# Picture Fuzzy Rough Set and Rough Picture Fuzzy Set on Two Different Universes and Their Applications

Authors: Dliouah Ahmed; Binxiang Dai
Journal: Journal of Mathematics
 (2020)

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## Abstract

The major concern of this article is to propose the notion of picture fuzzy rough sets (PFRSs) over two different universes which depend onδ,ζ,ϑ-cut of picture fuzzy relation ℛ on two different universes (i.e., by combining picture fuzzy sets (PFSs) with rough sets (RSs)). Then, we discuss several interesting properties and related results on the PFRSs. Furthermore, we define some notions related to PFRSs such as (Type-I/Type-II) graded PFRSs, the degree α and β with respect to ℛδ,ζ,ϑ on PFRSs, and (Type-I/Type-II) generalized PFRSs based on the degree α and β with respect to ℛδ,ζ,ϑ and investigate the basic properties of above notions. Finally, an approach based on the rough picture fuzzy approximation operators on two different universes in decision-making problem is introduced, and we give an example to show the validity of this approach.


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## Body

## 1. Introduction

In the past few years, Pawlak [1] proposed the notion of RS as a mathematical tool to handle with ambiguity and incomplete information systems. The lower/upper approximations (i.e., rough sets) are firstly described through the equivalence classes. That is to say, many datasets cannot be treated properly by way of classical rough sets. In mild of this, the graded rough sets [2], similarity or tolerance relations [3–5], arbitrary binary relation [6, 7], and variable precision rough sets [8, 9] are a few extensions of the classical rough sets. So, several researchers, for example, Dubois and Prade [10], presented the concept of fuzzy rough set (FRS) (i.e., the fuzzy set (FS) [11] and the RS). Many researchers have worked on fuzzy rough models (see [12–16]).Wong et al. [17] presented the notion of the RS model over two universes and its application. Several applications and the fundamental properties of the FRS model on two universes are studied [18–32]. Yao and Lin [33, 34] proposed the notion of graded rough sets (GRSs) on one universe. Zhang et al. [35] gave a comparative between the variable precision rough set (VPRS) and the GRS. In addition to the previous studies, Liu et al. [36] introduced the notion of GRSs on two universes. Yu et al. [37] presented the notion of a variable precision-graded rough set (VPGRSs) over two universes and Yu and Wang [38] presented a novel type of GRS with VP over two inconsistent universes.In this paper, we propose the notions of PFRSs and RPFSs over two different universes. The basic properties of PFRSs based onδ,ζ,ϑ-cut of picture fuzzy relation ℛ over two different universes are discussed. Meanwhile, we propose two types of graded picture fuzzy rough sets (GPFRSs) based on δ,ζ,ϑ-cut of ℛ on two different universes: type-I PFRS is according to the graded n with respect to ℛδ,ζ,ϑ and type-II PFRS is according to the graded n with respect to ℛδ,ζ,ϑ. The interesting properties of Type-I/Type-II PFRSs are investigated in detail. Furthermore, we define the notions of PFRS according to the degree α and β with respect to ℛδ,ζ,ϑ and Type-I/Type-II generalized PFRs according to the degree α and β with respect to ℛiδ,ζ,ϑ. The main results of the above notions are studied and explored. Finally, an application of RPFs model over two different universes is presented to solve the decision-making problem.Sections of this article are arranged as follows. In Section2, we gave the concepts of PFSs and picture fuzzy relations. In Section 3, we give the notion of PFRSs based on δ,ζ,ϑ-cut of picture fuzzy relation ℛ over two different universes and study some interesting properties on PFRSs. In Section 4, an algorithm is constructed and an application on RPFSs over two different universes in decision-making problem is explored. Lastly, conclusion is discussed in Section 5.


## 2. Preliminaries

### 2.1. Picture Fuzzy Sets and Picture Fuzzy Relations

Cuong [39–41] introduced the notion of PFS is an extension of fuzzy FS [42] and intuitionistic fuzzy set IFS [43]. Later on, many researchers defined some notions related to PFSs (e.g., [44–49]) and solved some problems related to PFSs (e.g., [50–57]).Definition 1 (cf. see [39–41]).
LetU=u1,u2,…,un be an n-element set (n is a natural number), and a PFS A∈IU is(1)A=p1∘Au1,p2∘Au1,p3∘Au1u1,p1∘Au2,p2∘Au2,p3∘Au2u2,…,p1∘Aun,p2∘Aun,p3∘Aunun.
Moreover,1−p1∘Au−p2∘Au−p3∘Au is called the refusal degree of uu∈U. A PFS A∈IU with refusal degree 0 at each point u∈U can be identified with an IFS on U and IU with the pointwise order ≤ is the set of all mappings from a set U (or an universe) to I=a1,a2,a3∈0,13|a1+a2+a3≤1. Then, each element A of IU is called an I-set or a PFS on U, p1∘Au (i.e., the degree of positive), p2∘Au (i.e., the degree of neutral), and p3∘Au (i.e., the degree of negative) of the element u∈U, where pi:0,13⟶0,1 (i.e., the ith projection from 0,13 to 0,1i=1,2,3).Definition 2 (cf. see [39–41]).
LetA,ℬ∈IU. Then,(1)
The complementAc of A∈IU is defined by(2)Ac=p3∘Au1,p2∘Au1,p1∘Au1u1,p3∘Au2,p2∘Au2,p1∘Au2u2,…,p3∘Aun,p2∘Aun,p1∘Aunun.(2)
The union∪k∈KAk (called also supremum ∨k∈KAk) and the intersection ∩k∈KAk (called also infimum ∧k∈KAk) of a family Akk∈K⊆IU can be defined by the following formulae:(3)∪k∈KAku=∨k∈Kp1∘Aku,∧k∈Kp2∘Aku,∧k∈Kp3∘Aku,∩k∈KAku=∧k∈Kp1∘Aku,∧k∈Kp2∘Aku,∨k∈Kp3∘Aku.(3)
A is a subset of ℬ if A≤ℬ (i.e., p1∘Au≤p1∘ℬu, p2∘Au≤p2∘ℬu, and p3∘Au≥p3∘ℬu, for each x∈X).(4)
A is an equal of ℬ if A≤ℬ (i.e., p1∘Au≤p1∘ℬu, p2∘Au≤p2∘ℬu, and p3∘Au≥p3∘ℬu) and A≥ℬ (i.e., p1∘Au≥p1∘ℬu, p2∘Au≥p2∘ℬu, and p3∘Au≥p3∘ℬu).Definition 3 (cf. see [39–41, 57]).
Letℛ=p1∘ℛ,p2∘ℛ,p3∘ℛ be a picture fuzzy relation, denoted by IU×V, where p1∘ℛ∈0,1U×V, p2∘ℛ∈0,1U×V, and p3∘ℛ∈0,1U×V satisfy 0≤p1∘ℛu,v+p2∘ℛu,v+p3∘ℛu,v≤1 for all u,v∈U×V and p1:ℛ2⟶ℛ,p2:ℛ2⟶ℛ, and ℛ3⟶ℛ are first projection, second projection, and third projection, respectively.Definition 4 (cf. see [39–41, 57]).
For two picture fuzzy relationsℛ∈IU×V and ℋ∈IV×W, the picture fuzzy relation ℋ∘ℛ∈IU×W, defined by 1(4)ℋ∘ℛu,w=∨v∈Vℛu,v∧ℋv,w,∀u,w∈U×W,is said to be a composition of ℛ and ℋ.


## 2.1. Picture Fuzzy Sets and Picture Fuzzy Relations

Cuong [39–41] introduced the notion of PFS is an extension of fuzzy FS [42] and intuitionistic fuzzy set IFS [43]. Later on, many researchers defined some notions related to PFSs (e.g., [44–49]) and solved some problems related to PFSs (e.g., [50–57]).Definition 1 (cf. see [39–41]).
LetU=u1,u2,…,un be an n-element set (n is a natural number), and a PFS A∈IU is(1)A=p1∘Au1,p2∘Au1,p3∘Au1u1,p1∘Au2,p2∘Au2,p3∘Au2u2,…,p1∘Aun,p2∘Aun,p3∘Aunun.
Moreover,1−p1∘Au−p2∘Au−p3∘Au is called the refusal degree of uu∈U. A PFS A∈IU with refusal degree 0 at each point u∈U can be identified with an IFS on U and IU with the pointwise order ≤ is the set of all mappings from a set U (or an universe) to I=a1,a2,a3∈0,13|a1+a2+a3≤1. Then, each element A of IU is called an I-set or a PFS on U, p1∘Au (i.e., the degree of positive), p2∘Au (i.e., the degree of neutral), and p3∘Au (i.e., the degree of negative) of the element u∈U, where pi:0,13⟶0,1 (i.e., the ith projection from 0,13 to 0,1i=1,2,3).Definition 2 (cf. see [39–41]).
LetA,ℬ∈IU. Then,(1)
The complementAc of A∈IU is defined by(2)Ac=p3∘Au1,p2∘Au1,p1∘Au1u1,p3∘Au2,p2∘Au2,p1∘Au2u2,…,p3∘Aun,p2∘Aun,p1∘Aunun.(2)
The union∪k∈KAk (called also supremum ∨k∈KAk) and the intersection ∩k∈KAk (called also infimum ∧k∈KAk) of a family Akk∈K⊆IU can be defined by the following formulae:(3)∪k∈KAku=∨k∈Kp1∘Aku,∧k∈Kp2∘Aku,∧k∈Kp3∘Aku,∩k∈KAku=∧k∈Kp1∘Aku,∧k∈Kp2∘Aku,∨k∈Kp3∘Aku.(3)
A is a subset of ℬ if A≤ℬ (i.e., p1∘Au≤p1∘ℬu, p2∘Au≤p2∘ℬu, and p3∘Au≥p3∘ℬu, for each x∈X).(4)
A is an equal of ℬ if A≤ℬ (i.e., p1∘Au≤p1∘ℬu, p2∘Au≤p2∘ℬu, and p3∘Au≥p3∘ℬu) and A≥ℬ (i.e., p1∘Au≥p1∘ℬu, p2∘Au≥p2∘ℬu, and p3∘Au≥p3∘ℬu).Definition 3 (cf. see [39–41, 57]).
Letℛ=p1∘ℛ,p2∘ℛ,p3∘ℛ be a picture fuzzy relation, denoted by IU×V, where p1∘ℛ∈0,1U×V, p2∘ℛ∈0,1U×V, and p3∘ℛ∈0,1U×V satisfy 0≤p1∘ℛu,v+p2∘ℛu,v+p3∘ℛu,v≤1 for all u,v∈U×V and p1:ℛ2⟶ℛ,p2:ℛ2⟶ℛ, and ℛ3⟶ℛ are first projection, second projection, and third projection, respectively.Definition 4 (cf. see [39–41, 57]).
For two picture fuzzy relationsℛ∈IU×V and ℋ∈IV×W, the picture fuzzy relation ℋ∘ℛ∈IU×W, defined by 1(4)ℋ∘ℛu,w=∨v∈Vℛu,v∧ℋv,w,∀u,w∈U×W,is said to be a composition of ℛ and ℋ.


