A comparison of three types of rough fuzzy sets based on two universal sets

Abstract

The extension of rough set model is an important research direction in rough set theory. This paper presents two new extensions of the rough set model over two different universes. By means of a binary relation between two universes of discourse, two pairs of rough fuzzy approximation operators are proposed. These models guarantee that the approximating sets and the approximated sets are on the same universes of discourse. Furthermore, some interesting properties are investigated, the connections between relations and rough fuzzy approximation operators are examined. Finally, the connections of these approximation operators are made, and conditions under which these approximation operators made equivalent are obtained.

1 Introduction

The theory of rough sets was originally proposed by Pawlak [25, 26] as a formal tool for modeling and processing incomplete information. The basic structure of the rough set theory is an approximation space consisting of a universe of discourse and an equivalence relation imposed on it. So equivalence relation is a key notion in Pawlak’s rough set model. However, the requirement of an equivalence relation seems to be a very restrictive condition that may limit the applications of rough set theory. Therefore, some interesting and meaningful extensions of Pawlak’s rough set model have been proposed in the literature. For example, the notions of approximation operators have been generalized by tolerance relations or similarity relations [2, 32, 34, 48], dominance relations [10, 11], general binary relations on the universe of discourse [15–17, 47, 48, 51], partitions and coverings of the universe of discourse [4, 27, 28], neighborhood systems and Boolean algebras [3, 19–21, 49], and general approximation spaces [32, 33, 35]. Based on the covering of the universe of discourse, Pomykala [27, 28], in particular, proposed two pairs of dual approximation operators. Yao [49, 50] discussed such extension by the notions of neighborhood and granulation. In addition, Yao and Lin [51] provided a systematic study on the constructive methods using binary relations.

On the other hand, the fuzzy generalization of rough sets is another topic receiving much attention in recent years [5, 7, 13, 22–24]. Based on an equivalence relation on the universe of discourse, Dubois and Prade [6] introduced the lower and upper approximations of fuzzy sets in the Pawlak approximation space to obtain an extended notion called rough fuzzy sets. Alternatively, a fuzzy similarity relation can be used to replace an equivalence relation, and the result is a deviation of rough set theory called fuzzy rough sets [6, 7]. In general, a rough fuzzy set is the approximation of a fuzzy set in a crisp approximation space [8, 12, 40]. Alternatively, a fuzzy rough set is the approximation of a crisp set or a fuzzy set in a fuzzy approximation space. Particular studies on rough fuzzy sets and fuzzy rough sets can be found in the literatures [9, 14, 29–31, 39, 41, 42, 45, 46, 48].

Rough fuzzy and fuzzy rough set models on two universes of discourse have been studied in the literature [18, 36, 39–44]. These models can be interpreted by the notions of interval structure [37, 38, 52] and generalized approximation space [32, 33, 35]. Though these models have different methods of computation, they start with almost the ‘‘same’’ framework (two universes of discourse with a compatibility relation). It should also be noted that the approximated sets and the approximating sets in these models always locate at two different universes of discourse. This is however un-natural and inconvenient for knowledge discovery by means of rough set theory. The main purpose of the present paper is to discuss this problem in the case of rough fuzzy sets on two universes. Many interesting properties of the new approximation operators, their associated relationships and the existing rough fuzzy approximation operators on two universes of discourse are examined. The new rough fuzzy approximation operators are then compared. The necessary and sufficient conditions for their equivalence are investigated.

2 Preliminaries

Let U be a finite and nonempty set called the universe. The class of all crisp subsets (respectively, fuzzy subsets) of U will be denoted by ${\mathcal{P}}(U)$ (respectively, by ${\mathcal{F}}(U)$).

Let U and V be two finite and nonempty universes and let R be a binary relation from U to V, the triple (U, V, R) is called (two-universe) approximation space. Then the relation R is called:

• (i) Serial if for all x ∊ U, there exists y ∊ V such that (x, y) ∊ R.

• (ii) Inverse serial if for all y ∊ V, there exists x ∊ U such that (x, y) ∊ R.

• (iii) Compatibility relation, if R is both serial and inverse serial.

If U = V, R is referred to as a binary relation on U and is called:

• (i) Reflexive if for all x ∊ U, (x, x) ∊ R.

• (ii) Symmetric if for all x, y ∊ U, (x, y) ∊ R implies that (y, x) ∊ R.

• (iii) Transitive if for all x, y, z ∊ U, (x, y) ∊ R and (y, z) ∊ R imply that (x, z) ∊ R.

• (iv) Euclidean if for all x, y, z ∊ U, (x, y) ∊ R and (x, z) ∊ R imply that (y, z) ∊ R.

Definition 2.1 [53]

Let (U, V, R) be a (two-universe) approximation space. Then, a set-valued mappings F and G representing the successor and predecessor neighborhood operators, respectively, defined as follows:

$$F:U \to {\mathcal{P}}\left( V \right), F\left( x \right) = \{ y \in V :(x,y) \in R\} ,$$

$$G:V \to {\mathcal{P}}\left( U \right), G\left( y \right) = \{ x \in U :(x,y) \in R\} .$$

A natural mappings ${\text{F}}$ and G can be introduced according to the following form

$$F:{\mathcal{P}}\left( U \right) \to {\mathcal{P}}\left( V \right), F\left( X \right) = \mathop {\cup }\nolimits \{ F\left( x \right) :x \in X\} ,$$

$$G:{\mathcal{P}}\left( V \right) \to {\mathcal{P}}\left( U \right), G\left( Y \right) = \mathop {\cup }\nolimits \{ G\left( y \right) :y \in Y\} .$$

Definition 2.2

Let (U, V, R) be a (two-universe) approximation space, an inverse serial relation $R \in {\mathcal{P}}(U \times V)$ is called strong inverse serial if for all $y_{1} , y_{2} \in V$, $G(y_{1} )\mathop {\cap }\nolimits G(y_{2} ) \ne \emptyset$ implies that $G\left( {y_{1} } \right) = G(y_{2} )$.

Lemma 2.1

Let $(U,V, R)$ be a (two-universe) approximation space, if R is strong inverse serial, then for all $x_{1} , x_{2} \in U$, $F(X_{1} )\mathop {\cap }\nolimits F(X_{2} ) \ne \emptyset$ implies that $F\left( {x_{1} } \right) = F(x_{2} )$.

Proof

Assume that $F\left( {x_{1} } \right) \ne F(x_{2} )$, then there exists $y_{1} \in F\left( {x_{1} } \right)$, $y_{1} \notin F(x_{2} )$. If $F(x_{2} ) \ne \emptyset$, then there exists $y_{2} \in F\left( {x_{2} } \right)$, i.e., $x_{1} \in G\left( {y_{1} } \right)$ and $x_{2} \in G\left( {y_{2} } \right)$ such that $G\left( {y_{1} } \right) \ne G(y_{2} )$. Since R is strong inverse serial, then $G\left( {y_{1} } \right)\mathop {\cap }\nolimits G\left( {y_{2} } \right) = \emptyset$. Moreover, $G\left( {y_{1} } \right) = G(F\left( {x_{1} } \right))$ and $G\left( {y_{2} } \right) = G(F\left( {x_{2} } \right)).$ Hence $G\left( {F\left( {x_{1} } \right)} \right)\mathop \cap \nolimits G\left( {F\left( {x_{2} } \right)} \right) \supseteq G\left( {F\left( {x_{1} } \right)\mathop \cap \nolimits F\left( {x_{2} } \right)} \right) = \emptyset$. Because $R$ is inverse serial, we get $F\left( {x_{1} } \right)\mathop \cap \nolimits F\left( {x_{2} } \right) = \emptyset$.

3 Rough fuzzy generalizations

A rough fuzzy set is a generalization of rough set, derived from the approximation of fuzzy set in a crisp approximation space.

Definition 3.1 [42]

Let $U$ and $V$ be two finite non-empty universes of discourse and $R \in {\mathcal{P}}(U \times V)$ a binary relation from U to V. The ordered triple (U, V, R) is called a (two-universe) approximation space. For any fuzzy set $Y \in {\mathcal{F}}(V)$, the lower and upper approximations of Y, ${\underline{R}}_{s} (Y)$ and $\bar{R}_{s} (Y)$, with respect to the approximation space are fuzzy sets of U whose membership functions, for each x ∊ U, are defined, respectively, by

$$\underline{R}_{s} \left( Y \right)\left( x \right) = {min} \left\{ {Y\left( y \right)\, | \, y \in F(x)} \right\},$$

$$\bar{R}_{s} \left( Y \right)\left( x \right) = {max} \left\{ {Y\left( y \right)\, | \,y \in F(x)} \right\}.$$

where F(x) is the successor neighborhood of x defined in Definition 2.1.

The ordered set-pair $(\underline{R}_{s} (Y) , \bar{R}_{s} (Y))$ is referred to as a generalized rough fuzzy set with respect to successor neighborhood, and $\underline{R}_{s}$ and $\bar{R}_{s} \! :{\mathcal{F}}(V) \to {\mathcal{F}}(U)$ are referred to as lower and upper generalized rough fuzzy approximation operators, respectively.

