Multigranulation roughness based on soft relations

Abstract

The original rough set model, developed by Pawlak depends on a single equivalence relation. Qian et al, extended this model and defined multigranulation rough sets by using finite number of equivalence relations. This model provide new direction to the research. Recently, Shabir et al. proposed a rough set model which depends on a soft relation from an universe V to an universe W . In this paper we are present multigranulation roughness based on soft relations. Firstly we approximate a non-empty subset with respect to aftersets and foresets of finite number of soft binary relations. In this way we get two sets of soft sets called the lower approximation and upper approximation with respect to aftersets and with respect to foresets. Then we investigate some properties of lower and upper approximations of the new multigranulation rough set model. It can be found that the Pawlak rough set model, Qian et al. multigranulation rough set model, Shabir et al. rough set model are special cases of this new multigranulation rough set model. Finally, we added two examples to illustrate this multigranulation rough set model.

Keywords

Rough set multigranulation rough set soft set soft relation and approximation by soft binary relation

1 Introduction

In our surroundings, there are many problems which contain uncertainty. These problems generally occur in engineering, social science, medical science, ecology, economics, biology and many other fields. It is very difficult to handle these problems, but there exist some mathematical approaches to deal with uncertainty, like probability theory, vague set theory [18] fuzzy set theory [58] intuitionistic fuzzy sets [7] and interval mathematics [19]. All these theories are well recognized and often beneficial approaches to describe uncertainty. But all these theories have their own limitations, as mentioned by Molodtsov [30]. For example probability theory only deals with stochastically stable phenomena with out any mathematical details, interval mathematics is useful in many cases to deal with uncertainties, but the method of this theory is not sufficient to deal with problems having different uncertainties, fuzzy set theory is the most suitable theory dealing with uncertainties, but there exists a difficulty how to set the membership function in each case.

To overcome these difficulties Molodtsov [30] initiated the idea of soft set theory, it is a completely new mathematical approach for modeling vagueness and uncertainty, which is free from difficulties which occur in existing mathematical approaches. The soft set theory have a very wide variety of applications in many directions, such as the smoothness of functions, game theory, operation research, Riemann integration and so on. Latter Maj et al. [28] discussed some properties of soft sets, Ali et al. [5] discussed some new operations in soft set theory. After then, many researchers discussed the properties, applications of soft sets. Maji et al. [28] successfully applied the soft sets theory to decision making problem. It attracted the attention of many researchers all over the world, who contributed in its development and applications [8 , 66]. Soft sets theory is applied successfully to algebraic structures [1 , 22] Ali et al. [4] discussed the fuzzy sets and fuzzy soft sets induced by soft sets. Feng et al. [16] combined soft sets with rough sets and fuzzy set, Yang et al. [51] introduced interval-valued fuzzy soft sets, Feng et al. [13] discussed soft relations.

Pawlak [31] introduced another interesting theory to deal with uncertainty and vagueness, called Rough set theory. Rough set theory is just a common approach to uncertainties. Like fuzzy set theory it is not an alternative to classic set theory but integrated into it. Uncertainties in this approach are expressed by a boundary region of a set, not by a partial membership, like in fuzzy set theory. Major advantage of rough set theory is that it helps to reduce the data without losing useful information. Rough set theory has been proposed for a very wide variety of applications. In particular, the rough set approach is important for artificial intelligence and cognitive sciences, especially in machine learning, knowledge discovery, data mining, expert systems, approximate reasoning and pattern recognition. Equivalence relation is one of the key notion in Pawlak rough set model. The equivalence classes are the building blocks for the construction of the lower and upper approximations. The classical rough set theory is based on equivalence relation, but in some situations, equivalence relations are not suitable for dealing with granularity and it may limit the applications of Pawlak rough set, thus many researchers replace equivalence relation with, tolerance relation by [40] similarity relation by [41] dominance relation [20] neighborhood system [45, 52] general binary relation [47, 65] Shaheen et al. [38] discussed dominance based soft rough sets and their application to decision analysis, Feng et al. [15] combined soft set with fuzzy set and rough set, the soft rough set has been investigated in [4 , 37] recently Sharma el al. [39] successfully applied rough sets to forecasting model, Alcantud et al. [3] discussed an N-soft sets pproach to sough sets, Li et al. [24] discussed rough set approximation based on soft binary relations and knowledge bases. The covering based rough set can be seen in [12 , 65]. In reality many practical problems contains different universe of objects, such as the symptoms of disease and drugs in disease diagnosis. The original rough set model deal with the problems of single universe. To tackle the problems of rough set on single-universe, Liu [27] and Yan et al. [54] generalized the rough set model over two universes in aspect of building connection between single-universe model and two universes model. Shabir et al. [36] investigated approximation by soft binary relation over two universe and their application in reduction of an information systems.

The Pawlak rough set model, which is mainly concerned with the approximations of set described by single equivalence relation on the universe, Qain et al. [32, 33] extended the Pawlak rough set model to a multi granulation rough set model MGRS, where the roughness of sets are defined by using multi equivalence relations on the universe. It give new direction to research in rough sets. MGRS model can be found in literature [23 , 55] Xu et al. [48, 50] replaced equivalence relation by tolerance and order relations and constructed two type of MGRS. Fuzzy MGRS can be found in [49, 56] covering pessimistic multigranulation rough set based on evidence investigated by You et al. [57] Ali et al. [6] present new type of dominance based MGRS and their application in conflict analysis problems, Sun et al. [42] introduce the MGRS rough set theory over two universes, Sun et al. [43] discussed MGRS fuzzy rough set over two universes and its application to decision making, Zhang et al. [62] generalized fuzzy rough set to two universes with interval-valued data. Recently Tan et al. [44] discussed Granulation selection and decision making with multigranulation rough set over two universes.