## 3. Picture Fuzzy Rough Sets over Two Different Universes

### 3.1. Picture Fuzzy Rough Sets Based onδ,ζ,ϑ-Cut of ℛ on Two Different Universes

We will begin by defining theδ,ζ,ϑ-cut of ℛ and will subsequently define a picture fuzzy rough set based on δ,ζ,ϑ-cut.Definition 5.
Letℛ∈IU×V and δ,ζ,ϑ∈I. Then,(i)
ℛδ,ζ,ϑ=u,v∈U×V|ℛu,v≥δ,ζ,ϑ=u,v∈U×V|p1∘ℛu,v≥δ,p2∘ℛu,v≤ζ,p3∘ℛu,v≤ϑ is called the δ,ζ,ϑ-cut of ℛ, and(5)ℛδ,ζ,ϑu=v∈V|ℛu,v≥δ,ζ,ϑ.(ii)
LetA∈2V. Then,(6)ℛδ,ζ,ϑ−A=u∈U|ℛδ,ζ,ϑu⊆A∧ℛδ,ζ,ϑu≠∅,(i.e., the lower approximation ofA) and(7)ℛδ,ζ,ϑ+A=u∈U|ℛδ,ζ,ϑu∩A≠∅∨ℛδ,ζ,ϑu=∅,(i.e., the upper approximation ofA).
The pairℛδ,ζ,ϑ−A,ℛδ,ζ,ϑ+A is a PFR approximation of A with respect to ℛδ,ζ,ϑ.
Now, we present some properties based on PFRS as follows.Theorem 1.
Letℛ∈IU×V, δ,ζ,ϑ∈I, and A∈2V. Then, the following holds:(1)
ℛδ,ζ,ϑ−∅=∅;ℛδ,ζ,ϑ+V=U(2)
ℛδ,ζ,ϑ−A⊆ℛδ,ζ,ϑ+A(3)
IfA⊆ℬ, then ℛδ,ζ,ϑ−A⊆ℛδ,ζ,ϑ−ℬ; ℛδ,ζ,ϑ+A⊆ℛδ,ζ,ϑ+ℬ(4)
ℛδ,ζ,ϑ−A∩ℬ=ℛδ,ζ,ϑ−A∩ℛδ,ζ,ϑ−ℬ; ℛδ,ζ,ϑ+A∪ℬ=ℛδ,ζ,ϑ+A∪ℛδ,ζ,ϑ+ℬ(5)
ℛδ,ζ,ϑ−A∪ℬ⊇ℛδ,ζ,ϑ−A∪ℛδ,ζ,ϑ−ℬ; ℛδ,ζ,ϑ+A∩ℬ⊆ℛδ,ζ,ϑ+A∩ℛδ,ζ,ϑ+ℬ(6)
ℛδ,ζ,ϑ−A=ℛδ,ζ,ϑ+Acc; ℛδ,ζ,ϑ+A=ℛδ,ζ,ϑ−AccProof.
We only prove (6).
ℛδ,ζ,ϑ+Acc=u∈U|ℛδ,ζ,ϑu∩Ac≠∅∨ℛδ,ζ,ϑu=∅c = u∈U|ℛδ,ζ,ϑu∩Ac=∅∧ℛδ,ζ,ϑu≠∅=u∈U|ℛδ,ζ,ϑu⊆A∧ℛδ,ζ,ϑu≠∅=ℛδ,ζ,ϑ−A, and ℛδ,ζ,ϑ−Acc=u∈U|ℛδ,ζ,ϑu⊆Ac∧ℛδ,ζ,ϑu≠∅c = u∈U|ℛδ,ζ,ϑu∩A=∅∧ℛδ,ζ,ϑu≠∅c = u∈U|ℛδ,ζ,ϑu∩A≠∅∨ℛδ,ζ,ϑu=∅=ℛδ,ζ,ϑ+A.
The equality of Theorem1 (5) does not hold as the following example.Example 1.
SupposeU=xi|i=1,2,3 and V=yi|i=1,2,3 be two three-element set, and ℛ∈IU×V is defined by(8)ℛ=0.2,0.5,0.3x1,y1,0.6,0.3,0x1,y2,0.4,0.3,0.1x1,y3,0.4,0.2,0.3x2,y1,0.3,0.3,0.2x2,y2,0.1,0.6,0.2x2,y3,0.5,0.2,0.1x3,y1,0.5,0.5,0x3,y2,0.6,0.1,0.1x3,y3.
Takeδ,ζ,ϑ=0.5,0.3,0.1. Then, ℛ0.5,0.3,0.1x1=y2, ℛ0.5,0.3,0.1x2=∅, and ℛ0.5,0.3,0.1x3=y1,y3.
LetA=y1 and ℬ=y2,y3. Thus, ℛ0.5,0.3,0.1−A=∅,ℛ0.5,0.3,0.1−ℬ=x1,ℛ0.5,0.3,0.1−A∪ℬ=x1,x3≠x1=ℛ0.5,0.3,0.1−A∪ℛ0.5,0.3,0.1−ℬ and ℛ0.5,0.3,0.1+A=x2,x3,ℛ0.5,0.3,0.1+ℬ=U,ℛ0.5,0.3,0.1+A∩ℬ=x2≠x2,x3=ℛ0.5,0.3,0.1+A∩ℛ0.5,0.3,0.1+ℬ.Remark 1.
In general,ℛδ,ζ,ϑ+∅=∅ and ℛδ,ζ,ϑ−V=U do not hold. For example, let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈IU×V is defined by(9)ℛ=0.2,0.4,0.4x1,y1,0.4,0.2,0.3x1,y2,0.5,0.3,0.2x2,y1,0.7,0.1,0.2x2,y2.
Takeδ,ζ,ϑ=0.5,0.1,0.2. Then, ℛ0.5,0.1,0.2x1=∅ and ℛ0.5,0.1,0.2x2=y2. Thus, ℛ0.5,0.1,0.2+∅=x1≠∅ and ℛ0.5,0.1,0.2−V=x2≠U.Remark 2.
It should be pointed that some conclusions in [58–60] which are similar to those in Theorem 1 are wrong. Firstly, the equality ℛδ,ζ−∅=∅ and ℛδ,ζ+V=U in Theorem 3.1(2) in [60] are incorrect. Let U=xi|i=1,2, V=yi|i=1,2, and ℛ∈JU×V be defined by(10)ℛ=0.5,0.2x1,y1,0.7,0.1x1,y2,0.5,0.3x2,y1,0.3,0.4x2,y2.
Takeδ,ζ=0.5,0.1. Then, ℛ0.5,0.1−∅ and ℛ0.5,0.1+V in the sense of [60] are not ∅ and U, but x2 and x1, respectively. Secondly, the inclusion ℛδ−A⊆ℛδ+A in Theorem 3.1(1) in [58] is incorrect. Let U=xi|i=1,2,3 and V=yi|i=1,2,3 be two three-element sets, and ℛ∈0,1U×V, is defined by(11)ℛ=0.5x1,y1,0.7x1,y2,0.4x1,y3,0.5x2,y1,0.3x2,y2,0.4x2,y3,0.8x3,y1,0.1x3,y2,0.6x3,y3.
Takeδ=0.6 and A=y1,y3. Then, ℛ0.6x1=y2, ℛ0.6x2=∅, and ℛ0.6x3=y1,y3, and thus ℛ0.6−A (resp., ℛ0.6+A) in the sense of [58] is x2,x3 (resp., x3). Therefore, ℛδ−A⊆ℛδ+A. Analogously, assertions ℛδ,ζ−A⊆ℛδ,ζ+A in Theorem 3.1(1) [59] and ℛδ,ζ−A⊆A⊆ℛδ,ζ+A in Theorem 3.1(1) [60] are incorrect. Let U=xi|i=1,2,3 and V=yi|i=1,2,3 be two three-element sets, and ℛ∈0,12U×V and also ℛ∈JU×V is defined by(12)ℛ=0.2,0.5x1,y1,0.6,0.3x1,y2,0.4,0.3x1,y3,0.4,0.2x2,y1,0.3,0.3x2,y2,0.1,0.6x2,y3,0.5,0.2x3,y1,0.5,0.5x3,y2,0.6,0.1x3,y3.
Takeδ,ζ=0.5,0.3, and A=y1,y3. Then, ℛ0.5,0.3x1=y2, ℛ0.5,0.3x2=∅, and ℛ0.5,0.3x3=y1,y3, and thus ℛ0.5,0.3−A=x2,x3⊆x3=ℛ0.5,0.3+A and ℛ0.5,0.3−A=x2,x3⊆A⊆x3=ℛ0.5,0.3+A.
To correct some results in [58–60] in above Remark 2, we will give new notations of lower and upper approximations as follow:Definition 6.
LetA∈2V. Then,(1)
Forℛ∈0,1U×V and δ∈0,1, we have(13)ℛδ−A=u∈U|ℛδu⊆A∧ℛδu≠∅,ℛδ+A=u∈U|ℛδu∩A≠∅∨ℛδu=∅.(2)
Forℛ∈0,12U×V and δ,ζ∈0,12, we have(14)ℛδ,ζ−A=u∈U|ℛδ,ζu⊆A∧ℛδ,ζu≠∅,ℛδ,ζ+A=u∈U|ℛδ,ζu∩A≠∅∨ℛδ,ζu=∅.(3)
Forℛ∈JU×V and δ,ζ∈J, we have (cf. [61])(15)ℛδ,ζ−A=u∈U|ℛδ,ζu⊆A∧ℛδ,ζu≠∅,ℛδ,ζ+A=u∈U|ℛδ,ζu∩A≠∅∨ℛδ,ζu=∅.Theorem 2.
Letℛ,S∈IU×V, δ,ζ,ϑ∈I, and A∈2V. If ℛ≤S, then ℛδ,ζ,ϑ−A⊆Sδ,ζ,ϑ−A and ℛδ,ζ,ϑ+A⊆Sδ,ζ,ϑ+A.Proof.
Ifℛ≤S, then p1∘ℛu,v≤p1∘Su,v,p2∘ℛu,v≤p2∘Su,v and p3∘ℛu,v≥p3∘Su,v for all u,v∈U×V. By Definition 5, we have ℛδ,ζ,ϑu=v∈V|ℛu,v≥δ,ζ,ϑ⊆v∈V|Su,v≥δ,ζ,ϑ=Sδ,ζ,ϑu. Therefore, ℛδ,ζ,ϑu⊆A∧ℛδ,ζ,ϑu≠∅⊆Sδ,ζ,ϑu⊆A∧Sδ,ζ,ϑu≠∅ for all u∈U. Thus, ℛδ,ζ,ϑ−A⊆Sδ,ζ,ϑ−A. Similarly, ℛδ,ζ,ϑ+A⊆ℛδ,ζ,ϑ+A.Corollary 1.
(1) Letℛ,S∈0,1U×V,δ∈0,1, and A∈2V. If ℛ≤S, then ℛδ−A⊆Sδ−A and ℛδ+A⊆Sδ+A.
(2) Letℛ,S∈0,12U×V,δ,ζ∈0,12, and A∈2V. If ℛ≤S, then ℛδ,ζ−A⊆Sδ,ζ−A and ℛδ,ζ+A⊆Sδ,ζ+A.
(3) Letℛ,S∈JU×V,δ,ζ∈J, and A∈2V. If ℛ≤S, then ℛδ,ζ−A⊆Sδ,ζ−A and ℛδ,ζ+A⊆Sδ,ζ+A.Proof.
It follows from Definition6.
Next, we give comparison between some properties by Definition 3.3 in [58], Definition 3.2 in [59], Definition 3.3 in [60] and 6 as shown in Tables 1–3:Table 1
Comparison between some properties by Definition 3.3 in [58] and 6 (1).
Definition 3.3 in [58]Definition6 (1)ℛδ−∅=∅×√ℛδ+∅=∅√×ℛδ−V=U√×ℛδ+V=U×√ℛδ−A⊆ℛδ+A×√ℛδ−A⊆Sδ−Aifℛ≤S×√ℛδ+A⊆Sδ+Aifℛ≤S×√ℛδ1−A⊆ℛδ2−Aifδ1≤δ2√×ℛδ2+A⊆ℛδ1+Aifδ1≤δ2√×(√) indicates that the property is satisfied.Table 2
Comparison between some properties by Definition 3.2 in [59] and 6 (2).
Definition 3.2 in [59]Definition6 (2)ℛδ,ζ−∅=∅×√ℛδ,ζ+∅=∅√×ℛδ,ζ−V=U√×ℛδ,ζ+V=U×√ℛδ,ζ−A⊆ℛδ,ζ+A×√ℛδ,ζ−A⊆Sδ,ζ−Aifℛ≤S×√ℛδ,ζ+A⊆Sδ,ζ+Aifℛ≤S×√ℛδ1,ζ1−A⊆ℛδ2,ζ2−Aifδ1≤δ2,ζ1≤ζ2√×ℛδ2,ζ2+A⊆ℛδ1,ζ1+Aifδ1≤δ2,ζ1≤ζ2√×(√) indicates that the property is satisfied.Table 3
Comparison between some properties by Definition 3.3 in [60] and 6 (3).
Definition 3.3 in [60]Definition6 (2)ℛδ,ζ−∅=∅×√ℛδ,ζ+∅=∅√×ℛδ,ζ−V=U√×ℛδ,ζ+V=U×√ℛδ,ζ−A⊆ℛδ,ζ+A×√ℛδ,ζ−A⊆Sδ,ζ−Aifℛ≤S×√ℛδ,ζ+A⊆Sδ,ζ+Aifℛ≤S×√ℛδ1,ζ1−A⊆ℛδ2,ζ2−Aifδ1≤δ2,ζ1≤ζ2√×ℛδ2,ζ2+A⊆ℛδ1,ζ1+Aifδ1≤δ2,ζ1≤ζ2√×(√) indicates that the property is satisfied.Remark 3.
Assertionsℛδ1−A⊆ℛδ2−Aifδ1≤δ2 and ℛδ2+A⊆ℛδ1+Aifδ1≤δ2 do not hold by Definition 6 (1), let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈0,1U×V is defined by(16)ℛ=0.6x1,y1,0.4x1,y2,1x2,y1,0.6x2,y2.
Takeδ1=0.55≤0.7=δ2 and A=y1. Then, ℛ0.55x1=y1,ℛ0.55x2=V,ℛ0.7x1=∅,andℛ0.7x2=y1, and thus ℛ0.55−A=x1≠x2=ℛ0.7−A. Also, if we take A=y2, ℛ0.55+A=x2≠x1=ℛ0.7+A.Remark 4.
Assertionsℛδ1,ζ1−A⊆ℛδ2,ζ2−Aifδ1≤δ2,ζ1≤ζ2 and ℛδ2,ζ2+A⊆ℛδ1,ζ1+Aifδ1≤δ2,ζ1≤ζ2 in Tables 2 and 3 do not hold by Definition 6 (2) and (3), and let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈0,12U×V and also ℛ∈JU×V is defined by(17)ℛ=0.6x1,y1,0.4x1,y2,1x2,y1,0.6x2,y2.
Takeδ1=0.55≤0.7=δ2, and A=y1. Then, ℛ0.55x1=y1,ℛ0.55x2=V,ℛ0.7x1=∅,andℛ0.7x2=y1, and thus ℛ0.55−A=x1≠x2=ℛ0.7−A Also, if we take A=y2, thus ℛ0.55+A=x2≠x1=ℛ0.7+A


