y$ .

Proof. Because $X \subseteq_{\underline{R}} Y$ , $\underline{R} (X) \subseteq \underline{R} (Y)$ holds. Let U/IND (R) = {E₁, E₂, ⋯, E_n} and U/IND (P) = {F₁, F₂, ⋯, F_m}. For any E_i ⊆ X, there must be a F_k (k = 1, 2, ⋯, m) which makes E_i ⊆ F_k, and then the relations between F_k and X have two cases.

F_k ⊆ X,it is clear that $F_{k} \subseteq \underline{P} (X)$ ;

F_k ⊄ X and F_k∩ X ≠ ∅, under this condition, $F_{k} \subseteq \bar{P} (X)$ .

Similarly, for any E_i′ ⊆ Y (i′ = 1, 2, ⋯, n), there must be a F_k′ (k′ = 1, 2, ⋯, m) which makes E_i′ ⊆ F_k′ when the domain U is partitioned by attribute subset P, and then the relations between F_k′ and Y have two cases that are same as the two relations between F_k and X. The equivalence class F_k which satisfies E_i ⊆ F_k is equal to the equivalence class F_k′ which satisfies E_i′ ⊆ F_k′ when E_i = E_i′. In addition, due to $\underline{R} (X) \subseteq \underline{R} (Y)$ , we have $\underset{E_{i} \cap X = E_{i}}{\cup} E_{i} \subseteq \underset{E_{i^{'}} \cap Y = E_{i^{'}}}{\cup} E_{i^{'}}$ , then $\underset{F_{j} \cap X = F_{k}}{\cup} F_{k} \subseteq \underset{F_{k^{'}} \cap Y = F_{k^{'}}}{\cup} F_{k^{'}}$ , that is, $\underline{P} (X) \subseteq \underline{P} (Y)$ . Thus, $X \subseteq_{\underline{P}} Y$ .□

Theorem 5. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and P ⊆ R ⊆ A. Then $X \subseteq_{\bar{R}} Y$ implies $X \subseteq_{\bar{P}} Y$ .

Proof. Beacuse $X \subseteq_{\bar{R}} Y$ , it is clear that $\bar{R} (X) \subseteq \bar{R} (Y)$ . Let U/IND (R) = {E₁, E₂, ⋯, E_n} and U/IND (P) = {F₁, F₂, ⋯, F_m}. For any $E_{i} \subseteq \bar{R} (X)$ , that is to say E_i∩ X ≠ ∅, there must be a F_k (k = 1, 2, ⋯, m) which makes E_i ⊆ F_j. It can be known that $\underset{F_{j} \cap X \neq \emptyset}{\cup} F_{j} = \bar{P} (X)$ because of (F_j∪ E_i) ∩ X ≠ ∅. Similarly, for any $E_{i^{'}} \subseteq \bar{R} (Y) (i^{'} = 1, 2, \dots, n)$ , there must be a F_j′ (j′ = 1, 2, ⋯, m) which makes E_i′ ⊆ F_j′, then $\underset{F_{j^{'}} \cap X \neq \emptyset}{\cup} F_{j^{'}} = \bar{P} (Y)$ .

Due to $\bar{R} (X) \subseteq \bar{R} (Y)$ , we have $\underset{E_{i} \cap X \neq \emptyset}{\cup} E_{i} \subseteq \underset{E_{i^{'}} \cap Y \neq \emptyset}{\cup} E_{i^{'}}$ and | {E_i|E_i ∩ X ≠ ∅} | ≤ | {E_i′|E_i′ ∩ Y ≠ ∅} |. And then $\underset{F_{j} \cap X \neq \emptyset}{\cup} F_{j} \subseteq \underset{F_{j^{'}} \cap Y \neq \emptyset}{\cup} F_{j^{'}}$ can be obtained, that is to say $\bar{P} (X) \subseteq \bar{P} (Y)$ . Therefore, $X \subseteq_{\bar{P}} Y$ .

Theorem 6. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and P ⊆ R ⊆ A. Then X ⊆ _RY implies X ⊆ _PY.

Proof. It is clear that Theorem 6 can be established by Theorem 4 and Theorem 5.

In the multi-granulation spaces, according to Theorem 6(Theorem 4, Theorem 5), it can be known that if the uncertain target set X is roughly included(roughly bottom-included, roughly top-included) in the uncertain target set Y in a finer granularity space, then X is roughly included(roughly bottom-included, roughly top-included) in Y in a coarser granularity space can be obtained, as shown in Fig. 11(Fig. 7, Fig. 9) and Fig. 12(Fig. 8, Fig. 10). Conversely, if X is roughly included(roughly bottom-included, roughly top-included) in Y in a coarser granularity space, it is not sure that X is roughly included(roughly bottom-included, roughly top-included) in Y in a finer granularity space.