The original rough set based on equivalence relation over a non empty universe of objects. Xu et al. [47] generalized this concept and replace equivalence relation by general binary relation over a non empty universe of objects and approximate a set with respect to aftersets and foresets of binary relation over universe of objects. On the other hand Shabir et al. [36] generalized these concepts and replace binary relation by soft binary relation from a non-empty universe V to universe W and approximate a non- empty subset of universe W in universe V, and a non- empty subset of universe V in universe W by using the aftersets and foresets of a soft binary relation over V × W. The main purpose of this article is to describe multigranulation rough set based on soft binary relations over two universes. Firstly we approximate a non-empty subset of a universe W in universe V and approximate a non-empty subset of a universe V in universe W by using the aftersets and foresets of a soft binary relations over V × W. After that, we will focus on some algebraic structural properties of proposed model.

The remaining article is organized as follows: in Section 2 we recall some fundamental concepts about rough sets, multigranulations rough sets, soft sets, roughness by soft relations and their basic properties. In Section 3 we discuss roughness of a set based on two soft binary relations and some interesting properties are proved. In Section 4 we discuss roughness of a set by multi soft binary relations and in Section 5 we give some examples about data classifications. In last section we conclude the paper.

2 Preliminaries

In this section we recall some basic notions which we will use in upcoming sections.

Definition 2.1. [31] Let V be a finite non-empty set and ξ be an equivalence relation over V . Then {u ∈ V| (v, u) ∈ ξ} is called an equivalence class of v with respect to ξ, we denote it by [v] _ξ . Let N ⊆ V . The Pawlak lower and upper approximations of N are defined as $\begin{matrix} \underline{ξ} (N) & = \cup {[v]_{ξ} | [v]_{ξ} \subseteq N} \\ \bar{ξ} (N) & = \cup {[v]_{ξ} | [v]_{ξ} \cap N \neq \emptyset} . \end{matrix}$ The pair (V, ξ) is called Pawlak approximation space.

Qian et al. [32, 33], extended the Pawlak rough set model to multigranulation rough set model, where the set approximations are defined by using multi equivalence relations on the universe.

Definition 2.2. [32] Let V be a non-empty set, $\hat{ξ_{1}}, \hat{ξ_{2}}$ be two equivalence relations on the universal set V and N ⊆ V. Then $\begin{matrix} {\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} & = {v \in V : [v]_{\hat{ξ_{1}}} \subseteq N or [v]_{\hat{ξ_{2}}} \subseteq N} \\ {\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} & = {v \in V : [v]_{\hat{ξ_{1}}} \cap N \neq \emptyset and [v]_{\hat{ξ_{2}}} \cap N \neq \emptyset} \end{matrix}$ are called the lower and upper approximations of N with respect to ξ₁ and ξ₂ .

The following properties are given in Qian et al.[32].

Proposition 2.1. [32] Let $\hat{ξ_{1}}, \hat{ξ_{2}}$ be two equivalence relations on a universal set V and N ⊆ V. Then the following properties hold.

${\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} \subseteq N \subseteq {\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}}$

${\underline{\emptyset}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} = \emptyset = {\bar{\emptyset}}^{\hat{ξ_{1}} + \hat{ξ_{2}}}$ and ${\underline{V}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} = V = {\bar{V}}^{\hat{ξ_{1}} + \hat{ξ_{2}}}$

${\underline{N^{c}}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} = ({\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}})^{c}$ and ${\bar{N^{c}}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} = ({\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}})^{c}$

${\underline{{\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}}}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} = {\bar{{\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}}}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} = {\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}}$

${\bar{{\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}}}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} = {\underline{{\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}}}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} = {\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}}$

${\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} = {\underline{N}}_{\hat{ξ_{1}}} \cup {\underline{N}}_{\hat{ξ_{2}}}$

${\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} = {\bar{N}}^{\hat{ξ_{1}}} \cap {\bar{N}}^{\hat{ξ_{2}}}$

${\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} = {\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{1}}}$ and ${\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} = {\bar{N}}^{\hat{ξ_{2}} + \hat{ξ_{1}}} .$

Proposition 2.2. [32] Let $\hat{ξ_{1}}, \hat{ξ_{2}}$ be two equivalence relations on a universal set V and M, N ⊆ V. Then the following properties hold.

${\underline{M \cap N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} = ({\underline{M}}_{\hat{ξ_{1}}} \cap {\underline{N}}_{\hat{ξ_{1}}}) \cup ({\underline{M}}_{\hat{ξ_{2}}} \cap {\underline{N}}_{\hat{ξ_{2}}})$

${\bar{M \cup N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} = ({\bar{M}}^{\hat{ξ_{1}}} \cup {\bar{N}}^{\hat{ξ_{1}}}) \cap ({\bar{M}}^{\hat{ξ_{2}}} \cup {\bar{N}}^{\hat{ξ_{2}}})$

${\underline{M \cap N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} \subseteq {\underline{M}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} \cap {\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}}$

${\underline{M \cup N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} \supseteq {\underline{M}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} \cup {\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}}$

${\bar{M \cup N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} \supseteq {\bar{M}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} \cup {\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}}$

${\bar{M \cap N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} \subseteq {\bar{M}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} \cap {\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}}$

If $M \subseteq N \Rightarrow {\underline{M}}_{\hat{ξ_{1}} + \hat{ξ_{2}}} \subseteq {\underline{N}}_{\hat{ξ_{1}} + \hat{ξ_{2}}}$

If $M \subseteq N \Rightarrow {\bar{M}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} \subseteq {\bar{N}}^{\hat{ξ_{1}} + \hat{ξ_{2}}} .$

Definition 2.3. [32] Let ${\hat{ξ}}_{1}, {\hat{ξ}}_{2}, \dots, {\hat{ξ}}_{m}$ be m equivalence relations on a universal set V and N ⊆ V . Then the lower and upper approximations of N are defined as $\begin{matrix} {\underline{N}}_{\sum_{i = 1}^{m} {\hat{ξ}}_{i}} & = {v \in V : [v]_{{\hat{ξ}}_{i}} \subseteq N for some i, 1 \leq i \leq m}, \\ {\bar{N}}^{\sum_{i = 1}^{m} {\hat{ξ}}_{i}} & = ({\underline{N^{c}}}_{\sum_{i = 1}^{m} {\hat{ξ}}_{i}})^{c} . \end{matrix}$

Molodtsov (1999) defined a soft set as:

Definition 2.4. [30] A pair (α, A) is called a soft set over V, where α is a mapping given by α : A ⟼ P (V) , V is a non-empty finite set and A is a subset of E (set of parameters).