### 3.2. Graded Picture Fuzzy Rough Sets Based onℛ on Two Different Universes

Definition 7.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N, and A∈2V. Then,(18)ℛδ,ζ,ϑ−InA=u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅,ℛδ,ζ,ϑ+InA=u∈U|ℛδ,ζ,ϑu∩A>n∨ℛδ,ζ,ϑu=∅,are called the Type-I lower approximation and the Type-I upper approximation of A according to the graded n with respect to ℛδ,ζ,ϑ on U and V, respectively, and ℛδ,ζ,ϑ−InA,ℛδ,ζ,ϑ+InA is called the Type-I picture fuzzy rough approximation of A according to the graded n with respect to ℛδ,ζ,ϑ (briefly, a Type-I picture fuzzy rough set according to the graded n with respect to ℛδ,ζ,ϑ).Theorem 3.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N, and A∈2V. Then,(1)
IfA⊆ℬ, then ℛδ,ζ,ϑ−InA⊆ℛδ,ζ,ϑ−Inℬ and ℛδ,ζ,ϑ+InA⊆ℛδ,ζ,ϑ+Inℬ(2)
ℛδ,ζ,ϑ−InA∩ℬ⊆ℛδ,ζ,ϑ−InA∩ℛδ,ζ,ϑ−Inℬ(3)
ℛδ,ζ,ϑ−InA∪ℬ⊇ℛδ,ζ,ϑ−InA∪ℛδ,ζ,ϑ−Inℬ; ℛδ,ζ,ϑ+InA∩ℬ⊆ℛδ,ζ,ϑ+InA∩ℛδ,ζ,ϑ+Inℬ(4)
ℛδ,ζ,ϑ−InA=ℛδ,ζ,ϑ+InAcc;ℛδ,ζ,ϑ+InA=ℛδ,ζ,ϑ−InAcc(5)
Ifm≤n, then ℛδ,ζ,ϑ−ImA⊆ℛδ,ζ,ϑ−InA and ℛδ,ζ,ϑ+InA⊆ℛδ,ζ,ϑ+ImAProof.
(1) By Definition7, we have ℛδ,ζ,ϑ−InA=u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅. Since A⊆ℬ, then we have u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅⊆u∈U|ℛδ,ζ,ϑu−ℬ≤n∧ℛδ,ζ,ϑu≠∅=ℛδ,ζ,ϑ−Inℬ. Therefore, ℛδ,ζ,ϑ−InA⊆ℛδ,ζ,ϑ−Ikℬ. Similarly, ℛδ,ζ,ϑ+nA⊆ℛδ,ζ,ϑ+Inℬ.
(2) and (3) Clear.
(4)ℛδ,ζ,ϑ+InAcc=u∈U|ℛδ,ζ,ϑu∩Ac>n∨ℛδ,ζ,ϑu=∅c=u∈U|ℛδ,ζ,ϑu−A>n∨ℛδ,ζ,ϑu=∅c=u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅=ℛδ,ζ,ϑ−InA and ℛδ,ζ,ϑ−InAcc=u∈U|ℛδ,ζ,ϑu−Ac≤n∧ℛδ,ζ,ϑu≠∅c=u∈U|ℛδ,ζ,ϑu∩A≤n∧ℛδ,ζ,ϑu≠∅c=u∈U|ℛδ,ζ,ϑu∩A>n∨ℛδ,ζ,ϑu=∅=ℛδ,ζ,ϑ+InA.
(5) By Definition7, we have ℛδ,ζ,ϑ−ImA=u∈U|ℛδ,ζ,ϑu−A≤m∧ℛδ,ζ,ϑu≠∅. If m≤n, then we have u∈U|ℛδ,ζ,ϑu−A≤m∧ℛδ,ζ,ϑu≠∅⊆u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅=ℛδ,ζ,ϑ−InA. Hence, ℛδ,ζ,ϑ−ImA⊆ℛδ,ζ,ϑ−InA. Similarly, ℛδ,ζ,ϑ+InA⊆ℛδ,ζ,ϑ+ImA.Remark 5.
The equalityℛδ,ζ,ϑ−In∅=ℛδ,ζ,ϑ+In∅\\=∅ and ℛδ,ζ,ϑ−InV=ℛδ,ζ,ϑ+InV=U does not hold. For example, let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈IU×V is defined by(19)ℛ=0.2,0.6,0.2x1,y1,0.4,0.2,0.3x1,y2,0.5,0.2,0.3x2,y1,0.6,0.1,0.2x2,y2.
Takeδ,ζ,ϑ=0.5,0.1,0.2 and n=2. Then, ℛ0.5,0.1,0.2x1=∅,ℛ0.5,0.1,0.2x2=y2. Thus, ℛ0.5,0.1,0.2−I2∅=x2≠∅,ℛ0.5,0.1,0.2+I2∅=x1≠∅,ℛ0.5,0.1,0.2−I2V=x2≠U, and ℛ0.5,0.1,0.2+I2V=x1≠U.Remark 6.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N, and A∈2V. Then, there does not exist inclusion relation between ℛδ,ζ,ϑ−InA and ℛδ,ζ,ϑ+InA. Consider U,V,δ,ζ,ϑ and ℛ∈IU×V are given in Remark 1. Let A=y1 and n=1. Then, ℛ0.5,0.1,0.2−I1A=x2 and ℛ0.5,0.1,0.2+I1A=x1. This show that there does not exist inclusion relation between ℛδ,ζ,ϑ−InA and ℛδ,ζ,ϑ+InA.
We study the notion of Type-II PFRS according to the gradedn with respect to ℛδ,ζ,ϑ satisfy ℛδ,ζ,ϑ−InA⊆ℛδ,ζ,ϑ+InA.Definition 8.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N s.t. n∈0,minℛδ,ζ,ϑu/2,A and A∈2V. Then,(20)ℛδ,ζ,ϑ−IInA=u∈U|ℛδ,ζ,ϑu−A≤n,ℛδ,ζ,ϑ+IInA=u∈U|ℛδ,ζ,ϑu∩A>n,are called the Type-II lower approximation and the Type-II upper approximation of A according to the graded n with respect to ℛδ,ζ,ϑ on U and V, respectively, and ℛδ,ζ,ϑ−IInA,ℛδ,ζ,ϑ+IInA is called the Type-II PFR approximation of A according to the graded n with respect to ℛδ,ζ,ϑ (briefly, a Type-II PFRS according to the graded n with respect to ℛδ,ζ,ϑ). The subsets ℛδ,ζ,ϑ−IInA, U−ℛδ,ζ,ϑ+IInA, and ℛδ,ζ,ϑ+IInA−ℛδ,ζ,ϑ−IInA-ℛδ,ζ,ϑ−IInA are called the ℛδ,ζ,ϑ-positive region ℛδ,ζ,ϑ−POS, the ℛδ,ζ,ϑ-negative region ℛδ,ζ,ϑ−NEG, and the ℛδ,ζ,ϑ-boundary region ℛδ,ζ,ϑ−BND of A with respect to graded n, respectively.Lemma 1.
Letℛ∈IU×V,δ,ζ,ϑ∈I,n∈N, and A∈2V. Then, the following holds:(1)
n∈0,minℛδ,ζ,ϑu/2,A(2)
The limits of the graden=max0,minℛδ,ζ,ϑu/2,AProof.
(1) For anyu∈U,ℛδ,ζ,ϑu−A∪ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu Then, ℛδ,ζ,ϑu−A+ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu implies that ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu−ℛδ,ζ,ϑu−A. If ℛδ,ζ,ϑu∩A>n, then ℛδ,ζ,ϑu−ℛδ,ζ,ϑu−A>n, and if ℛδ,ζ,ϑu−A≤n, then ℛδ,ζ,ϑu−n>n. Consequently, ℛδ,ζ,ϑu>2n and thus n<ℛδ,ζ,ϑu/2. Since n<ℛδ,ζ,ϑu∩A≤minA,ℛδ,ζ,ϑu and n<ℛδ,ζ,ϑu/2, then n<minℛδ,ζ,ϑu/2,A. Thus, n∈0,minℛδ,ζ,ϑu/2,A.
(2) Follows from (1) above.Theorem 4.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N s.t. n∈0,minℛδ,ζ,ϑu/2,A and A∈2V. Then, the following holds:(1)
ℛδ,ζ,ϑ−IInV=ℛδ,ζ,ϑ+IInV=U; ℛδ,ζ,ϑ−IIn∅=ℛδ,ζ,ϑ+IIn∅=0(2)
ℛδ,ζ,ϑ−IInA⊆ℛδ,ζ,ϑ+IInA(3)
IfA⊆ℬ, then ℛδ,ζ,ϑ−IInA⊆ℛδ,ζ,ϑ−IInℬ and ℛδ,ζ,ϑ+IInA⊆ℛδ,ζ,ϑ+IInℬ(4)
ℛδ,ζ,ϑ−IInA∩ℬ⊆ℛδ,ζ,ϑ−IInA∩ℛδ,ζ,ϑ−IInℬ and ℛδ,ζ,ϑ+IInA∪ℬ⊆ℛδ,ζ,ϑ+IInA∩ℛδ,ζ,ϑ+IInℬ(5)
ℛδ,ζ,ϑ−IInA∪ℬ⊇ℛδ,ζ,ϑ−IInA∪ℛδ,ζ,ϑ−IInℬ and ℛδ,ζ,ϑ+IInA∩ℬ⊆ℛδ,ζ,ϑ+IInA∩ℛδ,ζ,ϑ+IInℬ(6)
ℛδ,ζ,ϑ−IInA=ℛδ,ζ,ϑ+IInAcc and ℛδ,ζ,ϑ+IInA=ℛδ,ζ,ϑ−IInAcc(7)
Ifm≤n, then ℛδ,ζ,ϑ−IImA⊆ℛδ,ζ,ϑ−IInA and ℛδ,ζ,ϑ+IInA⊆ℛδ,ζ,ϑ+IImA.Proof.
We only prove (2). Let∃u∈U and u∈ℛδ,ζ,ϑ−IInA, but u∉ℛδ,ζ,ϑ+IInA. From Definition 8, ℛδ,ζ,ϑu−A≤n,ℛδ,ζ,ϑu∩A>n, and for any u∈U,ℛδ,ζ,ϑu−A∪ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu Since ℛδ,ζ,ϑu−A+ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu and ℛδ,ζ,ϑu−A=ℛδ,ζ,ϑu−ℛδ,ζ,ϑu∩A, then ℛδ,ζ,ϑu−ℛδ,ζ,ϑu∩A≤n and thus ℛδ,ζ,ϑu−n≤n. Consequently, ℛδ,ζ,ϑu>2n and thus n<ℛδ,ζ,ϑu/2. It is contradiction with the limits of n, so the element which only belongs to ℛδ,ζ,ϑ−IInA but does not belong to ℛδ,ζ,ϑ+IInA does not exist.Remark 7.
(1) The equalityℛδ,ζ−n∅=∅ and ℛδ,ζ+nV=U in Theorem 3.10 (2) in [60] are incorrect. (2) For example, let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈JU×V is defined by(21)ℛ=0.2,0.6x1,y1,0.4,0.2x1,y2,0.5,0.2x2,y1,0.8,0.1x2,y2.
Takeδ,ζ=0.5,0.1 and n=2. Then, ℛ0.5,0.1x1=∅,ℛ0.5,0.1x2=y2. Thus, ℛ0.5,0.1−2∅=U≠∅ and ℛ0.5,0.1+2V=∅≠U.
(2) The inclusions inℛδ,ζ−nA⊆ℛδ,ζ+nA and ℛδ,ζ−nA⊆A⊆ℛδ,ζ−nA in Theorem 3.10(1) in [60] are incorrect. For example, let U=xi|i=1,2,3 and V=yi|i=1,2,3 be two three-element sets, and ℛ∈JU×V is defined by(22)ℛ=0.5,0.2x1,y1,0.7,0.1x1,y2,0.4,0.3x1,y3,0.5,0.3x2,y1,0.3,0.4x2,y2,0.4,0.5x2,y3,0.8,0.1x3,y1,0.1,0.7x3,y2,0.6,0.2x3,y3.
Takeδ,ζ=0.6,0.2,n=1 and A=y1,y3. Then, ℛ0.6,0.2x1=y2,ℛ0.6,0.2x2=∅,andℛ0.6,0.2x3=y1,y3. Thus, ℛ0.6,0.2−1A=U⊆∅=ℛ0.6,0.2+1A and ℛ0.6,0.2−1A=U⊆A⊆∅=ℛ0.6,0.2+1A.
(3) The equalitiesℛδ,ζ−nA∩ℬ=ℛδ,ζ−nA∩ℛδ,ζ−nℬ and ℛδ,ζ+nA∪ℬ=ℛδ,ζ+nA∪ℛδ,ζ+nℬ in Theorem 3.10(3) in [60] are incorrect. Consider above example in (2), let A=y1 and ℬ=y2,y3. Then, ℛ0.6,0.2−1A=ℛ0.6,0.2−1ℬ=U and ℛ0.6,0.2+1A=ℛ0.6,0.2+1ℬ=∅. Thus, ℛ0.6,0.2−1A∩ℬ=x1,x2⊊U=ℛ0.6,0.2−1A∩ℛ0.6,0.2−1ℬ and ℛ0.6,0.2+1A∪ℬ=x3⊋∅=ℛ0.6,0.2+1A∪ℛ0.6,0.2+1ℬ.Definition 9.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then,(23)ℛδ,ζ,ϑ−αA=u∈U|ℛδ,ζ,ϑu∩Aℛδ,ζ,ϑu≥α,ℛδ,ζ,ϑ+βA=u∈U|ℛδ,ζ,ϑu∩Aℛδ,ζ,ϑu>β,are called the lower approximation and the upper approximation of A according to the degree α and β with respect to ℛδ,ζ,ϑ on U and V, respectively, and ℛδ,ζ,ϑ−αA,ℛδ,ζ,ϑ+βA is called the picture fuzzy rough approximation of A according to the degree α and β with respect to ℛδ,ζ,ϑ (briefly, a picture fuzzy rough set according to the degree α and β with respect to ℛδ,ζ,ϑ), where A denotes the cardinality of the set A. The subset bnℛδ,ζ,ϑαβA=u∈U|β<ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu<α is called the boundary region of A according to the degree α and β with respect to ℛδ,ζ,ϑ.
The main results are as follows.Theorem 5.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ℛδ,ζ,ϑ−α∅=ℛδ,ζ,ϑ+β∅=∅;ℛδ,ζ,ϑ−αV=ℛδ,ζ,ϑ+βV=U(2)
ℛδ,ζ,ϑ−αA⊆ℛδ,ζ,ϑ+βA(3)
ℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+1−αAcc;ℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ−1−βAcc,(4)
IfA⊆ℬ, then ℛδ,ζ,ϑ−αA⊆ℛδ,ζ,ϑ−αℬ and ℛδ,ζ,ϑ+βA⊆ℛδ,ζ,ϑ+βℬ(5)
ℛδ,ζ,ϑ−αA∩ℬ⊆ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ and ℛδ,ζ,ϑ+βA∩ℬ⊆ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ(6)
ℛδ,ζ,ϑ−αA∪ℬ⊇ℛδ,ζ,ϑ−αA∪ℛδ,ζ,ϑ−αℬ;ℛδ,ζ,ϑ+βA∪ℬ⊇ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ(7)
ℛδ,ζ,ϑ−α1A⊆ℛδ,ζ,ϑ−αA (if 0.5<α<α1≤1); ℛδ,ζ,ϑ+β1A⊆ℛδ,ζ,ϑ+βA (if 0≤β<β1<0.5)Proof.
(1) For anyu∈U,0≤β<α≤1, we have ℛδ,ζ,ϑu∩∅/ℛδ,ζ,ϑu=0≤β<α,u∉ℛδ,ζ,ϑ−α∅, and u∉ℛδ,ζ,ϑ+β∅. Therefore, ℛδ,ζ,ϑ−α∅=ℛδ,ζ,ϑ+β∅=∅. Also, ℛδ,ζ,ϑu∩V/ℛδ,ζ,ϑu=1≥β>α,u∈ℛδ,ζ,ϑ−αV, and u∈ℛδ,ζ,ϑ+βV. Thus, ℛδ,ζ,ϑ−αV=ℛδ,ζ,ϑ+βV=U.
(2)–(4) It follows from Definition9.
(5) Letu∈ℛδ,ζ,ϑ−αA∩ℬ. Then, ℛδ,ζ,ϑu∩A∩ℬ/ℛδ,ζ,ϑu≥α⟹ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α, ℛδ,ζ,ϑu∩ℬ/ℛδ,ζ,ϑu≥α⟹u∈ℛδ,ζ,ϑ−αA, and u∈ℛδ,ζ,ϑ−αℬ⟹u∈ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ. Similarily, we can obtain ℛδ,ζ,ϑ+βA∩ℬ⊆ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ.
(6) It is analogous to (5).
(7) For0.5<α<α1≤1, let u∈ℛδ,ζ,ϑ−α1A. Then, ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α1. Since α<α1, then ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α1>α⟹ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α. Therefore, u∈ℛδ,ζ,ϑ−αA Thus ℛδ,ζ,ϑ−α1A⊆ℛδ,ζ,ϑ−αA Similarly, ℛδ,ζ,ϑ+β1A⊆ℛδ,ζ,ϑ+βA (if 0≤β<β1<0.5).
Equality (5) and (6) of Theorem5 does not hold as the following example.Example 2 (continuation of Example1).
Letα=0.4,β=0.3,A=y1, and ℬ=y2,y3. Then,(1)
ℛ0.5,0.3,0.1−0.4A∩ℬ=ℛ0.5,0.3,0.1+0.3A∩ℬ=∅,ℛ0.5,0.3,0.1−0.4A=ℛ0.5,0.3,0.1+0.3A=x3,ℛ0.5,0.3,0.1−0.4ℬ=ℛ0.5,0.3,0.1+0.3ℬ=x1,x3 and thus ℛ0.5,0.3,0.1−0.4A∩ℬ=∅≠x3=ℛ0.5,0.3,0.1−0.4A∩ℛ0.5,0.3,0.1−0.4ℬ,ℛ0.5,0.3,0.1+0.3A∩ℬ=∅≠x3=ℛ0.5,0.3,0.1+0.3A∩ℛ0.5,0.3,0.1+0.3ℬ(2)
ℛ0.5,0.3,0.1−0.4A∪ℬ=ℛ0.5,0.3,0.1+0.3A∪ℬ=U, and thus ℛ0.5,0.3,0.1−0.4A∪ℬ=U≠x1,x3=ℛ0.5,0.3,0.1−0.4A∪ℛ0.5,0.3,0.1−0.4ℬ,ℛ0.5,0.3,0.1+0.3A∪ℬ=U≠x1,x3=ℛ0.5,0.3,0.1+0.3A∪ℛ0.5,0.3,0.1+0.3ℬ
The relationship among the regulation for lower approximation, upper approximation, and boundary region with errorα and β is discussed as the following theorems.Theorem 6.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0<r<1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the following holds:(1)
limα⟶r+ℛδ,ζ,ϑ−αA=∪α>rℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+rA(2)
limβ⟶r−ℛδ,ζ,ϑ+βA=∩β<rℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ−rA(3)
limβ⟶r−;α⟶r+bnδ,ζ,ϑαβA=∩β<r<αℛδ,ζ,ϑ+βA\ℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+rA\ℛδ,ζ,ϑ−rA=u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=rProof.
(1) From Definition9, when α>r, we have ℛδ,ζ,ϑ−αA=u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α⊆u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu>r=ℛδ,ζ,ϑ+rA is hold. In addition, by Theorem 5 (7), we have ℛδ,ζ,ϑ−αA increase with error α decrease. Thus, limα⟶r+ℛδ,ζ,ϑ−αA=∩α>rℛδ,ζ,ϑ−αA⊆ℛδ,ζ,ϑ+rA. Conversely, if there exists u0∈ℛδ,ζ,ϑ+rA\∪α>rℛδ,ζ,ϑ−αA, for any α>r, ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0>r and ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0<α. That is, ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0<r. It contradicts with ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0>r, i.e., u0∈∪α>rℛδ,ζ,ϑ−αA. Consequently, ∪α>rℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+rA. Thus, limα⟶r+ℛδ,ζ,ϑ−αA=∪α>rℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+rA hold.
(2) It is analogous to (1) above.
(3) We knowℛδ,ζ,ϑ+rA\ℛδ,ζ,ϑ−rA=u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=r⊆bnδ,ζ,ϑαβA is held. In addition, we know that when α decreased to r and β decreased to r, the boundary region bnδ,ζ,ϑαβA will decrease. Then, we have ℛδ,ζ,ϑ+rA\ℛδ,ζ,ϑ−rA=u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=r⊆limβ⟶r−;α⟶r+bnδ,ζ,ϑαβA=∩β<r<αℛδ,ζ,ϑ+βA\ℛδ,ζ,ϑ−αA. Conversely, if there exists u0∈∩β<r<αℛδ,ζ,ϑ+βA\ℛδ,ζ,ϑ−αA\u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=r, then for any β<r<α, we have ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0>β,ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0<α and ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0≠r. Furthermore, there is ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0≥r by ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0>β and β<r. Similarily, there is ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0≤r by ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0<α and α>r. This is a contradiction. Consequently, there is ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0=r. That is, u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=r=∩β<r<αℛδ,ζ,ϑ+βA\ℛδ,ζ,ϑ−αA.Theorem 7.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0<r<1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the following holds:(1)
limα⟶r−ℛδ,ζ,ϑ−αA=∩α<rℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ−rA(2)
limβ⟶r+ℛδ,ζ,ϑ+βA=∪β>rℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ+rAProof.
Obvious.Remark 8.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0<r<1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the following does not hold:(1)
limα⟶r+ℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ−rA (i.e., the lower approximation ℛδ,ζ,ϑ−αA is not right continuous with error α)(2)
limβ⟶r−ℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ+rA (i.e., the upper approximation ℛδ,ζ,ϑ+βA is not left continuous with error β)Example 3 (continuation of Example1).
LetA=y2,y3. Then, we can obtain the following relations:(1)
limα⟶0.5+ℛδ,ζ,ϑ−αA=x1,ℛδ,ζ,ϑ−0.5A=x1,x3. Consequently, limα⟶0.5−ℛδ,ζ,ϑ−αA≠ℛδ,ζ,ϑ−0.5A.(2)
limβ⟶0.5−ℛδ,ζ,ϑ+βA=x1,x3,ℛδ,ζ,ϑ+0.5A=x1. Consequently, limβ⟶0.5−ℛδ,ζ,ϑ+βA≠ℛδ,ζ,ϑ+0.5A.Definition 10.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the precision ρδ,ζ,ϑβαA and rough degree μδ,ζ,ϑβαA of A with error α and β are defined by(24)ρδ,ζ,ϑβαA=1−ℛδ,ζ,ϑ−αAℛδ,ζ,ϑ+βA,μδ,ζ,ϑβαA=1−ρδ,ζ,ϑβαA.
Obviously,0≤ρδ,ζ,ϑβαA≤1 and 0≤μδ,ζ,ϑβαA≤1.Lemma 2.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the following holds:(1)
ρδ,ζ,ϑβαA does not decrease with error α and does not increase with error β(2)
μδ,ζ,ϑβαA does not increase with error α and does not decrease with error βProof.
We only prove (1) and then the proof of (2) can be obtained using similar techniques.
(1) Sinceℛδ,ζ,ϑ−α1A⊆ℛδ,ζ,ϑ−αA0.5<α<α1≤1 by Theorem 5 (7), then there is ℛδ,ζ,ϑ−α1A/ℛδ,ζ,ϑ+βA≤ℛδ,ζ,ϑ−αA/ℛδ,ζ,ϑ+βA implies that 1−ℛδ,ζ,ϑ−α1A/ℛδ,ζ,ϑ+βA≥1−ℛδ,ζ,ϑ−αA/ℛδ,ζ,ϑ+βA. Therefore, ρδ,ζ,ϑβαA≤ρδ,ζ,ϑβα1A. Similarly, ρδ,ζ,ϑβ1αA≤ρδ,ζ,ϑβαA for ℛδ,ζ,ϑ+β1A⊆ℛδ,ζ,ϑ+βA0≤β<β1<0.5.Theorem 8.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, A,ℬ∈2V, and A⊆ℬ. Then, the following holds:(1)
Ifℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ−αℬ, there is ρδ,ζ,ϑβαA≤ρδ,ζ,ϑβαℬ and μδ,ζ,ϑβαℬ≤μδ,ζ,ϑβαA(2)
Ifℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ+βℬ, there is ρδ,ζ,ϑβαℬ≤ρδ,ζ,ϑβαA and μδ,ζ,ϑβαA≤μδ,ζ,ϑβαℬ(3)
Ifℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ−αℬ and ℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ+βℬ, there is ρδ,ζ,ϑβαA≤ρδ,ζ,ϑβαℬ and μδ,ζ,ϑβαA=μδ,ζ,ϑβαℬProof.
It follows from Theorem5 (4) and Definition 10.Theorem 9.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ρδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≤ρδ,ζ,ϑβαAℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬℛδ,ζ,ϑ+βℬ−ρδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ(2)
μδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≥μδ,ζ,ϑβαAℛδ,ζ,ϑ+βA+μδ,ζ,ϑβαℬℛδ,ζ,ϑ+βℬ−μδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬProof.
(1) From Definition10, we have ρδ,ζ,ϑβαA∪ℬ=1−ℛδ,ζ,ϑ−αA∪ℬ/ℛδ,ζ,ϑ+βA∪ℬ=1−ℛδ,ζ,ϑ−αA∪ℬ/ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ ≤ 1−ℛδ,ζ,ϑ−αA∪ℛδ,ζ,ϑ−αℬ/ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ. Consequently, ρδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≤ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA∪ℛδ,ζ,ϑ−αℬ. Similarly, ρδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βX∩ℛδ,ζ,ϑ+βℬ ≤ ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ. We know X∪Y=X+Y−X∩Y. Then,(25)ρδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≤ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA∪ℛδ,ζ,ϑ−αℬ=ℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA+ℛδ,ζ,ϑ−αℬ−ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ=ℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA−ℛδ,ζ,ϑ−αℬ−ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ≤ℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA−ℛδ,ζ,ϑ−αℬ−ρpβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ.
Also, by the relation ofρδ,ζ,ϑβαA=1−ℛδ,ζ,ϑ−αA/ℛδ,ζ,ϑ+βA and ρpβαℬ=1−ℛδ,ζ,ϑ−αℬ/ℛδ,ζ,ϑ+βℬ, then ℛδ,ζ,ϑ+βA−ℛδ,ζ,ϑ−αA=ρδ,ζ,ϑβαAℛδ,ζ,ϑ+βA and ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αℬ=ρδ,ζ,ϑβαℬℛδ,ζ,ϑ+βℬ. Thus, we have(26)ρδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≤ℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA−ℛδ,ζ,ϑ−αℬ−ρδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ=ρδ,ζ,ϑβαAℛδ,ζ,ϑ+βA+ρδ,ζ,ϑβαℬℛδ,ζ,ϑ+βℬ−ρδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ.
Hence, (1) is holding.
(2) It can be easily proved by the relationshipμδ,ζ,ϑβαA=1−ρδ,ζ,ϑβαA.