Because rough equality and rough inclusion are two special relations of two sets, the roughness of two sets under the special relations will be compared now.

Theorem 7. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and R ⊆ A. If $X \approx_{\underline{R}} Y$ and $X \subseteq_{\bar{R}} Y$ then β_R (X) ≤ β_R (Y).

Proof. Because $X \approx_{\underline{R}} Y$ and $X \subseteq_{\bar{R}} Y$ , we have $| \underline{R} (X) | = | \underline{R} (Y) |$ and $| \bar{R} (X) | \leq | \bar{R} (Y) |$ . Thus, we can obtain $\begin{matrix} β_{R} (X) - β_{R} (Y) = (1 - \frac{| \underline{R} (X) |}{| \bar{R} (X) |}) - (1 - \frac{| \underline{R} (Y) |}{| \bar{R} (Y) |}) \\ = \frac{| \bar{R} (X) | | \underline{R} (Y) | - | \underline{R} (X) | | \bar{R} (Y) |}{| \bar{R} (X) | | \bar{R} (Y) |} \\ = \frac{| \underline{R} (X) | | \bar{R} (X) | - | \bar{R} (Y) |}{| \bar{R} (X) | | \bar{R} (Y) |} \leq 0, \end{matrix}$ therefore, β_R (X) ≤ β_R (Y).□

Theorem 7 shows that if X and Y roughly bottom-equal with respect to R, and X roughly top-included in Y with respect to R, the roughness of X is less than or equal to the roughness of Y.

Theorem 8. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and R ⊆ A. If $X \approx_{\bar{R}} Y$ and $X \subseteq_{\underline{R}} Y$ , then β_R (Y) ≤ β_R (X).

Proof. Because $X \approx_{\bar{R}} Y$ and $X \subseteq_{\underline{R}} Y$ , we have $| \bar{R} (X) | = | \bar{R} (Y) |$ and $| \underline{R} (X) | \leq | \underline{R} (Y) |$ . Thus, we can obtain $\begin{matrix} β_{R} (Y) - β_{R} (X) = (1 - \frac{| \underline{R} (Y) |}{| \bar{R} (Y) |}) - (1 - \frac{| \underline{R} (X) |}{| \bar{R} (X) |}) \\ = \frac{| \underline{R} (X) | | \bar{R} (Y) | - | \bar{R} (X) | | \underline{R} (Y) |}{| \bar{R} (X) | | \bar{R} (Y) |} \\ = \frac{| \bar{R} (X) | (| \underline{R} (X) | - | \underline{R} (Y) |)}{| \bar{R} (X) | | \bar{R} (Y) |} \leq 0, \end{matrix}$ therefore, β_R (Y) ≤ β_R (X).

Theorem 8 shows that if X and Y roughly top-equal with respect to R, and X roughly bottom-included in Y with respect to R, the roughness of Y is less than or equal to the roughness of X.□

4 Similarity degree of sets that are rough equality

Rough equality is a special relation between two uncertain target sets X and Y. The formula $S (X, Y) = \frac{| X \cap Y |}{| X \cup Y |}$ can be used to describe the similarity degree of X and Y effectively, but it is a fixed value in multi-granulation spaces and cannot reflect the granularity of knowledge space. Because rough equality is a special relation between two uncertain target sets, we consider whether a formula can be established that can not only reflect the similarity degree of two sets which are roughly equal but also reflect the granularity of knowledge space. Based on this idea, the definition of similarity degree of two roughly equal sets will be given.

Definition 13. Given a decision information system S = (U, A, V, f), let X and Y be two subsets of domain U, R is a subset of A, which means X ⊆ U, Y ⊆ U, R ⊆ A. And X ≈ _RY, then the similarity degree of X and Y which are rough equality will be defined as follows: $\begin{matrix} D_{R} (X, Y) \\ = \frac{| X \cup^{} Y | - | \underline{R} (X) |}{| X \cup^{} Y |} \times \frac{| (X - \underline{R} (X)) \cap^{} (Y - \underline{R} (Y)) |}{| (X - \underline{R} (X)) \cup^{} (Y - \underline{R} (Y)) |} \\ + \frac{| \underline{R} (X) |}{| X \cup^{} Y |} \times \frac{| X \cap^{} Y |}{| X \cup^{} Y |} \\ = \frac{| X \cup^{} Y | \times | X \cap^{} Y | + | \underline{R} (X) | (| X \cap^{} Y | - | X \cup^{} Y |)}{| X \cup^{} Y |^{2}} . \end{matrix}$

If X and Y are precise, then D_R (X, Y) =1. Because X ≈ _RY, if X∩ Y = ∅, we have $\underline{R} (X) = \underline{R} (Y) = \emptyset$ , then D_R (X, Y) =0.