Proposition 3.1 [42]

In a (two-universe) approximation space (U, V, R) with compatibility relation R, the approximation operators satisfy the following properties for all $Y, Y_{1} , Y_{2} \in {\mathcal{F}}(V)$:

(L₁):

$\underline{R}_{s} \left( Y \right) = (\bar{R}_{s} (Y^{c} ))^{c}$, where Y ^c denotes the complement of the fuzzy subset Y in V

(L₂):

$\underline{R}_{s} \left( V \right) = U$

(L₃):

$\underline{R}_{s} \left( {Y_{1} \mathop \cap \nolimits Y_{2} } \right) = \underline{R}_{s} (Y_{1} ) \cap \underline{R}_{s} (Y_{2} )$

(L₄):

$\underline{R}_{s} \left( {Y_{1} \mathop \cup \nolimits Y_{2} } \right) \supseteq \underline{R}_{s} (Y_{1} ) \cup \underline{R}_{s} (Y_{2} )$

(L₅):

$Y_{1} \subseteq Y_{2} \Rightarrow \underline{R}_{s} (Y_{1} ) \subseteq \underline{R}_{s} \left( {Y_{2} } \right)$

(L₆):

$\underline{R}_{s} \left( \emptyset \right) = \emptyset$

(U₁):

$\bar{R}_{s} \left( Y \right) = (\underline{R}_{s} (Y^{c} ))^{c}$

(U₂):

$\bar{R}_{s} \left( \emptyset \right) = \emptyset$

(U₃):

$\bar{R}_{s} \left( {Y_{1} \mathop \cup \nolimits Y_{2} } \right) = \bar{R}_{s} (Y_{1} )\mathop \cup \nolimits \bar{R}_{s} (Y_{2} )$

(U₄):

$\bar{R}_{s} \left( {Y_{1} \mathop \cap \nolimits Y_{2} } \right) \subseteq \bar{R}_{s} (Y_{1} )\mathop \cap \nolimits \bar{R}_{s} (Y_{2} )$

(U₅):

$Y_{1} \subseteq Y_{2} \Rightarrow \bar{R}_{s} (Y_{1} ) \subseteq \bar{R}_{s} (Y_{2} )$

(U₆):

$\bar{R}_{s} \left( V \right) = U$

(LU):

$\underline{R}_{s} (Y) \subseteq \bar{R}_{s} (Y)$

Properties (L₁) and (U₁) show that the approximation operators $\underline{R}_{s}$ and $\bar{R}_{s}$ are dual to each other. Properties with the same number may be considered as dual properties. With respect to certain special types, say, reflexive, symmetric, transitive, and Euclidean binary relations on the universe U, the approximation operators have additional properties.

Proposition 3.2 [42]

Let $R \in {\mathcal{P}}(U \times U)$ be an arbitrary binary relation on U. Then $\forall X \in {\mathcal{F}}(U)$,

$$\begin{aligned} R\;is \, reflexive & \Leftrightarrow (L_{7} ) \underline{R}_{s} \left( X \right) \subseteq X, \hfill \\ & \Leftrightarrow (U_{7} ) X \subseteq \bar{R}_{s} (X). \hfill \\ \end{aligned}$$

$$\begin{aligned} R\;is \, symmetric\; & \Leftrightarrow (L_{8} )\bar{R}_{s} (\underline{R}_{s} \left( {X)} \right) \subseteq X, \hfill \\ & \Leftrightarrow (U_{8} ) X \subseteq \underline{R}_{s} (\bar{R}_{s} (X). \hfill \\ \end{aligned}$$

$$\begin{aligned} R is \, transitive & \Leftrightarrow \left( {L_{9} } \right) \underline{R}_{s} \left( X \right) \subseteq \underline{R}_{s} (\underline{R}_{s} \left( X \right)), \hfill \\ & \Leftrightarrow (U_{9} ) \bar{R}_{s} (\bar{R}_{s} \left( X \right)) \subseteq \bar{R}_{s} (X). \hfill \\ \end{aligned}$$

$$\begin{aligned} R \, is \, Euclidean & \Leftrightarrow (L_{10} )\bar{R}_{s} (\underline{R}_{s} \left( {X)} \right) \subseteq \underline{R}_{s} (X), \hfill \\ & \Leftrightarrow (U_{10} ) \bar{R}_{s} (X) \subseteq \underline{R}_{s} (\bar{R}_{s} (X)). \hfill \\ \end{aligned}$$

Definition 3.2

Let (U, V, R) be a (two-universe) approximation space. Then the lower and upper approximations of $X \in {\mathcal{F}}(U)$ are defined respectively as follows:

$$\underline{R}_{p} \left( X \right)\left( y \right) = {min} \left\{ {X\left( x \right)\, | \, x \in G(y)} \right\},$$

$$\bar{R}_{p} \left( X \right)\left( y \right) = {max} \left\{ {X\left( x \right)\, | \,x \in G(y)} \right\}.$$

where G(y) is the predecessor neighborhood of y in Definition 2.1.

The pair $(\underline{R}_{p} \left( X \right),\bar{R}_{p} \left( X \right))$ is referred to as a generalized rough fuzzy set with respect to predecessor neighborhood, and $\underline{R}_{p}$ and $\bar{R}_{p} :{\mathcal{F}}(U) \to {\mathcal{F}}(V)$ are referred to as lower and upper rough fuzzy approximation operators, respectively.

Proposition 3.3

In a (two-universe) approximation space (U, V, R) with a binary relation R, the approximation operators R _p and $\bar{R}_{p}$ satisfy the following properties for all $X, X_{1} , X_{2} \in {\mathcal{F}}(U)$:

(L₁):

$\underline{R}_{p} \left( X \right) = (\bar{R}_{p} (X^{c} ))^{c}$

(L₂):

$\underline{R}_{p} \left( U \right) = V$

(L₃):

$\underline{R}_{p} \left( {X_{1} \cap X_{2} } \right) = \underline{R}_{p} (X_{1} ) \cap \underline{R}_{p} (X_{2} )$

(L₄):

$\underline{R}_{p} \left( {X_{1} \cup X_{2} } \right) \supseteq \underline{R}_{p} (X_{1} ) \cup \underline{R}_{p} (X_{2} )$

(L₅):

$X_{1} \subseteq X_{2} \Rightarrow \underline{R}_{p} (X_{1} ) \subseteq \underline{R}_{p} \left( {X_{2} } \right)$

(U₁):

$\bar{R}_{p} \left( X \right) = (\underline{R}_{p} (X^{c} ))^{c}$

(U₂):

$\bar{R}_{p} \left( \emptyset \right) = \emptyset$

(U₃):

$\bar{R}_{p} \left( {X_{1} \mathop \cup \nolimits X_{2} } \right) = \bar{R}_{p} (X_{1} )\mathop \cup \nolimits \bar{R}_{p} (X_{2} )$

(U₄):

$\bar{R}_{p} \left( {X_{1} \mathop \cap \nolimits X_{2} } \right) \subseteq \bar{R}_{p} (X_{1} )\mathop \cap \nolimits \bar{R}_{p} (X_{2} )$

(U₅):

$X_{1} \subseteq X_{2} \Rightarrow \bar{R}_{p} (X_{1} ) \subseteq \bar{R}_{p} (X_{2} )$

Proof

By the duality of approximation operators, we only need to prove the properties L ₁ – L ₁₅.

(L₁) For all y ∊ V, according to Definition 3.2, we can obtain

$$\begin{aligned} (\bar{R}_{p} (X^{c} ))^{c} (y) & = 1 - \{ { {max} }\{ X^{c} (x):x \in G(y)\} \} \hfill \\ &= 1 - \{ { {max} }\{ 1 - X(x):x \in G(y)\} \} \hfill \\ &= 1 - \{ 1 - { {min} }\{ X(x):x \in G(y)\} \} \hfill \\ &= 1 - \{ {max} \{ X(x):x \in G(y)\} \hfill \\ &= { {min} }\{ X(x):x \in G(y)\} \hfill \\ &= \underline{R}_{p} \left( X \right)(y). \hfill \\ \end{aligned}$$

Therefore $\underline{R}_{p} \left( X \right) = (\bar{R}_{p} (X^{c} ))^{c}$.

(L₂) Since $\forall x \in U$, U(x) = 1 and $G(y) \subseteq U$, then ${ {min} }\{ U\left( x \right): x \in G(y)\} = 1$. Thus, $\underline{R}_{p} \left( U \right)(y) = { {min} }\{ U\left( x \right): x \in G(y)\} = 1$ for all y ∊ V. Therefore R _p (U) = V.