### 3.3. Generalized Picture Fuzzy Rough Sets Based onδ,ζ,ϑ-Cut of ℛ on Two Different Universes

Definition 11.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛ^iδ,ζ,ϑu≠∅, and A∈2V. Then,(1)
ℛ−∧inℛ^iδ,ζ,ϑIαA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α∧ℛ^2δ,ζ,ϑu∩A/ℛ^2δ,ζ,ϑu≥α∧⋯∧ℛ^iδ,ζ,ϑu∩A/ℛ^iδ,ζ,ϑu≥α and ℛ+∧inℛ^iδ,ζ,ϑIβA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu>β∧ℛ^2δ,ζ,ϑu∩A/ℛ^2δ,ζ,ϑu>β∧⋯∧ℛ^iδ,ζ,ϑu∩A/ℛ^iδ,ζ,ϑu>β are called the Type-I lower ℛ^iδ,ζ,ϑ-approximation and the Type-I upper ℛ^iδ,ζ,ϑ-approximation of A according to the degree α and β with respect to ℛiδ,ζ,ϑ on U and V, respectively, and ℛ−∧inℛ^iδ,ζ,ϑIαA,ℛ+∧inℛ^iδ,ζ,ϑIβA is called the Type-I generalized picture fuzzy rough approximation of A according to the degree α and β with respect to ℛiδ,ζ,ϑ (briefly, the Type-I generalized picture fuzzy rough set according to the degree α and β with respect to ℛiδ,ζ,ϑ).(2)
ℛ−∨inℛ^iδ,ζ,ϑIIαA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α∨ℛ^2δ,ζ,ϑu∩A/ℛ^2δ,ζ,ϑu≥α∨⋯∨ℛ^iδ,ζ,ϑu∩A/ℛ^iδ,ζ,ϑu≥α and ℛ+∨inℛ^iδ,ζ,ϑIIβA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu>β∨ℛ^2δ,ζ,ϑu∩A/ℛ^2δ,ζ,ϑu>β∨⋯∨ℛ^iδ,ζ,ϑu∩A/ℛ^iδ,ζ,ϑu>β are called the Type-II lower ℛ^iδ,ζ,ϑ-approximation and the Type-II upper ℛ^iδ,ζ,ϑ-approximation of A according to the degree α and β with respect to ℛiδ,ζ,ϑ on U and V, respectively, and ℛ−∨inℛ^iδ,ζ,ϑIIαA,ℛ+∨inℛ^iδ,ζ,ϑIIβA is called the Type-II generalized picture fuzzy rough approximation of A according to the degree α and β with respect to ℛiδ,ζ,ϑ (briefly, the Type-II generalized picture fuzzy rough set according to the degree α and β with respect to ℛiδ,ζ,ϑ).Remark 9.
(1) From Definition11, we have(i)
ℛ−∧inℛ^iδ,ζ,ϑIαA⊆ℛ−∨inℛ^iδ,ζ,ϑIIαA(ii)
ℛ+∧inℛ^iδ,ζ,ϑIβA⊆ℛ−∨inℛ^iδ,ζ,ϑIIβA
(2) If we takeα=1 and β=0 in Definition 11, then we can obtain in the following definitions:(i)
ℛ−∧inℛ^iδ,ζ,ϑIA=u∈U|ℛ^1δ,ζ,ϑu⊆A∧ℛ^2δ,ζ,ϑu⊆A∧⋯∧ℛ^iδ,ζ,ϑu⊆A and ℛ+∧inℛ^iδ,ζ,ϑIA=u∈U|ℛ^1δ,ζ,ϑu∩A≠∅∧ℛ^2δ,ζ,ϑu∩A≠∅∧⋯∧ℛ^iδ,ζ,ϑu∩A≠∅.(ii)
ℛ−∨inℛ^iδ,ζ,ϑIIA=u∈U|ℛ^1δ,ζ,ϑu⊆A∨ℛ^2δ,ζ,ϑu⊆A∨⋯∨ℛ^iδ,ζ,ϑu⊆A and ℛ+∨inℛ^iδ,ζ,ϑIIA=u∈U|ℛ^1δ,ζ,ϑu∩A≠∅∨ℛ^2δ,ζ,ϑu∩A≠∅∨⋯∨ℛ^iδ,ζ,ϑu∩A≠∅.
The main results are as follows.Theorem 10.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ℛ−∧inℛ^iδ,ζ,ϑIα∅=ℛ+∧inℛ^iδ,ζ,ϑIβ∅=∅; ℛ−∧inℛ^iδ,ζ,ϑIαU=ℛ+∧inℛ^iδ,ζ,ϑIβU=V(2)
ℛ−∧inℛ^iδ,ζ,ϑIαA⊆ℛ+∧inℛ^iδ,ζ,ϑIβA(3)
IfA⊆ℬ, then ℛ−∧inℛ^iδ,ζ,ϑIαA⊆ℛ−∧inℛ^iδ,ζ,ϑIαℬ;ℛ+∧inℛ^iδ,ζ,ϑIβA⊆ℛ+∧inℛ^iδ,ζ,ϑIβℬ(4)
ℛ−∧inℛ^iδ,ζ,ϑIαA∩ℬ⊆ℛ−∧inℛ^iδ,ζ,ϑIαA∩ℛ−∧inℛ^iδ,ζ,ϑIαℬ; ℛ+∧inℛ^iδ,ζ,ϑIβA∩ℬ⊆ℛ+∧inℛ^iδ,ζ,ϑIβA∩ℛ+∧inℛ^iδ,ζ,ϑIβℬ∩ℛ+∧inℛ^iδ,ζ,ϑIβℬ(5)
ℛ−∧inℛ^iδ,ζ,ϑIαA∪ℬ⊇ℛ−∧inℛ^iδ,ζ,ϑIαA∪ℛ−∧inℛ^iδ,ζ,ϑIαℬ; ℛ+∧inℛ^iδ,ζ,ϑIβA∪ℬ⊇ℛ+∧inℛ^iδ,ζ,ϑIβA∪ℛ+∧inℛ^iδ,ζ,ϑIβℬ(6)
ℛ−∧inℛ^iδ,ζ,ϑIα1A⊆ℛ−∧inℛ^iδ,ζ,ϑIαA (if 0.5<α<α1≤1); ℛ+∧inℛ^iδ,ζ,ϑIβ1A⊆ℛ+∧inℛ^iδ,ζ,ϑIβA (if 0≤β<β1<0.5)Proof.
By Definition11, the result can be similarly proven as Theorem 5.Remark 10.
LetA∈2V and 0≤β<α≤1. Then,(1)
ℛ−∨inℛ^iδ,ζ,ϑIαA=ℛ+∨inℛ^iδ,ζ,ϑI1−αAcc(2)
ℛ+∨inℛ^iδ,ζ,ϑIβA=ℛ−∨inℛ^iδ,ζ,ϑI1−βAccdoes not hold by the following example.Example 4.
LetU=xi|i=1,2 and V=yi|i=1,2 be two two-element set, and ℛ^1,ℛ^2,ℛ^3∈IU×V is defined by(27)ℛ^1=0.2,0.4,0.4x1,y1,0.4,0.2,0.3x1,y2,0.5,0.3,0.2x2,y1,0.7,0.1,0.2x2,y2,ℛ^2=0.3,0.2,0.4x1,y1,0.1,0.2,0.2x1,y2,0.5,0.3,0.1x2,y1,0.6,0.1,0.3x2,y2,ℛ^3=0.2,0.3,0.5x1,y1,0.6,0.1,0.2x1,y2,0.7,0.1,0.1x2,y1,0.4,0.3,0.3x2,y2.
Takeδ,ζ,ϑ=0.5,0.3,0.2. Then, ℛ^10.5,0.3,0.2x1=∅, ℛ^10.5,0.3,0.2x2=V,ℛ^20.5,0.3,0.2x1=∅,ℛ^20.5,0.3,0.2x2=y1, ℛ^30.5,0.3,0.2x1=y2,ℛ^30.5,0.3,0.2x2=y1. Let A=y1,α=0.6, and β=0.4. Thus,(1)
ℛ−∨inℛ^iδ,ζ,ϑIαA=∅≠x1=ℛ+∨inℛ^iδ,ζ,ϑI1−αAcc(2)
ℛ+∨inℛ^iδ,ζ,ϑIβA=x2≠U=ℛ−∨inℛ^iδ,ζ,ϑI1−βAcc
From Remark9 (2), we can conclude the following corollary.Corollary 2.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ℛ−∨inℛ^iδ,ζ,ϑI∅=ℛ+∨inℛ^iδ,ζ,ϑI∅=∅; ℛ−∨inℛ^iδ,ζ,ϑIU=ℛ+∨inℛ^iδ,ζ,ϑIU=V(2)
ℛ−∨inℛ^iδ,ζ,ϑIA⊆ℛ+∨inℛ^iδ,ζ,ϑIA(3)
IfA⊆ℬ, then ℛ−∨inℛ^iδ,ζ,ϑIA⊆ℛ−∨inℛ^iδ,ζ,ϑIℬ;ℛ+∨inℛ^iδ,ζ,ϑIA⊆ℛ+∨inℛ^iδ,ζ,ϑIℬ(4)
ℛ−∨inℛ^iδ,ζ,ϑIA∩ℬ⊆ℛ−∨inℛ^iδ,ζ,ϑIA∩ℛ−∨inℛ^iδ,ζ,ϑIℬ; ℛ+∨inℛ^iδ,ζ,ϑIA∩ℬ⊆ℛ+∨inℛ^iδ,ζ,ϑIA∩ℛ+∨inℛ^iδ,ζ,ϑIℬ(5)
ℛ−∨inℛ^iδ,ζ,ϑIA∪ℬ⊇ℛ−∨inℛ^iδ,ζ,ϑIA∪ℛ−∨inℛ^iδ,ζ,ϑIℬ; ℛ+∨inℛ^iδ,ζ,ϑIA∪ℬ⊇ℛ+∨inℛ^iδ,ζ,ϑIA∪ℛ+∨inℛ^iδ,ζ,ϑIℬTheorem 11.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V.
Then, the following holds:(1)
ℛ−∨inℛ^iδ,ζ,ϑIIα∅=ℛ+∨inℛ^iδ,ζ,ϑIIβ∅=∅; ℛ−∨inℛ^iδ,ζ,ϑIIαU=ℛ+∨inℛ^iδ,ζ,ϑIIβU=V(2)
ℛ−∨inℛ^iδ,ζ,ϑIIαA⊆ℛ+∨inℛ^iδ,ζ,ϑIIβA(3)
IfA⊆ℬ, then ℛ−∨inℛ^iδ,ζ,ϑIIαA⊆ℛ−∨inℛ^iδ,ζ,ϑIIβℬ; ℛ+∨inℛ^iδ,ζ,ϑIIβA⊆ℛ−∨inℛ^iδ,ζ,ϑIIβℬ(4)
ℛ−∨inℛ^iδ,ζ,ϑIIαA∩ℬ⊆ℛ−∨inℛ^iδ,ζ,ϑIIαA∩ℛ−∨inℛ^iδ,ζ,ϑIIαℬ; ℛ+∨inℛ^iδ,ζ,ϑIIβA∩ℬ⊆ℛ+∨inℛ^iδ,ζ,ϑIIβA∩ℛ+∨inℛ^iδ,ζ,ϑIIβℬ(5)
ℛ−∨inℛ^iδ,ζ,ϑIIαA∪ℬ⊇ℛ−∨inℛ^iδ,ζ,ϑIIαA∪ℛ−∨inℛ^iδ,ζ,ϑIIαℬ; ℛ+∨inℛ^iδ,ζ,ϑIIβA∪ℬ⊇ℛ+∨inℛ^iδ,ζ,ϑIIβA∪ℛ+∨inℛ^iδ,ζ,ϑIIβℬ(6)
ℛ−∨inℛ^iδ,ζ,ϑIIα1A⊆ℛ−∨inℛ^iδ,ζ,ϑIIαA (if 0.5<α<α1≤1); ℛ+∨inℛ^iδ,ζ,ϑIIβ1A⊆ℛ+∨inℛ^iδ,ζ,ϑIIβA (if 0≤β<β1<0.5)Proof.
It follows from Definition11 (2).
From Remark9 (2), we can conclude the following corollary.Corollary 3.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then the following holds:(1)
ℛ−∨inℛ^iδ,ζ,ϑII∅=ℛ+∨inℛ^iδ,ζ,ϑII∅=∅; ℛ−∨inℛ^iδ,ζ,ϑIIU=ℛ+∨inℛ^iδ,ζ,ϑIIU=V(2)
ℛ−∨inℛ^iδ,ζ,ϑIIA⊆ℛ+∨inℛ^iδ,ζ,ϑIIA(3)
IfA⊆ℬ, then ℛ−∨inℛ^iδ,ζ,ϑIIA⊆ℛ−∨inℛ^iδ,ζ,ϑIIℬ; ℛ+∨inℛ^iδ,ζ,ϑIIA⊆ℛ+∨inℛ^iδ,ζ,ϑIIℬ(4)
ℛ−∨inℛ^iδ,ζ,ϑIIA∩ℬ⊆ℛ−∨inℛ^iδ,ζ,ϑIIA∩ℛ−∨inℛ^iδ,ζ,ϑIIℬ; ℛ+∨inℛ^iδ,ζ,ϑIIA∩ℬ⊆ℛ+∨inℛ^iδ,ζ,ϑIIA∩ℛ+∨inℛ^iδ,ζ,ϑIIℬ(5)
ℛ−∨inℛ^iδ,ζ,ϑIIA∪ℬ⊇ℛ−∨inℛ^iδ,ζ,ϑIIA∪ℛ−∨inℛ^iδ,ζ,ϑIIℬ; ℛ+∨inℛ^iδ,ζ,ϑIIA∪ℬ⊇ℛ+∨inℛ^iδ,ζ,ϑIIA∪ℛ+∨inℛ^iδ,ζ,ϑIIℬTheorem 12.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ℛ−∨inℛ^iδ,ζ,ϑIαA=ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∩ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA∩⋯∩ℛδ,ζ,ϑ−ℛ^iδ,ζ,ϑαA(2)
ℛ+∨inℛ^iδ,ζ,ϑIβA=ℛδ,ζ,ϑ+ℛ^1δ,ζ,ϑβA∩ℛδ,ζ,ϑ+ℛ^2δ,ζ,ϑβA∩⋯∩ℛδ,ζ,ϑ+ℛ^iδ,ζ,ϑβA(3)
ℛ−∨inℛ^iδ,ζ,ϑIIαA=ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA∪⋯∪ℛδ,ζ,ϑ−ℛ^iδ,ζ,ϑαA(4)
ℛ+∨inℛ^iδ,ζ,ϑIIβA=ℛδ,ζ,ϑ+ℛ^1δ,ζ,ϑβA∪ℛδ,ζ,ϑ+ℛ^2δ,ζ,ϑβA∪⋯∪ℛδ,ζ,ϑ+ℛ^iδ,ζ,ϑβAProof.
We only prove (3), while rest can be proven similarly.
(3) We show thatℛ−∨i=12ℛ^iδ,ζ,ϑIIαA=ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA. Let u∈ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαAu∈U, where ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α, but u∈ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA. So, ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA⊆ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA. Similarly, ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA⊆ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA. Consequently, ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA⊆ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA. Conversely, for u∈ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA and by Definition 11, we have either ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α, then u∈ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA or ℛ^2δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α, and then u∈ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA. Hence, u∈ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA, and thus ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA⊆ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA. Therefore, (3) is holding.


