# Covering-based fuzzy rough sets

#### Abstract

Many researchers have combined rough set theory and fuzzy set theory in order to easily approach problems of imprecision and uncertainty. Covering-based rough sets are one of the important generalizations of classical rough sets. Naturally, covering-based fuzzy rough sets can be studied as a combination of covering-based rough set theory and fuzzy set theory. It is clear that Pawlak’s rough set model and fuzzy rough set model are special cases of the covering-based fuzzy rough set model. This paper investigates the properties of covering-based fuzzy rough sets. In addition, operations of intersection, union and complement on covering-based fuzzy rough sets are investigated. Finally, the corresponding algebraic properties are discussed in detail.

## 1Introduction

Rough set theory, first proposed by Pawlak [35], is an excellent tool with which to handle vagueness and uncertainty in data analysis. The theory has been applied to the fields of medical diagnosis, conflictanalysis, pattern recognition and data mining [7, 12, 17, 19].

Pawlak rough set theory is built on equivalence relations. However, an equivalence relation is restrictive for many real-world applications [8, 14, 22]. To overcome this limitation, there are two primary methods to generalize Pawlak rough set theory. Rough set theory has been generalized from the perspective of extending the equivalence relation to other binary relations, such as dominance relations, tolerance relations and similarity relations [15, 32]. In addition, one of the most important generalizations is to replace a partition obtained by the equivalence relation with a covering [3, 9–11, 24, 26–28, 30, 33, 34]. Zakowski, in 1983, first employed the covering of a universe to establish a covering-based generalized rough set. Since then, the study of covering-based rough set theory has attracted many researchers. Many kinds of lower and upper approximation operators have been proposed [18, 24, 25, 27, 29, 31]. Yao proposed approximation operators based on coverings produced by the predecessor and/or successor neighborhoods of serial or inverse serial binary relations [33]. Zhu, et al. studied six types of approximation operators and investigated the properties and relationships among them [26–28]. Qian, et al. simultaneously investigated five pairs of dual covering-based approximation operators by employing the notion of the neighborhood [13]. In addition, Yun, et al., also discussed covering rough sets and solved an open problem identified by Zhu and Wang [27]. To construct the lower and upper approximations of an arbitrary, Chen, et al. proposed a new covering-based on generalized rough set [3].

Alternatively, rough set theory was generalized by combining with other theories that deal with uncertain knowledge. The fuzzy rough set model which combines fuzzy set theory with rough set theory is one of the most important adaptations. It is well known that fuzzy set theory and rough set theory are complementary in terms of handling different kinds of uncertainty. Rough set theory deals with uncertainty resulting from ambiguity of information [1], while fuzzy set theory is adept at dealing with probabilistic uncertainty, connected to the imprecision of states, perceptions and preferences. The two theories can be encountered in many specific problems. Therefore, rough set theory has been generalized by combining it with fuzzy set theory. Many researchers have discussed the fuzzy rough set model from various perspectives [1, 4–6, 21]. Dubois and Prade proposed the concepts of the rough fuzzy set and the fuzzy rough set [2]. Morsi, et al. discussed some axioms of fuzzy rough sets [16]. Wu, et al. studied the (*I*, *T*)-fuzzy rough approximation operators [20]. Xu, et al. proposed the multi-granulation fuzzy rough set model and studied the properties of multi-granulation fuzzy rough sets [23]. However, it is still an open problem regarding the research of covering-based fuzzy rough sets.

In this paper, the primary objective is to investigate covering-based rough set theory when combined with fuzzy set theory. The paper is organized as follows. In Section 2, some basic concepts of Pawlak’s rough set theory and fuzzy rough set theory are described. Furthermore, the concept of the monotone covering is proposed. In Section 3, the properties of covering-based fuzzy approximation operators are investigated. In Section 4, the operations of intersection, union and complement on covering-based fuzzy rough sets are discussed, as are the algebraic properties of covering-based fuzzy rough sets. Finally, Section 5 concludes this study.

## 2Preliminaries

In this section, some basic concepts and notions according to Pawlak’s theory rough sets, fuzzy sets, and covering are described. Additional details can be found in various references [23, 25, 35].

(*U*, *R*) is referred to as an approximation space, in which *U* = {*x*
_{1}, *x*
_{2}, ⋯ , *x*
_{
n
}} is a non-empty finite set. *x*] _{
R
} = {*y*| (*x*, *y*) ∈ *R*}, *U*/*R* = {[*x*] _{
R
}|*x* ∈ *U*}; then, [*x*] _{
R
} is called the equivalence class of *x* and the quotient set *U*/*R* is called the equivalence class set of *U*.