Next, D_R (X, Y) will be checked whether it satisfies the four conditions in Definition 10.

Obviously, D_R (X, Y) satisfies condition(2) and (4) in Definition 10, here, we will prove D_R (X, Y) satisfies condition(1) and (3).

Proof. Condition(1): In the formula D_R (X, Y), because $| \underline{R} (X) | \leq | X \cap Y | \leq | X \cup Y |$ , we have |X ∪ Y| × |X ∩ Y| ≤ |X ∪ Y|² and $| \underline{R} (X) | \times (| X \cap Y | -$ |X ∪ Y|) ≤0, then $| X \cup Y | \times | X \cap Y | + | \underline{R} (X) | \times (| X \cap Y | - | X \cup Y |) \leq | X \cup Y |^{2}$ . That to say D_R (X, Y) ≤1. And $| X \cup Y | \times | X \cap Y | + | \underline{R} (X) | \times (| X \cap Y | - | X \cup Y |) = | X \cup Y | \times (| X \cap Y | - | \underline{R}$ $(X) |) + | \underline{R} (X) | \times | X \cap Y | \geq 0$ can be obtained, we have D_R (X, Y) ≥0. Therefore, 0 ≤ D_R (X, Y) ≤1.

Condition(3): If X ≈ _RY ≈ _RZ and X ⊆ Y ⊆ Z, then $| \underline{R} (X) | = | \underline{R} (Y) | = | \underline{R} (Z) | \leq | X | \leq | Y | \leq | Z |$ . $\begin{array}{l} D_{R} (X, Y) - D_{R} (X, Z) \\ = \frac{| X \cup Y | \times | X \cap Y | + | \underline{R} (X) | (| X \cap Y | - | X \cup Y |}{{| X \cup Y |}^{2}} \\ - \frac{| X \cup Y | \times | X \cap Y | + | \underline{R} (X) | (| X \cap Y | - | X \cup Y |}{{| X \cup Y |}^{2}} \\ = \frac{| X | - | \underline{R} (X) |}{| Y |} + \frac{| X | \times | \underline{R} (X) |}{{| Y |}^{2}} - \frac{| X | - | \underline{R} (X) |}{| Z |} \\ - \frac{| X | \times | \underline{R} (X) |}{{| Z |}^{2}} \geq 0, \\ D_{R} (Y, Z) - D_{R} (X, Z) \\ = \frac{| Y \cup Z | \times | Y \cap Z | + | \underline{R} (X) | (| X \cup Z | - | X \cup Z |)}{{| Y \cup Z |}^{2}} \end{array}$

$\begin{array}{l} = \frac{| Y | - | \underline{R} (X) |}{| Z |} + \frac{| Y | \times | \underline{R} (X) |}{| Z |^{2}} - \frac{| X | - | \underline{R} (X) |}{| Z |} \\ - \frac{| X | \times | \underline{R} (X) |}{| Z |^{2}} \geq 0. \end{array}$

So, D_R (X, Z) ≤ D_R (X, Y) and D_R (X, Z) ≤ D_R (Y, Z).□

Therefore, the similarity degree of two roughly equal sets can be calculated by the formula D_R (X, Y) in a certain granulation space with respect to R. From Theorem 3, it can be learned that two sets must be roughly equal in a coarser granularity space if they are roughly equal in a finer granularity space. Then the relationship between D_R (X, Y) and S (X, Y) as well as the change rules of D_R (X, Y) in multi-granulation spaces will be discussed in the following.

Theorem 9. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and R ⊆ A. If X ≈ _RY, then D_R (X, Y) ≤ S (X, Y).

Proof. Because X ≈ _RY, $\underline{R} (X) = \underline{R} (Y)$ holds, and then $\begin{matrix} D_{R} (X, Y) - S (X, Y) \\ = \frac{| X \cup^{} Y | \times | X \cap^{} Y | + | \underline{R} (X) | (| X \cap^{} Y | - | X \cup^{} Y |)}{| X \cup^{} Y |^{2}} \\ - \frac{| X \cap^{} Y |}{| X \cup^{} Y |} \\ = \frac{| \underline{R} (X) | (| X \cap^{} Y | - | X \cup^{} Y |)}{| X \cup^{} Y |^{2}} \leq 0. \end{matrix}$ Therefore, D_R (X, Y) - S (X, Y) ≤0, that is to say, D_R (X, Y) ≤ S (X, Y).□

Theorem 10. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U. If X ≈ _RY and P ⊆ R ⊆ A, then D_R (X, Y) ≤ D_P (X, Y).