A soft set (α, A) is a soft subset of a soft set (β, B) over a common universe V, if A ⊆ B and for all a ∈ A, α (a) ⊆ β (a) . Two soft sets over a common universe are equal if they are soft subsets of each other.

Feng et al. (2013) defined soft binary relation on a universe U as following.

Definition 2.5. [13] Let (α, A) be a soft set over U × U . Then (α, A) is called a soft binary relation over U. In fact (α, A) is a parameterized collection of binary relations over U, that is, we have a binary relation α (e) on U for each parameter e ∈ A.

We shall denote the collection of all soft binary relations over U by SBr (U) .

If each α (e) is reflexive then we say that (α, A) is a soft reflexive relation.

If each α (e) is symmetric then we say that (α, A) is a soft symmetric relation.

If each α (e) is transitive then we say that (α, A) is a soft transitive relation.

If each α (e) is equivalence then we say that (α, A) is a soft equivalence relation.

Li et al. [24], generalized the definition of soft binary relation over a set U to soft binary relation from V to W, as following.

Definition 2.6. [24] If (α, A) is a soft set over V × W, that is α : A ⟼ P (V × W) , then (α, A) is said to be a soft binary relation (SB-relation) from V to W . In fact (α, A) is a parameterized collection of binary relations from V to W . We shall denote the collection of all soft binary relations from V to W by SBr (V, W) .

Shabir et al. [36] defined lower and upper approximations of a set by using soft binary relations as follows:

Definition 2.7. [36] If (α, A) is a SB-relation from V to W and N ⊆ W, then we define two soft sets over V, by $\begin{matrix} {\underline{α}}^{N} (e) & = {v \in V : \emptyset \neq v α (e) \subseteq N}, \\ {\bar{α}}^{N} (e) & = {v \in V : v α (e) \cap N \neq \emptyset}, \end{matrix}$ where vα (e) = {w ∈ W : (v, w) ∈ α (e)} for each e ∈ A and is called afterset of v corresponding to parameter e.

Moreover, ${\underline{α}}^{N} : A \mapsto P (V)$ and ${\bar{α}}^{N} : A \mapsto P (V)$ and we say (V, W, α) a generalized soft approximation space.

If M ⊆ V, then we can define two soft sets over W, by $^{M} \underline{α} (e) = {w \in W : \emptyset \neq α (e) w \subseteq M}$ $^{M} \bar{α} (e) = {w \in W : α (e) w \cap M \neq \emptyset}$ where α (e) w = {v ∈ V : (v, w) ∈ α (e)} for each e ∈ A and is called foreset of w corresponding to parameter e.

Moreover, $^{M} \underline{α} : A \mapsto P (V)$ and $^{M} \bar{α} : A \mapsto P (V) .$

Example 2.1. [36] Suppose that Mr : X wants to buy a shirt for his own use. Let U = = {u₁, u₂, u₃, u₄} be the set of all shirts under consideration and V = {bright, cheap, warm} = {b, c, w} and the set of attributes be A = {e₁, e₂} = {outlook, stuff}.

Define J : A ↦ P (U × V) by $\begin{matrix} J (e_{1}) & = {(u_{1}, b), (u_{1}, c), (u_{2}, w)}, \\ J (e_{2}) & = {(u_{2}, b), (u_{2}, w), (u_{4}, b)} . \end{matrix}$ Let N = {b, w} ⊆ V and M = {u₂, u₃} ⊆ U . Then $\begin{matrix} u_{1} J (e_{1}) & = {b, c}, & u_{1} J (e_{2}) & = \emptyset, \\ u_{2} J (e_{1}) & = {w}, & u_{2} J (e_{2}) & = {b, w}, \\ u_{3} J (e_{1}) & = \emptyset, & u_{3} J (e_{2}) & = \emptyset, \\ u_{4} J (e_{1}) & = \emptyset, & u_{4} J (e_{2}) & = {b}, \end{matrix}$ and $\begin{matrix} J (e_{1}) b & = {u_{1}}, & J (e_{2}) b & = {u_{2}, u_{4}} \\ J (e_{1}) c & = {u_{1}}, & J (e_{2}) c & = \emptyset \\ J (e_{1}) w & = {u_{2}}, & J (e_{2}) w & = {u_{2}} . \end{matrix}$ Therefore, $\begin{matrix} {\underline{J}}^{N} (e_{1}) & = {u_{2}, u_{3}, u_{4}}, & {\underline{J}}^{N} (e_{2}) & = {u_{1}, u_{2}, u_{3}, u_{4}}, \\ {\bar{J}}^{N} (e_{1}) & = {u_{1}, u_{2}}, & {\bar{J}}^{N} (e_{2}) & = {u_{2}, u_{4}} . \end{matrix}$ And $^{M} \underline{J} (e_{1}) = {w},^{M} \underline{J} (e_{2}) = {w},$ $^{M} \bar{J} (e_{1}) = {w},^{M} \bar{J} (e_{2}) = {w} .$

Proposition 2.3. [36] If (α, A) is a SB-relation from V to W . For N₁, N₂ ⊆ W, the following properties hold.

$N_{1} \subseteq N_{2} \Rightarrow {\underline{α}}^{N_{1}} \subseteq {\underline{α}}^{N_{2}}$

$N_{1} \subseteq N_{2} \Rightarrow {\bar{α}}^{N_{1}} \subseteq {\bar{α}}^{N_{2}}$

${\underline{α}}^{N_{1} \cap N_{2}} = {\underline{α}}^{N_{1}} \cap {\underline{α}}^{N_{2}}$

${\underline{α}}^{N_{1} \cup N_{2}} \supseteq {\underline{α}}^{N_{1}} \cup {\underline{α}}^{N_{2}}$

${\bar{α}}^{N_{1} \cup N_{2}} = {\bar{α}}^{N_{1}} \cup {\bar{α}}^{N_{2}}$

${\bar{α}}^{N_{1} \cap N_{2}} \subseteq {\bar{α}}^{N_{1}} \cap {\bar{α}}^{N_{2}}$

${\underline{α}}^{W} (e) = V,$ for all e ∈ A

${\bar{α}}^{W} (e) \subseteq V,$ for all e ∈ A

${\underline{α}}^{N} (e) = ({\bar{α}}^{N^{c}} (e))^{c}$

${\bar{α}}^{N} (e) = ({\underline{α}}^{N^{c}} (e))^{c} .$

3 Roughness of a set by two soft relations

In this section we discuss the roughness of a set by two soft relations. We use soft relations from V to W and approximate subsets of V in W and subsets of W in V . In this way we get two soft sets corresponding to a subset of V (W) , called lower approximation and upper approximation with respect to the foresets (aftersets).