(L₃) Since $\forall y \in V$

$$\begin{aligned} \underline{R}_{p} \left( {X_{1} \mathop \cap \nolimits X_{2} } \right)(y) &= { {min} }\{ (X_{1} \mathop \cap \nolimits X_{2} )(x):x \in G(y)\} \hfill \\ &= { {min} }\left\{ {{min} \left\{ {X_{1} \left( x \right),X_{2} \left( x \right)} \right\}:x \in G\left( y \right)} \right\} \hfill \\ &= { {min} }\{ X_{1} (x):x \in G(y)\} \wedge { {min} }\{ X_{2} (x):x \in G(y)\} \hfill \\ &= {\text{min}}\{ \underline{R}_{p} \left( {X_{1} } \right)\left( y \right) \wedge \underline{R}_{p} \left( {X_{2} } \right)\left( y \right)\} \hfill \\ &= \underline{R}_{p} (X_{1} ) \cap \underline{R}_{p} (X_{2} )(y). \hfill \\ \end{aligned}$$

Therefore $\underline{R}_{p} \left( {X_{1} \cap X_{2} } \right) = \underline{R}_{p} (X_{1} ) \cap \underline{R}_{p} (X_{2} )$.

(L₄) For all y ∊ V, we can have

$$\begin{aligned} \underline{R}_{p} \left( {X_{1} \cup X_{2} } \right)(y) &= { {min} }\{ (X_{1} \cup X_{2} )(x):x \in G(y)\} \hfill \\ &= { {min} }\{ {max} \left\{ {X_{1} \left( x \right),X_{2} \left( x \right)} \right\}:x \in G(y)\} \hfill \\ & \ge { {max} }\{ { {min} }\left\{ {X_{1} \left( x \right):x \in G\left( y \right)} \right\},{ {min} }\{ X_{2} (x):x \in G(y)\} \hfill \\ & = { {max} }\{ \underline{R}_{p} \left( {X_{1} } \right)(y),\underline{R}_{p} \left( {X_{2} } \right)(y)\} \hfill \\ & = (\underline{R}_{p} (X_{1} ) \cup \underline{R}_{p} (X_{2} ))(y). \hfill \\ \end{aligned}$$

Hence $\underline{R}_{p} \left( {X_{1} \cup X_{2} } \right) \supseteq \underline{R}_{p} (X_{1} ) \cup \underline{R}_{p} (X_{2} )$.

(L₅) Since $X_{1} \subseteq X_{2}$, then $\forall x \in U, X_{1} (x) \le X_{2} (x)$. Thus $\underline{R}_{p} \left( {X_{1} } \right)\left( y \right) = { {min} }\{ X_{1} (x):x \in G(y)\} \le { {min} }\{ X_{2} \left( x \right):x \in G(y)\} = \underline{R}_{p} \left( {X_{2} } \right)(y)$

Therefore $\underline{R}_{p} (X_{1} ) \subseteq \underline{R}_{p} \left( {X_{2} } \right)$.

Proposition 3.4

Let $R \in {\mathcal{P}}(U \times V)$ be an arbitrary binary relation on. Then $\forall X \in {\mathcal{F}}(U)$

$$\begin{aligned}R \, is \, inverse \, serial \, & \Leftrightarrow (L_{6} )\underline{R}_{p} \left( \emptyset \right) = \emptyset \\ & \Leftrightarrow (U_{6} ) \bar{R}_{p} \left( U \right) = V, \\ & \Leftrightarrow (LU)\underline{R}_{p} (X) \subseteq \bar{R}_{p} (X). \end{aligned}$$

Proof

Suppose that R is an inverse serial relation, for any y ∊ V, we have $G(y) \ne \emptyset$. Then $min\left\{ {\emptyset \left( x \right): x \in G\left( y \right)} \right\} = 0 \, \forall x \in U$. Therefore, $\underline{R}_{p} \left( \emptyset \right) = \emptyset$.

Conversely, assume that $\underline{R}_{p} \left( \emptyset \right) = \emptyset$, i.e., $min\left\{ {\emptyset \left( x \right): x \in G\left( y \right)} \right\} = 0 \, \forall x \in U$. If there exists $y_{ \circ } \in V$ such that $G\left( {y_{ \circ } } \right) = \emptyset$, then $min\left\{ {\emptyset \left( x \right): x \in G\left( {y_{ \circ } } \right)} \right\} = 1$ which contradicts the assumption. Thus $G\left( y \right) \ne \emptyset \, \forall y \in V$. i.e., R is inverse serial. We can prove that R is inverse serial$\Leftrightarrow$ $(U_{6} ) \bar{R}_{p} \left( U \right) = V$ by the duality of approximation operators. For the third part, R is inverse serial $\Leftrightarrow (LU)\underline{R}_{p} (X) \subseteq \bar{R}_{p} (X)$, let R is inverse serial, then $G(y) \ne \emptyset$ for every $y \in V$. So ${ {min} }\{ X(x):x \in G(y)\} \le { {max} }\{ X(x):x \in G(y)\}$. Therefore $\underline{R}_{p} (X) \subseteq \bar{R}_{p} (X)$.

On the other hand, if we assume that $({\text{LU}}) \underline{\text{R}}_{\text{U}} ({\text{X}}) \subseteq {\bar{\text{R}}}_{\text{U}} ({\text{X}})$ holds, let X = U, then by Proposition 2.3 we have ${\bar{\text{R}}}_{\text{p}} \left( {\text{U}} \right) \supseteq \underline{\text{R}}_{\text{p}} \left( {\text{U}} \right) = {\text{V}}$, which follows that R is fuzzy inverse serial.

Proposition 3.5

Let (U, V, R) be a (two-universe) approximation space, then the following hold for all $X \in {\mathcal{F}}(U)$ and $Y \in {\mathcal{F}}(V)$:

• (i) $\bar{R}_{s} (\underline{R}_{p} \left( {X)} \right) \subseteq X,X \subseteq \underline{R}_{s} (\bar{R}_{p} \left( X \right))$.

• (ii) $\bar{R}_{p} (\underline{R}_{s} \left( {Y)} \right) \subseteq Y,Y \subseteq \underline{R}_{p} (\bar{R}_{s} \left( Y \right))$.

• (iii) $\underline{R}_{s} \left( Y \right) = \underline{R}_{s} (\bar{R}_{p} (\underline{R}_{s} \left( {Y)} \right))$.

• (iv) $\bar{R}_{s} \left( Y \right) = \bar{R}_{s} (\underline{R}_{p} (\bar{R}_{s} \left( Y \right)))$.

• (v) $\underline{R}_{p} \left( X \right) = \underline{R}_{p} (\bar{R}_{s} (\underline{R}_{p} \left( {X)} \right))$.

• (vi) $\bar{R}_{p} \left( X \right) = \bar{R}_{p} (\underline{R}_{s} (\bar{R}_{p} \left( X \right)))$.

Proof

(i) Since for every x ∊ U, we have either $F\left( x \right) = \emptyset$ or $F\left( x \right) \ne \emptyset$.

If $F\left( x \right) = \emptyset$, then $\bar{R}_{s} (\underline{R}_{p} \left( {X)} \right)\left( x \right) = {max} \left\{ {min\left\{ {X\left( z \right):z \in G\left( y \right)} \right\}:y \in F\left( x \right)} \right\} = 0$ and hence $\bar{R}_{s} (\underline{R}_{p} \left( {X)} \right) \subseteq X$. If $F\left( x \right) \ne \emptyset$, then we have $x \in G(y)$ $\forall y \in F\left( x \right)$. Thus ${max} \left\{ {min\left\{ {X\left( z \right):z \in G\left( y \right)} \right\}:y \in F\left( x \right)} \right\} \le X(x)$, and hence $\bar{R}_{s} (\underline{R}_{p} \left( {X)} \right) \subseteq X$.

We can easily prove the other part by the duality of approximation operators.

(ii) Similarly as (i).

(iii)–(vi) can be proved by properties (i) and (ii).

Proposition 3.6

Let (U, V, R) be a (two-universe) approximation space with strong inverse serial relation, then the following hold for all $X \in {\mathcal{F}}(U)$ and $Y \in {\mathcal{F}}(V)$:

$$({\text{i}})\,\bar{R}_{p} (\underline{R}_{s} \left( {Y)} \right) = \underline{R}_{p} (\underline{R}_{s} \left( Y \right)),$$

$$({\text{ii}})\;\underline{R}_{p} (\bar{R}_{s} \left( Y \right)) = \bar{R}_{p} (\bar{R}_{s} \left( Y \right)).$$

Proof

The proof comes directly from Definition 2.2 and Lemma 2.1.

Because reflexivity, symmetry and transitivity are meaningless for binary relations from U to V, the properties $\left( {{\text{L}}_{7} } \right) - ({\text{L}}_{10} )$ and (U₇) – (U₁₀) which are true in various generalized rough fuzzy set models do not hold in two-universe models. However, In the above model for generalized rough fuzzy sets, fuzzy subsets of the universe V (resp. U) are approximated by fuzzy subsets of the other universe U (resp. V). This seems to be very unreasonable. Thus a more natural form for rough fuzzy sets on two universes is proposed so that the approximations of subsets of the universe are subsets of the same universe. Therefore we will modify these models in the next sections.