Proof. Owing to X ≈ _RY and P ⊆ R, it can be known that X ≈ _PY from Theorem 3, and then $\begin{matrix} D_{R} (X, Y) - D_{P} (X, Y) \\ = \frac{| X \cup^{} Y | \times | X \cap^{} Y | + | \underline{R} (X) | (| X \cap^{} Y | - | X \cup^{} Y |)}{| X \cup^{} Y |^{2}} \\ - \frac{| X \cup^{} Y | \times | X \cap^{} Y | + | \underline{P} (X) | (| X \cap^{} Y | - | X \cup^{} Y |)}{| X \cup^{} Y |^{2}} \\ = \frac{(| \underline{R} (X) | - | \underline{P} (X) |) \times (| X \cap^{} Y | - | X \cup^{} Y |)}{| X \cup^{} Y |^{2}} . \end{matrix}$

Because P ⊆ R, U/IND (R) ≺ U/IND (P) holds. That is to say, the granularity of the knowledge with respect to R on U is finer than the granularity of the knowledge with respect to P on U. Here, BND_R (X) ⊆ BND_P (X). We have $0 \leq | \underline{P} (X) | \leq | \underline{R} (X) | \leq | X \cap Y | \leq | X \cup Y |$ , so D_R (X, Y) - D_P (X, Y) ≤0. That is to say D_R (X, Y) ≤ D_P (X, Y).

It can be seen from Theorem 10, if two uncertain target sets are roughly equal in different granularity spaces, then the similarity degree of two roughly equal sets in a finer granularity space is not greater than the similarity degree of two roughly equal sets in a coarser granularity space. It also reflects that the similarity degree of two roughly equal sets is proportional to the knowledge granularity, that is to say, if D_R (X, Y) ≤ D_P (X, Y), then β_R (X) ≤ β_P (X) can be obtained.□

5 Optimistic λ-approximation set and pessimistic λ-approximation set

In the previous work, the membership function describes the membership degree of element x (x ∈ U) belongs to target set X. And combining with cut sets, an approximation set of target set X is constructed. Then the knowledge granules in a certain granularity space can be used to realize the approximate description of target set X. This work facilitated the development of rough sets in a way. However, for two uncertain target sets which are roughly equal, owing to they have same upper approximation set and lower approximation set, we consider that whether an approximation set can be used to describe the two sets at the same time.

Given a decision information system S = (U, A, V, f), let X and Y be two subsets of domain U, R be a subset of A, and X ≈ _RY. Obviously, $μ_{X}^{R} (x)$ and $μ_{Y}^{R} (x)$ may be different when [x] _R ⊆ BND_R (X) and [x] _R ⊆ BND_R (Y). There are three cases of the two values, that is, $μ_{X}^{R} (x) < μ_{Y}^{R} (x)$ , $μ_{X}^{R} (x) = μ_{Y}^{R} (x)$ and $μ_{X}^{R} (x) > μ_{Y}^{R} (x)$ . In [18], the basic idea of λ-approximation set translate a target set of rough sets into a fuzzy set. Combining with cut sets, giving a threshold λ, the equivalence class will be judged that whether it can be used to describe the λ-approximation set by compare λ and its membership degree. So, the λ-approximation set of X and Y may be different. In order to describe X and Y approximately by a same λ-approximation set, some rules should be given. For an equivalence class [x] _R, if $μ_{X}^{R} (x) < μ_{Y}^{R} (x) < λ$ , [x] _R cannot be a part of λ-approximation set; if $λ \leq μ_{X}^{R} (x) \leq μ_{Y}^{R} (x)$ or $λ \leq μ_{Y}^{R} (x) \leq μ_{X}^{R} (x)$ , [x] _R can be a part of λ-approximation set. But whether [x] _R can be a part of λ-approximation set when $μ_{Y}^{R} (x) < λ \leq μ_{X}^{R} (x)$ or $μ_{X}^{R} (x) < λ \leq μ_{Y}^{R} (x)$ ? For this question, we can treat it from optimistic view and pessimistic view. From optimistic view, [x] _R can be a part of λ-approximation set of X and Y when $λ \leq μ_{X}^{R} (x)$ or $λ \leq μ_{Y}^{R} (x)$ ; from pessimistic view, [x] _R can be a part of λ-approximation set of X and Y if and only if $λ \leq μ_{X}^{R} (x)$ and $λ \leq μ_{Y}^{R} (x)$ . Next, the definitions of optimistic λ-approximation set and pessimistic λ-approximation set of two sets which are rough equality will be given.

Definition 14. (Optimistic λ-approximation set) Given a decision information system S = (U, A, V, f), let X and Y be two subsets of domain U, R is a subset of A, which means X ⊆ U, Y ⊆ U and R ⊆ A, let $R_{λ}^{O} (X, Y) = {x \in U | μ_{X}^{R} (x) \geq λ \lor μ_{Y}^{R} (x) \geq λ},$ and 0 < λ ≤ 1, then $R_{λ}^{O} (X, Y)$ is called the rough equality optimistic λ-approximation set of uncertain target sets X and Y.