Definition 3.1. Let V≠ ∅ , W ≠ ∅ be two universal sets, (ξ₁, A) , (ξ₂, A) be two soft relations from V to W and N ⊆ W, where A ⊆ E(set of parameters). Then we define two soft sets over V by $\begin{matrix} \underline{ξ_{1} + ξ_{2}}^{N} (e) = {v \in V ∣ \emptyset \neq v ξ_{1} (e) \subseteq N or \emptyset \neq v ξ_{2} (e) \subseteq N} for all e \in A \bar{ξ_{1} + ξ_{2}} N (e) = {v \in V ∣ v ξ_{1} (e) \cap N \neq \emptyset and v ξ_{2} (e) \cap N \neq \emptyset} for all e \in A \end{matrix}$ called the lower and upper approximations of N with respect to the aftersets. We denote these soft sets by $({\underline{ξ_{1} + ξ_{2}}}^{N} (e), A), ({\bar{ξ_{1} + ξ_{2}}}^{N} (e), A),$ respectively.

If N ⊆ V then we define two soft sets over W by $^{N} \underline{ξ_{1} + ξ_{2}} (e) = {w \in W ∣ \emptyset \neq ξ_{1} (e) w \subseteq N$ $or \emptyset \neq ξ_{2} (e) w \subseteq N} for all e \in A$ $^{N} \bar{ξ_{1} + ξ_{2}} (e) = {w \in W ∣ ξ_{1} (e) w \cap N \neq \emptyset$ $and ξ_{2} (e) w \cap N \neq \emptyset} for all e \in A$ called the lower and upper approximations of N with respect to the foresets. We denote these soft sets by $(^{N} \underline{ξ_{1} + ξ_{2}} (e), A), (^{N} \bar{ξ_{1} + ξ_{2}} (e), A),$ respectively.

Remark 3.1.

From above definition it can be seen that the lower approximation given in Definition 3.1 is defined by using the two aftersets induced by two independent soft relations, where as the lower approximation defined in Definition 2.7 via one afterset induced by only one soft relation.

In fact, if we perform union operation between the lower approximations obtained by Definition 2.7 then it is equal to the lower approximation given by Definition 3.1. On the otherhand the upper approximation defined in above Definition 3.1 is the intersection of the upper approximations defined in Definition 2.7.

If we take V = W in above definition and N ⊆ V, even then

$({\underline{ξ_{1} + ξ_{2}}}^{N} (e), A) \neq (^{N} \underline{ξ_{1} + ξ_{2}} (e), A)$ and $({\bar{ξ_{1} + ξ_{2}}}^{N} (e), A) \neq (^{N} \bar{ξ_{1} + ξ_{2}} (e), A) .$

This is due to the fact that vξ₁ ≠ ξ₁v and vξ₂ ≠ ξ₂v .

In order to explain the above definitions we give the following example.

Example 3.1. Let V = {1, 2, 3} , W = {a, b, c} and A = {e₁, e₂}.

Let N = {a, b} ⊆ W, M = {1, 2} ⊆ V and (ξ₁, A) , (ξ₂, A) be two soft relations from V to W defined by ξ₁ (e₁) = {(1, a) , (1, b) , (2, a)}, ξ₁ (e₂) = {(2, b) , (3, a)} and ξ₂ (e₁) = {(2, b) , (2, c) , (3, a)} , ξ₂ (e₂) = {(1, c) , (3, b) , (3, c)}.

Then their aftersets and foresets are $\begin{matrix} \begin{matrix} 1 ξ_{1} (e_{1}) = {a, b}, & 1 ξ_{2} (e_{1}) = \emptyset, & ξ_{1} (e_{1}) a = {1, 2}, & ξ_{2} (e_{1}) a = {3} \\ 2 ξ_{1} (e_{1}) = {a}, & 2 ξ_{2} (e_{1}) = {b, c}, & ξ_{1} (e_{1}) b = {1}, & ξ_{2} (e_{1}) b = {2} \\ 3 ξ_{1} (e_{1}) = \emptyset, & 3 ξ_{2} (e_{1}) = {a}, & ξ_{1} (e_{1}) c = \emptyset, & ξ_{2} (e_{1}) c = {2} \\ 1 ξ_{1} (e_{2}) = \emptyset, & 1 ξ_{2} (e_{2}) = {c}, & ξ_{1} (e_{2}) a = {3}, & ξ_{2} (e_{2}) a = \emptyset \\ 2 ξ_{1} (e_{2}) = {b}, & 2 ξ_{2} (e_{2}) = \emptyset, & ξ_{1} (e_{2}) b = {2}, & ξ_{2} (e_{2}) b = {3} \\ 3 ξ_{1} (e_{2}) = {a}, & 3 ξ_{2} (e_{2}) = {b, c}, & ξ_{1} (e_{2}) c = \emptyset, & ξ_{2} (e_{2}) c = {1, 3} \end{matrix} \end{matrix}$ Then ${\underline{ξ_{1}}}^{N} (e_{1}) = {1, 2}, {\underline{ξ_{1}}}^{N} (e_{2}) = {2, 3}, {\bar{ξ_{1}}}^{N} (e_{1}) = {1, 2}, {\bar{ξ_{1}}}^{N} (e_{2}) = {2, 3} {\underline{ξ_{2}}}^{N} (e_{1}) = {3}, {\underline{ξ_{2}}}^{N} (e_{2}) = \emptyset, {\bar{ξ_{2}}}^{N} (e_{1}) = {2, 3}, {\bar{ξ_{2}}}^{N} (e_{2}) = {3}$ ${\underline{ξ_{1} + ξ_{2}}}^{N} (e_{1}) = {1, 2, 3} = {\underline{ξ_{1}}}^{N} (e_{1}) \cup {\underline{ξ_{2}}}^{N} (e_{1}) {\underline{ξ_{1} + ξ_{2}}}^{N} (e_{2}) = {2, 3} = {\underline{ξ_{1}}}^{N} (e_{2}) \cup {\underline{ξ_{2}}}^{N} (e_{2}),$ ${\bar{ξ_{1} + ξ_{2}}}^{N} (e_{1}) = {2} = {\bar{ξ_{1}}}^{N} (e_{1}) \cap {\bar{ξ_{2}}}^{N} (e_{1}), {\bar{ξ_{1} + ξ_{2}}}^{N} (e_{2}) = {3} = {\bar{ξ_{1}}}^{N} (e_{2}) \cap {\bar{ξ_{2}}}^{N} (e_{2}) .$