4 Revised rough fuzzy sets

Definition 4.1 [2]

Let (U, V, R) be a (two-universe) approximation space. Then we can define a set valued mapping G ^* from V to ${\mathcal{P}}(V)$ induced by R as follows:

$$G^{*} :V \to {\mathcal{P}}\left( V \right), G^{*} \left( y \right) = \left\{ {\begin{array}{ll} {\mathop {\cap }\nolimits F\left( x \right) \quad if \exists x \in U : \left\{ y \right\} \subseteq F\left( x \right),} \\ {\emptyset} \quad \quad otherwise. \\ \end{array} } \right.$$

Definition 4.2 [1]

Let (U, V, R) be a (two-universe) approximation space. Then the lower and upper approximations of ${\text{Y}} \in {\mathcal{F}}({\text{V}})$ are defined respectively as follows:

$$\begin{gathered} \underline{R}^{*} \left( Y \right)\left( y \right) = {min} \left\{ {Y\left( z \right)\, | \,z \in G^{*} (y)} \right\} \hfill \\ \bar{R}^{*} \left( Y \right)\left( y \right) = {max} \left\{ {Y\left( z \right)\, | \,z \in G^{*} (y)} \right\} . \hfill \\ \end{gathered}$$

The pair $(\underline{R}^{*} \left( Y \right),\bar{R}^{*} \left( Y \right))$ is referred to as a revised rough fuzzy set, and R ^* and $\bar{R}^{*} :{\mathcal{F}}(V) \to {\mathcal{F}}(V)$ are referred to as revised lower and upper rough fuzzy approximation operators, respectively.

Proposition 4.1 [1]

In a (two-universe) approximation space $(U ,V,R)$ , the approximation operators have the following properties for all $Y, Y_{1} , Y_{2} \in {\mathcal{F}}(V)$:

(L₁):

$\underline{R}^{*} \left( Y \right) = (\bar{R}^{*} (Y^{c} ))^{c}$

(L₂):

$\underline{R}^{*} \left( V \right) = V$

(L₃):

$\underline{R}^{*} \left( {Y_{1} \cap Y_{2} } \right) = \underline{R}^{*} (Y_{1} ) \cap \underline{R}^{*} (Y_{2} )$

(L₄):

$\underline{R}^{*} \left( {Y_{1} \cup Y_{2} } \right) \supseteq \underline{R}^{*} (Y_{1} ) \cup \underline{R}^{*} (Y_{2} )$

(L₅):

$Y_{1} \subseteq Y_{2} \Rightarrow \underline{R}^{*} (Y_{1} ) \subseteq \underline{R}^{*} \left( {Y_{2} } \right)$

(L₉):

$\underline{R}^{*} \left( Y \right) \subseteq \underline{R}^{*} (\underline{R}^{*} \left( Y \right))$

(U₁):

$\bar{R}^{*} \left( Y \right) = (\underline{R}^{*} (Y^{c} ))^{c}$

(U₂):

$\bar{R}^{*} \left( \emptyset \right) = \emptyset$

(U₃):

$\bar{R}^{*} \left( {Y_{1} \mathop \cup \nolimits Y_{2} } \right) = \bar{R}^{*} (Y_{1} )\mathop \cup \nolimits \bar{R}^{*} (Y_{2} )$

(U₄):

$\bar{R}^{*} \left( {Y_{1} \mathop \cap \nolimits Y_{2} } \right) \subseteq \bar{R}^{*} (Y_{1} )\mathop \cap \nolimits \bar{R}^{*} (Y_{2} )$

(U₅):

$Y_{1} \subseteq Y_{2} \Rightarrow \bar{R}^{*} (Y_{1} ) \subseteq \bar{R}^{*} (Y_{2} )$

(U₉):

$\bar{R}^{*} (\bar{R}^{*} \left( Y \right)) \subseteq \bar{R}^{*} (Y)$

Proposition 4.2 [1]

In a (two-universe) approximation space $(U ,V,R)$ with inverse serial relation R, the approximation operators have the following properties for all $Y \in {\mathcal{F}}(V)$:

(L₆):

$\underline{R}^{*} \left( \emptyset \right) = \emptyset$

(L₇):

$\underline{R}^{*} \left( Y \right) \subseteq Y$

(U₆):

$\bar{R}^{*} \left( V \right) = V$

(U₇):

$Y \subseteq \bar{R}^{*} (Y)$

(LU):

$\underline{R}^{*} (Y) \subseteq \bar{R}^{*} (Y)$

Proposition 4.3 [1]

In a (two-universe) approximation space $(U ,V,R)$ with strong inverse serial relation R, the approximation operators have the following properties for all $Y \in {\mathcal{F}}(V)$:

(L₈):

$Y \subseteq \underline{R}^{*} (\bar{R}^{*} (Y))$

(L₁₀):

$\bar{R}^{*} (Y) \subseteq \underline{R}^{*} (\bar{R}^{*} (Y))$

(U₈):

$\bar{R}^{*} (\underline{R}^{*} \left( {Y)} \right) \subseteq Y$

(U₁₀):

$\bar{R}^{*} (\underline{R}^{*} \left( {Y)} \right) \subseteq \underline{R}^{*} (Y)$

5 Another two new generalizations of rough fuzzy sets

Definition 5.1

Let (U, V, R) be a (two-universe) approximation space. Then the lower and upper approximations of ${\text{Y}} \in {\mathcal{F}}({\text{V}})$ are defined respectively as follows:

$$\underline{R}^{'} \left( Y \right)(y) = {max} \{ {min} \{ Y\left( z \right): z \in F(x)\} :x \in G(y)\}$$

$$\bar{R}'\left( Y \right)\left( y \right) = min\{ {max} \left\{ {Y\left( z \right): z \in F\left( x \right)} \right\}:y \in G(y)\} .$$

The pair $(\underline{R}^{'} \left( Y \right),\bar{R}'\left( Y \right))$ is referred to as a weak rough fuzzy set, and $\underline{R}^{'}$ and $\bar{R} ':{\mathcal{F}}(V) \to {\mathcal{F}}(V)$ are referred to as weak lower and upper rough fuzzy approximation operators, respectively.

Proposition 5.1

Let (U, V, R) be a (two-universe) approximation space. Then

$$\underline{R}^{'} \left( Y \right) = \bar{R}_{p} (\underline{R}_{s} \left( {Y)} \right),$$

$$\bar{R}'\left( Y \right) = \underline{R}_{p} (\bar{R}_{s} \left( Y \right)).$$

Proof

Straightforward.

Proposition 5.2

In a (two-universe) approximation space $(U ,V,R)$ , the approximation operators have the following properties for all $Y, Y_{1} , Y_{2} \in {\mathcal{F}}(V)$:

(L₁):

$\underline{R}^{'} \left( Y \right) = (\bar{R}'(Y^{c} ))^{c}$

(L₄):

$\underline{R}^{'} \left( {Y_{1} \cup Y_{2} } \right) \supseteq \underline{R}^{'} (Y_{1} ) \cup \underline{R}^{'} (Y_{2} )$

(L₅):

$Y_{1} \subseteq Y_{2} \Rightarrow \underline{R}^{'} (Y_{1} ) \subseteq \underline{R}^{'} \left( {Y_{2} } \right)$

(L₆):

$\underline{R}^{'} \left( \emptyset \right) = \emptyset$

(L₇):

$\underline{R}^{'} \left( Y \right) \subseteq Y$

(L₉):

$\underline{R}^{'} \left( Y \right) \subseteq \underline{R}^{'} (\underline{R}^{'} \left( Y \right))$

(U₁):

$\bar{R}'\left( Y \right) = (\underline{R}^{'} (Y^{c} ))^{c}$

(U₄):

$\bar{R}'\left( {Y_{1} \mathop \cap \nolimits Y_{2} } \right) \subseteq \bar{R}'(Y_{1} )\mathop \cap \nolimits \bar{R}'(Y_{2} )$

(U₅):

$Y_{1} \subseteq Y_{2} \Rightarrow \bar{R}'(Y_{1} ) \subseteq \bar{R}'(Y_{2} )$

(U₆):

$\bar{R}'\left( V \right) = V$

(U₇):

$Y \subseteq \bar{R}'(Y)$

(U₉):

$\bar{R}'(\bar{R}'\left( Y \right)) \subseteq \bar{R}'(Y)$

(LU):

$\underline{R}^{'} (Y) \subseteq \bar{R}'(Y)$

Proof

By the duality of approximation operators, we only need to prove the properties $\left( {{\text{L}}_{1} } \right),\left( {{\text{L}}_{4} } \right) - ({\text{L}}_{7} )$ and (L₉).