Let λ = 0.5, the rough equality optimistic 0.5-approximation set of target sets X and Y described by the red dotted line, as shown in Fig. 13.

Fig.13

Optimistic 0.5-approximation set.fig13.png

Definition 15. (Pessimistic λ-approximation set) Given a decision information system S = (U, A, V, f), let X and Y be two subsets of domain U, R is a subset of A, which means X ⊆ U, Y ⊆ U and R ⊆ A, let $R_{λ}^{P} (X, Y) = {x \in U | μ_{X}^{R} (x) \geq λ \land μ_{Y}^{R} (x) \geq λ},$ and 0 < λ ≤ 1, then $R_{λ}^{P} (X, Y)$ is called the rough equality pessimistic λ-approximation set of uncertain target sets X and Y.

Let λ = 0.5, the rough equality pessimistic 0.5-approximation set of target sets X and Y described by the red dotted line, as shown in Fig. 14.

Fig.14

Pessimistic 0.5-approximation set.fig14.png

In order to show the definitions of optimistic λ-approximation set and pessimistic λ-approximation set more clearly, an example will be given as follows.

When we evaluate ”Triple-A student”(who is good in attitude, study and health), the composite score of a student will be given from three aspects: ideology and morality, academic record and health. Suppose there are sixteen students take part in the evaluation of ”Triple-A student”, the students are marked as x_i (i = 1, 2, ⋯, 16), let U = {x₁, x₂, ⋯, x₁₆}, and the composite score of students are shown as Table 1. Then there will be two teachers X and Y select ”Triple-A student” according to the score in Table 1. As we all know, there is no absolute equity when people make a decision, which is usually mixed up with personal emotions. There exist definite subjectivity when the teacher choose ”Triple-A student”, suppose teacher X will choose the student as ”Triple-A student” who has high score in ideology and morality, and teacher Y will choose the student as ”Triple-A student” who has high score in study. And then supposing X = {x₁, x₄, x₇, x₁₂, x₁₄, x₁₅}, Y = {x₁, x₂, x₅, x₇, x₁₂, x₁₃, x₁₄, x₁₅}. Composite score will be regarded as the basis to evaluate ”Triple-A student”, and S (compositescore) = $\frac{1}{3} (S (ideology and morality) + S (academic record) +$ S (health)). And it is easy to know U / S (compositescore) = {{x₁, x₂, x₃}, {x₄, x₅, x₆, x₇}, {x₈, x₉, x₁₀, x₁₁, x₁₂, x₁₃}, {x₁₄, x₁₅}, {x₁₆}}.

Let E₁ = {x₁, x₂, x₃}, E₂ = {x₄, x₅, x₆, x₇}, E₃ = {x₈, x₉, x₁₀, x₁₁, x₁₂, x₁₃}, E₄ = {x₁₄, x₁₅} and E₅ = {x₁₆}.

Table 1

Student score

Student	Ideology and morality	Academic record	Health	Composite score
x ₁	9	9	8.7	8.9
x ₂	8.9	9.3	8.5	8.9
x ₃	8.8	8.9	9	8.9
x ₄	9.1	8.9	9.6	9.2
x ₅	8.7	9.3	9.6	9.2
x ₆	8.8	8.9	9.9	9.2
x ₇	9.1	9.1	9.4	9.2
x ₈	8.5	8.6	8.4	8.5
x ₉	8.8	8.3	8.4	8.5
x ₁₀	8.9	8.4	8.2	8.5
x ₁₁	8.4	8.7	8.4	8.5
x ₁₂	9	9.4	7.1	8.5
x ₁₃	8.8	9.2	7.5	8.5
x ₁₄	9.5	9.5	9.5	9.5
x ₁₅	9.5	9.5	9.5	9.5
x ₁₆	7.1	6.5	9.2	7.6