## 3.1. Picture Fuzzy Rough Sets Based onδ,ζ,ϑ-Cut of ℛ on Two Different Universes

We will begin by defining theδ,ζ,ϑ-cut of ℛ and will subsequently define a picture fuzzy rough set based on δ,ζ,ϑ-cut.Definition 5.
Letℛ∈IU×V and δ,ζ,ϑ∈I. Then,(i)
ℛδ,ζ,ϑ=u,v∈U×V|ℛu,v≥δ,ζ,ϑ=u,v∈U×V|p1∘ℛu,v≥δ,p2∘ℛu,v≤ζ,p3∘ℛu,v≤ϑ is called the δ,ζ,ϑ-cut of ℛ, and(5)ℛδ,ζ,ϑu=v∈V|ℛu,v≥δ,ζ,ϑ.(ii)
LetA∈2V. Then,(6)ℛδ,ζ,ϑ−A=u∈U|ℛδ,ζ,ϑu⊆A∧ℛδ,ζ,ϑu≠∅,(i.e., the lower approximation ofA) and(7)ℛδ,ζ,ϑ+A=u∈U|ℛδ,ζ,ϑu∩A≠∅∨ℛδ,ζ,ϑu=∅,(i.e., the upper approximation ofA).
The pairℛδ,ζ,ϑ−A,ℛδ,ζ,ϑ+A is a PFR approximation of A with respect to ℛδ,ζ,ϑ.
Now, we present some properties based on PFRS as follows.Theorem 1.
Letℛ∈IU×V, δ,ζ,ϑ∈I, and A∈2V. Then, the following holds:(1)
ℛδ,ζ,ϑ−∅=∅;ℛδ,ζ,ϑ+V=U(2)
ℛδ,ζ,ϑ−A⊆ℛδ,ζ,ϑ+A(3)
IfA⊆ℬ, then ℛδ,ζ,ϑ−A⊆ℛδ,ζ,ϑ−ℬ; ℛδ,ζ,ϑ+A⊆ℛδ,ζ,ϑ+ℬ(4)
ℛδ,ζ,ϑ−A∩ℬ=ℛδ,ζ,ϑ−A∩ℛδ,ζ,ϑ−ℬ; ℛδ,ζ,ϑ+A∪ℬ=ℛδ,ζ,ϑ+A∪ℛδ,ζ,ϑ+ℬ(5)
ℛδ,ζ,ϑ−A∪ℬ⊇ℛδ,ζ,ϑ−A∪ℛδ,ζ,ϑ−ℬ; ℛδ,ζ,ϑ+A∩ℬ⊆ℛδ,ζ,ϑ+A∩ℛδ,ζ,ϑ+ℬ(6)
ℛδ,ζ,ϑ−A=ℛδ,ζ,ϑ+Acc; ℛδ,ζ,ϑ+A=ℛδ,ζ,ϑ−AccProof.
We only prove (6).
ℛδ,ζ,ϑ+Acc=u∈U|ℛδ,ζ,ϑu∩Ac≠∅∨ℛδ,ζ,ϑu=∅c = u∈U|ℛδ,ζ,ϑu∩Ac=∅∧ℛδ,ζ,ϑu≠∅=u∈U|ℛδ,ζ,ϑu⊆A∧ℛδ,ζ,ϑu≠∅=ℛδ,ζ,ϑ−A, and ℛδ,ζ,ϑ−Acc=u∈U|ℛδ,ζ,ϑu⊆Ac∧ℛδ,ζ,ϑu≠∅c = u∈U|ℛδ,ζ,ϑu∩A=∅∧ℛδ,ζ,ϑu≠∅c = u∈U|ℛδ,ζ,ϑu∩A≠∅∨ℛδ,ζ,ϑu=∅=ℛδ,ζ,ϑ+A.
The equality of Theorem1 (5) does not hold as the following example.Example 1.
SupposeU=xi|i=1,2,3 and V=yi|i=1,2,3 be two three-element set, and ℛ∈IU×V is defined by(8)ℛ=0.2,0.5,0.3x1,y1,0.6,0.3,0x1,y2,0.4,0.3,0.1x1,y3,0.4,0.2,0.3x2,y1,0.3,0.3,0.2x2,y2,0.1,0.6,0.2x2,y3,0.5,0.2,0.1x3,y1,0.5,0.5,0x3,y2,0.6,0.1,0.1x3,y3.
Takeδ,ζ,ϑ=0.5,0.3,0.1. Then, ℛ0.5,0.3,0.1x1=y2, ℛ0.5,0.3,0.1x2=∅, and ℛ0.5,0.3,0.1x3=y1,y3.
LetA=y1 and ℬ=y2,y3. Thus, ℛ0.5,0.3,0.1−A=∅,ℛ0.5,0.3,0.1−ℬ=x1,ℛ0.5,0.3,0.1−A∪ℬ=x1,x3≠x1=ℛ0.5,0.3,0.1−A∪ℛ0.5,0.3,0.1−ℬ and ℛ0.5,0.3,0.1+A=x2,x3,ℛ0.5,0.3,0.1+ℬ=U,ℛ0.5,0.3,0.1+A∩ℬ=x2≠x2,x3=ℛ0.5,0.3,0.1+A∩ℛ0.5,0.3,0.1+ℬ.Remark 1.
In general,ℛδ,ζ,ϑ+∅=∅ and ℛδ,ζ,ϑ−V=U do not hold. For example, let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈IU×V is defined by(9)ℛ=0.2,0.4,0.4x1,y1,0.4,0.2,0.3x1,y2,0.5,0.3,0.2x2,y1,0.7,0.1,0.2x2,y2.
Takeδ,ζ,ϑ=0.5,0.1,0.2. Then, ℛ0.5,0.1,0.2x1=∅ and ℛ0.5,0.1,0.2x2=y2. Thus, ℛ0.5,0.1,0.2+∅=x1≠∅ and ℛ0.5,0.1,0.2−V=x2≠U.Remark 2.
It should be pointed that some conclusions in [58–60] which are similar to those in Theorem 1 are wrong. Firstly, the equality ℛδ,ζ−∅=∅ and ℛδ,ζ+V=U in Theorem 3.1(2) in [60] are incorrect. Let U=xi|i=1,2, V=yi|i=1,2, and ℛ∈JU×V be defined by(10)ℛ=0.5,0.2x1,y1,0.7,0.1x1,y2,0.5,0.3x2,y1,0.3,0.4x2,y2.
Takeδ,ζ=0.5,0.1. Then, ℛ0.5,0.1−∅ and ℛ0.5,0.1+V in the sense of [60] are not ∅ and U, but x2 and x1, respectively. Secondly, the inclusion ℛδ−A⊆ℛδ+A in Theorem 3.1(1) in [58] is incorrect. Let U=xi|i=1,2,3 and V=yi|i=1,2,3 be two three-element sets, and ℛ∈0,1U×V, is defined by(11)ℛ=0.5x1,y1,0.7x1,y2,0.4x1,y3,0.5x2,y1,0.3x2,y2,0.4x2,y3,0.8x3,y1,0.1x3,y2,0.6x3,y3.
Takeδ=0.6 and A=y1,y3. Then, ℛ0.6x1=y2, ℛ0.6x2=∅, and ℛ0.6x3=y1,y3, and thus ℛ0.6−A (resp., ℛ0.6+A) in the sense of [58] is x2,x3 (resp., x3). Therefore, ℛδ−A⊆ℛδ+A. Analogously, assertions ℛδ,ζ−A⊆ℛδ,ζ+A in Theorem 3.1(1) [59] and ℛδ,ζ−A⊆A⊆ℛδ,ζ+A in Theorem 3.1(1) [60] are incorrect. Let U=xi|i=1,2,3 and V=yi|i=1,2,3 be two three-element sets, and ℛ∈0,12U×V and also ℛ∈JU×V is defined by(12)ℛ=0.2,0.5x1,y1,0.6,0.3x1,y2,0.4,0.3x1,y3,0.4,0.2x2,y1,0.3,0.3x2,y2,0.1,0.6x2,y3,0.5,0.2x3,y1,0.5,0.5x3,y2,0.6,0.1x3,y3.
Takeδ,ζ=0.5,0.3, and A=y1,y3. Then, ℛ0.5,0.3x1=y2, ℛ0.5,0.3x2=∅, and ℛ0.5,0.3x3=y1,y3, and thus ℛ0.5,0.3−A=x2,x3⊆x3=ℛ0.5,0.3+A and ℛ0.5,0.3−A=x2,x3⊆A⊆x3=ℛ0.5,0.3+A.
To correct some results in [58–60] in above Remark 2, we will give new notations of lower and upper approximations as follow:Definition 6.
LetA∈2V. Then,(1)
Forℛ∈0,1U×V and δ∈0,1, we have(13)ℛδ−A=u∈U|ℛδu⊆A∧ℛδu≠∅,ℛδ+A=u∈U|ℛδu∩A≠∅∨ℛδu=∅.(2)
Forℛ∈0,12U×V and δ,ζ∈0,12, we have(14)ℛδ,ζ−A=u∈U|ℛδ,ζu⊆A∧ℛδ,ζu≠∅,ℛδ,ζ+A=u∈U|ℛδ,ζu∩A≠∅∨ℛδ,ζu=∅.(3)
Forℛ∈JU×V and δ,ζ∈J, we have (cf. [61])(15)ℛδ,ζ−A=u∈U|ℛδ,ζu⊆A∧ℛδ,ζu≠∅,ℛδ,ζ+A=u∈U|ℛδ,ζu∩A≠∅∨ℛδ,ζu=∅.Theorem 2.
Letℛ,S∈IU×V, δ,ζ,ϑ∈I, and A∈2V. If ℛ≤S, then ℛδ,ζ,ϑ−A⊆Sδ,ζ,ϑ−A and ℛδ,ζ,ϑ+A⊆Sδ,ζ,ϑ+A.Proof.
Ifℛ≤S, then p1∘ℛu,v≤p1∘Su,v,p2∘ℛu,v≤p2∘Su,v and p3∘ℛu,v≥p3∘Su,v for all u,v∈U×V. By Definition 5, we have ℛδ,ζ,ϑu=v∈V|ℛu,v≥δ,ζ,ϑ⊆v∈V|Su,v≥δ,ζ,ϑ=Sδ,ζ,ϑu. Therefore, ℛδ,ζ,ϑu⊆A∧ℛδ,ζ,ϑu≠∅⊆Sδ,ζ,ϑu⊆A∧Sδ,ζ,ϑu≠∅ for all u∈U. Thus, ℛδ,ζ,ϑ−A⊆Sδ,ζ,ϑ−A. Similarly, ℛδ,ζ,ϑ+A⊆ℛδ,ζ,ϑ+A.Corollary 1.
(1) Letℛ,S∈0,1U×V,δ∈0,1, and A∈2V. If ℛ≤S, then ℛδ−A⊆Sδ−A and ℛδ+A⊆Sδ+A.
(2) Letℛ,S∈0,12U×V,δ,ζ∈0,12, and A∈2V. If ℛ≤S, then ℛδ,ζ−A⊆Sδ,ζ−A and ℛδ,ζ+A⊆Sδ,ζ+A.
(3) Letℛ,S∈JU×V,δ,ζ∈J, and A∈2V. If ℛ≤S, then ℛδ,ζ−A⊆Sδ,ζ−A and ℛδ,ζ+A⊆Sδ,ζ+A.Proof.
It follows from Definition6.
Next, we give comparison between some properties by Definition 3.3 in [58], Definition 3.2 in [59], Definition 3.3 in [60] and 6 as shown in Tables 1–3:Table 1
Comparison between some properties by Definition 3.3 in [58] and 6 (1).
Definition 3.3 in [58]Definition6 (1)ℛδ−∅=∅×√ℛδ+∅=∅√×ℛδ−V=U√×ℛδ+V=U×√ℛδ−A⊆ℛδ+A×√ℛδ−A⊆Sδ−Aifℛ≤S×√ℛδ+A⊆Sδ+Aifℛ≤S×√ℛδ1−A⊆ℛδ2−Aifδ1≤δ2√×ℛδ2+A⊆ℛδ1+Aifδ1≤δ2√×(√) indicates that the property is satisfied.Table 2
Comparison between some properties by Definition 3.2 in [59] and 6 (2).
Definition 3.2 in [59]Definition6 (2)ℛδ,ζ−∅=∅×√ℛδ,ζ+∅=∅√×ℛδ,ζ−V=U√×ℛδ,ζ+V=U×√ℛδ,ζ−A⊆ℛδ,ζ+A×√ℛδ,ζ−A⊆Sδ,ζ−Aifℛ≤S×√ℛδ,ζ+A⊆Sδ,ζ+Aifℛ≤S×√ℛδ1,ζ1−A⊆ℛδ2,ζ2−Aifδ1≤δ2,ζ1≤ζ2√×ℛδ2,ζ2+A⊆ℛδ1,ζ1+Aifδ1≤δ2,ζ1≤ζ2√×(√) indicates that the property is satisfied.Table 3
Comparison between some properties by Definition 3.3 in [60] and 6 (3).
Definition 3.3 in [60]Definition6 (2)ℛδ,ζ−∅=∅×√ℛδ,ζ+∅=∅√×ℛδ,ζ−V=U√×ℛδ,ζ+V=U×√ℛδ,ζ−A⊆ℛδ,ζ+A×√ℛδ,ζ−A⊆Sδ,ζ−Aifℛ≤S×√ℛδ,ζ+A⊆Sδ,ζ+Aifℛ≤S×√ℛδ1,ζ1−A⊆ℛδ2,ζ2−Aifδ1≤δ2,ζ1≤ζ2√×ℛδ2,ζ2+A⊆ℛδ1,ζ1+Aifδ1≤δ2,ζ1≤ζ2√×(√) indicates that the property is satisfied.Remark 3.
Assertionsℛδ1−A⊆ℛδ2−Aifδ1≤δ2 and ℛδ2+A⊆ℛδ1+Aifδ1≤δ2 do not hold by Definition 6 (1), let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈0,1U×V is defined by(16)ℛ=0.6x1,y1,0.4x1,y2,1x2,y1,0.6x2,y2.
Takeδ1=0.55≤0.7=δ2 and A=y1. Then, ℛ0.55x1=y1,ℛ0.55x2=V,ℛ0.7x1=∅,andℛ0.7x2=y1, and thus ℛ0.55−A=x1≠x2=ℛ0.7−A. Also, if we take A=y2, ℛ0.55+A=x2≠x1=ℛ0.7+A.Remark 4.
Assertionsℛδ1,ζ1−A⊆ℛδ2,ζ2−Aifδ1≤δ2,ζ1≤ζ2 and ℛδ2,ζ2+A⊆ℛδ1,ζ1+Aifδ1≤δ2,ζ1≤ζ2 in Tables 2 and 3 do not hold by Definition 6 (2) and (3), and let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈0,12U×V and also ℛ∈JU×V is defined by(17)ℛ=0.6x1,y1,0.4x1,y2,1x2,y1,0.6x2,y2.
Takeδ1=0.55≤0.7=δ2, and A=y1. Then, ℛ0.55x1=y1,ℛ0.55x2=V,ℛ0.7x1=∅,andℛ0.7x2=y1, and thus ℛ0.55−A=x1≠x2=ℛ0.7−A Also, if we take A=y2, thus ℛ0.55+A=x2≠x1=ℛ0.7+A