So we get two soft sets over $V, ({\underline{ξ_{1} + ξ_{2}}}^{N} (e), A) = {{1, 2, 3}, {2, 3}}$ and $({\bar{ξ_{1} + ξ_{2}}}^{N} (e), A) = {{2}, {3}} .$

Similarly, $^{M} \underline{ξ_{1}} (e_{1}) = {a, b},^{M} \underline{ξ_{1}} (e_{2}) = {b},^{M} \bar{ξ_{1}} (e_{1}) = {a, b},^{M} \bar{ξ_{1}} (e_{2}) = {b},^{M} \underline{ξ_{2}} (e_{1}) = {b, c},^{M} \underline{ξ_{2}} (e_{2}) = \emptyset,^{M} \bar{ξ_{2}} (e_{1}) = {b, c},^{M} \bar{ξ_{2}} (e_{2}) = {c}^{M} \underline{ξ_{1} + ξ_{2}} (e_{1}) = {a, b, c} =^{M} \underline{ξ_{1}} (e_{1}) \cup^{M} \underline{ξ_{2}} (e_{1}),^{M} \underline{ξ_{1} + ξ_{2}} (e_{2}) = {b} =^{M} \underline{ξ_{1}} (e_{2}) \cup^{M} \underline{ξ_{2}} (e_{2}) .^{M} \bar{ξ_{1} + ξ_{2}} (e_{1}) = {b} =^{M} \bar{ξ_{1}} (e_{1}) \cap^{M} \bar{ξ_{2}} (e_{1}),^{M} \bar{ξ_{1} + ξ_{2}} (e_{2}) = \emptyset =^{M} \bar{ξ_{1}} (e_{2}) \cap^{M} \bar{ξ_{2}} (e_{2}) .$

So we ge two soft sets over $W, ({\underline{ξ_{1} + ξ_{2}}}^{N} (e), A) = {{a, b, c}, {b}}$ and $({\bar{ξ_{1} + ξ_{2}}}^{N} (e), A) = {{b}, \emptyset} .$

Lemma 3.1. Let V≠ ∅ , W ≠ ∅ be two universal sets and (ξ₁, A) , (ξ₂, A) be two soft relations from V to W and N ⊆ W . Then

${\bar{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq ({\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c}$

${\underline{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq ({\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c}$ for all e ∈ A .

Proof.

Let $v \in {\bar{ξ_{1} + ξ_{2}}}^{N} (e) .$ Then vξ₁ (e)∩ N ≠ ∅ and vξ₂ (e) ∩ N ≠ ∅ . These imply that ∅ ≠ vξ₁ (e) ⊈ N^c and ∅ ≠ vξ₂ (e) ⊈ N^c . Thus $v \notin {\underline{ξ_{1} + ξ_{2}}}^{N^{c}} .$ That is $v \in ({\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c}$ . Thus ${\bar{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq ({\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c} .$

Let $v \in {\underline{ξ_{1} + ξ_{2}}}^{N} (e) .$ Then ∅ ≠ vξ₁ (e) ⊆ N or ∅ ≠ vξ₂ (e) ⊆ N . These imply that vξ₁ (e)∩ N^c = ∅ or vξ₂ (e) ∩ N^c = ∅ . Thus $v \notin {\bar{ξ_{1} + ξ_{2}}}^{N^{c}} .$ That is $v \in ({\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c}$ . Thus ${\underline{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq ({\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c} .$

In the next example, we show that the inverse inclusions in above lemma do not necessarily hold.

Example 3.2. Let ξ₁, ξ₂ be two soft relations from V to W as defined in Example 3.1. Let N = {a, b} then N^c = {c} .

Now ${\bar{ξ_{1} + ξ_{2}}}^{N} (e_{1}) = {2} ⊉ {1, 2, 3} = ({\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e_{1}))^{c} .$

If N = {a, c} then N^c = {b} .

Now ${\underline{ξ_{1} + ξ_{2}}}^{N} (e_{1}) = {2, 3} ⊉ {1, 2, 3} = ({\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e_{1}))^{c} .$

Remark 3.2. If vξ₁ (e)≠ ∅ and vξ₂ (e)≠ ∅ for all e ∈ A and for all v ∈ V, then ${\bar{ξ_{1} + ξ_{2}}}^{N} (e) = ({\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c}$ and ${\underline{ξ_{1} + ξ_{2}}}^{N} (e) = ({\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c} .$

Proof. Let $v \in ({\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c} .$ Then $v \notin {\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e) .$ This implies that vξ₁ (e) ⊈ N^c and vξ₂ (e) ⊈ N^c . Since vξ₁ (e) , vξ₂ (e) , are non-empty so vξ₁ (e)∩ N ≠ ∅ and vξ₂ (e) ∩ N ≠ ∅ , that is $v \in {\bar{ξ_{1} + ξ 2}}^{N} (e) .$ Then ${\bar{ξ_{1} + ξ_{2}}}^{N} (e) = ({\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c} .$

Now, let $v \in ({\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c} .$ Thus $v \notin {\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e) .$ This implies that vξ₁ (e)∩ N^c = ∅ or vξ₂ (e) ∩ N^c = ∅ .