(L₁) Since $\forall {\text{y}} \in {\text{V}}$

$$\begin{aligned} (\bar{R}'(Y^{c} ))^{c} (y) &= 1 - \{ min\{ {max} \left\{ {Y^{c} \left( z \right) :z \in F\left( x \right)} \right\}:x \in G(y)\} \} \hfill \\ &= 1 - \{ min\{ {max} \left\{ {1 - Y\left( z \right) :z \in F\left( x \right)} \right\}:x \in G(y)\} \} \hfill \\ &= 1 - \{ min\{ 1 - min\{ Y\left( z \right):z \in F\left( x \right)\} :x \in G(y)\} \} \hfill \\ &= 1 - \{ 1 - max\{ min\{ Y\left( z \right):z \in F\left( x \right)\} :x \in G(y)\} \} \hfill \\ &= max\{ min\{ Y\left( z \right):z \in F\left( x \right)\} :x \in G(y)\} \hfill \\ &= \underline{R}^{'} (Y)(y). \hfill \\ \end{aligned}$$

Therefore $\underline{R}^{'} \left( Y \right) = (\bar{R}'(Y^{c} ))^{c}$.

(L₄) $\forall y \in V$, we can have

$$\begin{aligned} \underline{R}^{'} \left( {Y_{1} \cup Y_{2} } \right)(y) & = max\{ min\{ (Y_{1} \cup Y_{2} )(z):z \in F\left( x \right)\} :x \in G(y)\} \hfill \\ & = max\{ min\{ max\{ Y_{1} \left( z \right),Y_{2} \left( z \right)\} :z \in F\left( x \right)\} :x \in G(y)\} \hfill \\ &\ge max\!\{ max\left\{ {min\left\{ {Y_{1} \left( z \right):z \in F\left( x \right)} \right\}:x \in G\left( y \right)} \right\}, \hfill \\ & \quad max\left\{ {min\left\{ {Y_{2} \left( z \right):z \in F\left( x \right)} \right\}:x \in G\left( y \right)} \right\}\} \hfill \\ & = max\{ \underline{R}^{'} \left( {Y_{1} } \right)\left( y \right),\underline{R}^{'} \left( {Y_{2} } \right)\left( y \right)\} \hfill \\ & = (\underline{R}^{'} (Y_{1} ) \cup \underline{R}^{'} (Y_{2} ))(y). \hfill \\ \end{aligned}$$

Hence $\underline{R}^{'} \left( {Y_{1} \cup Y_{2} } \right) \supseteq \underline{R}^{'} (Y_{1} ) \cup \underline{R}^{'} (Y_{2} )$.

(L₅) Since $Y_{1} \subseteq Y_{2}$, then $\forall y \in V, Y_{1} (y) \le Y_{2} (y)$.

Thus

$$\begin{aligned} \underline{R}^{'} \left( {Y_{1} } \right)\left( y \right) &= max\{ min\{ Y_{1} (z):z \in F\left( x \right)\} :x \in G(y)\} \hfill \\ &\le\!{max} \left\{ {min\left\{ {Y_{2} \left( z \right):z \in F\left( x \right)} \right\}:x \in G\left( y \right)} \right\} \hfill \\ &= \underline{R}^{'} \left( {Y_{2} } \right)(y) \hfill \\ \end{aligned}$$

Therefore $\underline{R}^{'} (Y_{1} ) \subseteq \underline{R}^{'} \left( {Y_{2} } \right)$.

(L₇) and (L₉) Obvious from Propositions 3.5 and 5.1

(L₆) Comes from (L₇) and the fact that empty set is a subset from any set.

Remark 5.1

If $R \in {\mathcal{P}}(U \times V)$ is a binary relation in a (two-universe) approximation space $(U ,V,R)$, then the following properties do not hold for all $Y, Y_{1} , Y_{2} \in {\mathcal{F}}(V)$:

(L₂):

$\underline{R}^{'} \left( V \right) = V$

(L₃):

$\underline{R}^{'} \left( {Y_{1} \cap Y_{2} } \right) = \underline{R}^{'} (Y_{1} ) \cap \underline{R}^{'} (Y_{2} )$

(L₈):

$Y \subseteq \underline{R}^{'} (\bar{R}'(Y))$

(L₁₀):

$\bar{R}'(Y) \subseteq \underline{R}^{'} (\bar{R}'(Y))$

(U₂):

$\bar{R}'\left( \emptyset \right) = \emptyset$

(U₃):

$\bar{R}'\left( {Y_{1} \mathop \cup \nolimits Y_{2} } \right) = \bar{R}'(Y_{1} )\mathop \cup \nolimits \bar{R}'(Y_{2} )$

(U₈):

$\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \subseteq Y$

(U₁₀):

$\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \subseteq \underline{R}^{'} (Y)$

The following example shows Remark 5.1.

Example 5.1

Let ${\text{U}} = \{ {\text{x}}_{1} , {\text{x}}_{2} , {\text{x}}_{3} , {\text{x}}_{4} ,{\text{x}}_{5} , {\text{x}}_{6} , {\text{x}}_{7} \}$, ${\text{V}} = \{ {\text{y}}_{1} , {\text{y}}_{2} , {\text{y}}_{3} , {\text{y}}_{4} , {\text{y}}_{5} , {\text{y}}_{6} \}$ and ${\text{R}} \in {\mathcal{P}}({\text{U}} \times {\text{V}})$ be a binary relation defined as:

R	y 1	y 2	y 3	y 4	y 5	y 6
x 1	0	1	1	0	0	0
x 2	1	0	1	0	1	0
x 3	0	0	0	0	0	0
x 4	0	1	1	0	0	1
x 5	1	0	0	0	1	0
x 6	0	1	0	0	1	1
x 7	1	0	0	0	0	1

If Y and Z are two fuzzy subsets of V defined as:

$$\begin{gathered} Y\left( {y_{1} } \right) = 0.5, Y\left( {y_{2} } \right) = 0.3, Y\left( {y_{3} } \right) = 0.7, Y\left( {y_{4} } \right) = 0.1, Y\left( {y_{5} } \right) = 0.8, Y(y_{6} ) = 0.4, \hfill \\ Z\left( {y_{1} } \right) = 0.3, Z\left( {y_{2} } \right) = 0.4, Z\left( {y_{3} } \right) = 0.9, Z\left( {y_{4} } \right) = 0.2, Z\left( {y_{5} } \right) = 0.1, Z(y_{6} ) = 0.6, {\text{then we have}}\hfill \\ \end{gathered}$$

	y 1	y 2	y 3	y 4	y 5	y 6
$\underline{R}^{'} (Y)(y)$	0.5	0.3	0.5	0	0.5	0.4
$\bar{R}'(\underline{R}^{'} \left( {Y)} \right)(y)$	0.5	0.5	0.5	1	0.5	0.5
$\bar{R}'(Y)(y)$	0.5	0.7	0.7	1	0.8	0.5
$\underline{R}^{'} (\bar{R}'(Y))(y)$	0.5	0.7	0.7	0	0.5	0.5
$\bar{R}'(\emptyset )(y)$	0	0	0	1	0	0
$\underline{R}^{'} (V)(y)$	1	1	1	0	1	1
$\underline{R}^{'} (Z)(y)$	0.3	0.4	0.4	0	0.1	0.4
$\underline{R}^{'} (Y \cap Z)(y)$	0.3	0.3	0.3	0	0.1	0.3
$\bar{R}'(Z)(y)$	0.3	0.6	0.9	1	0.3	0.6
$\bar{R}'(Y \cup Z)(y)$	0.6	0.8	0.9	1	0.8	0.6

Hence we have $\underline{R}^{'} (V) \ne V$, $\bar{R}'(\emptyset ) \ne \emptyset$, $\underline{R}^{'} (Y \cap Z) \ne \underline{R}^{'} (Y) \cap \underline{R}^{'} (Z)$, $\bar{R}'(Y \cup Z) \ne \bar{R}'(Y) \cup \bar{R}'(Z)$, $Y \not\subset \underline{R}^{'} (\bar{R}'(Y))$, $\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \not\subset Y$, $\bar{R}'(Y) \not\subset \underline{R}^{'} (\bar{R}'(Y))$ and $\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \not\subset \underline{R}^{'} (Y)$, i.e., L₂, U₂, L₃, U₃, L₈, U₈, L₁₀ and U₁₀ do not hold.

Proposition 5.3

In a (two-universe) approximation space $(U ,V,R)$ with inverse serial relation R, the approximation operators have the following properties for all $Y \in {\mathcal{F}}(V)$:

(L₂):

$\underline{R}^{'} \left( V \right) = V$

(U₂):

$\bar{R}'\left( \emptyset \right) = \emptyset$

Proof

Obvious.

