On Covering-Based Rough Intuitionistic Fuzzy Sets

Abstract

Intuitionistic Fuzzy Sets ($IF{S}_s$) and rough sets depending on covering are important theories for dealing with uncertainty and inexact problems. We think the neighborhood of an element is more realistic than any cluster in the processes of classification and approximation. So, we introduce intuitionistic fuzzy sets on the space of rough sets based on covering by using the concept of the neighborhood. Three models of intuitionistic fuzzy set approximation space based on covering are defined by using the concept of neighborhood. In the first and second model, we approximate IFS by rough set based on one covering (C) by defining membership and non-membership degree depending on the neighborhood. In the third mode, we approximate IFS by rough set based on family of covering ($C_i$) by defining membership and non-membership degree depending on the neighborhood. We employ the notion of the neighborhood to prove the definitions and the features of these models. Finlay, we give an illustrative example for the new covering rough $IF$ approximation structure.

1. Introduction

The equivalence relation is the main tool for classification algorithms by Pawlak rough theory [1]. Pawlak’s model is suitable for discrete data but not continuous data, as the cost of computation becomes too high. So, many extensions of Pawlak’s model have been studied by many researchers, such as probabilistic rough sets [2], fuzzy rough set models [3], tolerance relations [4], similarity rough sets [5], decision-theoretic rough sets [6], rough sets based on covering [7] and probabilistic rough sets [2]. One of the most important extensions of rough set models are neighborhood rough set systems [8]. These systems can deal with discrete and continuous data by using the $\theta$-neighborhood relation. Neighborhood rough set models have many application in feature selection [9,10,11,12].

X. Zhan-Ao et al. [13] combined rough sets depending on covering and IFSs using the concept of minimal description. In our approach, we use the neighborhood concept in the approximation of IFS defined on rough sets depending on covering. Many scholars, such as Goguen [14], Gau et al. [15] and Nada et al. [16], extended fuzzy set concepts in various directions after Zadeh [17] discovered them in 1965. Since then, fuzzy set concepts have been used in topology, analysis and other branches of mathematics. The definition of $IF{S}_s$ is one of the important generalizations of fuzzy sets, which has been presented by Atanassov [18], and it is a group that is important for the importance of its applications. $IFS$ theories, such as in [19,20], are more convenient and are also applied in decision making in various fields, including programming, health [11,21,22] and decision-making problems [23,24,25]. Dubois and Prade extended the fuzzy lower ($L_{app}$) and fuzzy upper approximation ($U_{app}$) in the approximation structure of Pawlak to obtain a rough fuzzy set [26]. Although $IFSs$ is a generalization of classical fuzzy sets and rough sets depending on covering, it is a generalization of classical rough sets. The combination of rough sets depending on covering and $IF{S}_s$ has gained the attention of many researchers, and many studies have been conducted using single-granulation methods [27,28,29,30,31,32]. Pawlak’s rough sets depending on single-granulation have been generalized to rough sets depending on multi-granulation (MGRSs) [33]. Zhan-Ao et al. [13] combined rough sets depending on covering and $IF{S}_s$ using the concept of minimal description. In this paper, we introduce this combination by using the concepts of neighborhood for each element of the universe. We think that the neighborhood of an element is more realistic than its description. So, it will be the best in classification processes and decision making. We process this combination for both single-granulation and multi-granulation. We introduce many basic concepts related to $IFS$, fuzzy set, rough $IFS$ and $co{v}_{app}$ structure.

2. Preliminaries

Definition 1

([17]). Assume that $X_1$ is a non-empty set. A fuzzy set H is characterized by a membership function ${\mu }_H\left(x_1\right)$ from $X_1$ to $I=]0,1[$ and totally characterizing the fuzzy set H is $H=\{(X_1,{\mu }_H\left(x_1\right)):x_1\in X_1\}$. In what comes next, let us assume that U, the universe of discourse, is finite.

Definition 2

([18]). An $IFS$ on a universe $X_1$ is an object of the form $H=\{(x_1,{\mu }_H\left(x_1\right),{\nu }_H\left(x_1\right)):x_1\in X_1\}$. Such that ${\mu }_H\left(x_1\right)\in [0,1]$ is said to be "degree of membership of $x_1$ in H ", and ${\nu }_H\left(x_1\right)\in [0,1]$ is called the "degree of non-membership of $x_1$ in H" since $\forall x_1\in X_1:{\mu }_H\left(x_1\right)+{\nu }_H\left(x_1\right)⩽1,{\pi }_H\left(x_1\right)=1-({\mu }_H\left(x_1\right)+{\nu }_H\left(x_1\right))$, such that ${\pi }_H\left(x_1\right)$ is called the hesitation part. The next example illustrates a collection of faction plants and the presence or absence of yellow cards. Next is an example of faction plants and yellow cards or not. $H=\left\{\right(Grourds,0.6,0.2),(Citrus,0.1,0.6)$, $(Legumes,0.7,0.25),(Gramineous,0.9,0.1)\}$.

Definition 3

([18]). AIf $H,K$ two IFS${}_s$ on$X_1$, then the IFS $H\subseteq K$ iff, $\forall x_1\in X_1,{\mu }_H\left(x_1\right)⩽{\mu }_K\left(x_1\right)$ and ${\nu }_H\left(x_1\right)⩾{\nu }_K\left(x_1\right)$.