Since vξ₁ (e) , vξ₂ (e) , are non-empty so vξ₁ (e) ⊆ N or ∅ ≠ vξ₂ (e) ⊆ N, that is $\emptyset \neq v \in {\underline{ξ_{1} + ξ 2}}^{N} (e) .$ Thus ${\underline{ξ_{1} + ξ_{2}}}^{N} (e) = ({\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e))^{c} .$

Lemma 3.2. Let V≠ ∅ , W ≠ ∅ be two universal sets and (ξ₁, A) , (ξ₂, A) be two soft relations from V to W and N ⊆ V . Then

$^{N} \bar{ξ_{1} + ξ_{2}} (e) \subseteq {(^{N^{c}} \underline{ξ_{1} + ξ_{2}} (e))}^{c},$

$^{N} \underline{ξ_{1} + ξ_{2}} (e) \subseteq {(^{N^{c}} \bar{ξ_{1} + ξ_{2}} (e))}^{c}$ for all e ∈ A .

Proof. The proof is similar to that of Lemma 3.1.

Remark 3.3. If ξ₁ (e) w≠ ∅ and ξ₂ (e) w≠ ∅ for all e ∈ A and for all w ∈ W, then $^{N} \bar{ξ_{1} + ξ_{2}} (e) = (^{N^{c}} \underline{ξ_{1} + ξ_{2}} (e))^{c}$ and $^{N} \underline{ξ_{1} + ξ_{2}} (e) = (^{N^{c}} \bar{ξ_{1} + ξ_{2}} (e))^{c} .$

Proof. The proof is similar to that of Remark 3.2.

Proposition 3.1. Let V≠ ∅ , W ≠ ∅ be two universal sets, (ξ₁, A) , (ξ₂, A) be two soft relations from V to W and N ⊆ W. Then the following properties hold.

${\underline{ξ_{1} + ξ_{2}}}^{N} (e) = {\underline{ξ_{1}}}^{N} (e) \cup {\underline{ξ_{2}}}^{N} (e)$

${\bar{ξ_{1} + ξ_{2}}}^{N} (e) = {\bar{ξ_{1}}}^{N} (e) \cap {\bar{ξ_{2}}}^{N} (e)$

${\underline{ξ_{1} + ξ_{2}}}^{W} (e) \subseteq V$ and ${\bar{ξ_{1} + ξ_{2}}}^{W} (e) \subseteq V$

${\underline{ξ_{1} + ξ_{2}}}^{\emptyset} (e) = \emptyset$ and ${\bar{ξ_{1} + ξ_{2}}}^{\emptyset} (e) = \emptyset$

${\underline{ξ_{1} + ξ_{2}}}^{N} (e) = {\underline{ξ_{2} + ξ_{1}}}^{N} (e)$ and ${\bar{ξ_{1} + ξ_{2}}}^{N} (e) = {\bar{ξ_{2} + ξ_{1}}}^{N} (e)$

${\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e) \subseteq ({\bar{ξ_{1} + ξ_{2}}}^{N} (e))^{c}$ and ${\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e) \subseteq ({\underline{ξ_{1} + ξ_{2}}}^{N} (e))^{c}$

Proof.

Let $v \in {\underline{ξ_{1} + ξ_{2}}}^{N} (e)$ . Then ∅ ≠ vξ₁ (e) ⊆ N or ∅ ≠ vξ₂ (e) ⊆ N . These imply that $v \in {\underline{ξ_{1}}}^{N} (e)$ or $v \in {\underline{ξ_{2}}}^{N} (e) .$ That is $v \in {\underline{ξ_{1}}}^{N} (e) \cup {\underline{ξ_{2}}}^{N} (e) .$ Thus ${\underline{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq {\underline{ξ_{1}}}^{N} (e) \cup {\underline{ξ_{2}}}^{N} (e)$ .

Conversely, let $v \in {\underline{ξ_{1}}}^{N} (e) \cup {\underline{ξ_{2}}}^{N} (e) .$ Then $v \in {\underline{ξ_{1}}}^{N} (e)$ or $v \in {\underline{ξ_{2}}}^{N} (e) .$ Thus ∅ ≠ vξ₁ (e) ⊆ N or ∅ ≠ vξ₂ (e) ⊆ N . These imply that $v \in {\underline{ξ_{1} + ξ_{2}}}^{N} (e) .$ Thus ${\underline{ξ_{1}}}^{N} (e) \cup {\underline{ξ_{2}}}^{N} (e) \subseteq {\underline{ξ_{1} + ξ_{2}}}^{N} (e) .$ Hence ${\underline{ξ_{1} + ξ_{2}}}^{N} (e) = {\underline{ξ_{1}}}^{N} (e) \cup {\underline{ξ_{2}}}^{N} (e) .$

Let $v \in {\bar{ξ_{1} + ξ_{2}}}^{N} (e)$ . Then vξ₁ (e)∩ N ≠ ∅ and vξ₂ (e) ∩ N ≠ ∅ . These imply that $v \in {\bar{ξ_{1}}}^{N} (e)$ and $v \in {\bar{ξ_{2}}}^{N} (e),$ that is $v \in ({\bar{ξ_{1}}}^{N} (e) \cap {\bar{ξ_{2}}}^{N} (e)) .$ Thus ${\bar{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq {\bar{ξ_{1}}}^{N} (e) \cap {\bar{ξ_{2}}}^{N} (e)$ .

Conversely, let $v \in {\bar{ξ_{1}}}^{N} (e) \cap {\bar{ξ_{2}}}^{N} (e) .$ Then $v \in {\bar{ξ_{1}}}^{N} (e)$ and $v \in {\bar{ξ_{2}}}^{N} (e)$ , that is vξ₁ (e)∩ N ≠ ∅ and vξ₂ (e) ∩ N ≠ ∅ . These imply that $v \in {\bar{ξ_{1} + ξ_{2}}}^{N} (e) .$ Thus ${\bar{ξ_{1}}}^{N} (e) \cap {\bar{ξ_{2}}}^{N} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N} (e),$ Hence ${\bar{ξ_{1} + ξ_{2}}}^{N} (e) = {\bar{ξ_{1}}}^{N} (e) \cap {\bar{ξ_{2}}}^{N} (e) .$

(a) By definition ${\underline{ξ_{1} + ξ_{2}}}^{W} (e) = {v \in V ∣ \emptyset \neq v ξ_{1} (e) \subseteq W$ or ∅ ≠ vξ₂ (e) ⊆ W} ⊆ V .