Remark 5.2

If $R \in {\mathcal{P}}(U \times V)$ is an inverse serial relation in a (two-universe) approximation space $(U ,V,R)$, then the following properties do not hold for all $Y \in {\mathcal{F}}(V)$:

(L₃):

$\underline{R}^{'} \left( {Y_{1} \cap Y_{2} } \right) = \underline{R}^{'} (Y_{1} ) \cap \underline{R}^{'} (Y_{2} )$

(L₈):

$Y \subseteq \underline{R}^{'} (\bar{R}'(Y))$

(L₁₀):

$\bar{R}'(Y) \subseteq \underline{R}^{'} (\bar{R}'(Y))$

(U₃):

$\bar{R}'\left( {Y_{1} \cup Y_{2} } \right) = \bar{R}'(Y_{1} ) \cup \bar{R}'(Y_{2} )$

(U₈):

$\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \subseteq Y$

(U₁₀):

$\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \subseteq \underline{R}^{'} (Y)$

The following example shows Remark 5.2.

Example 5.2

Let $U = \{ x_{1} , x_{2} , x_{3} , x_{4} ,x_{5} , x_{6} , x_{7} \}$, $V = \{ y_{1} , y_{2} , y_{3} , y_{4} , y_{5} , y_{6} \}$ and $R \in {\mathcal{P}}(U \times V)$ be an inverse serial relation defined as:

R	y 1	y 2	y 3	y 4	y 5	y 6
x 1	0	1	1	0	0	0
x 2	1	0	0	1	1	0
x 3	0	0	0	0	0	0
x 4	0	1	1	1	0	0
x 5	1	0	0	0	1	0
x 6	0	1	0	0	1	1
x 7	0	0	0	1	0	1

If Y and Z are two fuzzy subsets of V defined as:

$$\begin{gathered} Y\left( {y_{1} } \right) = 0.2, Y\left( {y_{2} } \right) = 0.7, Y\left( {y_{3} } \right) = 0.3, Y\left( {y_{4} } \right) = 0.9, Y\left( {y_{5} } \right) = 0.5, Y(y_{6} ) = 0.8, \hfill \\ Z\left( {y_{1} } \right) = 0.9, Z\left( {y_{2} } \right) = 0.5, Z\left( {y_{3} } \right) = 0.6, Z\left( {y_{4} } \right) = 0.8, Z\left( {y_{5} } \right) = 0.1, Z(y_{6} ) = 0.3, {\text{\,then we have}}\hfill \\ \end{gathered}$$

	y 1	y 2	y 3	y 4	y 5	y 6
$\underline{R}^{'} (Y)(y)$	0.2	0.5	0.3	0.8	0.5	0.8
$\bar{R}'(\underline{R}^{'} \left( {Y)} \right)(y)$	0.5	0.5	0.5	0.8	0.5	0.8
$\bar{R}'(Y)(y)$	0.5	0.7	0.7	0.9	0.5	0.8
$\underline{R}^{'} (\bar{R}'(Y))(y)$	0.5	0.7	0.7	0.8	0.5	0.8
$\underline{R}^{'} (Z)(y)$	0.1	0.5	0.5	0.5	0.1	0.3
$\underline{R}^{'} (Y \cap Z)(y)$	0.1	0.3	0.3	0.3	0.1	0.3
$\bar{R}'(Z)(y)$	0.9	0.5	0.6	0.8	0.5	0.5
$\bar{R}'(Y\mathop \cup \nolimits Z)(y)$	0.9	0.7	0.7	0.9	0.8	0.8

Hence we have $\underline{R}^{'} (Y \cap Z) \ne \underline{R}^{'} (Y) \cap \underline{R}^{'} (Z)$, $\bar{R}'(Y\mathop \cup \nolimits Z) \ne \bar{R}'(Y)\mathop \cup \nolimits \bar{R}'(Z)$, $Y \not\subset \underline{R}^{'} (\bar{R}'(Y))$, $\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \not\subset Y$, $\bar{R}'(Y) \not\subset \underline{R}^{'} (\bar{R}'(Y))$ and $\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \not\subset \underline{R}^{'} (Y)$, i.e., L₃, U₃, L₈, U₈, L₁₀ and U₁₀ do not hold.

Proposition 5.4

In a (two-universe) approximation space $(U ,V,R)$ with strong inverse serial relation R, the approximation operators have the following properties for all $Y \in {\mathcal{F}}(V)$:

(L₃):

$\underline{R}^{'} \left( {Y_{1} \cap Y_{2} } \right) = \underline{R}^{'} (Y_{1} ) \cap \underline{R}^{'} (Y_{2} )$

(L₈):

$Y \subseteq \underline{R}^{'} (\bar{R}'(Y))$

(L₁₀):

$\bar{R}'(Y) \subseteq \underline{R}^{'} (\bar{R}'(Y))$

(U₃):

$\bar{R}'\left( {Y_{1} \mathop \cup \nolimits Y_{2} } \right) = \bar{R}'(Y_{1} )\mathop \cup \nolimits \bar{R}'(Y_{2} )$

(U₈):

$\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \subseteq Y$

(U₁₀):

$\bar{R}'(\underline{R}^{'} \left( {Y)} \right) \subseteq \underline{R}^{'} (Y)$

Proof

(L₃) Assume that R is strong inverse serial. Then by Proposition 3.6, we have $\forall x \in G(y)$, $max\left\{ {min\left\{ {Y\left( z \right):z \in F\left( x \right)} \right\}:x \in G\left( y \right)} \right\}\,=\,min\{ min\{ Y\left( z \right):z \in F\left( x \right)\} :x \in G(y)\}$. Thus,

$$\begin{aligned} \underline{R}^{'} \left( {Y_{1} \cap Y_{2} } \right)(y) & = max\{ min\{ (Y_{1} \cap Y_{2} )(z):z \in F\left( x \right)\} :x \in G(y)\} \hfill \\ & = min\{ min\{ min\{ Y_{1} \left( z \right),Y_{2} \left( z \right)\} :z \in F\left( x \right)\} :x \in G(y)\} \hfill \\ & = min\{ min\left\{ {min\left\{ {Y_{1} \left( z \right):z \in F\left( x \right)} \right\}:x \in G\left( y \right)} \right\}, \hfill \\ & \quad min\left\{ {min\left\{ {Y_{2} \left( z \right):z \in F\left( x \right)} \right\}:x \in G\left( y \right)} \right\}\} \hfill \\ & = min\{ max\left\{ {min\left\{ {Y_{1} \left( z \right):z \in F\left( x \right)} \right\}:x \in G\left( y \right)} \right\}, \hfill \\ & \quad max\left\{ {min\left\{ {Y_{2} \left( z \right):z \in F\left( x \right)} \right\}:x \in G\left( y \right)} \right\}\} \hfill \\ & = min\{ \underline{R}^{'} \left( {Y_{1} } \right)\left( y \right),\underline{R}^{'} \left( {Y_{2} } \right)\left( y \right)\} \hfill \\ & = (\underline{R}^{'} (Y_{1} ) \cap \underline{R}^{'} (Y_{2} ))(y). \hfill \\ \end{aligned}$$

(L₁₀) Since R is a strong inverse serial. Then by Proposition 3.6, $\forall y \in V$ , we can have

$$\begin{aligned} \underline{R} '(\bar{R}'\left( {Y)} \right)\left( y \right) &= max \{ {min} \{ {min} \{ {max} \left\{ {Y\left( u \right): u \in F\left( w \right)} \right\}:w \in G(z)\} : z \in F(x)\} :x \in G(y)\} \hfill \\ & = {max} \{ {min} \{ {max} \{ {max} \left\{ {Y\left( u \right): u \in F\left( w \right)} \right\}:w \in G(z)\} : z \in F(x)\} :x \in G(y)\} . \hfill \\ \end{aligned}$$

In terms of Proposition 3.5 and Proposition 3.6 we have

$$\underline{R} '(\bar{R}'\left( {Y)} \right) = \bar{R}_{p} (\underline{R}_{s} (\bar{R}_{p} (\bar{R}_{s} \left( Y \right)))) = \bar{R}_{p} (\bar{R}_{s} \left( {Y)} \right) = \underline{R}_{p} (\bar{R}_{s} \left( {Y)} \right) = \bar{R}'\left( Y \right).$$

(L₈) The proof follows from (U₉) of Proposition 5.2 and (L₁₀) of Proposition 5.4. In the same manner we can also prove (U₃), (U₈) and (U₁₀).

Definition 5.2

Let (U, V, R) be a (two-universe) approximation space. Then the lower and upper approximations of $Y \in {\mathcal{F}}(V)$ are defined respectively as follows:

$$\underline{R} ''\left( Y \right)(y) = {min} \{ {min} \{ Y\left( z \right): z \in F(x)\} :x \in G(y)\}$$

$$\bar{R}''\left( Y \right)\left( y \right) = max\{ {max} \left\{ {Y\left( z \right): z \in F\left( x \right)} \right\}:y \in G(y)\} .$$

The pair $(\underline{R} ''\left( Y \right),\bar{R}''\left( Y \right))$ is referred to as a strong rough fuzzy set, and $\underline{R}^{''}$ and $\bar{R}^{\prime\prime}: {\mathcal{F}}(V)\to {\mathcal{F}}(V)$ are referred to as strong lower and upper rough fuzzy approximation operators, respectively.