Definition 4

([18]). An $IFSH$ equal to $IFSK$ iff $\forall x_1\in X_1,{\mu }_H\left(V\right)={\mu }_K\left(x_1\right)$ and ${\nu }_H\left(x_1\right)={\nu }_K\left(x_1\right)$.

Definition 5

([23]). If $H,K$ are two IFS${}_s$ on$X_1$, then the IFS $H\cap K=\left\{\right(x_1,min({\mu }_H\left(x_1\right),{\mu }_K\left(x_1\right))$, $max({\nu }_H\left(x_1\right),{\nu }_K\left(x_1\right))):x_1\in X_1\}$.

Definition 6

([23]). If $H,K$ two IFS${}_s$ on$X_1$, then the IFS $H\cup K=\{(x_1,max({\mu }_H\left(x_1\right),{\mu }_K\left(x_1\right)),min({\nu }_H\left(x_1\right),{\nu }_K\left(x_1\right))):x_1\in X_1\}$.

Definition 7

([34,35]). Let U be a universe and C be a family of subsets of U whenever no subset in C is empty and ∪$C=U$. Then, C is called a covering of U. Clearly, a partition of U is also a covering of U, hence the notion of a covering extends the notion of a partition.

Definition 8

([34,35]). If U is a non-empty set, and C a covering of U, then the ordered pair $\prec U,C\succ$ is called a covering approximation ($co{v}_{app}$) structure.

Definition 9

([34,35]). If $\prec U,C\succ$ is a $co{v}_{app}$ structure, $x_1\in U$, then the set family $Md\left(x_1\right)$ is called the minimal description of $x_1$ such that $Md\left(x_1\right)=\{L\in C:x_1\in L\wedge (\forall S\in C\wedge x_1\in S\wedge S\subseteq L\Rightarrow L=S)\}$.

Definition 10

([36]). If U is a non-null set, R is an equivalence relation on U and A is an $IFS$ in U with the membership function ${\mu }_A$ and non-membership function ${\nu }_A$, then $L_{app}$ and the $U_{app}$s $R_1\left(A\right)$ and $R_2\left(A\right)$, respectively, of the $IFS$ A are $IF{S}_s$ of the quotient set $U/R$ with:

(i) Membership function is defined as

${\mu }_{R_1\left(A\right)}\left(x\right)=inf\{{\mu }_A\left(x\right):x\in U\}$,    ${\mu }_{R_2\left(A\right)}\left(x\right)=sup\{{\mu }_A\left(x\right):x\in U\}$

(ii) Non-membership function is defined by

${\nu }_{R_1\left(A\right)}\left(x\right)=sup\{{\nu }_A\left(x\right):x\in U\}$,    ${\nu }_{R_2\left(A\right)}\left(x\right)=inf\{{\nu }_A\left(x\right):x\in U\}$

The rough $IFS$ of A is $R\left(A\right)$ given by the pair $R\left(A\right)=(R_1\left(A\right),R_2\left(A\right))$.

Example 1.

Let A be an $IFS$, $A=\{(y_1,0.8,0.1),(y_2,0.7,0.2),(y_3,0.6,0.1),(y_4,0.9,0.1),$$(y_5,0.8,0.2)\}$ and IF relation R as in

Table 1. IF relation R.

R	${\mathit{y}}_1$	${\mathit{y}}_2$	${\mathit{y}}_3$	${\mathit{y}}_4$	${\mathit{y}}_5$
$R\left(y_1\right)$	(0.9, 0.0)	(0.7, 0.1)	(0.6, 0.2)	(0.5, 0.1)	(0.3, 0.2)
$R\left(y_2\right)$	(0.8, 0.1)	(0.4, 0.4)	(0.8, 0.1)	(0.7, 0.1)	(0.1, 0.0)
$R\left(y_3\right)$	(0.7, 0.2)	(0.3, 0.1)	(0.0, 0.6)	(0.2, 0.2)	(0.6, 0.2)
$R\left(y_4\right)$	(0.6, 0.1)	(0.5, 0.5)	(0.4, 0.4)	(0.7, 0.2)	(0.3, 0.4)
$R\left(y_5\right)$	(0.9, 0.0)	(0.0, 0.1)	(0.1, 0.1)	(0.5, 0.5)	(0.3, 0.3)

.

$R_1\left(A\right)=\{(y_1,0.3,0.2),(y_2,0.1,0.4),(y_3,0.0,0.6),(y_4,0.3,0.5),(y_5,0.0,0.5)\}$

$R_2\left(A\right)=\{(y_1,0.9,0.0),(y_2,0.8,0.0),(y_3,0.7,0.1),(y_4,0.7,0.1),(y_5,0.9,0.0)\}$

Definition 11

([37]). If $(U,C)$ is a $co{v}_{app}$ structure, $X_1\in U$, then

$\cup \{K:x_1\in K\in C\}$ is called the discernible neighborhood of $x_1$ and is denoted as a friend of $x_1$.

$\cap \{K:x_1\in K\in C\}$ is called the neighborhood of $x_1$ and defined as $N_C\left(X_1\right)$. We eliminate the lowercase C when there is no confusion.

This article’s sections are arranged as follows: in Section 2, we define the first type of covering-based $IFS$ and its related properties. In Section 3, we introduce the second type of covering rough $IF$-based neighborhood. In Section 4, we present the third type as a multi-granulation covering rough $IF{S}_s$.

3. Covering Rough ${\mathit{I}\mathit{F}\mathit{S}}_{\mathbf{s}}$ of the First Type

Definition 12.