For simplicity, we used R to mark S (compositescore). Obviously, X ≈ _RY. And according to the membership function $μ_{X}^{R} (x) = \frac{| {[x]}_{R} \cap X |}{| {[x]}_{R} |}$ , two fuzzy sets can be obtained as follows: $\begin{array}{l} F_{X} (U) = \\ {\frac{1 / 3}{x_{1}}, \frac{1 / 3}{x_{2}}, \frac{1 / 3}{x_{3}}, \frac{1 / 2}{x_{4}}, \frac{1 / 2}{x_{5}}, \frac{1 / 2}{x_{6}}, \frac{1 / 2}{x_{7}}, \frac{1 / 6}{x_{8}}, \\ \frac{1 / 6}{x_{9}}, \frac{1 / 6}{x_{10}}, \frac{1 / 6}{x_{11}}, \frac{1 / 6}{x_{12}}, \frac{1 / 6}{x_{13}}, \frac{1}{x_{14}}, \frac{1}{x_{15}}, \frac{0}{x_{16}}}, \\ F_{Y} (U) = \\ {\frac{2 / 3}{x_{1}}, \frac{2 / 3}{x_{2}}, \frac{2 / 3}{x_{3}}, \frac{1 / 2}{x_{4}}, \frac{1 / 2}{x_{5}}, \frac{1 / 2}{x_{6}}, \frac{1 / 2}{x_{7}}, \frac{1 / 3}{x_{8}}, \\ \frac{1 / 3}{x_{9}}, \frac{1 / 3}{x_{10}}, \frac{1 / 3}{x_{11}}, \frac{1 / 3}{x_{12}}, \frac{1 / 3}{x_{13}}, \frac{1}{x_{14}}, \frac{1}{x_{15}}, \frac{0}{x_{16}}} . \end{array}$

Let λ = 0.5, then $R_{0.5}^{O} (X, Y) = {E_{1}, E_{2}, E_{4}} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{14}, x_{15}}$ , and $R_{0.5}^{P} (X, Y) = {E_{2}, E_{4}} = {x_{4}, x_{5}, x_{6}, x_{7}, x_{14}, x_{15}}$ .

For any two uncertain target sets X, Y (X ⊆ U, Y ⊆ U), if they are rough equality, combining with the rough equality optimistic λ-approximation set and the rough equality pessimistic λ-approximation set of X and Y, some basic properties will be established as follows:

$\underline{R} (X) = \underline{R} (Y) \subseteq R_{λ}^{P} (X, Y) \subseteq R_{λ}^{o} (X, Y) \subseteq \bar{R} (X) = \bar{R} (Y)$ ;

$R_{λ}^{o} (X, Y) = R_{λ} (X) \cup R_{λ} (Y)$ ;

$R_{λ}^{P} (X, Y) = R_{λ} (X) \cap R_{λ} (Y)$ ;

$R_{λ}^{P} (X, Y) \subseteq R_{λ}^{P} (X, X \cup Y) \subseteq R_{λ}^{O} (X, X \cup Y)$ , $R_{λ}^{P} (X, X \cap Y) \subseteq R_{λ}^{O} (X, X \cap Y) \subseteq R_{λ}^{O} (X, Y)$ .

Proof. (1), (2) and (3) are established clearly.

(4) Because Y ⊆ X ∪ Y, it can be known that $R_{λ}^{P} (X, Y) = R_{λ} (X) \cap R_{λ} (Y) \subseteq R_{λ} (X) \cap R_{λ} (X \cup Y) = R_{λ}^{P} (X, X \cup Y)$ . And then from property (1), $R_{λ}^{P} (X, X \cup Y) \subseteq R_{λ}^{O} (X, X \cup Y)$ ; therefore, $R_{λ}^{P} (X, Y) \subseteq R_{λ}^{P} (X, X \cup Y) \subseteq R_{λ}^{O} (X, X \cup Y)$ . Similarly, $R_{λ}^{P} (X, X \cap Y) \subseteq R_{λ}^{O} (X, X \cap Y) \subseteq R_{λ}^{O} (X, Y)$ .□

Theorem 11. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and R ⊆ A. If X ≈ _RY and X ⊆ Y, then $R_{λ}^{o} (X, Y) = R_{λ} (Y)$ and $R_{λ}^{P} (X, Y) = R_{λ} (X)$ .

Proof. Owing to X ≈ _RY and X ⊆ Y, for every x ∈ U, we have $μ_{X}^{R} (x) \leq μ_{Y}^{R} (x)$ .

So, it can be known that ${x | μ_{X}^{R} (x) \geq λ} \subseteq {x | μ_{Y}^{R} (x) \geq λ}$ , that is to say R_λ (X) ⊆ R_λ (Y). Therefore, $R_{λ}^{O} (X, Y) = R_{λ} (X) \cup R_{λ} (Y) = R_{λ} (Y)$ , $R_{λ}^{P} (X, Y) = R_{λ} (X) \cap R_{λ} (Y) = R_{λ} (X)$ . The theorem is proved.□

Theorem 12. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and R ⊆ A. If X ≈ _RY and λ = 1, then $\underline{R} (X) = \underline{R} (Y) = R_{λ}^{o} (X, Y) = R_{λ}^{P} (X, Y)$ .