**Definition 2.1.** Let *R* be an equivalence relation. For any *X* ⊆ *U*,*φ*}.

These are the Pawlak lower and upper approximations of *X* with respect to equivalence relation *R*, respectively.

Let *U* represent a non-empty finite set. A fuzzy set *X* is a mapping from *U* into the unit interval [0, 1]; *X* : *U* → [0, 1], where each *X* (*x*) is the membership degree of *x* in *X*. The set of all the fuzzy sets defined on *U* is denoted by *F* (*U*).

**Definition 2.2.** Let *R* be an equivalence relation. For any *X* ⊆ *U*, denote

*X* with respect to equivalence relation *R*, where ∧ represents “min” and ∨ represents “max”.

**Definition 2.3.** Let *U* be the universe and *U*. If *U*. The ordered pair

Let

**Definition 2.4.** Let *U*. For any*x* ∈ *U* and *K*
_{
x
1
}, *K*
_{
x
2
}, ⋯ , *K*
_{
x
n
} can be reordered to *K*
_{
x
i1
}, *K*
_{
x
i2
}, ⋯ , *K*
_{
x
in
} such that *K*
_{
x
i1
} ⊆ *K*
_{
x
i2
} ⊆ ⋯ *K*
_{
x
in
}, then *U*, and

Furthermore, we denote *i* = 1, 2, ⋯ , *m*}, where

(1) For any *m* ;

(2) For any *i*≠ *j*, *i*, *j* = 1, 2, ⋯ , *m* ;

(3) For any

**Remark 2.1.** For any *x* ∈ *K*, by the construction of

**Example 2.1.** Let *U* = {*x*
_{1}, *x*
_{2}, ⋯ , *x*
_{8}} , *K*
_{1} = {*x*
_{1}, *x*
_{2}, *x*
_{3}, *x*
_{4}, *x*
_{5}} , *K*
_{2} = {*x*
_{3}, *x*
_{4}, *x*
_{5}} , *K*
_{3} = {*x*
_{5}} , *K*
_{4} = {*x*
_{6}, *x*
_{7}, *x*
_{8}} , and *K*
_{5} = {*x*
_{8}} , *U*. Moreover,

**Definition 2.5.** Let *X* ∈ *F* (*U*), denote

*X*, respectively. The pair *X* and

**Remark 2.2.** In a special case, when

**Example 2.2.** Let *U* = {*x*
_{1}, *x*
_{2}, *x*
_{3}, *x*
_{4}} , *K*
_{1} = {*x*
_{1}, *x*
_{2}}, *K*
_{2} = {*x*
_{2}} , *K*
_{3} = {*x*
_{1}, *x*
_{3}} , *K*
_{4} = {*x*
_{3}, *x*
_{4}}, and *U*. For fuzzy set *X* = (0.3, 0.4, 0.2, 0.5) ,

## 3Covering-based fuzzy approximation operators

In the section, the properties of the lower and upper covering fuzzy approximation operators in a covering approximation space are considered.

**Proposition 3.1.** *Let* *be a covering approximation space and* *X*, *Y* ∈ *F* (*U*). *Then, the following properties hold:*

**Proof:** The properties can be easily proved by Definition 2.5.□

**Example 3.1.** (Continued from Example 2.2) Foranother fuzzy set *Y* = (0.3, 0.2, 0.4, 0.1) ,

If *U*, then the following properties are observed.

**Proposition 3.2.** *Let* *be a monotone covering approximation space and* *X*, *Y* ∈ *F* (*U*). *Then the following properties hold:*

**Proof:** (1) (⇒ :) can be proved easily by Proposition 3.1.

(2) The item can be proved similarly to (1).□

**Proposition 3.3.** *Let* *be a monotone covering approximation space and* *X* ∈ *F* (*U*). *Then the following properties hold:*

**Proof:** (1) (⇒ :) can be proved easily by Proposition 3.1.

(2) The item can be proved similarly to (1). □

**Proposition 3.4.** *Let* *be a monotone covering approximation space and* *X*, *Y* ∈ *F* (*U*). *Then the following properties hold:*

**Proof:** (1) It is clear according to Proposition 3.1 and Proposition 3.2.

(2) (⇒ :) is clear according to Proposition 3.1.

(3) This item can be proved similarly to (2).

(4) This item can be proved similarly to (1). □

**Proposition 3.5.** *Let* *be a monotone covering approximation space . For any* *x* ∈ *U* *and* *X* ∈ *F* (*U*), *if* *then*

**Proof:** Clear according to Definition 2.5.□

**Proposition 3.6.** *Let* *be a monotone covering approximation space*. *For* ∀*x*, *y* ∈ *U* *and* *X* ∈ *F* (*U*), *if* *the following properties hold:*

**Proof:** Clear according to Definition 2.5.□

## 4Covering-based fuzzy rough sets

### 4.1Operations on covering-based fuzzy rough sets

In this section, the operations of intersection, union and complement on covering-based fuzzy rough sets are investigated. We first propose the concepts of intersection, union and complement of covering-based fuzzy rough sets.