If $(U,C)$ is a $co{v}_{app}$ structure, $X_1\in IFS\left(U\right)$, then $C_{*}\left(X_1\right)=\{(x_1,\alpha ,\beta ):x_1\in U\}$ is called covering rough intuitionistic $L_{app}$ of $X_1$. Where,

$\alpha =\wedge \{\mu \left(y\right):y\in N\left(x_1\right)\}$    $\beta =\vee \{\nu \left(y\right):y\in N\left(x_1\right)\}$

and $C^{*}\left(X_1\right)=\{(x_1,\Psi ,\omega ):x_1\in U$ is called covering rough intuitionistic $U_{app}$ of $X_1$. Where,

$\Psi =\vee \{\mu \left(y\right):y\in N\left(x_1\right)\}$    $\omega =\wedge \{\nu \left(y\right):y\in N\left(x_1\right)\}$

We call this model type-I covering rough $IF{S}_s$.

Example 2.

Consider $U=\{u_a,u_b,u_c,u_d,u_e,u_f,u_g,u_h\}$, $C=\{long,average,short\}=\{\{u_a,u_b,u_d\},$$\{u_c,u_d,u_e,u_g\},\{u_f,u_g,u_h\}\}$, and $X_1=\{(u_a,0.7,0.2),(u_b,0.3,0.6),(u_c,0.0,0.9),(u_d,0.9,0.1),(u_e,0.1,0.9),$$(u_f,0.0,0.8),(u_g,0.2,0.7),(u_h,0.9,0.1)\}$. As such, we compute the $L_{app}$ and $U_{app}$ of $X_1$ based on the model we presented in Definitions 9 and 12.

$N\left(u_a\right)=N\left(u_b\right)=\{u_a,u_b,u_d\}$, $N\left(u_c\right)=N\left(u_e\right)=\{u_c,u_d,u_e,u_g\}$, $N\left(u_d\right)=\left\{u_d\right\}$, $N\left(u_f\right)=N\left(u_h\right)=\{u_f,u_g,u_h\}$, $N\left(u_g\right)=\left\{u_g\right\}$. Then $C_{*}\left(X_1\right)=\{(u_a,0.3,0.6),(u_b,0.3,0.6),(u_c,0.0,0.9),$

$(u_d,0.9,0.1),(u_e,0.0,0.9),(u_f,0.0,0.8),(u_g,0.0,0.9),(u_h,0.0,0.8),\}$, and $C^{*}\left(X_1\right)=\left\{\right(u_a,0.9,$$0.1),$$(u_b,0.9,0.1),(u_c,0.9,0.1),(u_d,0.9,0.1),(u_e,0.9,0.1),(u_f,0.9,0.1),(u_g,0.2,0.7),(u_h,0.9,$$0.1\left)\right\}$

Proposition 1.

If $(U,C)$ is a $co{v}_{app}$ approximation structure, then $\forall X_1,Y\in IFS\left(U\right)$; the next properties hold:

(1) $C_{*}\left(U\right)=C^{*}\left(U\right)=U$    (2) $C_{*}\left(\Phi \right)=C^{*}\left(\Phi \right)=\Phi$

(3) $C_{*}\left(X_1\right)\subseteq X_1\subseteq C^{*}\left(X_1\right)$    (4) Let $X_1\subseteq Y$. Therefore, $C_{*}\left(X_1\right)\subseteq C_{*}\left(Y\right)$ and $C^{*}\left(X_1\right)\subseteq C^{*}\left(Y\right)$

Proof. 

From Definition 12, it is clear. □

4. Covering Rough ${\mathit{IFS}}_{\mathbf{s}}$ of the Second Type

Definition 13.

Let U be non-empty and C be a covering of U. For $X_1\in IFS\left(U\right)$, $x_1\in U$ covering rough $IF$ membership and the non-membership degrees of $x_1$ depending on the neighborhood of $X_1$ are described as: ${\displaystyle {\mu }_{X_1}^{\prime }\left(x_1\right)=\sum _{y\in N\left(x_1\right)}{\mu }_{X_1}\left(y\right)∕\left|N\left(x_1\right)\right|}$,    ${\displaystyle {\nu }_{X_1}^{\prime }\left(x_1\right)=\sum _{y\in N\left(x_1\right)}{\nu }_{X_1}\left(y\right)∕\left|N\left(x_1\right)\right|}$.

The covering rough $IF$ $L_{app}$ of $X_1$ is described as

$$C{N}_{*}\left(X_1\right)=\{(x_1,{\mu }_{C{N}_{*}\left(x_1\right)},{\nu }_{C{N}_{*}\left(x_1\right)}):x_1\in U\}$$

We define the covering rough $IF$$U_{app}$ of $X_1$ in terms of

$$C{N}^{*}\left(X_1\right)=\{(x_1,{\mu }_{C{N}^{*}\left(x_1\right)},{\nu }_{C{N}^{*}\left(x_1\right)}):x_1\in U\}$$

Since ${\mu }_{C{N}_{*}\left(x_1\right)}=min({\mu }_{X_1}\left(x_1\right),{\mu }_{X_1}^{\prime }\left(x_1\right))$, ${\nu }_{C{N}_{*}\left(x_1\right)}=max({\nu }_{X_1}\left(x_1\right),{\nu }_{X_1}^{\prime }\left(x_1\right))$,

${\mu }_{C{N}^{*}\left(x_1\right)}=max({\mu }_{X_1}\left(x_1\right),{\mu }_{X_1}^{\prime }\left(x_1\right))$ and ${\nu }_{C{N}^{*}\left(x_1\right)}=min({\nu }_{X_1}\left(x_1\right),{\nu }_{X_1}^{\prime }\left(x_1\right))$.