## 3.2. Graded Picture Fuzzy Rough Sets Based onℛ on Two Different Universes

Definition 7.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N, and A∈2V. Then,(18)ℛδ,ζ,ϑ−InA=u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅,ℛδ,ζ,ϑ+InA=u∈U|ℛδ,ζ,ϑu∩A>n∨ℛδ,ζ,ϑu=∅,are called the Type-I lower approximation and the Type-I upper approximation of A according to the graded n with respect to ℛδ,ζ,ϑ on U and V, respectively, and ℛδ,ζ,ϑ−InA,ℛδ,ζ,ϑ+InA is called the Type-I picture fuzzy rough approximation of A according to the graded n with respect to ℛδ,ζ,ϑ (briefly, a Type-I picture fuzzy rough set according to the graded n with respect to ℛδ,ζ,ϑ).Theorem 3.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N, and A∈2V. Then,(1)
IfA⊆ℬ, then ℛδ,ζ,ϑ−InA⊆ℛδ,ζ,ϑ−Inℬ and ℛδ,ζ,ϑ+InA⊆ℛδ,ζ,ϑ+Inℬ(2)
ℛδ,ζ,ϑ−InA∩ℬ⊆ℛδ,ζ,ϑ−InA∩ℛδ,ζ,ϑ−Inℬ(3)
ℛδ,ζ,ϑ−InA∪ℬ⊇ℛδ,ζ,ϑ−InA∪ℛδ,ζ,ϑ−Inℬ; ℛδ,ζ,ϑ+InA∩ℬ⊆ℛδ,ζ,ϑ+InA∩ℛδ,ζ,ϑ+Inℬ(4)
ℛδ,ζ,ϑ−InA=ℛδ,ζ,ϑ+InAcc;ℛδ,ζ,ϑ+InA=ℛδ,ζ,ϑ−InAcc(5)
Ifm≤n, then ℛδ,ζ,ϑ−ImA⊆ℛδ,ζ,ϑ−InA and ℛδ,ζ,ϑ+InA⊆ℛδ,ζ,ϑ+ImAProof.
(1) By Definition7, we have ℛδ,ζ,ϑ−InA=u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅. Since A⊆ℬ, then we have u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅⊆u∈U|ℛδ,ζ,ϑu−ℬ≤n∧ℛδ,ζ,ϑu≠∅=ℛδ,ζ,ϑ−Inℬ. Therefore, ℛδ,ζ,ϑ−InA⊆ℛδ,ζ,ϑ−Ikℬ. Similarly, ℛδ,ζ,ϑ+nA⊆ℛδ,ζ,ϑ+Inℬ.
(2) and (3) Clear.
(4)ℛδ,ζ,ϑ+InAcc=u∈U|ℛδ,ζ,ϑu∩Ac>n∨ℛδ,ζ,ϑu=∅c=u∈U|ℛδ,ζ,ϑu−A>n∨ℛδ,ζ,ϑu=∅c=u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅=ℛδ,ζ,ϑ−InA and ℛδ,ζ,ϑ−InAcc=u∈U|ℛδ,ζ,ϑu−Ac≤n∧ℛδ,ζ,ϑu≠∅c=u∈U|ℛδ,ζ,ϑu∩A≤n∧ℛδ,ζ,ϑu≠∅c=u∈U|ℛδ,ζ,ϑu∩A>n∨ℛδ,ζ,ϑu=∅=ℛδ,ζ,ϑ+InA.
(5) By Definition7, we have ℛδ,ζ,ϑ−ImA=u∈U|ℛδ,ζ,ϑu−A≤m∧ℛδ,ζ,ϑu≠∅. If m≤n, then we have u∈U|ℛδ,ζ,ϑu−A≤m∧ℛδ,ζ,ϑu≠∅⊆u∈U|ℛδ,ζ,ϑu−A≤n∧ℛδ,ζ,ϑu≠∅=ℛδ,ζ,ϑ−InA. Hence, ℛδ,ζ,ϑ−ImA⊆ℛδ,ζ,ϑ−InA. Similarly, ℛδ,ζ,ϑ+InA⊆ℛδ,ζ,ϑ+ImA.Remark 5.
The equalityℛδ,ζ,ϑ−In∅=ℛδ,ζ,ϑ+In∅\\=∅ and ℛδ,ζ,ϑ−InV=ℛδ,ζ,ϑ+InV=U does not hold. For example, let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈IU×V is defined by(19)ℛ=0.2,0.6,0.2x1,y1,0.4,0.2,0.3x1,y2,0.5,0.2,0.3x2,y1,0.6,0.1,0.2x2,y2.
Takeδ,ζ,ϑ=0.5,0.1,0.2 and n=2. Then, ℛ0.5,0.1,0.2x1=∅,ℛ0.5,0.1,0.2x2=y2. Thus, ℛ0.5,0.1,0.2−I2∅=x2≠∅,ℛ0.5,0.1,0.2+I2∅=x1≠∅,ℛ0.5,0.1,0.2−I2V=x2≠U, and ℛ0.5,0.1,0.2+I2V=x1≠U.Remark 6.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N, and A∈2V. Then, there does not exist inclusion relation between ℛδ,ζ,ϑ−InA and ℛδ,ζ,ϑ+InA. Consider U,V,δ,ζ,ϑ and ℛ∈IU×V are given in Remark 1. Let A=y1 and n=1. Then, ℛ0.5,0.1,0.2−I1A=x2 and ℛ0.5,0.1,0.2+I1A=x1. This show that there does not exist inclusion relation between ℛδ,ζ,ϑ−InA and ℛδ,ζ,ϑ+InA.
We study the notion of Type-II PFRS according to the gradedn with respect to ℛδ,ζ,ϑ satisfy ℛδ,ζ,ϑ−InA⊆ℛδ,ζ,ϑ+InA.Definition 8.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N s.t. n∈0,minℛδ,ζ,ϑu/2,A and A∈2V. Then,(20)ℛδ,ζ,ϑ−IInA=u∈U|ℛδ,ζ,ϑu−A≤n,ℛδ,ζ,ϑ+IInA=u∈U|ℛδ,ζ,ϑu∩A>n,are called the Type-II lower approximation and the Type-II upper approximation of A according to the graded n with respect to ℛδ,ζ,ϑ on U and V, respectively, and ℛδ,ζ,ϑ−IInA,ℛδ,ζ,ϑ+IInA is called the Type-II PFR approximation of A according to the graded n with respect to ℛδ,ζ,ϑ (briefly, a Type-II PFRS according to the graded n with respect to ℛδ,ζ,ϑ). The subsets ℛδ,ζ,ϑ−IInA, U−ℛδ,ζ,ϑ+IInA, and ℛδ,ζ,ϑ+IInA−ℛδ,ζ,ϑ−IInA-ℛδ,ζ,ϑ−IInA are called the ℛδ,ζ,ϑ-positive region ℛδ,ζ,ϑ−POS, the ℛδ,ζ,ϑ-negative region ℛδ,ζ,ϑ−NEG, and the ℛδ,ζ,ϑ-boundary region ℛδ,ζ,ϑ−BND of A with respect to graded n, respectively.Lemma 1.
Letℛ∈IU×V,δ,ζ,ϑ∈I,n∈N, and A∈2V. Then, the following holds:(1)
n∈0,minℛδ,ζ,ϑu/2,A(2)
The limits of the graden=max0,minℛδ,ζ,ϑu/2,AProof.
(1) For anyu∈U,ℛδ,ζ,ϑu−A∪ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu Then, ℛδ,ζ,ϑu−A+ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu implies that ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu−ℛδ,ζ,ϑu−A. If ℛδ,ζ,ϑu∩A>n, then ℛδ,ζ,ϑu−ℛδ,ζ,ϑu−A>n, and if ℛδ,ζ,ϑu−A≤n, then ℛδ,ζ,ϑu−n>n. Consequently, ℛδ,ζ,ϑu>2n and thus n<ℛδ,ζ,ϑu/2. Since n<ℛδ,ζ,ϑu∩A≤minA,ℛδ,ζ,ϑu and n<ℛδ,ζ,ϑu/2, then n<minℛδ,ζ,ϑu/2,A. Thus, n∈0,minℛδ,ζ,ϑu/2,A.
(2) Follows from (1) above.Theorem 4.
Letℛ∈IU×V,δ,ζ,ϑ∈I, n∈N s.t. n∈0,minℛδ,ζ,ϑu/2,A and A∈2V. Then, the following holds:(1)
ℛδ,ζ,ϑ−IInV=ℛδ,ζ,ϑ+IInV=U; ℛδ,ζ,ϑ−IIn∅=ℛδ,ζ,ϑ+IIn∅=0(2)
ℛδ,ζ,ϑ−IInA⊆ℛδ,ζ,ϑ+IInA(3)
IfA⊆ℬ, then ℛδ,ζ,ϑ−IInA⊆ℛδ,ζ,ϑ−IInℬ and ℛδ,ζ,ϑ+IInA⊆ℛδ,ζ,ϑ+IInℬ(4)
ℛδ,ζ,ϑ−IInA∩ℬ⊆ℛδ,ζ,ϑ−IInA∩ℛδ,ζ,ϑ−IInℬ and ℛδ,ζ,ϑ+IInA∪ℬ⊆ℛδ,ζ,ϑ+IInA∩ℛδ,ζ,ϑ+IInℬ(5)
ℛδ,ζ,ϑ−IInA∪ℬ⊇ℛδ,ζ,ϑ−IInA∪ℛδ,ζ,ϑ−IInℬ and ℛδ,ζ,ϑ+IInA∩ℬ⊆ℛδ,ζ,ϑ+IInA∩ℛδ,ζ,ϑ+IInℬ(6)
ℛδ,ζ,ϑ−IInA=ℛδ,ζ,ϑ+IInAcc and ℛδ,ζ,ϑ+IInA=ℛδ,ζ,ϑ−IInAcc(7)
Ifm≤n, then ℛδ,ζ,ϑ−IImA⊆ℛδ,ζ,ϑ−IInA and ℛδ,ζ,ϑ+IInA⊆ℛδ,ζ,ϑ+IImA.Proof.
We only prove (2). Let∃u∈U and u∈ℛδ,ζ,ϑ−IInA, but u∉ℛδ,ζ,ϑ+IInA. From Definition 8, ℛδ,ζ,ϑu−A≤n,ℛδ,ζ,ϑu∩A>n, and for any u∈U,ℛδ,ζ,ϑu−A∪ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu Since ℛδ,ζ,ϑu−A+ℛδ,ζ,ϑu∩A=ℛδ,ζ,ϑu and ℛδ,ζ,ϑu−A=ℛδ,ζ,ϑu−ℛδ,ζ,ϑu∩A, then ℛδ,ζ,ϑu−ℛδ,ζ,ϑu∩A≤n and thus ℛδ,ζ,ϑu−n≤n. Consequently, ℛδ,ζ,ϑu>2n and thus n<ℛδ,ζ,ϑu/2. It is contradiction with the limits of n, so the element which only belongs to ℛδ,ζ,ϑ−IInA but does not belong to ℛδ,ζ,ϑ+IInA does not exist.Remark 7.
(1) The equalityℛδ,ζ−n∅=∅ and ℛδ,ζ+nV=U in Theorem 3.10 (2) in [60] are incorrect. (2) For example, let U=xi|i=1,2 and V=yi|i=1,2 be two two-element sets, and ℛ∈JU×V is defined by(21)ℛ=0.2,0.6x1,y1,0.4,0.2x1,y2,0.5,0.2x2,y1,0.8,0.1x2,y2.
Takeδ,ζ=0.5,0.1 and n=2. Then, ℛ0.5,0.1x1=∅,ℛ0.5,0.1x2=y2. Thus, ℛ0.5,0.1−2∅=U≠∅ and ℛ0.5,0.1+2V=∅≠U.
(2) The inclusions inℛδ,ζ−nA⊆ℛδ,ζ+nA and ℛδ,ζ−nA⊆A⊆ℛδ,ζ−nA in Theorem 3.10(1) in [60] are incorrect. For example, let U=xi|i=1,2,3 and V=yi|i=1,2,3 be two three-element sets, and ℛ∈JU×V is defined by(22)ℛ=0.5,0.2x1,y1,0.7,0.1x1,y2,0.4,0.3x1,y3,0.5,0.3x2,y1,0.3,0.4x2,y2,0.4,0.5x2,y3,0.8,0.1x3,y1,0.1,0.7x3,y2,0.6,0.2x3,y3.
Takeδ,ζ=0.6,0.2,n=1 and A=y1,y3. Then, ℛ0.6,0.2x1=y2,ℛ0.6,0.2x2=∅,andℛ0.6,0.2x3=y1,y3. Thus, ℛ0.6,0.2−1A=U⊆∅=ℛ0.6,0.2+1A and ℛ0.6,0.2−1A=U⊆A⊆∅=ℛ0.6,0.2+1A.
(3) The equalitiesℛδ,ζ−nA∩ℬ=ℛδ,ζ−nA∩ℛδ,ζ−nℬ and ℛδ,ζ+nA∪ℬ=ℛδ,ζ+nA∪ℛδ,ζ+nℬ in Theorem 3.10(3) in [60] are incorrect. Consider above example in (2), let A=y1 and ℬ=y2,y3. Then, ℛ0.6,0.2−1A=ℛ0.6,0.2−1ℬ=U and ℛ0.6,0.2+1A=ℛ0.6,0.2+1ℬ=∅. Thus, ℛ0.6,0.2−1A∩ℬ=x1,x2⊊U=ℛ0.6,0.2−1A∩ℛ0.6,0.2−1ℬ and ℛ0.6,0.2+1A∪ℬ=x3⊋∅=ℛ0.6,0.2+1A∪ℛ0.6,0.2+1ℬ.Definition 9.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then,(23)ℛδ,ζ,ϑ−αA=u∈U|ℛδ,ζ,ϑu∩Aℛδ,ζ,ϑu≥α,ℛδ,ζ,ϑ+βA=u∈U|ℛδ,ζ,ϑu∩Aℛδ,ζ,ϑu>β,are called the lower approximation and the upper approximation of A according to the degree α and β with respect to ℛδ,ζ,ϑ on U and V, respectively, and ℛδ,ζ,ϑ−αA,ℛδ,ζ,ϑ+βA is called the picture fuzzy rough approximation of A according to the degree α and β with respect to ℛδ,ζ,ϑ (briefly, a picture fuzzy rough set according to the degree α and β with respect to ℛδ,ζ,ϑ), where A denotes the cardinality of the set A. The subset bnℛδ,ζ,ϑαβA=u∈U|β<ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu<α is called the boundary region of A according to the degree α and β with respect to ℛδ,ζ,ϑ.
The main results are as follows.Theorem 5.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ℛδ,ζ,ϑ−α∅=ℛδ,ζ,ϑ+β∅=∅;ℛδ,ζ,ϑ−αV=ℛδ,ζ,ϑ+βV=U(2)
ℛδ,ζ,ϑ−αA⊆ℛδ,ζ,ϑ+βA(3)
ℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+1−αAcc;ℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ−1−βAcc,(4)
IfA⊆ℬ, then ℛδ,ζ,ϑ−αA⊆ℛδ,ζ,ϑ−αℬ and ℛδ,ζ,ϑ+βA⊆ℛδ,ζ,ϑ+βℬ(5)
ℛδ,ζ,ϑ−αA∩ℬ⊆ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ and ℛδ,ζ,ϑ+βA∩ℬ⊆ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ(6)
ℛδ,ζ,ϑ−αA∪ℬ⊇ℛδ,ζ,ϑ−αA∪ℛδ,ζ,ϑ−αℬ;ℛδ,ζ,ϑ+βA∪ℬ⊇ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ(7)
ℛδ,ζ,ϑ−α1A⊆ℛδ,ζ,ϑ−αA (if 0.5<α<α1≤1); ℛδ,ζ,ϑ+β1A⊆ℛδ,ζ,ϑ+βA (if 0≤β<β1<0.5)Proof.
(1) For anyu∈U,0≤β<α≤1, we have ℛδ,ζ,ϑu∩∅/ℛδ,ζ,ϑu=0≤β<α,u∉ℛδ,ζ,ϑ−α∅, and u∉ℛδ,ζ,ϑ+β∅. Therefore, ℛδ,ζ,ϑ−α∅=ℛδ,ζ,ϑ+β∅=∅. Also, ℛδ,ζ,ϑu∩V/ℛδ,ζ,ϑu=1≥β>α,u∈ℛδ,ζ,ϑ−αV, and u∈ℛδ,ζ,ϑ+βV. Thus, ℛδ,ζ,ϑ−αV=ℛδ,ζ,ϑ+βV=U.
(2)–(4) It follows from Definition9.
(5) Letu∈ℛδ,ζ,ϑ−αA∩ℬ. Then, ℛδ,ζ,ϑu∩A∩ℬ/ℛδ,ζ,ϑu≥α⟹ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α, ℛδ,ζ,ϑu∩ℬ/ℛδ,ζ,ϑu≥α⟹u∈ℛδ,ζ,ϑ−αA, and u∈ℛδ,ζ,ϑ−αℬ⟹u∈ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ. Similarily, we can obtain ℛδ,ζ,ϑ+βA∩ℬ⊆ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ.
(6) It is analogous to (5).
(7) For0.5<α<α1≤1, let u∈ℛδ,ζ,ϑ−α1A. Then, ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α1. Since α<α1, then ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α1>α⟹ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α. Therefore, u∈ℛδ,ζ,ϑ−αA Thus ℛδ,ζ,ϑ−α1A⊆ℛδ,ζ,ϑ−αA Similarly, ℛδ,ζ,ϑ+β1A⊆ℛδ,ζ,ϑ+βA (if 0≤β<β1<0.5).
Equality (5) and (6) of Theorem5 does not hold as the following example.Example 2 (continuation of Example1).
Letα=0.4,β=0.3,A=y1, and ℬ=y2,y3. Then,(1)
ℛ0.5,0.3,0.1−0.4A∩ℬ=ℛ0.5,0.3,0.1+0.3A∩ℬ=∅,ℛ0.5,0.3,0.1−0.4A=ℛ0.5,0.3,0.1+0.3A=x3,ℛ0.5,0.3,0.1−0.4ℬ=ℛ0.5,0.3,0.1+0.3ℬ=x1,x3 and thus ℛ0.5,0.3,0.1−0.4A∩ℬ=∅≠x3=ℛ0.5,0.3,0.1−0.4A∩ℛ0.5,0.3,0.1−0.4ℬ,ℛ0.5,0.3,0.1+0.3A∩ℬ=∅≠x3=ℛ0.5,0.3,0.1+0.3A∩ℛ0.5,0.3,0.1+0.3ℬ(2)
ℛ0.5,0.3,0.1−0.4A∪ℬ=ℛ0.5,0.3,0.1+0.3A∪ℬ=U, and thus ℛ0.5,0.3,0.1−0.4A∪ℬ=U≠x1,x3=ℛ0.5,0.3,0.1−0.4A∪ℛ0.5,0.3,0.1−0.4ℬ,ℛ0.5,0.3,0.1+0.3A∪ℬ=U≠x1,x3=ℛ0.5,0.3,0.1+0.3A∪ℛ0.5,0.3,0.1+0.3ℬ
The relationship among the regulation for lower approximation, upper approximation, and boundary region with errorα and β is discussed as the following theorems.Theorem 6.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0<r<1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the following holds:(1)
limα⟶r+ℛδ,ζ,ϑ−αA=∪α>rℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+rA(2)
limβ⟶r−ℛδ,ζ,ϑ+βA=∩β<rℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ−rA(3)
limβ⟶r−;α⟶r+bnδ,ζ,ϑαβA=∩β<r<αℛδ,ζ,ϑ+βA\ℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+rA\ℛδ,ζ,ϑ−rA=u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=rProof.
(1) From Definition9, when α>r, we have ℛδ,ζ,ϑ−αA=u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu≥α⊆u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu>r=ℛδ,ζ,ϑ+rA is hold. In addition, by Theorem 5 (7), we have ℛδ,ζ,ϑ−αA increase with error α decrease. Thus, limα⟶r+ℛδ,ζ,ϑ−αA=∩α>rℛδ,ζ,ϑ−αA⊆ℛδ,ζ,ϑ+rA. Conversely, if there exists u0∈ℛδ,ζ,ϑ+rA\∪α>rℛδ,ζ,ϑ−αA, for any α>r, ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0>r and ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0<α. That is, ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0<r. It contradicts with ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0>r, i.e., u0∈∪α>rℛδ,ζ,ϑ−αA. Consequently, ∪α>rℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+rA. Thus, limα⟶r+ℛδ,ζ,ϑ−αA=∪α>rℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ+rA hold.
(2) It is analogous to (1) above.
(3) We knowℛδ,ζ,ϑ+rA\ℛδ,ζ,ϑ−rA=u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=r⊆bnδ,ζ,ϑαβA is held. In addition, we know that when α decreased to r and β decreased to r, the boundary region bnδ,ζ,ϑαβA will decrease. Then, we have ℛδ,ζ,ϑ+rA\ℛδ,ζ,ϑ−rA=u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=r⊆limβ⟶r−;α⟶r+bnδ,ζ,ϑαβA=∩β<r<αℛδ,ζ,ϑ+βA\ℛδ,ζ,ϑ−αA. Conversely, if there exists u0∈∩β<r<αℛδ,ζ,ϑ+βA\ℛδ,ζ,ϑ−αA\u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=r, then for any β<r<α, we have ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0>β,ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0<α and ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0≠r. Furthermore, there is ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0≥r by ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0>β and β<r. Similarily, there is ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0≤r by ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0<α and α>r. This is a contradiction. Consequently, there is ℛδ,ζ,ϑu0∩A/ℛδ,ζ,ϑu0=r. That is, u∈U|ℛδ,ζ,ϑu∩A/ℛδ,ζ,ϑu=r=∩β<r<αℛδ,ζ,ϑ+βA\ℛδ,ζ,ϑ−αA.Theorem 7.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0<r<1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the following holds:(1)
limα⟶r−ℛδ,ζ,ϑ−αA=∩α<rℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ−rA(2)
limβ⟶r+ℛδ,ζ,ϑ+βA=∪β>rℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ+rAProof.
Obvious.Remark 8.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0<r<1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the following does not hold:(1)
limα⟶r+ℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ−rA (i.e., the lower approximation ℛδ,ζ,ϑ−αA is not right continuous with error α)(2)
limβ⟶r−ℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ+rA (i.e., the upper approximation ℛδ,ζ,ϑ+βA is not left continuous with error β)Example 3 (continuation of Example1).
LetA=y2,y3. Then, we can obtain the following relations:(1)
limα⟶0.5+ℛδ,ζ,ϑ−αA=x1,ℛδ,ζ,ϑ−0.5A=x1,x3. Consequently, limα⟶0.5−ℛδ,ζ,ϑ−αA≠ℛδ,ζ,ϑ−0.5A.(2)
limβ⟶0.5−ℛδ,ζ,ϑ+βA=x1,x3,ℛδ,ζ,ϑ+0.5A=x1. Consequently, limβ⟶0.5−ℛδ,ζ,ϑ+βA≠ℛδ,ζ,ϑ+0.5A.Definition 10.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the precision ρδ,ζ,ϑβαA and rough degree μδ,ζ,ϑβαA of A with error α and β are defined by(24)ρδ,ζ,ϑβαA=1−ℛδ,ζ,ϑ−αAℛδ,ζ,ϑ+βA,μδ,ζ,ϑβαA=1−ρδ,ζ,ϑβαA.
Obviously,0≤ρδ,ζ,ϑβαA≤1 and 0≤μδ,ζ,ϑβαA≤1.Lemma 2.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A∈2V. Then, the following holds:(1)
ρδ,ζ,ϑβαA does not decrease with error α and does not increase with error β(2)
μδ,ζ,ϑβαA does not increase with error α and does not decrease with error βProof.
We only prove (1) and then the proof of (2) can be obtained using similar techniques.
(1) Sinceℛδ,ζ,ϑ−α1A⊆ℛδ,ζ,ϑ−αA0.5<α<α1≤1 by Theorem 5 (7), then there is ℛδ,ζ,ϑ−α1A/ℛδ,ζ,ϑ+βA≤ℛδ,ζ,ϑ−αA/ℛδ,ζ,ϑ+βA implies that 1−ℛδ,ζ,ϑ−α1A/ℛδ,ζ,ϑ+βA≥1−ℛδ,ζ,ϑ−αA/ℛδ,ζ,ϑ+βA. Therefore, ρδ,ζ,ϑβαA≤ρδ,ζ,ϑβα1A. Similarly, ρδ,ζ,ϑβ1αA≤ρδ,ζ,ϑβαA for ℛδ,ζ,ϑ+β1A⊆ℛδ,ζ,ϑ+βA0≤β<β1<0.5.Theorem 8.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, A,ℬ∈2V, and A⊆ℬ. Then, the following holds:(1)
Ifℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ−αℬ, there is ρδ,ζ,ϑβαA≤ρδ,ζ,ϑβαℬ and μδ,ζ,ϑβαℬ≤μδ,ζ,ϑβαA(2)
Ifℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ+βℬ, there is ρδ,ζ,ϑβαℬ≤ρδ,ζ,ϑβαA and μδ,ζ,ϑβαA≤μδ,ζ,ϑβαℬ(3)
Ifℛδ,ζ,ϑ−αA=ℛδ,ζ,ϑ−αℬ and ℛδ,ζ,ϑ+βA=ℛδ,ζ,ϑ+βℬ, there is ρδ,ζ,ϑβαA≤ρδ,ζ,ϑβαℬ and μδ,ζ,ϑβαA=μδ,ζ,ϑβαℬProof.
It follows from Theorem5 (4) and Definition 10.Theorem 9.
Letℛ∈IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ρδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≤ρδ,ζ,ϑβαAℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬℛδ,ζ,ϑ+βℬ−ρδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ(2)
μδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≥μδ,ζ,ϑβαAℛδ,ζ,ϑ+βA+μδ,ζ,ϑβαℬℛδ,ζ,ϑ+βℬ−μδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬProof.
(1) From Definition10, we have ρδ,ζ,ϑβαA∪ℬ=1−ℛδ,ζ,ϑ−αA∪ℬ/ℛδ,ζ,ϑ+βA∪ℬ=1−ℛδ,ζ,ϑ−αA∪ℬ/ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ ≤ 1−ℛδ,ζ,ϑ−αA∪ℛδ,ζ,ϑ−αℬ/ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ. Consequently, ρδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≤ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA∪ℛδ,ζ,ϑ−αℬ. Similarly, ρδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βX∩ℛδ,ζ,ϑ+βℬ ≤ ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ. We know X∪Y=X+Y−X∩Y. Then,(25)ρδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≤ℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA∪ℛδ,ζ,ϑ−αℬ=ℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA+ℛδ,ζ,ϑ−αℬ−ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ=ℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA−ℛδ,ζ,ϑ−αℬ−ℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA∩ℛδ,ζ,ϑ−αℬ≤ℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA−ℛδ,ζ,ϑ−αℬ−ρpβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ.
Also, by the relation ofρδ,ζ,ϑβαA=1−ℛδ,ζ,ϑ−αA/ℛδ,ζ,ϑ+βA and ρpβαℬ=1−ℛδ,ζ,ϑ−αℬ/ℛδ,ζ,ϑ+βℬ, then ℛδ,ζ,ϑ+βA−ℛδ,ζ,ϑ−αA=ρδ,ζ,ϑβαAℛδ,ζ,ϑ+βA and ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αℬ=ρδ,ζ,ϑβαℬℛδ,ζ,ϑ+βℬ. Thus, we have(26)ρδ,ζ,ϑβαA∪ℬℛδ,ζ,ϑ+βA∪ℛδ,ζ,ϑ+βℬ≤ℛδ,ζ,ϑ+βA+ℛδ,ζ,ϑ+βℬ−ℛδ,ζ,ϑ−αA−ℛδ,ζ,ϑ−αℬ−ρδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ=ρδ,ζ,ϑβαAℛδ,ζ,ϑ+βA+ρδ,ζ,ϑβαℬℛδ,ζ,ϑ+βℬ−ρδ,ζ,ϑβαA∩ℬℛδ,ζ,ϑ+βA∩ℛδ,ζ,ϑ+βℬ.
Hence, (1) is holding.
(2) It can be easily proved by the relationshipμδ,ζ,ϑβαA=1−ρδ,ζ,ϑβαA.


