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

. The major concern of this article is to propose the notion of picture fuzzy rough sets (PFRSs) over two diﬀerent universes which depend on ( δ , 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 R [( δ , ζ , ϑ )] on PFRSs, and (Type-I/Type-II) generalized PFRSs based on the degree α and β with respect to R [( δ , 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.


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. e lower/upper approximations (i.e., rough sets) are firstly described through the equivalence classes.
at 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][4][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][13][14][15][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][19][20][21][22][23][24][25][26][27][28][29][30][31][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. e basic properties of PFRSs based on (δ, 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 R [(δ,ζ,ϑ)] and type-II PFRS is according to the graded n with respect to R [(δ,ζ,ϑ)] . e 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 R [(δ,ζ,ϑ)] and Type-I/Type-II generalized PFRs according to the degree α and β with respect to R i [(δ,ζ,ϑ) ] . e 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 Section 2, 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.  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][45][46][47][48][49]) and solved some problems related to PFSs (e.g., [50][51][52][53][54][55][56][57]).

Picture Fuzzy Sets and Picture Fuzzy
Definition 1 (cf. see [39][40][41]). Let U � u 1 , u 2 , . . . , u n be an n-element set (n is a natural number), and a PFS A ∈ I U is . . , p 1 ∘ A u n , p 2 ∘ A u n , p 3 ∘ A u n u n . (1) is called the refusal degree of u (u ∈ U). A PFS A ∈ I U with refusal degree 0 at each point u ∈ U can be identified with an IFS on U and I U with the pointwise order ≤ is the set of all mappings from a set U (or an universe) to I � (a 1 , a 2 , a 3 ) ∈ [0, 1] 3 |a 1 + a 2 + a 3 ≤ 1}. en, each element A of I U is called an I-set or a PFS on U, p 1 ∘ A(u) (i.e., the degree of positive), p 2 ∘ A(u) (i.e., the degree of neutral), and Definition 2 (cf. see [39][40][41]). Let A, B ∈ I U . en, (2) e union k∈K A k (called also supremum ∨ k∈K A k ) and the intersection ∩ k∈K A k (called also infimum ∧ k∈K A k ) of a family A k k∈K ⊆ I U can be defined by the following formulae: Definition 3 (cf. see [39][40][41]57]). Let R � (p 1 ∘ R, p 2 ∘ R, p 3 ∘ R) be a picture fuzzy relation, denoted by I U×V , where ⟶ R are first projection, second projection, and third projection, respectively.

Picture Fuzzy Rough Sets
Based on (δ, ζ, ϑ)-Cut of R on Two Different Universes. We will begin by defining the (δ, ζ, ϑ)-cut of R and will subsequently define a picture fuzzy rough set based on (δ, ζ, ϑ)-cut.
Now, we present some properties based on PFRS as follows.
. e equality of eorem 1 (5) does not hold as the following example. Example 1. Suppose U � x i |i � 1, 2, 3 and V � y i |i � 1, 2, 3} be two three-element set, and R ∈ I U×V is defined by Let A � y 1 and B � y 2 , y 3   us,

Graded Picture Fuzzy Rough Sets Based on R on Two Different Universes
Definition 7. Let R ∈ I U×V , (δ, ζ, ϑ) ∈ I, n ∈ N, and A ∈ 2 V . en, are called the Type-I lower approximation and the Type-I upper approximation of A according to the graded n with respect to R (δ,ζ,ϑ) on U and V, respectively, and is called the Type-I picture fuzzy rough approximation of A according to the graded n with respect to R [(δ,ζ,ϑ)] (briefly, a Type-I picture fuzzy rough set according to the graded n with respect to R [(δ,ζ,ϑ)] ).
indicates that the property is satisfied.

Journal of Mathematics
For example, let U � x i |i � 1, 2 and V � y i |i � 1, 2 be two two-element sets, and R ∈ I U×V is defined by us, We study the notion of Type-II PFRS according to the graded n with respect to are called the Type-II lower approximation and the Type-II upper approximation of A according to the graded n with respect to R [(δ,ζ,ϑ)] on U and V, respectively, and is called the Type-II PFR approximation of A according to the graded n with respect to R [(δ,ζ,ϑ)] (briefly, a Type-II PFRS according to the graded n with respect to en, the following holds: us, n ∈ (0, min((|R [(δ,ζ,ϑ)] (u)|/2), |A|)).
(2) It can be easily proved by the relationship

Generalized Picture Fuzzy Rough Sets Based on (δ, ζ, ϑ)-Cut of R on Two Different Universes
(2) If we take α � 1 and β � 0 in Definition 11, then we can obtain in the following definitions: e main results are as follows.
en, the following holds: Journal of Mathematics Proof. By Definition 11, the result can be similarly proven as eorem 5.

Conclusions
In this paper, we suggest novel notion of picture fuzzy rough sets (PFRSs) over two different universes which depend on (δ, 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-IIgraded PFRSs, the degree α and β with respect to R [(δ,ζ,ϑ)] on PFRSs, and Type-I-/Type-II-generalized PFRSs based on the degree α and β with respect to R [(δ,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 decisionmaking problem is introduced, and we present an example to show the validity of this approach.

Data Availability
All data required for this paper are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.