The following example is an illustrative example.

Example 3.

Consider $(U,C)$ is a $co{v}_{app}$ structure, $U=\{u_a,u_b,u_c,u_d,u_e,u_f,u_g,u_h\}$, $C=\{\{u_a,u_b\},\{u_c,u_d,u_e\},\{u_d,u_e,u_g\},\{u_f,u_g,u_h\}\}$

and $X_1\in IFS\left(U\right)$, such that $X_1=\{(u_a,0.2,0.3),(u_b,0.7,0.2),(u_c,0.5,0.2),(u_d,0.3,0.6),(u_e,0.2,0.7),(u_f,0.0,0.6)$, $(u_g,0.5,0.4),(u_h,0.1,0.5)\}$.

Accordingly, we compute the $L_{a}pp$ and $U_{a}pp$ of $X_1$ based on the model presented in Definitions 9 and 13

$N\left(u_a\right)=N\left(u_b\right)=\{u_a,u_b\}$, $N\left(u_c\right)=\{u_c,u_d,u_e\}$, $Nd\left(u_d\right)=N\left(u_e\right)=\{u_d,u_e\}$, $N\left(u_f\right)=N\left(u_h\right)=\{u_f,u_g,u_h\}$, $N\left(u_g\right)=\left\{u_g\right\}$.Then,${\mu }_{X_1}^{\prime }\left(x_1\right)={\mu }_{X_1}^{\prime }\left(x_2\right)=0.45$, ${\mu }_{X_1}^{\prime }\left(x_3\right)=0.33$, ${\mu }_{X_1}^{\prime }\left(x_4\right)={\mu }_{X_1}^{\prime }\left(x_5\right)=0.25$, ${\mu }_{X_1}^{\prime }\left(x_6\right)={\mu }_{X_1}^{\prime }\left(x_8\right)=0.3$, ${\mu }_{X_1}^{\prime }\left(x_7\right)=0.5$, and ${\nu }_{X_1}^{\prime }\left(x_1\right)={\nu }_X^{\prime }\left(x_2\right)=0.4$, ${\nu }_{X_1}^{\prime }\left(x_3\right)=0.5$, ${\nu }_{X_1}^{\prime }\left(x_4\right)={\nu }_{X_1}^{\prime }\left(x_5\right)=0.65$, ${\nu }_{X_1}^{\prime }\left(x_6\right)={\nu }_{X_1}^{\prime }\left(x_8\right)=0.5$, ${\nu }_{X_1}^{\prime }\left(x_7\right)=0.4$. From Definition 13, we have ${\mu }_{C{N}_{*}\left(x_1\right)}=min(0.2,0.45)=0.2$, ${\mu }_{C{N}_{*}\left(x_2\right)}=min(0.7,0.45)=0.45$,

${\mu }_{C{N}_{*}\left(x_3\right)}=min(0.5,0.33)=0.33$, ${\mu }_{C{N}_{*}\left(x_4\right)}=min(0.3,0.25)=0.25$, ${\mu }_{C{N}_{*}\left(x_5\right)}=min(0.2,0.25)$$=0.2$, ${\mu }_{C{N}_{*}\left(x_6\right)}=min(0.0,0.3)=0.0$, ${\mu }_{C{N}_{*}N\left(x_7\right)}=min(0.5,0.5)=0.5$, ${\mu }_{C{N}_{*}\left(x_8\right)}=min(0.1,0.3)$ = $0.1$, and ${\nu }_{C{N}_{*}\left(x_1\right)}=max(0.3,0.4)=0.4$, ${\nu }_{C{N}_{*}\left(x_2\right)}=max(0.2,0.4)=0.4$, ${\nu }_{C{N}_{*}\left(x_3\right)}=max(0.2,0.5)=0.5$, ${\nu }_{C{N}_{*}\left(x_4\right)}=max(0.6,0.65)=0.65$, ${\nu }_{C{N}_{*}\left(x_5\right)}=max(0.7,0.65)=0.7$, ${\nu }_{C{N}_{*}\left(x_6\right)}=max(0.6,0.5)=0.6$, ${\nu }_{C{N}_{*}\left(x_7\right)}=max(0.4,0.4)=0.4$, ${\nu }_{C{N}_{*}\left(x_8\right)}=max(0.5,0.5)$$=0.5$. We have $C{N}_{*}\left(x\right)=\{(x_1,0.2,0.4),(x_2,0.45,0.4),(x_3,0.33,0.5)$, $(x_4,0.25,0.65),(x_5,0.2,0.7),(x_6,0.0,0.6)$, $(x_7,0.5,0.4),(x_8,0.1,0.5)\}$. Likewise, from Definition 13, we can obtain $C{N}^{*}\left(x\right)=\{(x_1,0.45,0.3),(x_2,0.7,0.2)$, $(x_3,0.5,0.2),(x_4,0.3,0.6),(x_5,0.25,0.65),(x_6,0.3,0.5),(x_7,0.5,0.4),(x_8,0.3,0.5)\}$.

Proposition 2.

