Abstract
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.
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 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. 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]). 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 ).
Definition 3 (cf. see [39–41, 57]). Let be a picture fuzzy relation, denoted by , where , , and satisfy for all and , and are first projection, second projection, and third projection, respectively.
Definition 4 (cf. see [39–41, 57]). For two picture fuzzy relations and , the picture fuzzy relation , defined by 1is said to be a composition of 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 [60] are incorrect. Let , , and be defined byTake . Then, and in the sense of [60] are not and , but and , respectively. Secondly, the inclusion in Theorem 3.1(1) in [58] is incorrect. Let and be two three-element sets, and , is defined byTake and . Then, , , and , and thus (resp., ) in the sense of [58] is (resp., ). Therefore, . Analogously, assertions in Theorem 3.1(1) [59] and in Theorem 3.1(1) [60] 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. [61])
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 .
Proof. It follows from Definition 6.
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:
Remark 3. Assertions and do not hold by Definition 6 (1), let and be two two-element sets, and is defined byTake and . Then, , and thus Also, if we take , .
Remark 4. Assertions and in Tables 2 and 3 do not hold by Definition 6 (2) and (3), and let and be two two-element sets, and and also is defined byTake and . Then, , and thus Also, if we take , thus
3.2. Graded Picture Fuzzy Rough Sets Based on on Two Different Universes
Definition 7. Let , and . Then,are called the Type-I lower approximation and the Type-I upper approximation of according to the graded with respect to on and , respectively, and is called the Type-I picture fuzzy rough approximation of according to the graded with respect to (briefly, a Type-I picture fuzzy rough set according to the graded with respect to ).
Theorem 3. Let , , and . Then,(1)If then and (2)(3); (4) (5)If then and
Proof. (1) By Definition 7, we have . Since then we have . Therefore, . Similarly, .
(2) and (3) Clear.
(4) and .
(5) By Definition 7, we have . If then we have . Hence, . Similarly, .
Remark 5. The equality and does not hold. For example, let and be two two-element sets, and is defined byTake and . Then, Thus, and .
Remark 6. Let , , and . Then, there does not exist inclusion relation between and . Consider and are given in Remark 1. Let and Then, and . This show that there does not exist inclusion relation between and .
We study the notion of Type-II PFRS according to the graded with respect to satisfy .
Definition 8. Let , s.t. and . Then,are called the Type-II lower approximation and the Type-II upper approximation of according to the graded with respect to on and , respectively, and is called the Type-II PFR approximation of according to the graded with respect to (briefly, a Type-II PFRS according to the graded with respect to ). The subsets , , and - are called the -positive region , the -negative region , and the -boundary region of with respect to graded respectively.
Lemma 1. Let and Then, the following holds:(1)(2)The limits of the grade
Proof. (1) For any Then, implies that . If then , and if , then . Consequently, and thus . Since and , then . Thus, .
(2) Follows from (1) above.
Theorem 4. Let , s.t. and . Then, the following holds:(1); (2)(3)If then and (4) and (5) and (6) and (7)If then and .
Proof. We only prove (2). Let and , but . From Definition 8, and for any Since and , then and thus . Consequently, and thus . It is contradiction with the limits of so the element which only belongs to but does not belong to does not exist.
Remark 7. (1) The equality and in Theorem 3.10 (2) in [60] are incorrect. (2) For example, let and be two two-element sets, and is defined byTake and . Then, . Thus, and .
(2) The inclusions in and in Theorem 3.10(1) in [60] are incorrect. For example, let and be two three-element sets, and is defined byTake and . Then, . Thus, and .
(3) The equalities and in Theorem 3.10(3) in [60] are incorrect. Consider above example in (2), let and . Then, and . Thus, and .
Definition 9. Let , , , and . Then,are called the lower approximation and the upper approximation of according to the degree and with respect to on and , respectively, and is called the picture fuzzy rough approximation of according to the degree and with respect to (briefly, a picture fuzzy rough set according to the degree and with respect to ), where denotes the cardinality of the set . The subset is called the boundary region of according to the degree and with respect to .
The main results are as follows.
Theorem 5. Let , , , and . Then, the following holds:(1) (2)(3) (4)If then and (5) and (6) (7) (if ); (if )
Proof. (1) For any we have and . Therefore, Also, and . Thus, .
(2)–(4) It follows from Definition 9.
(5) Let Then, , , and Similarily, we can obtain .
(6) It is analogous to (5).
(7) For let . Then, Since then Therefore, Thus Similarly, (if ).
Equality (5) and (6) of Theorem 5 does not hold as the following example.
Example 2 (continuation of Example 1). Let , and . Then,(1) and thus (2) and thus The relationship among the regulation for lower approximation, upper approximation, and boundary region with error and is discussed as the following theorems.
Theorem 6. Let , , , and . Then, the following holds:(1)(2)(3)
Proof. (1) From Definition 9, when , we have is hold. In addition, by Theorem 5 (7), we have increase with error decrease. Thus, . Conversely, if there exists , for any , and . That is, . It contradicts with , i.e., . Consequently, . Thus, hold.
(2) It is analogous to (1) above.
(3) We know is held. In addition, we know that when decreased to and decreased to , the boundary region will decrease. Then, we have . Conversely, if there exists , then for any , we have and . Furthermore, there is by and . Similarily, there is by and . This is a contradiction. Consequently, there is . That is, .
Theorem 7. Let , , , and . Then, the following holds:(1)(2)
Proof. Obvious.
Remark 8. Let , , , and . Then, the following does not hold:(1) (i.e., the lower approximation is not right continuous with error )(2) (i.e., the upper approximation is not left continuous with error )
Example 3 (continuation of Example 1). Let . Then, we can obtain the following relations:(1). Consequently, .(2). Consequently, .