Proposition 5.5

Let $(U,V,R)$ be a (two-universe) approximation space. Then

$$\underline{R} ''\left( Y \right) = \underline{R}_{p} (\underline{R}_{s} \left( {Y)} \right),$$

$$\bar{R}''\left( Y \right) = \bar{R}_{p} (\bar{R}_{s} \left( Y \right)).$$

Proposition 5.6

In a (two-universe) approximation space $(U ,V,R)$ , the approximation operators have the following properties for all $Y, Y_{1} , Y_{2} \in {\mathcal{F}}(V)$:

(L₁):

$\underline{R} ''\left( Y \right) = (\bar{R}''(Y^{c} ))^{c}$

(L₂):

$\underline{R} ''\left( V \right) = V$

(L₃):

$\underline{R} ''\left( {Y_{1} \cap Y_{2} } \right) = \underline{R} ''(Y_{1} ) \cap \underline{R} ''(Y_{2} )$

(L₄):

$\underline{R} ''\left( {Y_{1} \cup Y_{2} } \right) \supseteq \underline{R} ''(Y_{1} ) \cup \underline{R} ''(Y_{2} )$

(L₅):

$Y_{1} \subseteq Y_{2} \Rightarrow \underline{R} ''(Y_{1} ) \subseteq \underline{R} ''\left( {Y_{2} } \right)$

(L₈):

$Y \subseteq \underline{R} ''(\bar{R}''(Y))$

(U₁):

$\bar{R}''\left( Y \right) = (\underline{R} ''(Y^{c} ))^{c}$

(U₂):

$\bar{R}''\left( \emptyset \right) = \emptyset$

(U₃):

$\bar{R}''\left( {Y_{1} \mathop \cup \nolimits Y_{2} } \right) = \bar{R}''(Y_{1} )\mathop \cup \nolimits \bar{R}''(Y_{2} )$

(U₄):

$\bar{R}''\left( {Y_{1} \mathop \cap \nolimits Y_{2} } \right) \subseteq \bar{R}''(Y_{1} )\mathop \cap \nolimits \bar{R}''(Y_{2} )$

(U₅):

$Y_{1} \subseteq Y_{2} \Rightarrow \bar{R}''(Y_{1} ) \subseteq \bar{R}''(Y_{2} )$

(U₈):

$\bar{R}''(\underline{R} ''\left( {Y)} \right) \subseteq Y$

Proof

We can obtain them according to Propositions 3.1, 3.3, 3.5 and 5.5.

Remark 5.3

If $R \in {\mathcal{P}}(U \times V)$ is a binary relation in a (two-universe) approximation space $(U ,V,R)$, then the following properties do not hold for all $Y, Y_{1} , Y_{2} \in {\mathcal{F}}(V)$:

(L₆):

$\underline{R} ''\left( \emptyset \right) = \emptyset$

(L₇):

$\underline{R} ''\left( Y \right) \subseteq Y$

(L₉):

$\underline{R} ''\left( Y \right) \subseteq \underline{R} ''(\underline{R} ''\left( Y \right))$

(L₁₀):

$\bar{R}''(Y) \subseteq \underline{R} ''(\bar{R}''(Y))$

(U₆):

$\bar{R}''\left( V \right) = V$

(U₇):

$Y \subseteq \bar{R}''(Y)$

(U₉):

$\bar{R}''(\bar{R}''\left( Y \right)) \subseteq \bar{R}''(Y)$

(U₁₀):

$\bar{R}''(\underline{R} ''\left( {Y)} \right) \subseteq \underline{R} ''(Y)$

(LU):

$\underline{R} ''(Y) \subseteq \bar{R}''(Y)$

The following example shows Remark 5.3.

Example 5.3

In Example 5.1, if Y and Z are two fuzzy subsets of V defined as:

$$\begin{gathered} Y\left( {y_{1} } \right) = 0.5, Y\left( {y_{2} } \right) = 0.3, Y\left( {y_{3} } \right) = 0.7, Y\left( {y_{4} } \right) = 0.1, Y\left( {y_{5} } \right) = 0.8, Y(y_{6} ) = 0.4, \hfill \\ Z\left( {y_{1} } \right) = 0.8, Z\left( {y_{2} } \right) = 0.9, Z\left( {y_{3} } \right) = 0.2, Z\left( {y_{4} } \right) = 0.3, Z\left( {y_{5} } \right) = 0.6, Z(y_{6} ) = 0.5, {\text{then we have}}\hfill \\ \end{gathered}$$

	y 1	y 2	y 3	y 4	y 5	y 6
$\underline{R} ''(Y)(y)$	0.4	0.3	0.3	1	0.3	0.3
$\bar{R}''(\underline{R} ''\left( {Y)} \right)(y)$	0.4	0.3	0.4	0	0.4	0.4
$\underline{R} ''(\underline{R} ''\left( {Y)} \right)(y)$	0.3	0.3	0.3	1	0.3	0.3
$\bar{R}''(Y)(y)$	0.8	0.8	0.8	0	0.8	0.8
$\underline{R} ''(\emptyset )(y)$	0	0	0	1	0	0
$\bar{R}''(V)(y)$	1	1	1	0	1	1
$\bar{R}''(Z)(y)$	0.8	0.9	0.9	0	0.9	0.9
$\underline{R} ''(\bar{R}''(Z))(y)$	0.8	0.9	0.8	1	0.8	0.8
$\bar{R}''(\bar{R}''(Z))(y)$	0.9	0.9	0.9	0	0.9	0.9

Hence we have $\underline{R} ''(\emptyset ) \ne \emptyset$, $\bar{R}''(V) \ne V$, $\underline{R}^{''} (Y) \not\subset Y$, $Z \not\subset \bar{R}''(Z)$, $\underline{R}^{''} (Y) \not\subset \underline{R} ''(\underline{R} ''\left( {Y)} \right)$, $\bar{R}''(\bar{R}''(Z)) \not\subset \bar{R}''(Z)$, $\bar{R}''(Z) \not\subset \underline{R} ''(\bar{R}''(Z))$, $\bar{R}''(\underline{R} ''\left( {Y)} \right) \not\subset \underline{R} ''(Y)$ and $\underline{R} ''(Y) \subseteq \bar{R}''(Y)$ i.e., L₆, U₆, L₇, U₇, L₉, U₉, L₁₀, U₁₀ and LU do not hold.

Proposition 5.7

In a (two-universe) approximation space $(U ,V,R)$ with inverse serial relation R, the approximation operators have the following properties for all $Y \in {\mathcal{F}}(V)$:

(L₆):

$\underline{R} ''\left( \emptyset \right) = \emptyset$

(L₇):

$\underline{R} ''\left( Y \right) \subseteq Y$

(U₆):

$\bar{R}''\left( V \right) = V$

(U₇):

$Y \subseteq \bar{R}''(Y)$

(LU):

$\underline{R} ''(Y) \subseteq \bar{R}''(Y)$

Proof

(L₇) Since R is an inverse serial. Then $\forall y \in V$, we can have

$$\begin{aligned} \underline{R} ''(Y)\left( y \right) & = {min} \{ {min} \{ Y\left( z \right): z \in F(x)\} :x \in G(y)\} . \hfill \\ & \le\,max \{ {min} \{ Y\left( z \right): z \in F(x)\} :x \in G(y)\} . \hfill \\ &= \bar{R}_{p} (\underline{R}_{s} \left( {Y)} \right)(y). \hfill \\ \end{aligned}$$

Then from Proposition 3.4, $\underline{R} ''\left( Y \right) \subseteq Y$.

(L₆) follows directly from (L₇).

(U₆) and (U₇) can be proved by the duality of approximation operators.

(LU) comes from (L₇) and (U₇).

Remark 5.4

If $R \in {\mathcal{P}}(U \times V)$ is an inverse serial relation in a (two-universe) approximation space $(U ,V,R)$, then the following properties do not hold for all $Y \in {\mathcal{F}}(V)$:

(L₉):

$\underline{R} ''\left( Y \right) \subseteq \underline{R} ''(\underline{R} ''\left( Y \right))$

(L₁₀):

$\bar{R}''(Y) \subseteq \underline{R} ''(\bar{R}''(Y))$

(U₉):

$\bar{R}''(\bar{R}''\left( Y \right)) \subseteq \bar{R}''(Y)$

(U₁₀):

$\bar{R}''(\underline{R} ''\left( {Y)} \right) \subseteq \underline{R} ''(Y)$

The following example shows Remark 5.4.