If $(U,C)$ is a $co{v}_{app}$ structure. The set family operators $C_{*}$, and $C^{*}$, $\forall X_1,Y\in IFS\left(U\right)$, then satisfy the next properties:

(1) $C{N}_{*}\left(U\right)=C{N}^{*}\left(U\right)=U$    (2) $C{N}_{*}\left(\Phi \right)=C{N}^{*}\left(\Phi \right)=\Phi$

(3) $C{N}_{*}\left(X_1\right)\subseteq X_1\subseteq C{N}^{*}\left(X_1\right)$    (4) Let $X_1\subseteq Y$. Thus, $C{N}_{*}\left(X_1\right)\subseteq C{N}_{*}\left(Y\right)$ and $C{N}^{*}\left(X_1\right)\subseteq C{N}^{*}\left(Y\right)$

Proof. 

(1) If we take $X_1=U$, then the membership degree =1 and the non-membership degree=0 for each $x\in X_1$. Then, ${\mu }_U\left(x_1\right)=1$, ${\nu }_U\left(x_1\right)=0$, so from Definition 13${\mu }_U^{\prime }\left(x_1\right)=1$, ${\nu }_U^{\prime }\left(x_1\right)=0$. By Definition 13$,{\mu }_{C{N}_{*}}\left(x_1\right)=1,{\nu }_{C{N}_{*}}\left(x_1\right)=0$ and ${\mu }_{C{N}^{*}}\left(x_1\right)=1$, ${\nu }_{C{N}^{*}}\left(x\right)=0$ hence $C{N}_{*}\left(U\right)=C{N}^{*}\left(U\right)=U$.

(2) Similar to (1).

(3) From Definition 13 $\forall x_1\in U$, $X_1$ is $IFS$, ${\mu }_{X_1}\left(x_1\right)\le {\mu }_{X_1}^{\prime }\left(x_1\right)$ or ${\mu }_{X_1}\left(x_1\right)\ge {\mu }_{X_1}^{\prime }\left(x_1\right)$ and ${\nu }_{X_1}\left(x_1\right)\le {\nu }_{X_1}^{\prime }\left(x_1\right)$ or ${\nu }_{X_1}\left(x_1\right)\ge {\nu }_{X_1}^{\prime }\left(x_1\right)$. Then, there are four cases:

(i) ${\mu }_{X_1}\left(x_1\right)\le {\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\le {\nu }_{X_1}^{\prime }\left(x_1\right)$    (ii) ${\mu }_{X_1}\left(x_1\right)\ge {\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\le {\nu }_{X_1}^{\prime }\left(x_1\right)$

(iii) ${\mu }_{X_1}\left(x_1\right)\le {\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\ge {\nu }_{X_1}^{\prime }\left(x_1\right)$    (iv) ${\mu }_{X_1}\left(x_1\right)\ge {\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\ge {\nu }_{X_1}^{\prime }\left(x_1\right)$ □

Proof. 

(i) If ${\mu }_{X_1}\left(x_1\right)\le {\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\le {\nu }_{X_1}^{\prime }\left(x_1\right)$, then ${\mu }_{C{N}_{*}}={\mu }_{X_1}\left(x_1\right)$, ${\mu }_{C{N}^{*}}={\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\nu }_{C{N}_{*}}={\nu }_{X_1}^{\prime }\left(x_1\right)$, ${\nu }_{C{N}^{*}}={\nu }_{X_1}\left(x_1\right)$. Therefor, ${\mu }_{C{N}_{*}}\left(x_1\right)\le {\mu }_{X_1}\left(x_1\right)\le {\mu }_{C{N}^{*}}$, ${\nu }_{C{N}_{*}}\left(x_1\right)\ge {\nu }_{X_1}\left(x_1\right)\ge {\nu }_{C{N}^{*}}$. □

Proof. 

(ii) If ${\mu }_{X_1}\left(x_1\right)\ge {\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\le {\nu }_{X_1}^{\prime }\left(x_1\right)$. Then, ${\mu }_{C{N}_{*}}={\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\mu }_{C{N}^{*}}={\mu }_{X_1}\left(x_1\right)$, ${\nu }_{C{N}_{*}}={\nu }_{X_1}^{\prime }\left(x_1\right)$, ${\nu }_{C{N}^{*}}={\nu }_{X_1}\left(x_1\right)$. Therefore, ${\mu }_{C{N}_{*}}\left(x_1\right)\le {\mu }_{X_1}\left(x_1\right)\le {\mu }_{C{N}^{*}}\left(x_1\right)$, and ${\nu }_{C{N}_{*}}\left(x_1\right)\ge {\nu }_{X_1}\left(x_1\right)\ge {\mu }_{C{N}^{*}}\left(x_1\right)$. □

The proofs (iii) and (iv) are the same, hence $C{N}_{*}\le X_1\le C{N}^{*}$.

(4) If $X_1\subseteq Y$, $\forall x_1\in U$, then ${\mu }_{X_1}\left(x_1\right)\le {\mu }_Y\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\ge {\nu }_Y\left(x_1\right)$, ${\mu }_X^{\prime }\left(x_1\right)\le {\mu }_Y^{\prime }\left(x_1\right)$, ${\nu }_{X_1}^{\prime }\left(x_1\right)\ge {\nu }_Y^{\prime }\left(x_1\right)$. Therefore, there exist four types of relationships for the membership and non-membership degrees of the $L_{app}$ and $U_{app}$s.