**Definition 4.1.** Let

**Definition 4.2.** Let

Here, a question is raised: do all the covering-based fuzzy rough sets satisfy the operations of intersection, union and complement as defined above? The following will employ an example to illustrate the question.

**Example 4.1.** Let *U* = {*x*
_{1}, *x*
_{2}, *x*
_{3}} , *K*
_{1} = {*x*
_{1}, *x*
_{2}}, *K*
_{2} = {*x*
_{2}, *x*
_{3}} , *U* . For fuzzy sets *X* = (0.1, 0.2, 0.3) and *Y* = (0.3, 0.2, 0.1), we have *V* ∈ *F* (*U*) such that

Example 4.1 indicates that all the covering-based fuzzy rough sets do not meet the operation of intersection.

Let

For any *x* ∈ *U*, either *y* ∈ *U* such that

Thus, we can present the following proposition.

**Proposition 4.1.** *Let* *be a monotone covering approximation space and* *satis fy the condition* (*). *For any* *X*, *Y* ∈ *F* (*U*), *the following properties hold:*

**Proof:** (1) Suppose that

For *x*
_{11} ∈ *K*
_{1} randomly. According to Remark 2.1,

(1_{
K
1
}) If

Denote *x*
_{11}) . For *x*).

(2_{
K
1
}) If

(3_{
K
1
}) If

⋯⋯

(*n*
_{
K
1
}) If

For _{
K
1
} to *n*
_{
K
1
}, respectively. Finally, a fuzzy set *V* ∈ *F* (*U*) is obtained such that

(2) The property can be proved according to (1).□

**Remark 4.1.** For any two covering-based fuzzy roughsets *V*, *W* ∈ *F* (*U*) can be obtained such that

**Example 4.2.** (Continued from Example 2.1) It is clear that *X* = (0.3, 0.1, 0.2, 0.4, 0.5, 0.4, 0.3, 0.4) and *Y* = (0.2, 0.3, 0.4, 0.2, 0.6, 0.3, 0.2, 0.1), fuzzy sets *V* = (0.1, 0.5, 0.2, 0.5, 0.5, 0.1, 0.3, 0.1) and *W* = (0.2, 0.6, 0.2, 0.6, 0.6, 0.3, 0.4, 0.4) can be obtained such that

Moreover, for *Z* = (0.2, 0.3, 0.4, 0.5, 0.4, 0.6, 0.1, 0.3), Let *C* = (0.2, 0.5, 0.4, 0.5, 0.5, 0.1, 0.6, 0.3), *D* = (0.2, 0.5, 0.2, 0.5, 0.4, 0.1, 0.4, 0.3), then

### 4.2Algebraic properties of covering-based fuzzy rough sets

In this section, the algebraic properties of covering-based fuzzy rough sets are investigated. Suppose that

**Proposition 4.3.**

**Proof:** For any

Hence,

Let (*φ*, *φ*) =0, (*U*, *U*) =1, then the following conclusion is obtained.

**Proposition 4.4.** *is a soft algebra*.

**Proof:** (1) For any

Thus, 0 and 1 are the minimal and maximal element of

Similarly,

Hence,

## 5Conclusion

To easily deal with problems of uncertainty and imprecision, Xu, et al. proposed the multi-granulation fuzzy rough set model based on equivalence relations [23]. The model is a meaningful contribution toward the generalization of the classical rough set model.

It is well known that a multi-granulation rough set is a generalization of a Pawlak rough set. Covering-based rough sets are also an important generalization of classical rough sets. In this paper, we proposed the covering-based fuzzy rough set model and discussed its corresponding properties. Although many researchers have studied many properties of rough sets, the operations of intersection, union and complement on rough sets have yet to be investigated. In this paper, we proposed the concept of monotone covering and researched the operations of intersection, union and complement on covering-based fuzzy rough sets. Thus, the construction of the covering-based fuzzy rough set model is a meaningful generalization of rough set theory.

## Acknowledgments

This work is supported by the Natural Science Foundation of China (No. 11161003, 11261006, 61472463 and 61402064) and the Science-Technology Foundation of the Education Department of Fujian Province, China (No. JA15281).