Definition 10. Let , , , and . Then, the precision and rough degree of with error and are defined byObviously, and .
Lemma 2. Let , , , and . Then, the following holds:(1) does not decrease with error and does not increase with error (2) does not increase with error and does not decrease with error
Proof. We only prove (1) and then the proof of (2) can be obtained using similar techniques.
(1) Since by Theorem 5 (7), then there is implies that . Therefore, . Similarly, for .
Theorem 8. Let , , , , and . Then, the following holds:(1)If , there is and (2)If , there is and (3)If and , there is and
Proof. It follows from Theorem 5 (4) and Definition 10.
Theorem 9. Let , , , and . Then, the following holds:(1)(2)
Proof. (1) From Definition 10, we have ≤ . Consequently, . Similarly, ≤ . We know . Then,Also, by the relation of and , then and . Thus, we haveHence, (1) is holding.
(2) It can be easily proved by the relationship .
3.3. Generalized Picture Fuzzy Rough Sets Based on -Cut of on Two Different Universes
Definition 11. Let , , , and . Then,(1) and are called the Type-I lower -approximation and the Type-I upper -approximation of according to the degree and with respect to on and , respectively, and is called the Type-I generalized picture fuzzy rough approximation of according to the degree and with respect to (briefly, the Type-I generalized picture fuzzy rough set according to the degree and with respect to ).(2) and are called the Type-II lower -approximation and the Type-II upper -approximation of according to the degree and with respect to on and , respectively, and is called the Type-II generalized picture fuzzy rough approximation of according to the degree and with respect to (briefly, the Type-II generalized picture fuzzy rough set according to the degree and with respect to ).
Remark 9. (1) From Definition 11, we have(i)(ii)(2) If we take and in Definition 11, then we can obtain in the following definitions:(i) and .(ii) and .The main results are as follows.
Theorem 10. Let , , , and . Then, the following holds:(1); (2)(3)If , then (4); (5); (6) (if ); (if )
Proof. By Definition 11, the result can be similarly proven as Theorem 5.
Remark 10. Let and . Then,(1)(2) does not hold by the following example.
Example 4. Let and be two two-element set, and is defined byTake . Then, , , . Let , and . Thus,(1)(2)From Remark 9 (2), we can conclude the following corollary.
Corollary 2. Let , , and . Then, the following holds:(1); (2)(3)If , then (4); (5);
Theorem 11. Let , , , and .
Then, the following holds:(1); (2)(3)If , then ; (4); (5); (6) (if ); (if )
Proof. It follows from Definition 11 (2).
From Remark 9 (2), we can conclude the following corollary.
Corollary 3. Let , , and . Then the following holds:(1); (2)(3)If , then ; (4); (5);
Theorem 12. Let , , , and . Then, the following holds:(1)(2)(3)(4)
Proof. We only prove (3), while rest can be proven similarly.
(3) We show that . Let , where , but . So, . Similarly, . Consequently, . Conversely, for and by Definition 11, we have either , then or , and then . Hence, , and thus . 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. Let and . Then, (i.e., the lower approximation of ) and(i.e., the upper approximation of ).
A pair is said to be the rough picture fuzzy approximation of with respect to .
Example 5. Let and be three-element sets, defined byand defined byBy Definition 12, we obtainSimilarly, , , , , and .
Therefore,
Definition 13. Suppose that be a set and . The sum , is defined bywhere is defined in Definition 3 in [50].
Definition 14. (cf. see [62])Let be an -element set and . A cosine similarity measure is defined byNext, we present the RPFS decision-making medical diagnosis problem over two different universes as indicated below.
Suppose that (a -element set) be the set of a disease, the set of symptoms (where ), and be picture fuzzy relation from to . For any , it represents the picture fuzzy relation between the disease and the symptom (i.e., by a doctor in advance). For (i.e., any patient set) who has some symptoms in , where is PFS on symptom , that is,where (i.e., the degree of positive membership) to the symptom of , (i.e., the degree of neutral membership) to the symptom of , and (i.e., the degree of negative membership) to the symptom of (clearly, , ). By Definition 12, we compute the lower and the upper RPF approximations and , respectively, of . Then, by Definition 14, we obtain as follows:From Definition 14, we calculate the cosine similarity measure between the corresponding to and the ideal . Finally, we determine . Hence, the best choice is to select , that is, we can conclude that patient 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 Algorithm 1.
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Example 6. Assume that five diseases (), where stands for the “viral fever,” stands for the “malaria,” stands for the “typhoid,” stands for the “stomach problem,” and stands for the “chest problem.” The five symptoms are in clinic (let ), where stands for the “temperature,” stands for the “headache,” stands for the “stomach Pain,” stands for the “cough,” and stands for the “Chest-Pain.” Let be a picture fuzzy relation from to , where is a medical knowledge statistic data of the relationship of the disease and the symptom (where ). The statistic data are given in Table 4.
By the Step 2 of Algorithm 1, suppose the symptoms of a patient are defined by a PFS on , andThen, by Definition 12, we obtain on the and of patient in Step 3 of Algorithm 1, respectively, as follows:By Step 4 of Algorithm 1, we obtainThen, by Step 4 of Algorithm 1, we can get the cosine similarity measure between the corresponding to and the ideal as follows:Thus, according to Step 6, we conclude the maximum value is . Thus, the patient is suffering from the disease chest problem .
5. Conclusions
In this paper, we suggest novel notion of picture fuzzy rough sets (PFRSs) over two different universes which depend on -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 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. 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.
Data Availability
All data required for this paper are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (no. 11871475).