(i) If ${\mu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)={\mu }_{X_1}\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)={\mu }_{X_1}^{\prime }\left(x_1\right)$ and ${\nu }_{C{N}_{*}\left(X\right)}\left(x\right)={\nu }_X\left(x\right)$, ${\nu }_{C{N}^{*}\left(X\right)}\left(x\right)$$={\nu }_{X_1}^{\prime }\left(x_1\right)$

(ii) If ${\mu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)={\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)={\mu }_{X_1}\left(x_1\right)$ and ${\nu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)={\nu }_{X_1}\left(x_1\right)$,

${\nu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)={\nu }_X^{\prime }\left(x_1\right)$

(iii) If ${\mu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)={\mu }_{X_1}\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)={\mu }_{X_1}^{\prime }\left(x_1\right)$ and ${\nu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)={\nu }_{X_1}^{\prime }\left(x_1\right)$,

${\nu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)={\nu }_{X_1}\left(x_1\right)$

(iv) If ${\mu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)={\mu }_{X_1}^{\prime }\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)={\mu }_{X_1}\left(x_1\right)$ and ${\nu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)={\nu }_{X_1}^{\prime }\left(x_1\right)$,

${\nu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)={\nu }_{X_1}\left(x_1\right)$

For the proof of (i) there are four cases:

Case 1: Let ${\mu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)={\mu }_Y\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)={\mu }_Y^{\prime }\left(x_1\right)$ and ${\nu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)={\nu }_Y\left(x_1\right)$, ${\nu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)={\nu }_Y^{\prime }\left(x_1\right)$.

Then, ${\mu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\le {\mu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(X\right)}\left(x_1\right)\le {\mu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)$, and ${\nu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{C{N}_{*}\left(Y\right)}$$\left(x_1\right)$, ${\nu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)$.

Case 2: If ${\mu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)={\mu }_Y^{\prime }\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)={\mu }_Y\left(x_1\right)$ and ${\nu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)={\nu }_Y\left(x\right)$, ${\nu }_{C{N}^{*}\left(Y\right)}$$\left(x_1\right)={\nu }_Y^{\prime }\left(x_1\right)$.

Since ${\mu }_{X_1}\left(x_1\right)\le {\mu }_{X_1}^{\prime }\left(x_1\right)\le {\mu }_Y^{\prime }\left(x_1\right)$, ${\mu }_{X_1}^{\prime }\left(x_1\right)\le {\mu }_Y^{\prime }\left(x_1\right)\le {\mu }_Y\left(x_1\right)$, then ${\mu }_{X_1}\left(x_1\right)\le {\mu }_Y^{\prime }\left(x_1\right)$, ${\mu }_{X_1}^{\prime }\left(x_1\right)\le {\mu }_Y\left(x_1\right)$, hence ${\mu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\le {\mu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(X\right)}\left(x\right)\le {\mu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)$, and ${\nu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{CN*\left(Y\right)}\left(x_1\right)$, ${\nu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)$.

Case 3: Let ${\mu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)={\mu }_Y\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)={\mu }_Y^{\prime }\left(x_1\right)$, and ${\nu }_{C{N}_{*}\left(Y\right)}\left(x\right)={\nu }_Y^{\prime }\left(x_1\right)$, ${\nu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)={\nu }_Y\left(x_1\right)$. Then ${\mu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\le {\mu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)\le {\mu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)$, and ${\nu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)$, ${\nu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)$. Since ${\nu }_{X_1}\left(x_1\right)\ge {\nu }_{X_1}^{\prime }\left(x_1\right)\ge {\mu }_Y^{\prime }\left(x_1\right)$, ${\nu }_{X_1}^{\prime }\left(x\right)\ge {\nu }_Y^{\prime }\left(x_1\right)\ge {\nu }_Y\left(x_1\right)$, hence ${\nu }_{X_1}\left(x_1\right)\ge {\nu }_Y^{\prime }\left(x_1\right)$, ${\nu }_{X_1}^{\prime }\left(x_1\right)\ge {\nu }_Y\left(x_1\right)$. Then ${\mu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\le {\mu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)\le {\mu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)$, ${\nu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{C{N}_{*}\left(Y\right)}\left(X_1\right)$, ${\nu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{C{N}^{*}\left(Y\right)}\left(X_1\right)$.

Case 4: Let ${\mu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)={\mu }_Y^{\prime }\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)={\mu }_Y\left(x_1\right)$, and ${\nu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)={\nu }_Y^{\prime }\left(x_1\right)$, ${\nu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)={\nu }_Y\left(x_1\right)$. Then ${\mu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\le {\mu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)$, ${\mu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)\le {\mu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)$, and ${\nu }_{C{N}_{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{C{N}_{*}\left(Y\right)}\left(x_1\right)$, ${\nu }_{C{N}^{*}\left(X_1\right)}\left(x_1\right)\ge {\nu }_{C{N}^{*}\left(Y\right)}\left(x_1\right)$. The proofs of (i), (iii) and (iv) are the same. Then, $C{N}_{*}\left(X_1\right)\subseteq C{N}_{*}\left(Y\right)$ and $C{N}^{*}\left(X_1\right)\subseteq C{N}^{*}\left(X_1\right)$.

5. Multi-Granulation Covering Rough ${\mathit{IFS}}_{\mathbf{s}}$ of the Third Type

In this part, we introduce a novel technique for multi-granulation covering rough $IF{S}_s$, in short MGCRIFSs.

Definition 14.

Let $(U,C_1)$ be a $co{v}_{app}$ structure, $C_1=\{C_i:1\le i\le n\}$ be a family of coverings of U and $C_i$ be a covering of U, $|C_1|=n$, $x_1\in U$. The neighborhood is $\widehat{N}\left(x_1\right)=\cap \{K\in C_i:x_1\in K\}$.

Definition 15.