Proof. Obviously, Theorem 12 is established.□

Theorem 13. Given a decision information system S = (U, A, V, f), let X ⊆ U, Y ⊆ U and R ⊆ A. If X ≈ _RY and λ₁ ≤ λ₂ ≤ ⋯ ≤ λ_n, then $R_{λ_{1}}^{P} (X, Y) \supseteq R_{λ_{2}}^{P} (X, Y) \supseteq \dots \supseteq R_{λ_{n}}^{P} (X, Y)$ and $R_{λ_{1}}^{O} (X, Y) \supseteq R_{λ_{2}}^{O} (X, Y) \supseteq \dots \supseteq R_{λ_{n}}^{O} (X, Y)$ .

Proof. Obviously, Theorem 13 is established.□

Theorem 13 shows that the rough equality optimistic λ-approximation set and the rough equality pessimistic λ-approximation set of X and Y will be increasingly approximate to their lower approximation set with the growth of λ.

6 Conclusions

Rough set theory is a mathematical tool which can process imprecise, inconsistent and uncertain knowledge, and it has been widely used in many fields. In this paper, the change rules of rough equality and rough inclusion of two uncertain target sets in multi-granulation spaces were studied, for two target sets X and Y, if X and Y are roughly equal(bottom-equal, top-equal) or X is roughly included(bottom-included, top-included) in Y in a finer granularity space, then X and Y must be roughly equal(bottom-equal, top-equal) or X is roughly included(bottom-included, top-included) in Y in a coarse granularity space. And then, the similarity degree of two roughly equal sets is given. On the one hand, the similarity of two sets can be calculated by it, on the other hand, the granularity of knowledge can be reflected by it. In order to describe two roughly equal sets approximately at the same time, the optimistic λ-approximation set and pessimistic λ-approximation set were proposed from the points of optimism and pessimism, and several properties were given. These research results probably are useful in knowledge discovery fields.

Footnotes

Acknowledgement

This work is supported by National Natural Science Foundation of P.R. China (No.61876201, No. 61472056) and Science and Technology Support Plan Project of Sichuan Province (No. 2015GZ0079).

References

Aprianti

, Mukhlash

, The application of rough set and fuzzy rough set based algorithm to classify incomplete meteorological data, In International Conference on Data and Software Engineering, 2014, pp. 1–6.

Azam

, Yao

J.T.

, Analyzing uncertainties of probabilistic rough set regions with game-theoretic rough sets, International Journal of Approximate Reasoning 55(1) (2014), 142–155.

K.W.

, Lai

V.S.

, A survey of the application of fuzzy set theory in production and operations management: 1998–2009, International Journal of Production Economics 129(1) (2011), 157–168.

Didier

, Henri

, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17(2-3) (1990), 191–209.

D”untsch

, Gediga

, Uncertainty measures of rough set prediction, Artificial Intelligence 106(1) (1998), 109–137.

Dou

H.L.

, Yang

X.B.

, Song

X.N.

, Yu

H.L.

, Wu

W.Z.

, Yang

J.Y.

, Decision-theoretic rough set, Knowledge-Based Systems 91(C) (2016), 71–83.

Fan

T.F.

, Liu

D.R.

, Tzeng

G.H.

, Rough set-based logics for multicriteria decision analysis, European Journal of Operational Research 182(1) (2007), 340–355.

Feng

, Zhang

S.P.

, Mi

J.S.

, The reduction and fusion of fuzzy covering systems based on the evidence theory, International Journal of Approximate Reasoning 53(1) (2012), 87–103.

Guiffrida

A.L.

, Nagi

, Fuzzy set theory applications in production management research: A literature survey, Journal of Intelligent Manufacturing 9(1) (1998), 39–56.

10.

Hassan

Y.F.

, Deep learning architecture using rough sets and rough neural networks, Kybernetes 46(4) (2017), 693–705.

11.

Herbert

J.P.

, Yao

J.T.

, Analysis of data-driven parameters in game-theoretic rough sets, In International Conference on Rough Sets and Knowledge Technology 2011, pp. 447–456.

12.

Herbert

J.P.

, Yao

J.T.

, Game-theoretic rough sets, Fundamenta Informaticae 108(3-4) (2011), 267–286.

13.

Jensen

, Shen

, Fuzzy-rough sets assisted attribute selection, IEEE Transactions on Fuzzy Systems 15(1) (2007), 73–89.

14.

Jeon

, Kim

, Jeong

, Rough sets attributes reduction based expert system in interlaced video sequences, IEEE Transactions on Consumer Electronics 52(4) (2006), 1348–1355.

15.

D.Y.

, Ch ,

Liu ,

Du and

Han , Artificial intelligence with uncertainty, Journal of Software 15(11) (2004), 1583–1594.

16.

D.Y.

, Meng

H.J.

, Shi

X.M.

, Membership cloud and membership cloud generators, Journal of Computer Research and Development (1995).

17.

Liang

J.Y.

, Chin

K.S.

, Dang

C.Y.

, Yam

R.C.M.