## References

1 | Radzikowska AM, Kerre EE 2002 A comparative study of fuzzy rough sets Fuzzy Sets and Systems 126 137 155 |

2 | Dubois D, Prade H 1992 Putting rough sets and fuzzy ests together Intelligent Decision Support: Handbook of Applications and Advances of the Sets Theory Slowinski R 203 232 Kluwer Dordrecht |

3 | Chen DG, Wang CZ, Hu QH 2007 A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets InformationSciences 177 3500 3518 |

4 | Chen DG, Zhao SY 2010 Local reduction of decision system with fuzzy rough sets Fuzzy Sets and Systems 161 1871 1883 |

5 | Feng F, Liu XY, Leoreanu-Fotea V, Jun YB 2011 Soft sets and soft rough sets Information Sciences 181 1125 1137 |

6 | Feng F, Li CX, Davvaz B, Ali MI 2010 Soft sets combined with fuzzy sets and rough sets; A tentative approach? Soft Computing 14 899 911 |

7 | Jeon G, Kim D, Jeong J 2006 Rough sets attributes reduction based expert system in interlaced video sequences IEEE Transactions on Consumer Electronics 52 1348 1355 |

8 | Liu G 2010 Closures and topological closures in quasi-discrete closure Applied Mathematics Letters 23 772 776 |

9 | Liu GL, Zhu W 2008 The algebraic structures of generalized rough set theory Information Sciences 178 4105 4113 |

10 | Liu GL, Sai Y 2009 A comparison of two types of rough sets induced by coverings International Journal of Approximate Reasoning 50 521 528 |

11 | Pomkala JA 1987 Approximation operations in approximation space Bulletin of the Polish Academy of Sciences 9–10 653 662 |

12 | Li JH, Mei CL, Lv YJ 2012 Knowledge reduction in real decision formal contexts Information Sciences 189 191 207 |

13 | Qin K, Gao Y, Pei Z 2007 On covering rough sets The Second International Conference on Rough Sets and Knowledge Technology (RSKT 2007), Lecture Notes in Computer Science 4481 34 41 |

14 | Kondo M 2006 On the structure of generalized rough sets Information Sciences 176 589 600 |

15 | Kryaskiewics M 1998 Rough set approach to incomplete information systems Information Sciences 112 39 49 |

16 | Morsi NN, Yakout MM 1998 Axiomation for fuzzy rough sets Fuzzy Sets and Systems 100 327 342 |

17 | Swiniarski RW, Skowron A 2003 Rough set method in feature selection and recognition Pattern Recognition Letter 24 833 849 |

18 | Yang T, Li Q 2010 Reduction about approximation spaces of covering generalized rough sets International Journal of Approximate Reasoning 51 335 345 |

19 | Ananthanarayana VS, Narasimha MM, Subramanian DK 2003 Tree structure for efficient data mining usingrough sets Pattern Recognition Letter 24 851 862 |

20 |
Wu WZ, Leung Y, Mi JS
2005
On characterizations of ( |

21 | Wu WZ, Mi JS, Zhang WX 2003 Generalized fuzzy rough sets Information Sciences 151 263 282 |

22 | Wei W, Liang J, Qian Y 2012 A comparative study of rough sets for hybrid data Information Sciences 190 1 16 |

23 | Xu WH, Wang QR, Luo SQ 2014 Multi-granulation fuzzy rough sets Journal of Intelligent and Fuzzy Systems 26 1323 1340 |

24 | Zakowski W 1983 Approximations in the space (μ, π) Demonstration Mathematica 16 761 769 |

25 | Zhu W 2007 Basics concepts in covering-based rough sets Third International Conference on Natural Computation 5 283 286 |

26 | Zhu W 2007 Topological approaches to covering rough sets Information Sciences 177 6 1499 1508 |

27 | Zhu W, Wang FY 2007 On three types of covering rough sets IEEE Transactions on Knowledge Data Engineering 19 8 1131 1144 |

28 | Zhu W 2007 Generalized rough sets based on relations Information Sciences 177 22 4997 5001 |

29 | Zhu W, Wang FY 2003 Reduction and axiomatization of covering generalized rough sets Information Sciences 152 217 230 |

30 | Ge X, Li Z 2011 Definable subsets in covering approximation spaces International Journal Computing Mathematical Sciences 5 1 31 34 |

31 | Yao YY 1998 Generalized rough sets models In: Knowledge Discovery Polkowski L, Skowron A 284 318 Physica-Verlag Heidelberg |

32 | Yao YY 2010 Three-way decisions with probabilistic rough sets Information Sciences 180 3 341 353 |

33 | Yao YY, Yao BX 2012 Covering based rough set approximations Information Sciences 200 1 91 107 |

34 | Bonikowski Z, Brynirski E, Wybraniec U 1998 Extensions and intentions in the rough set theory Information Sciences 107 149 167 |

35 | Pawlak Z 1982 Rough sets International Journal of Computer and Information Sciences 11 5 341 356 |