Let $(U,C_1)$ be a $co{v}_{app}$ structure, $C_1=\{C_i:1\le i\le n\}$ be a family of coverings of U and $C_i$ be a covering of U, $|C_1|=n$, $x_1\in U$. Then, multi-granulation $IF$ covering rough membership and non-membership degrees of $x_1$ depending on the neighborhood of $X_1$ are defined, respectively, as

${\displaystyle {\widehat{\mu }}_{X_1}\left(x_1\right)=\sum _{y\in \widehat{N}\left(x_1\right)}{\mu }_{X_1}\left(y\right)∕\left|\widehat{N}\left(x_1\right)\right|}$,    ${\displaystyle {\widehat{\nu }}_{X_1}\left(x_1\right)=\sum _{y\in \widehat{N}\left(x_1\right)}{\nu }_{X_1}\left(y\right)∕\left|\widehat{N}\left(x_1\right)\right|}$

The $L_{app}$ is $C{\widehat{N}}_{*}\left(X_1\right)=\{(x_1,\gamma ,\delta ):x_1\in U\}$, where $\gamma =min({\mu }_{X_1}\left(x_1\right),{\widehat{\mu }}_{X_1}\left(x_1\right))$,

$\delta =max({\nu }_{X_1}\left(x_1\right),{\widehat{\nu }}_{X_1}\left(x_1\right))$, and the $U_{app}$ is $C{\widehat{N}}^{*}\left(X_1\right)=\{(x_1,\theta ,\lambda ):x_1\in u_a\}$, where $\theta =max({\mu }_{X_1}\left(x_1\right),{\widehat{\mu }}_{X_1}\left(x_1\right))$, $\lambda =min({\nu }_{X_1}\left(x_1\right),{\widehat{\nu }}_{X_1}\left(x_1\right))$.

The following example is an illustrative example of the above definition.

Example 4.

Let $(U,C_1)$ be a $co{v}_{app}$ structure, $U=\{u_a,u_b,u_c,u_d,u_e,u_f,u_g,u_h\}$, $C=\{C_1,C_2,$$C_3\}$, $C_1=\{\{u_a,u_c,u_d\},\{u_a,u_g\},\{u_b,u_e,u_f,u_h\}\}$, $C_2=\{\{u_b,u_a,u_c,u_f\},\{u_a,u_d\},\{u_e,u_g,$$u_h\left\}\right\}$, $C_3=\{\{u_a,u_c,u_e,u_f\},\{u_b,u_d\}$,

$\{u_e,u_g,u_h\}\}$. The $IFS$ is: $X=\{(u_a,0.2,0.3),(u_b,0.7,0.2),(u_c,0.5,0.2),(u_d,0.3,0.6),(u_e,0.2,$$0.7)$,

$(u_f,0.0,0.6)$, $(u_g,0.5,0.4),(u_h,0.1,0.5)\}$. We can calculate the $L_{app}$ and $U_{app}$s of X. Using Definitions 14 and 15

$\widehat{N}\left(u_a\right)=\left\{u_a\right\}$, $\widehat{N}\left(u_b\right)=\left\{u_b\right\}$, $\widehat{N}\left(u_c\right)=\{u_a,u_c\}$, $\widehat{N}\left(u_d\right)=\left\{u_d\right\}$, $\widehat{N}\left(u_e\right)=\left\{u_e\right\}$, $\widehat{N}\left(u_f\right)=\left\{u_f\right\}$, $\widehat{N}\left(u_g\right)=\left\{u_g\right\}$, $\widehat{N}\left(u_h\right)=\{u_e,u_h\}$.

We can calculate the membership degrees as follows

${\displaystyle {\widehat{\mu }}_X\left(u_a\right)=0.2}$, ${\displaystyle {\widehat{\mu }}_X\left(u_b\right)=0.7}$, ${\displaystyle {\widehat{\mu }}_X\left(u_c\right)=0.35}$, ${\displaystyle {\widehat{\mu }}_U\left(u_d\right)=0.3}$, ${\displaystyle {\widehat{\mu }}_X\left(u_e\right)=0.2}$, ${\displaystyle {\widehat{\mu }}_X\left(u_f\right)=0.0}$, ${\displaystyle {\widehat{\mu }}_X\left(u_g\right)=0.5}$, ${\displaystyle {\widehat{\mu }}_X\left(u_h\right)=0.15}$.

Additionally, calculate the non-membership degrees as follows:

${\displaystyle {\widehat{\nu }}_X\left(u_a\right)=0.3}$, ${\displaystyle {\widehat{\nu }}_X\left(u_b\right)=0.2}$, ${\displaystyle {\widehat{\nu }}_X\left(u_c\right)=0.25}$, ${\displaystyle {\widehat{\nu }}_X\left(u_d\right)=0.6}$, ${\displaystyle {\widehat{\nu }}_X\left(u_e\right)=0.7}$, ${\displaystyle {\widehat{\nu }}_X\left(u_f\right)=0.6}$, ${\displaystyle {\widehat{\nu }}_X\left(u_g\right)=0.4}$, ${\displaystyle {\widehat{\nu }}_U\left(u_h\right)=0.6}$,.

The $L_{app}$ is

$C{\widehat{N}}_{*}\left(X\right)=\{(u_a,0.2,0.3),(u_b,0.7,0.2),(u_c,0.35,0.25),(u_d,0.3,0.6),(u_e,0.2,0.7)$, $(u_f,0.0,0.6),(u_g,0.5,0.4),(u_h,0.1,0.6)\}$.