(b) By definition ${\bar{ξ_{1} + ξ_{2}}}^{W} (e) = {v \in V ∣ v ξ_{1} (e) \cap W \neq \emptyset$ and vξ₂ (e) ∩ W ≠ ∅} ⊆ V.

(a)By definition ${\underline{ξ_{1} + ξ_{2}}}^{\emptyset} (e) = {v \in V ∣ \emptyset \neq v ξ_{1} (e) \subseteq \emptyset$ or ∅ ≠ vξ₂ (e) ⊆ ∅} = ∅ .

(b) On contrary suppose that ${\bar{ξ_{1} + ξ_{2}}}^{\emptyset} (e) \neq \emptyset .$ Then there exists v ∈ V such that $v \in {\bar{ξ_{1} + ξ_{2}}}^{\emptyset} (e) .$ Thus vξ₁ (e)∩ ∅ ≠ ∅ and vξ₂ (e)∩ ∅ ≠ ∅ which is a contradiction.

Hence ${\bar{ξ_{1} + ξ_{2}}}^{\emptyset} (e) = \emptyset .$

(a) Obvious. (b) Obvious.

(a) Let $v \in {\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e) .$ This implies that ∅ ≠ vξ₁ (e) ⊆ N^c or ∅ ≠ vξ₂ (e) ⊆ N^c .

These imply that vξ₁ (e)⋂ N = ∅ or vξ₂ (e) ∩ N = ∅ . These imply that $v \notin {\bar{ξ_{1} + ξ_{2}}}^{N} (e),$ that is $v \in ({\bar{ξ_{1} + ξ_{2}}}^{N} (e))^{c} .$ Thus ${\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e) \subseteq ({\bar{ξ_{1} + ξ_{2}}}^{N} (e))^{c} .$

(b) Let $v \in {\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e) .$ Then vξ₁ (e)∩ N^c ≠ ∅ and vξ₂ (e) ∩ N^c ≠ ∅ .

These imply that ∅ ≠ vξ₁ (e) ⊈ N and ∅ ≠ vξ₂ (e) ⊈ N . These imply that $v \notin {\underline{ξ_{1} + ξ_{2}}}^{N} (e),$ that is $v \in ({\underline{ξ_{1} + ξ_{2}}}^{N} (e))^{c} .$ Thus ${\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e) \subseteq ({\underline{ξ_{1} + ξ_{2}}}^{N} (e))^{c} .$

Corollary 3.1.

${\underline{ξ_{1} + ξ_{2}}}^{W} (e) = V$ if vξ₁ (e) and vξ₂ (e) are non-empty for all e ∈ A and for all v ∈ V .

${\bar{ξ_{1} + ξ_{2}}}^{W} (e) = V$ if vξ₁ (e) and vξ₂ (e) are non-empty for all e ∈ A and for all v ∈ V .

If vξ₁ (e)≠ ∅ and vξ₂ (e)≠ ∅ for all e ∈ A and v ∈ V, then ${\underline{ξ_{1} + ξ_{2}}}^{N^{c}} (e) = ({\bar{ξ_{1} + ξ_{2}}}^{N} (e))^{c}$ and ${\bar{ξ_{1} + ξ_{2}}}^{N^{c}} (e) = ({\underline{ξ_{1} + ξ_{2}}}^{N} (e))^{c} .$

Proposition 3.2. Let V≠ ∅ , W ≠ ∅ be two universal sets, (ξ₁, A) , (ξ₂, A) be two soft relations from V to W and N ⊆ V. Then the following properties hold.

$^{N} \underline{ξ_{1} + ξ_{2}} (e) =^{N} \underline{ξ_{1}} (e) \cup^{N} \underline{ξ_{2}} (e),$

$^{N} \bar{ξ_{1} + ξ_{2}} (e) =^{N} \bar{ξ_{1}} (e) \cap^{N} \bar{ξ_{2}} (e),$

$^{V} \underline{ξ_{1} + ξ_{2}} (e) \subseteq W$ and $^{V} \bar{ξ_{1} + ξ_{2}} (e) \subseteq W$

$\emptyset =^{\emptyset} \underline{ξ_{1} + ξ_{2}} (e) =$ and $^{\emptyset} \bar{ξ_{1} + ξ_{2}} (e) = \emptyset$

$^{N} \underline{ξ_{1} + ξ_{2}} (e) =^{N} \underline{ξ_{2} + ξ_{1}} (e)$ and $^{N} \bar{ξ_{1} + ξ_{2}} (e) =^{N} \bar{ξ_{2} + ξ_{1}} (e)$

$^{N^{c}} \underline{ξ_{1} + ξ_{2}} (e) \subseteq (^{N} \bar{ξ_{1} + ξ_{2}} (e))^{c}$ and $^{N^{c}} \bar{ξ_{1} + ξ_{2}} (e) \subseteq (^{N} \underline{ξ_{1} + ξ_{2}} (e))^{c} .$

Proof. The proof is similar to that of Proposition 3.1.

Corollary 3.2.

$^{V} \underline{ξ_{1} + ξ_{2}} (e) = W$ if ξ₁ (e) w and ξ₂ (e) w are non-empty, for all e ∈ A and for w ∈ W .

$^{V} \bar{ξ_{1} + ξ_{2}} (e) = W$ if ξ₁ (e) w and ξ₂ (e) w are non-empty, for all e ∈ A and for all w ∈ W .