, A new method for measuring uncertainty and fuzziness in rough set theory, International Journal of General Systems 31(4) (2002), 331–342.

18.

Moshkov

, Zielosko

, Combinatorial Machine Learning: A Rough Set Approach, Springer Publishing Company, Incorporated, 2013

19.

Park

K.S.

, Fuzzy-set theoretic interpretation of economic order quantity, IEEE Transactions on Systems, Man and Cybernetics 17(6) (1987), 1082–1084.

20.

Pattaraintakorn

, Cercone

, Integrating rough set theory and medical applications, Applied Mathematics Letters 21(4) (2008), 400–403.

21.

Pawlak

, Rough sets, International Journal of Computer Information Sciences 11(5) (1982), 341–356.

22.

Pawlak

, Rough sets and decision analysis, Infor: Information Systems and Operational Research 38(3) (2000), 132–144.

23.

Pawlak

, Skowron

, Rough membership functions: A tool for reasoning with uncertainty, Banach Center Publications 28 (1993), 135–150.

24.

Pawlak

, Skowron

, Rudiments of rough sets, Information Sciences 177(1) (2007), 3–27.

25.

Pawlak

, Slowinski

, Rough set approach to multiattribute decision analysis, European Journal of Operational Research 72(3) (1994), 443–459.

26.

Pedrycz

, Fuzzy set technology in knowledge discovery, Fuzzy Sets and Systems 98(3) (1998), 279–290.

27.

Shen

, Chouchoulas

, A rough-fuzzy approach for generating classification rules, Pattern Recognition 35(11) (2002), 2425–2438.

28.

Swiniarski

R.W.

, Skowron

, Rough set methods in feature selection and recognition, Pattern Recognition Letters 24(6) (2003), 833–849.

29.

Vluymans

, DEer

, Saeys

, Cornelis

, Applications of fuzzy rough set theory in machine learning: A survey, Fundamenta Informaticae 142(1-4) (2015), 53–86.

30.

Wang

G.Y.

Rough Set Theory and Knowledge Discovery, Xian Jiaotong University Press, 2001.

31.

Wang

G.Y.

, Yao

Y.Y.

, Yu

, A survey on rough set theory and applications, Chinese Journal of Computers 32(7) (2009), 1229–1246.

32.

Yang

H.H.

, Wu

C.L.

, Rough sets to help medical diagnosisevidence from a taiwans clinic, Expert Systems with Applications 36(5) (2009), 9293–9298.

33.

Yang

L.B.

, Gao

Y.Y.

Fuzzy set theory and its applications, South China University of Technology Press, 2011.

34.

Yao

Y.Y.

, Decision-theoretic rough set models, In International Conference on Rough Sets and Knowledge Technology 2007, pp. 1–12.

35.

Yao

Y.Y.

, Greco

, Sowiski

Probabilistic Rough Sets, Springer Handbook of Computational Intelligence, 2015.

36.

Yao

Y.Y.

, She

Y.H.

, Rough set models in multigranulation spaces, Information Sciences 327 (2016), 40–56.

37.

Yao

Y.Y.

, Wong

S.K.M.

, A decision theoretic framework for approximating concepts, International Journal of Manmachine Studies 37(6) (1992), 793–809.

38.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

39.

Zeng

, Pan

, Zheng

Q.L.

, Peng

, Knowledge acquisition based on rough set theory and principal component analysis, IEEE Intelligent Systems 21(2) (2006), 78–85.

40.

Zeng

W.Y.

, Li

H.X.

, Inclusion measures, similarity measures, and the fuzziness of fuzzy sets and their relations, International Journal of Intelligent Systems 21(6) (2006), 639–653.

41.

Zhang

, Zhang

Theory and Applications of Problem Solving, Elsevier Science Inc, 1992.

42.

Zhang

, Zhang

, The quotient space theory of problem solving, In International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing (2004), pp. 11–15.

43.

Zhang

Q.H.

, Wang

G.Y.

, Xiao

, Approximation sets of rough sets, Journal of Software 23(7) (2012), 1745–1759.

44.

Zhang

Q.H.

, Yang

J.J.

, Yao

L.Y.

, Attribute reduction based on rough approximation set in algebra and information views, IEEE Access 4 (2016), 5399–5407.

45.

Zhang

Q.H.

, Zhang

, Wang

G.Y.

, Research on approximation set of rough set based on fuzzy similarity, Journal of Intelligent & Fuzzy Systems 32(3) (2017), 2549–2562.

46.

Zhang

Z.W.

, Shi

, Gao

G.X.

, A rough set-based multiple criteria linear programming approach for the medical diagnosis and prognosis, Expert Systems with Applications 36(5) (2009), 8932–8937.

47.

Ziarko

, Variable precision rough set model, Journal of Computer and System Sciences 46(1) (1993), 39–59.