Additionally, the $U_{app}$ is

$C{\widehat{N}}^{*}\left(X\right)=\{(u_a,0.2,0.3),(u_b,0.7,0.2),(u_c,0.5,0.2),(u_d,0.3,0.6),(u_e,0.25,0.5)$, $(u_f,0.0,0.6),(u_g,0.5,0.4),(u_h,0.15,0.5)\}$.

Proposition 3.

Assume that $(U,C)$ is a $co{v}_{app}$ structure. The next properties are satisfied:

(1) $C{\widehat{N}}_{*}\left(U\right)=C{\widehat{N}}^{*}\left(U\right)=U$    (2) $C{\widehat{N}}_{*}\left(\Phi \right)=C{\widehat{N}}^{*}\left(\Phi \right)=\Phi$

(3) $C{\widehat{N}}_{*}\left(X_1\right)\subseteq X_1\subseteq C{\widehat{N}}^{*}\left(X_1\right)$    (4) If $X_1\subseteq Y_1$, then $C{\widehat{N}}_{*}\left(X_1\right)\subseteq C{\widehat{N}}_{*}\left(Y_1\right)$, $C{\widehat{N}}^{*}\left(X_1\right)\subseteq C{\widehat{N}}^{*}\left(Y_1\right)$

Proof. 

(1) Suppose that U is $IFS$. Then, ${\mu }_U\left(x_1\right)=1$, ${\nu }_U\left(x_1\right)=0$, $\forall x\in U$ by Definition 15, ${\widehat{\mu }}_U\left(x_1\right)=1$, ${\widehat{\nu }}_U\left(x_1\right)=0$. Then $\gamma =\theta =1$, $\delta =\lambda =0$. Therefore, $C\widehat{N_{*}}\left(U\right)=C\widehat{N^{*}}\left(U\right)=U$.

(2) Similar to (1)

(3) From Definition 15, ${\mu }_{X_1}\left(x_1\right)\le {\widehat{\mu }}_{X_1}\left(x_1\right)$ or ${\mu }_{X_1}\left(x_1\right)\ge {\widehat{\mu }}_{X_1}\left(x_1\right)$, and ${\nu }_{X_1}\left(x_1\right)\ge {\widehat{\nu }}_{X_1}\left(x_1\right)$ or ${\nu }_{X_1}\left(x_1\right)\le {\widehat{\nu }}_{X_1}\left(x_1\right)$. Therefore, there are four cases.

(i) ${\mu }_{X_1}\left(x_1\right)\le {\widehat{\mu }}_{X_1}\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\le {\widehat{\nu }}_{X_1}\left(x_1\right)$ (ii) ${\mu }_{X_1}\left(x_1\right)\ge {\widehat{\mu }}_{X_1}\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\le {\widehat{\nu }}_{X_1}\left(x_1\right)$

(iii) ${\mu }_{X_1}\left(x_1\right)\le {\widehat{\mu }}_{X_1}\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\ge {\widehat{\nu }}_{X_1}\left(x_1\right)$ (iv) ${\mu }_{X_1}\left(x_1\right)\ge {\widehat{\mu }}_{X_1}\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\ge {\widehat{\nu }}_{X_1}\left(x_1\right)$

We prove (i) and (ii)

(i) Let ${\mu }_{X_1}\left(x_1\right)\le {\widehat{\mu }}_{X_1}\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\le {\widehat{\nu }}_{X_1}\left(x_1\right)$. Then, $\gamma ={\mu }_{X_1}\left(x_1\right)$, $\theta ={\widehat{\mu }}_X\left(x\right)$, $\delta ={\widehat{\nu }}_{X_1}\left(x_1\right)$, $\lambda ={\nu }_{X_1}\left(x_1\right)$, hence $\gamma \le {\widehat{\mu }}_{X_1}\left(x_1\right)\le \theta$, and $\delta \ge {\nu }_X\left(x_1\right)\ge \lambda$.

(ii) Let ${\mu }_{X_1}\left(x_1\right)\ge {\widehat{\mu }}_{X_1}\left(x_1\right)$, ${\nu }_{X_1}\left(x_1\right)\le {\widehat{\nu }}_{X_1}\left(x_1\right)$. Then, $\gamma ={\widehat{\mu }}_{X_1}\left(x_1\right)$, $\theta ={\mu }_{X_1}\left(x_1\right)$, $\delta ={\widehat{\nu }}_{X_1}\left(x_1\right)$, $\lambda ={\nu }_{X_1}\left(x_1\right)$, hence $\gamma \le {\mu }_{X_1}\left(x_1\right)\le \theta$, and $\delta \ge {\nu }_X\left(x_1\right)\ge \lambda$, from (i) and (ii), we obtain $C{\widehat{N}}_{*}\left(X_1\right)\subseteq X_1\subseteq C{\widehat{N}}^{*}\left(X_1\right)$.

(4) The proof is similar to (3). □

6. Conclusions

Covering-based rough set theory was combined with $IF{S}_s$ in dealing with uncertainty and decision making process. We approximated IFS on the covering rough set space via the neighborhood of each element of the universe. Three models of the approximation of IFS are introduced by the neighborhood concept. Some examples were used to prove and explain the properties of these approximation structures. We think these models will be useful in the decision-making process. In the future, we will use the concept of the core of the neighborhood of the element in generating a new covering rough $IF$ approximation structure. In the future, we will using the IFS in new different applications (DNA mutation–repair mutation).