Example 5.4

In Example 5.2, we have

	y 1	y 2	y 3	y 4	y 5	y 6
$\underline{R} ''(Z)(y)$	0.1	0.1	0.5	0.1	0.1	0.1
$\bar{R}''(\underline{R} ''\left( {Z)} \right)(y)$	0.1	0.5	0.5	0.5	0.1	0.1
$\underline{R} ''(\underline{R} ''\left( {Z)} \right)(y)$	0.1	0.1	0.1	0.1	0.1	0.1
$\bar{R}''(Z)(y)$	0.9	0.8	0.8	0.9	0.9	0.8
$\underline{R} ''(\bar{R}''(Z))(y)$	0.9	0.8	0.8	0.8	0.8	0.8
$\bar{R}''(\bar{R}''(Z))(y)$	0.9	0.9	0.9	0.9	0.9	0.9

Hence we have $\underline{R}^{''} (Z) \not\subset \underline{R} ''(\underline{R} ''\left( {Z)} \right)$, $\bar{R}''(\bar{R}''(Z)) \not\subset \bar{R}''(Z)$, $\bar{R}''(Z) \not\subset \underline{R} ''(\bar{R}''(Z))$ and $\bar{R}''(\underline{R} ''\left( {Z)} \right) \not\subset \underline{R} ''(Z)$, i.e., L₉, U₉, L₁₀ and U₁₀ do not hold.

Proposition 5.8

In a (two-universe) approximation space $(U ,V,R)$ with strong inverse serial relation R, the approximation operators have the following properties for all $Y \in {\mathcal{F}}(V)$:

(L₉):

$\underline{R} ''\left( Y \right) \subseteq \underline{R} ''(\underline{R} ''\left( Y \right))$

(L₁₀):

$\bar{R}''(Y) \subseteq \underline{R} ''(\bar{R}''(Y))$

(U₉):

$\bar{R}''(\bar{R}''\left( Y \right)) \subseteq \bar{R}''(Y)$

(U₁₀):

$\bar{R}''(\underline{R} ''\left( {Y)} \right) \subseteq \underline{R} ''(Y)$

Proof

The proof is similar to Proposition 5.4.

In Table 1 we compare the properties that are satisfied by the different definitions of rough set.

Table 1 comparison between the properties of rough fuzzy sets depending on Definitions 3.2, 4.1, 5.1 and 5.2 by using binary, inverse serial and strong inverse serial relations

6 Connections of the rough fuzzy approximation operators

Proposition 6.1

Let $R \in {\mathcal{P}}(U \times V)$ be a binary relation from U to V. Then $\forall Y \in {\mathcal{F}}(V)$,

$$(1)\;\underline{R}^{'} \left( Y \right) \subseteq \underline{R}^{*} \left( Y \right),\bar{R}^{*} (Y) \subseteq \bar{R}'\left( Y \right).$$

$$(2)\;\underline{R} ''\left( Y \right) \subseteq \underline{R} .^{*} \left( Y \right),\bar{R}^{*} (Y) \subseteq \bar{R}''\left( Y \right).$$

Proof

By duality of approximation operators we only need to prove the first part of each property.

• (1) Since for every y ∈ Y, we have

$$\begin{aligned} \underline{R}^{*} \left( Y \right)\left( y \right) & = {min} \{ Y\left( z \right) : z \in G^{*} (y)\} = {min} \{ Y\left( z \right) : z \in \cap F(x), x \in G(y)\} \hfill \\ & \ge {max} \{ {min} \{ Y\left( z \right): z \in F(x)\} :x \in G(y)\} \hfill \\ & = \underline{R}^{'} \left( Y \right)(y) \hfill \\ \end{aligned}$$

Hence $\underline{R}^{'} \left( Y \right) \subseteq \underline{R}^{*} \left( Y \right)$.

• (2) the proof is similar as (1)

Remark 6.1

Let $R \in {\mathcal{P}}(U \times V)$ be a binary relation from U to V. Then Definitions 5.1 and 5.2 are independent.

The following example shows Remark 6.1. Moreover, the inclusion in Proposition 6.1 can not be replaced by equality,

Example 6.1

From Example 5.1, we get:

	y 1	y 2	y 3	y 4	y 5	y 6
$\underline{R}^{'} (Y)(y)$	0.5	0.3	0.5	0	0.5	0.4
$\underline{R} ''\left( Y \right)(y)$	0.4	0.3	0.3	1	0.3	0.3
$\underline{R}^{*} \left( Y \right)(y)$	0.5	0.3	0.7	1	0.8	0.4
$\bar{R}'(Y)(y)$	0.5	0.7	0.7	1	0.8	0.5
$\bar{R}''\left( Y \right)(y)$	0.8	0.8	0.8	0	0.8	0.8
$\bar{R}^{*} (Y)(y)$	0.5	0.3	0.7	0	0.8	0.4

Proposition 6.2

Let $R \in {\mathcal{P}}(U \times V)$ be an inverse serial relation from U to V. Then $\forall Y \in {\mathcal{F}}(V)$,

$$\underline{R} ''\left( Y \right) \subseteq \underline{R}^{'} \left( Y \right) \subseteq \underline{R}^{*} \left( Y \right) \subseteq Y \subseteq \bar{R}^{*} (Y) \subseteq \bar{R}'\left( Y \right) \subseteq \bar{R}''\left( Y \right).$$

Proof

Obvious.

We can introduce an example to show that the converse of Proposition 6.2 is not true in general.

Example 6.2

Let $U = \{ x_{1} , x_{2} , x_{3} , x_{4} ,x_{5} , x_{6} \}$, $V = \{ y_{1} , y_{2} , y_{3} ,$ $y_{4} , y_{5} , y_{6} , y_{7} \}$ and $R \in {\mathcal{P}}(U \times V)$ be an inverse serial relation defined as:

R	y1	y2	y3	y4	y5	y6	y7
x1	0	1	0	1	1	0	0
x2	1	1	0	0	1	1	0
x3	0	0	1	1	0	0	0
x4	1	0	0	1	0	0	1
x5	0	0	1	1	1	0	0
x6	0	0	0	0	0	0	0

If Y is a fuzzy subset of V defined as:

$Y\left( {y_{1} } \right) =\,0.2, Y\left( {y_{2} } \right) =\,0.5, Y\left( {y_{3} } \right) =\,0.1, Y\left( {y_{4} } \right) =\,0.7, Y\left( {y_{5} } \right) =\,0.3, Y(y_{6} ) =\,0.8, Y(y_{7} ) =\,0.4$, then we have

	y 1	y 2	y 3	y 4	y 5	y 6	y 7
$\underline{R}^{'} (Y)(y)$	0.2	0.3	0.1	0.3	0.3	0.2	0.2
$\underline{R} ''\left( Y \right)(y)$	0.2	0.2	0.1	0.1	0.1	0.2	0.2
$\underline{R}^{*} \left( Y \right)(y)$	0.2	0.3	0.1	0.7	0.3	0.2	0.2
$\bar{R}'(Y)(y)$	0.7	0.7	0.7	0.7	0.7	0.8	0.7
$\bar{R}''\left( Y \right)(y)$	0.8	0.8	0.7	0.7	0.8	0.8	0.7
$\bar{R}^{*} (Y)(y)$	0.2	0.5	0.7	0.7	0.3	0.8	0.7

Proposition 6.3

Three pairs of lower and upper approximation operators in Definition 4.1, Definition 5.1 and Definition 5.2 are equivalent if R is a strong inverse serial relation.

Proof

For a strong inverse serial relation, by Propositions 3.6, 5.1 and 5.5 we have $\underline{\text{R}}^{''} \left( {\text{Y}} \right) = \underline{\text{R}} '\left( {\text{Y}} \right)$ and ${\bar{\text{R}}}^{''} \left( {\text{Y}} \right) = {\bar{\text{R}}}^{'} \left( {\text{Y}} \right)$. We only need to show $\underline{\text{R}}^{ *} \left( {\text{Y}} \right) \subseteq \underline{\text{R}}^{ '} \left( {\text{Y}} \right)$. The other relation ${\bar{\text{R}}}^{'} \left( {\text{Y}} \right) \subseteq {\bar{\text{R}}}^{ *} ({\text{Y}})$ can be obtained by duality. Since for every ${\text{y}} \in {\text{Y}}$, we have

$$\begin{aligned} \underline{R}^{*} \left( Y \right)\left( y \right) & = {min} \{ Y\left( z \right) : z \in G^{*} (y)\} = {min} \{ Y\left( z \right) : z \in \cap F(x), x \in G(y)\} \hfill \\ &= {min} \{ Y\left( z \right) : z \in F(x), x \in G(y)\} \hfill \\ & \le {max} \left\{ {{min} \{ Y\left( z \right) : z \in F\left( x \right)} \right\}, x \in G(y)\} \hfill \\ & = \underline{\text{R}}^{ '} \left( {\text{Y}} \right)({\text{y}}) \hfill \\ \end{aligned}$$

7 Conclusion

In this paper we presented two new definitions of the lower approximation and upper approximation operators on two universes through the combination of successor and predecessor neighborhood operators. It should be pointed out that the approximating sets and the approximated sets in these rough set models are on the same universe of discourse V, and each type of the approximation operator captures different aspects of approximating a subset of the universe of discourse. All the properties of rough sets have been simulated by employing these notions, and the relationships between some of them and the existing rough fuzzy approximation operators on two universes of discourse have also been examined. By comparing these approximation operators, some conditions on the relation R under which all of the rough fuzzy approximation operators made equivalent are identified.