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Dliouah Ahmed, Binxiang Dai, "Picture Fuzzy Rough Set and Rough Picture Fuzzy Set on Two Different Universes and Their Applications", Journal of Mathematics, vol. 2020, Article ID 8823580, 17 pages, 2020. https://doi.org/10.1155/2020/8823580
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 -cut of picture fuzzy relation 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 and with respect to on PFRSs, and (Type-I/Type-II) generalized PFRSs based on the degree and with respect to 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.
In the past few years, Pawlak  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 , 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 , presented the concept of fuzzy rough set (FRS) (i.e., the fuzzy set (FS)  and the RS). Many researchers have worked on fuzzy rough models (see [12–16]).
Wong et al.  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.  gave a comparative between the variable precision rough set (VPRS) and the GRS. In addition to the previous studies, Liu et al.  introduced the notion of GRSs on two universes. Yu et al.  presented the notion of a variable precision-graded rough set (VPGRSs) over two universes and Yu and Wang  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 on -cut of picture fuzzy relation over two different universes are discussed. Meanwhile, we propose two types of graded picture fuzzy rough sets (GPFRSs) based on -cut of on two different universes: type-I PFRS is according to the graded with respect to and type-II PFRS is according to the graded with respect to . The interesting properties of Type-I/Type-II PFRSs are investigated in detail. Furthermore, we define the notions of PFRS according to the degree and with respect to and Type-I/Type-II generalized PFRs according to the degree and with respect to . 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 Section 2, we gave the concepts of PFSs and picture fuzzy relations. In Section 3, we give the notion of PFRSs based on -cut of picture fuzzy relation 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.
2.1. Picture Fuzzy Sets and Picture Fuzzy Relations
Cuong [39–41] introduced the notion of PFS is an extension of fuzzy FS  and intuitionistic fuzzy set IFS . 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]). Let be an -element set ( is a natural number), and a PFS isMoreover, is called the refusal degree of . A PFS with refusal degree 0 at each point can be identified with an IFS on and with the pointwise order is the set of all mappings from a set (or an universe) to . Then, each element of is called an -set or a PFS on , (i.e., the degree of positive), (i.e., the degree of neutral), and (i.e., the degree of negative) of the element , where (i.e., the th projection from to ).
Definition 2 (cf. see [39–41]). Let . Then,(1)The complement of is defined by(2)The union (called also supremum ) and the intersection (called also infimum ) of a family can be defined by the following formulae:(3) is a subset of if (i.e., , , and , for each ).(4) is an equal of if (i.e., , , and ) and (i.e., , , and ).
3. Picture Fuzzy Rough Sets over Two Different Universes
3.1. Picture Fuzzy Rough Sets Based on -Cut of on Two Different Universes
We will begin by defining the -cut of and will subsequently define a picture fuzzy rough set based on -cut.
Definition 5. Let and . Then,(i) is called the -cut of , and(ii)Let . Then, (i.e., the lower approximation of ) and (i.e., the upper approximation of ).The pair is a PFR approximation of with respect to .
Now, we present some properties based on PFRS as follows.
Theorem 1. Let , , and . Then, the following holds:(1)(2)(3)If , then ; (4); (5); (6);
Proof. We only prove (6).
= , and = = .
The equality of Theorem 1 (5) does not hold as the following example.
Example 1. Suppose and be two three-element set, and is defined byTake . Then, , , and .
Let and . Thus, and .
Remark 1. In general, and do not hold. For example, let and be two two-element sets, and is defined byTake . Then, and . Thus, and .
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 and in Theorem 3.1(2) in  are incorrect. Let , , and be defined byTake . Then, and in the sense of  are not and , but and , respectively. Secondly, the inclusion in Theorem 3.1(1) in  is incorrect. Let and be two three-element sets, and , is defined byTake and . Then, , , and , and thus (resp., ) in the sense of  is (resp., ). Therefore, . Analogously, assertions in Theorem 3.1(1)  and in Theorem 3.1(1)  are incorrect. Let and be two three-element sets, and and also is defined byTake , and . Then, , , and , and thus and .
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. Let . Then,(1)For and , we have(2)For and , we have(3)For and , we have (cf. )
Theorem 2. Let , , and . If , then and .
Proof. If , then and for all . By Definition 5, we have . Therefore, for all . Thus, Similarly, .
Corollary 1. (1) Let , and If then and .
(2) Let and If then and .
(3) Let and If then and .