## 3.3. Generalized Picture Fuzzy Rough Sets Based onδ,ζ,ϑ-Cut of ℛ on Two Different Universes

Definition 11.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛ^iδ,ζ,ϑu≠∅, and A∈2V. Then,(1)
ℛ−∧inℛ^iδ,ζ,ϑIαA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α∧ℛ^2δ,ζ,ϑu∩A/ℛ^2δ,ζ,ϑu≥α∧⋯∧ℛ^iδ,ζ,ϑu∩A/ℛ^iδ,ζ,ϑu≥α and ℛ+∧inℛ^iδ,ζ,ϑIβA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu>β∧ℛ^2δ,ζ,ϑu∩A/ℛ^2δ,ζ,ϑu>β∧⋯∧ℛ^iδ,ζ,ϑu∩A/ℛ^iδ,ζ,ϑu>β are called the Type-I lower ℛ^iδ,ζ,ϑ-approximation and the Type-I upper ℛ^iδ,ζ,ϑ-approximation of A according to the degree α and β with respect to ℛiδ,ζ,ϑ on U and V, respectively, and ℛ−∧inℛ^iδ,ζ,ϑIαA,ℛ+∧inℛ^iδ,ζ,ϑIβA is called the Type-I generalized picture fuzzy rough approximation of A according to the degree α and β with respect to ℛiδ,ζ,ϑ (briefly, the Type-I generalized picture fuzzy rough set according to the degree α and β with respect to ℛiδ,ζ,ϑ).(2)
ℛ−∨inℛ^iδ,ζ,ϑIIαA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α∨ℛ^2δ,ζ,ϑu∩A/ℛ^2δ,ζ,ϑu≥α∨⋯∨ℛ^iδ,ζ,ϑu∩A/ℛ^iδ,ζ,ϑu≥α and ℛ+∨inℛ^iδ,ζ,ϑIIβA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu>β∨ℛ^2δ,ζ,ϑu∩A/ℛ^2δ,ζ,ϑu>β∨⋯∨ℛ^iδ,ζ,ϑu∩A/ℛ^iδ,ζ,ϑu>β are called the Type-II lower ℛ^iδ,ζ,ϑ-approximation and the Type-II upper ℛ^iδ,ζ,ϑ-approximation of A according to the degree α and β with respect to ℛiδ,ζ,ϑ on U and V, respectively, and ℛ−∨inℛ^iδ,ζ,ϑIIαA,ℛ+∨inℛ^iδ,ζ,ϑIIβA is called the Type-II generalized picture fuzzy rough approximation of A according to the degree α and β with respect to ℛiδ,ζ,ϑ (briefly, the Type-II generalized picture fuzzy rough set according to the degree α and β with respect to ℛiδ,ζ,ϑ).Remark 9.
(1) From Definition11, we have(i)
ℛ−∧inℛ^iδ,ζ,ϑIαA⊆ℛ−∨inℛ^iδ,ζ,ϑIIαA(ii)
ℛ+∧inℛ^iδ,ζ,ϑIβA⊆ℛ−∨inℛ^iδ,ζ,ϑIIβA
(2) If we takeα=1 and β=0 in Definition 11, then we can obtain in the following definitions:(i)
ℛ−∧inℛ^iδ,ζ,ϑIA=u∈U|ℛ^1δ,ζ,ϑu⊆A∧ℛ^2δ,ζ,ϑu⊆A∧⋯∧ℛ^iδ,ζ,ϑu⊆A and ℛ+∧inℛ^iδ,ζ,ϑIA=u∈U|ℛ^1δ,ζ,ϑu∩A≠∅∧ℛ^2δ,ζ,ϑu∩A≠∅∧⋯∧ℛ^iδ,ζ,ϑu∩A≠∅.(ii)
ℛ−∨inℛ^iδ,ζ,ϑIIA=u∈U|ℛ^1δ,ζ,ϑu⊆A∨ℛ^2δ,ζ,ϑu⊆A∨⋯∨ℛ^iδ,ζ,ϑu⊆A and ℛ+∨inℛ^iδ,ζ,ϑIIA=u∈U|ℛ^1δ,ζ,ϑu∩A≠∅∨ℛ^2δ,ζ,ϑu∩A≠∅∨⋯∨ℛ^iδ,ζ,ϑu∩A≠∅.
The main results are as follows.Theorem 10.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ℛ−∧inℛ^iδ,ζ,ϑIα∅=ℛ+∧inℛ^iδ,ζ,ϑIβ∅=∅; ℛ−∧inℛ^iδ,ζ,ϑIαU=ℛ+∧inℛ^iδ,ζ,ϑIβU=V(2)
ℛ−∧inℛ^iδ,ζ,ϑIαA⊆ℛ+∧inℛ^iδ,ζ,ϑIβA(3)
IfA⊆ℬ, then ℛ−∧inℛ^iδ,ζ,ϑIαA⊆ℛ−∧inℛ^iδ,ζ,ϑIαℬ;ℛ+∧inℛ^iδ,ζ,ϑIβA⊆ℛ+∧inℛ^iδ,ζ,ϑIβℬ(4)
ℛ−∧inℛ^iδ,ζ,ϑIαA∩ℬ⊆ℛ−∧inℛ^iδ,ζ,ϑIαA∩ℛ−∧inℛ^iδ,ζ,ϑIαℬ; ℛ+∧inℛ^iδ,ζ,ϑIβA∩ℬ⊆ℛ+∧inℛ^iδ,ζ,ϑIβA∩ℛ+∧inℛ^iδ,ζ,ϑIβℬ∩ℛ+∧inℛ^iδ,ζ,ϑIβℬ(5)
ℛ−∧inℛ^iδ,ζ,ϑIαA∪ℬ⊇ℛ−∧inℛ^iδ,ζ,ϑIαA∪ℛ−∧inℛ^iδ,ζ,ϑIαℬ; ℛ+∧inℛ^iδ,ζ,ϑIβA∪ℬ⊇ℛ+∧inℛ^iδ,ζ,ϑIβA∪ℛ+∧inℛ^iδ,ζ,ϑIβℬ(6)
ℛ−∧inℛ^iδ,ζ,ϑIα1A⊆ℛ−∧inℛ^iδ,ζ,ϑIαA (if 0.5<α<α1≤1); ℛ+∧inℛ^iδ,ζ,ϑIβ1A⊆ℛ+∧inℛ^iδ,ζ,ϑIβA (if 0≤β<β1<0.5)Proof.
By Definition11, the result can be similarly proven as Theorem 5.Remark 10.
LetA∈2V and 0≤β<α≤1. Then,(1)
ℛ−∨inℛ^iδ,ζ,ϑIαA=ℛ+∨inℛ^iδ,ζ,ϑI1−αAcc(2)
ℛ+∨inℛ^iδ,ζ,ϑIβA=ℛ−∨inℛ^iδ,ζ,ϑI1−βAccdoes not hold by the following example.Example 4.
LetU=xi|i=1,2 and V=yi|i=1,2 be two two-element set, and ℛ^1,ℛ^2,ℛ^3∈IU×V is defined by(27)ℛ^1=0.2,0.4,0.4x1,y1,0.4,0.2,0.3x1,y2,0.5,0.3,0.2x2,y1,0.7,0.1,0.2x2,y2,ℛ^2=0.3,0.2,0.4x1,y1,0.1,0.2,0.2x1,y2,0.5,0.3,0.1x2,y1,0.6,0.1,0.3x2,y2,ℛ^3=0.2,0.3,0.5x1,y1,0.6,0.1,0.2x1,y2,0.7,0.1,0.1x2,y1,0.4,0.3,0.3x2,y2.
Takeδ,ζ,ϑ=0.5,0.3,0.2. Then, ℛ^10.5,0.3,0.2x1=∅, ℛ^10.5,0.3,0.2x2=V,ℛ^20.5,0.3,0.2x1=∅,ℛ^20.5,0.3,0.2x2=y1, ℛ^30.5,0.3,0.2x1=y2,ℛ^30.5,0.3,0.2x2=y1. Let A=y1,α=0.6, and β=0.4. Thus,(1)
ℛ−∨inℛ^iδ,ζ,ϑIαA=∅≠x1=ℛ+∨inℛ^iδ,ζ,ϑI1−αAcc(2)
ℛ+∨inℛ^iδ,ζ,ϑIβA=x2≠U=ℛ−∨inℛ^iδ,ζ,ϑI1−βAcc
From Remark9 (2), we can conclude the following corollary.Corollary 2.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ℛ−∨inℛ^iδ,ζ,ϑI∅=ℛ+∨inℛ^iδ,ζ,ϑI∅=∅; ℛ−∨inℛ^iδ,ζ,ϑIU=ℛ+∨inℛ^iδ,ζ,ϑIU=V(2)
ℛ−∨inℛ^iδ,ζ,ϑIA⊆ℛ+∨inℛ^iδ,ζ,ϑIA(3)
IfA⊆ℬ, then ℛ−∨inℛ^iδ,ζ,ϑIA⊆ℛ−∨inℛ^iδ,ζ,ϑIℬ;ℛ+∨inℛ^iδ,ζ,ϑIA⊆ℛ+∨inℛ^iδ,ζ,ϑIℬ(4)
ℛ−∨inℛ^iδ,ζ,ϑIA∩ℬ⊆ℛ−∨inℛ^iδ,ζ,ϑIA∩ℛ−∨inℛ^iδ,ζ,ϑIℬ; ℛ+∨inℛ^iδ,ζ,ϑIA∩ℬ⊆ℛ+∨inℛ^iδ,ζ,ϑIA∩ℛ+∨inℛ^iδ,ζ,ϑIℬ(5)
ℛ−∨inℛ^iδ,ζ,ϑIA∪ℬ⊇ℛ−∨inℛ^iδ,ζ,ϑIA∪ℛ−∨inℛ^iδ,ζ,ϑIℬ; ℛ+∨inℛ^iδ,ζ,ϑIA∪ℬ⊇ℛ+∨inℛ^iδ,ζ,ϑIA∪ℛ+∨inℛ^iδ,ζ,ϑIℬTheorem 11.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V.
Then, the following holds:(1)
ℛ−∨inℛ^iδ,ζ,ϑIIα∅=ℛ+∨inℛ^iδ,ζ,ϑIIβ∅=∅; ℛ−∨inℛ^iδ,ζ,ϑIIαU=ℛ+∨inℛ^iδ,ζ,ϑIIβU=V(2)
ℛ−∨inℛ^iδ,ζ,ϑIIαA⊆ℛ+∨inℛ^iδ,ζ,ϑIIβA(3)
IfA⊆ℬ, then ℛ−∨inℛ^iδ,ζ,ϑIIαA⊆ℛ−∨inℛ^iδ,ζ,ϑIIβℬ; ℛ+∨inℛ^iδ,ζ,ϑIIβA⊆ℛ−∨inℛ^iδ,ζ,ϑIIβℬ(4)
ℛ−∨inℛ^iδ,ζ,ϑIIαA∩ℬ⊆ℛ−∨inℛ^iδ,ζ,ϑIIαA∩ℛ−∨inℛ^iδ,ζ,ϑIIαℬ; ℛ+∨inℛ^iδ,ζ,ϑIIβA∩ℬ⊆ℛ+∨inℛ^iδ,ζ,ϑIIβA∩ℛ+∨inℛ^iδ,ζ,ϑIIβℬ(5)
ℛ−∨inℛ^iδ,ζ,ϑIIαA∪ℬ⊇ℛ−∨inℛ^iδ,ζ,ϑIIαA∪ℛ−∨inℛ^iδ,ζ,ϑIIαℬ; ℛ+∨inℛ^iδ,ζ,ϑIIβA∪ℬ⊇ℛ+∨inℛ^iδ,ζ,ϑIIβA∪ℛ+∨inℛ^iδ,ζ,ϑIIβℬ(6)
ℛ−∨inℛ^iδ,ζ,ϑIIα1A⊆ℛ−∨inℛ^iδ,ζ,ϑIIαA (if 0.5<α<α1≤1); ℛ+∨inℛ^iδ,ζ,ϑIIβ1A⊆ℛ+∨inℛ^iδ,ζ,ϑIIβA (if 0≤β<β1<0.5)Proof.
It follows from Definition11 (2).
From Remark9 (2), we can conclude the following corollary.Corollary 3.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then the following holds:(1)
ℛ−∨inℛ^iδ,ζ,ϑII∅=ℛ+∨inℛ^iδ,ζ,ϑII∅=∅; ℛ−∨inℛ^iδ,ζ,ϑIIU=ℛ+∨inℛ^iδ,ζ,ϑIIU=V(2)
ℛ−∨inℛ^iδ,ζ,ϑIIA⊆ℛ+∨inℛ^iδ,ζ,ϑIIA(3)
IfA⊆ℬ, then ℛ−∨inℛ^iδ,ζ,ϑIIA⊆ℛ−∨inℛ^iδ,ζ,ϑIIℬ; ℛ+∨inℛ^iδ,ζ,ϑIIA⊆ℛ+∨inℛ^iδ,ζ,ϑIIℬ(4)
ℛ−∨inℛ^iδ,ζ,ϑIIA∩ℬ⊆ℛ−∨inℛ^iδ,ζ,ϑIIA∩ℛ−∨inℛ^iδ,ζ,ϑIIℬ; ℛ+∨inℛ^iδ,ζ,ϑIIA∩ℬ⊆ℛ+∨inℛ^iδ,ζ,ϑIIA∩ℛ+∨inℛ^iδ,ζ,ϑIIℬ(5)
ℛ−∨inℛ^iδ,ζ,ϑIIA∪ℬ⊇ℛ−∨inℛ^iδ,ζ,ϑIIA∪ℛ−∨inℛ^iδ,ζ,ϑIIℬ; ℛ+∨inℛ^iδ,ζ,ϑIIA∪ℬ⊇ℛ+∨inℛ^iδ,ζ,ϑIIA∪ℛ+∨inℛ^iδ,ζ,ϑIIℬTheorem 12.
Letℛ=ℛ^iδ,ζ,ϑ,i∈N⊆IU×V,δ,ζ,ϑ∈I, 0≤β<α≤1, ℛ^iδ,ζ,ϑu≠∅, and A,ℬ∈2V. Then, the following holds:(1)
ℛ−∨inℛ^iδ,ζ,ϑIαA=ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∩ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA∩⋯∩ℛδ,ζ,ϑ−ℛ^iδ,ζ,ϑαA(2)
ℛ+∨inℛ^iδ,ζ,ϑIβA=ℛδ,ζ,ϑ+ℛ^1δ,ζ,ϑβA∩ℛδ,ζ,ϑ+ℛ^2δ,ζ,ϑβA∩⋯∩ℛδ,ζ,ϑ+ℛ^iδ,ζ,ϑβA(3)
ℛ−∨inℛ^iδ,ζ,ϑIIαA=ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA∪⋯∪ℛδ,ζ,ϑ−ℛ^iδ,ζ,ϑαA(4)
ℛ+∨inℛ^iδ,ζ,ϑIIβA=ℛδ,ζ,ϑ+ℛ^1δ,ζ,ϑβA∪ℛδ,ζ,ϑ+ℛ^2δ,ζ,ϑβA∪⋯∪ℛδ,ζ,ϑ+ℛ^iδ,ζ,ϑβAProof.
We only prove (3), while rest can be proven similarly.
(3) We show thatℛ−∨i=12ℛ^iδ,ζ,ϑIIαA=ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA. Let u∈ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαAu∈U, where ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA=u∈U|ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α, but u∈ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA. So, ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA⊆ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA. Similarly, ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA⊆ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA. Consequently, ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA⊆ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA. Conversely, for u∈ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA and by Definition 11, we have either ℛ^1δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α, then u∈ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA or ℛ^2δ,ζ,ϑu∩A/ℛ^1δ,ζ,ϑu≥α, and then u∈ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA. Hence, u∈ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA, and thus ℛ−∨i=12ℛ^iδ,ζ,ϑIIαA⊆ℛδ,ζ,ϑ−ℛ^1δ,ζ,ϑαA∪ℛδ,ζ,ϑ−ℛ^2δ,ζ,ϑαA. Therefore, (3) is holding.


## 4. An Application of Rough Picture Fuzzy Sets on Two Different Universes

We will present the notion of RPFSs over two different universes and also introduce an application of rough picture fuzzy sets on two different universes.Definition 12.
Letℛ∈IU×V and A∈IV. Then, ∀u∈U(28)ℛ−Au=∧v∈Vp3∘ℛu,v∨p1∘Av,∧v∈Vp2∘ℛu,v∨p2∘Av,∨v∈Vp1∘ℛu,v∧p3∘Av(i.e., the lower approximation of A) and(29)ℛ+Au=∧v∈Vp1∘ℛu,v∧p1∘Av,∧v∈Vp2∘ℛu,v∨p2∘Av,∨v∈Vp3∘ℛu,v∨p3∘Av(i.e., the upper approximation of A).
A pairℛ−A,ℛ+A is said to be the rough picture fuzzy approximation of A with respect to ℛ.Example 5.
LetU=xi|i=1,2,3 and V=yi|i=1,2,3 be three-element sets, ℛ∈IU×V defined by(30)ℛ=0.2,0.5,0.2x1,y1,0.6,0.3,0.1x1,y2,0.4,0.3,0.2x1,y3,0.4,0.2,0.3x2,y1,0.3,0.3,0.4x2,y2,0.1,0.6,0.2x2,y3,0.5,0.2,0.3x3,y1,0.5,0.4,0.1x3,y2,0.4,0.1,0.2x3,y3,and A∈IV defined by(31)A=0.4,0.3,0.2y1,0.2,0.1,0.5y2,0.3,0.5,0.2y3.
By Definition12, we obtain(32)ℛ−Ax1=∧v∈Vp3∘ℛx1,v∨p1∘Av,∧v∈Vp2∘ℛx1,v∨p2∘Av,∨v∈Vp1∘ℛx1,v∧p3∘Av=0.2∨0.4∧0.1∨0.2∧0.2∨0.3,0.5∨0.3∧0.3∨0.1∧0.3∨0.5,0.2∧0.2∨0.6∧0.5∨0.4∧0.2=0.2,0.3,0.5.
Similarly,ℛ−Ax2=0.3,0.3,0.3, ℛ−Ax3=0.2,0.3,0.5, ℛ+Ax1=0.3,0.3,0.2, ℛ+Ax2=0.4,0.3,0.2, and ℛ+Ax3=0.4,0.3,0.2.
Therefore,(33)ℛ−A=0.2,0.3,0.5x1,0.3,0.3,0.3x2,0.2,0.3,0.5x3,ℛ+A=0.3,0.3,0.2x1,0.4,0.3,0.2x2,0.4,0.3,0.2x3.Definition 13.
Suppose thatU be a set and A,ℬ∈IU. The sum A⊕^ℬ, is defined by(34)A⊕^ℬ=Au+ℬu\u|u∈U,where Au+ℬu is defined in Definition 3 in [50].Definition 14.
(cf. see [62])Let U=u1,u2,…,un be an n-element set and A,A∗∈IU. A cosine similarity measure is defined by(35)CA,A∗=1n∑i=1np1∘Aui⋅p1∘A∗ui+p2∘Aui⋅p2∘A∗ui+p3∘Aui⋅p3∘A∗uip1∘Aui2+p2∘Aui2+p3∘Aui2⋅p1∘A∗ui2+p2∘A∗ui2+p3∘A∗ui2.
Next, we present the RPFS decision-making medical diagnosis problem over two different universes as indicated below.
Suppose thatU=u1,u2,…,up (a p-element set) be the set of a disease, V=v1,v2,…,vq the set of symptoms (where p,q∈N), and ℛ∈IU×V be picture fuzzy relation from U to V. For any ℛup,vq, it represents the picture fuzzy relation between the disease upup∈U and the symptom vqvq∈V (i.e., by a doctor in advance). For A∈IV (i.e., any patient set) who has some symptoms in V, where A is PFS on symptom V, that is,(36)A=p1∘Av1,p2∘Av1,p3∘Av1v1,p1∘Av2,p2∘Av2,p3∘Av2v2,…,p1∘Avq,p2∘Avq,p3∘Avqvq,where p1∘Avq∈0,1 (i.e., the degree of positive membership) to the symptom vq∈V of A, p2∘Avq∈0,1 (i.e., the degree of neutral membership) to the symptom vq∈V of A, and p3∘Avq (i.e., the degree of negative membership) to the symptom vq∈V of A (clearly, ∀vq∈V, p1∘Avq+p2∘Avq+p3∘Avq≤1). By Definition 12, we compute the lower and the upper RPF approximations ℛ−A and ℛ+A, respectively, of A. Then, by Definition 14, we obtain ℛ−A⊕ℛ+A as follows:(37)ℛ−A⊕^ℛ+A=ℛ−Au+ℛ+Au\u|u∈U.
From Definition14, we calculate the cosine similarity measure between the ℛ−A⊕^ℛ+Aup corresponding to up and the ideal ℛ−A⊕^ℛ+A∗. Finally, we determine x^=argmaxCℛ−A⊕^ℛ+Au1,ℛ−A⊕^ℛ+A∗,Cℛ−A⊕^ℛ+Au2,ℛ−A⊕^ℛ+A∗,…,Cℛ−A⊕^ℛ+Aup,ℛ−A⊕^ℛ+A∗. Hence, the best choice is to select up, that is, we can conclude that patient A is suffering from the diseases.
The corresponding algorithm is as follows.
Now, we introduce the following an example (i.e., an application of RPFSs over two different universes) by using an Algorithm1.Algorithm 1:Determine the best choice of rough picture fuzzy soft sets over two different universes.
Step 1. Input the picture fuzzy relation ℛ∈IU×V, where ℛup,vq represents the picture fuzzy relation between the disease upup∈U and the symptom vqvq∈V, which is evaluated by a doctor in advanceStep 2. Define A (patient set) is PFS on symptom V, that is, A=p1∘Av1,p2∘Av1,p3∘Av1/v1,p1∘Av2,p2∘Av2,p3∘Av2/v2,…,p1∘Avq,p2∘Avq,p3∘Avq/vqStep 3. Compute and write ℛ−A and ℛ+A, respectively, of AStep 4. By Definition 13, we calculate ℛ−A⊕ℛ+A as ℛ−A⊕^ℛ+A=ℛ−Au+ℛ+Au∖u|u∈UStep 5. By Definition 14, we compute the cosine similarity measure between the ℛ−A⊕^ℛ+Aup corresponding to up and the ideal ℛ−A⊕^ℛ+A∗Step 6. Get the best choice to select up, that is, we can conclude that patient A is suffering from the diseasesExample 6.
Assume that five diseases (U=xi|i=1,2,3,4,5), where x1 stands for the “viral fever,” x2 stands for the “malaria,” x3 stands for the “typhoid,” x4 stands for the “stomach problem,” and x5 stands for the “chest problem.” The five symptoms are in clinic (let V=yi|i=1,2,3,4,5), where y1 stands for the “temperature,” y2 stands for the “headache,” y3 stands for the “stomach Pain,” y4 stands for the “cough,” and y5 stands for the “Chest-Pain.” Let ℛ∈IU×V be a picture fuzzy relation from U to V, where ℛ is a medical knowledge statistic data of the relationship of the disease xpxp∈U and the symptom yqyq∈V (where p,q=1,2,3,4,5). The statistic data are given in Table 4.
By the Step 2 of Algorithm1, suppose the symptoms of a patient A are defined by a PFS on V, and(38)A=0.4,0.3,0.2y1,0.2,0.1,0.5y2,0.3,0.5,0.2y3,0.6,0.2,0.1y4,0.4,0.4,0.2y5.
Then, by Definition12, we obtain on the ℛ−A and ℛ+A of patient A in Step 3 of Algorithm 1, respectively, as follows:(39)ℛ−A=0.4,0.3,0.3x1,0.3,0.2,0.5x2,0.3,0.1,0.5x3,0.2,0.3,0.5x4,0.2,0.1,0.4x5,ℛ+A=0.5,0.3,0.2x1,0.4,0.2,0.3x2,0.5,0.1,0.2x3,0.4,0.3,0.1x4,0.4,0.1,0.1x5.
By Step 4 of Algorithm1, we obtain(40)ℛ−A⊕^ℛ+A=0.7,0.09,0.06x1,0.58,0.04,0.15x2,0.65,0.01,0.1x3,0.52,0.09,0.05x4,0.52,0.01,0.04x5.
Then, by Step 4 of Algorithm1, we can get the cosine similarity measure between the ℛ−A⊕^ℛ+Aup corresponding to up and the ideal ℛ−A⊕^ℛ+A∗ as follows:(41)Cℛ−A⊕^ℛ+Ax1,ℛ−A⊕^ℛ+A∗=0.141,Cℛ−A⊕^ℛ+Ax2,ℛ−A⊕^ℛ+A∗=0.181,Cℛ−A⊕^ℛ+Ax3,ℛ−A⊕^ℛ+A∗=0.198,Cℛ−A⊕^ℛ+Ax4,ℛ−A⊕^ℛ+A∗=0.196,Cℛ−A⊕^ℛ+Ax5,ℛ−A⊕^ℛ+A∗=0.199.
Thus, according to Step 6, we conclude the maximum value isCℛ−A⊕^ℛ+Ax5,ℛ−A⊕^ℛ+A∗=0.199. Thus, the patient is suffering from the disease chest problem x5.Table 4
Picture fuzzy relationℛ∈IU×V between the diseases and symptoms.
ℛTemperaturey1Headachey2Stomach painy3Coughy4Chest-painy5Viral feverx1(0.1, 0.5, 0.4)(0.3, 0.3, 0.4)(0.3, 0.2, 0.4)(0.5, 0.3, 0.2)(0.1, 0.4, 0.4)Malariax2(0.2, 0.4, 0.3)(0.5, 0.2, 0.3)(0.5, 0.1, 0.3)(0.2, 0.3, 0.5)(0.4, 0.2, 0.4)Typhoidx3(0.7, 0.2, 0.1)(0.6, 0.1, 0.3)(0.3, 0.6, 0.1)(0.5, 0.1, 0.4)(0.7, 0.2, 0.1)Stomach problemx4(0.4, 0.4, 0.2)(0.7, 0.3, 0)(0.8, 0.1, 0.1)(0.2, 0.3, 0.1)(0.6, 0.2, 0.2)Chest problemx5(0.1, 0.8, 0.1)(0.4, 0.1, 0.2)(0.4, 0.2, 0.3)(0.1, 0.8, 0.1)(0.9, 0, 0.1)


## 5. Conclusions

In this paper, we suggest novel notion of picture fuzzy rough sets (PFRSs) over two different universes which depend onδ,ζ,ϑ-cut. Also, we discussed some interesting properties and related results on the PFRSs. Furthermore, we presented several notions related to PFRSs such as Type-I-/Type-II-graded PFRSs, the degree α and β with respect to ℛδ,ζ,ϑ on PFRSs, and Type-I-/Type-II-generalized PFRSs based on the degree α and β with respect to ℛδ,ζ,ϑ and investigate the basic properties of above notions. Lastly, we gave an approach based on the rough picture fuzzy approximation RPFA operators on two different universes in decision-making problem is introduced, and we present an example to show the validity of this approach.



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