If ξ₁ (e) w≠ ∅ and ξ₂ (e) w≠ ∅ for all e ∈ A and for all w ∈ W, then $^{N^{c}} \underline{ξ_{1} + ξ_{2}} (e) = (^{N} \bar{ξ_{1} + ξ_{2}} (e))^{c}$ and $^{N^{c}} \bar{ξ_{1} + ξ_{2}} (e) = (^{N} \underline{ξ_{1} + ξ_{2}} (e))^{c} .$

Proposition 3.3. Let V≠ ∅ , W ≠ ∅ be two universal sets, N₁ ⊆ W, N₂ ⊆ W and (ξ₁, A) , (ξ₁, A) be two soft relations from V to W . Then the following properties hold, for all e ∈ A .

${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) = ({\underline{ξ_{1}}}^{N_{1}} (e) \cap {\underline{ξ_{1}}}^{N_{2}} (e)) \cup ({\underline{ξ_{2}}}^{N_{1}} (e) \cap {\underline{ξ_{2}}}^{N_{2}} (e)),$

${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) = ({\bar{ξ_{1}}}^{N_{1}} (e) \cup {\bar{ξ_{1}}}^{N_{2}} (e)) \cap ({\bar{ξ_{2}}}^{N_{1}} (e) \cup {\bar{ξ_{2}}}^{N_{2}} (e)),$

$N_{1} \subseteq N_{2} \Rightarrow {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \subseteq {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$

$N_{1} \subseteq N_{2} \Rightarrow {\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$

${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) \subseteq {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cap {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$

${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) \supseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cup {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$

${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) = {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cup {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$

${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cap {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e) .$

Proof.

By part (1) of Proposition 3.1, we have ${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) = {\underline{ξ_{1}}}^{N_{1} \cap N_{2}} (e) \cup {\underline{ξ_{2}}}^{N_{1} ⋂ N_{2}} (e)$ so ${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) = ({\underline{ξ_{1}}}^{N_{1}} (e) \cap {\underline{ξ_{1}}}^{N_{2}} (e)) \cup ({\underline{ξ_{2}}}^{N_{1}} (e) \cap {\underline{ξ_{2}}}^{N_{2}} (e)) .$

By part (2) of Proposition 3.1, we have ${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) = {\bar{ξ_{1}}}^{N_{1} \cup N_{2}} (e) \cap {\bar{ξ_{2}}}^{N_{1} \cup N_{2}} (e)$ so ${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) = ({\bar{ξ_{1}}}^{N_{1}} (e) \cup {\bar{ξ_{1}}}^{N_{2}} (e)) \cap ({\bar{ξ_{2}}}^{N_{1}} (e) \cup {\bar{ξ_{2}}}^{N_{2}} (e)) .$

Let $v \in {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) .$ Then ∅ ≠ vξ₁ (e) ⊆ N₁ or ∅ ≠ vξ₂ (e) ⊆ N₁.

Since N₁ ⊆ N₂ so ∅ ≠ vξ₁ (e) ⊆ N₁ ⊆ N₂ or ∅ ≠ vξ₂ (e) ⊆ N₁ ⊆ N₂ .

Thus $v \in {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e) .$ Hence ${\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \subseteq {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e) .$

Let $v \in {\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e) .$ Then vξ₁ (e)∩ N₁ ≠ ∅ and vξ₂ (e)⋂ N₁ ≠ ∅.

Since N₁ ⊆ N₂ so vξ₁ (e)∩ N₂ ≠ ∅ and vξ₂ (e) ∩ N₂ ≠ ∅ . Thus $v \in {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e) .$ This implies that ${\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e) .$

Since N₁ ⊇ N₁ ∩ N₂, N₂ ⊇ N₁ ∩ N₂, we have from (3) ${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) \subseteq {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e)$ and ${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) \subseteq {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$ these imply that ${\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cap {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e) \supseteq {\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) .$

Since N₁ ⊆ N₁ ∪ N₂, N₂ ⊆ N₁ ∪ N₂, we have from (4) ${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) \supseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e)$ and ${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) \supseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$ these imply that ${\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cup {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) .$

Since N₁ ⊆ N₁ ∪ N₂, N₂ ⊆ N₁ ∪ N₂, we have from (3) ${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) \supseteq {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e)$ and ${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) \supseteq {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$ these imply that ${\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) \supseteq {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cup {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) .$

Conversely, let $v \in {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cup {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$ that is $v \in {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e)$ or $v \in {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$ these imply that ∅ ≠ vξ₁ (e) ⊆ N₁ or ∅ ≠ vξ₂ (e) ⊆ N₁ or ∅ ≠ vξ₁ (e) ⊆ N₂ or ∅ ≠ vξ₂ (e) ⊆ N₂, these imply that ∅ ≠ vξ₁ (e) ⊆ N₁ ∪ N₂ or ∅ ≠ vξ₂ (e) ⊆ N₁ ∪ N₂, that is $v \in {\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) .$

Thus ${\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cup {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e) = {\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e) .$

Since N₁ ⊇ N₁ ∩ N₂, N₂ ⊇ N₁ ∩ N₂, we have from (4) ${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e)$ and ${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e),$ these imply that ${\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e) \cap {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e) .$

The following example shows that the inverse inclusions in parts (5) and (6) do not necessarily hold.

Example 3.3. Let V = {1, 2, 3, 4} , W = {a, b, c, d} and A = {e₁, e₂} .

Let N₁ = {a, c} ⊆ W, N₂ = {c, d} ⊆ W and (ξ₁, A) , (ξ₂, A) be two soft relations from V to W such that ξ₁ (e₁) = {(1, b) , (3, d) , (4, b) , (4, d)}, ξ₁ (e₂) = {(1, a) , (2, b) , (3, d) , (4, c)}, ξ₂ (e₁) = {(1, c) , (1, d) , (2, c) , (2, d) , (4, c)}, ξ₂ (e₂) = {(2, b) , (3, c) , (3, d) , (4, a)}. Then $\begin{matrix} \begin{matrix} 1 ξ_{1} (e_{1}) = {b}, & 1 ξ_{1} (e_{2}) = {a}, & 1 ξ_{2} (e_{1}) = {c, d}, & 1 ξ_{2} (e_{2}) = \emptyset, \\ 2 ξ_{1} (e_{1}) = \emptyset, & 2 ξ_{1} (e_{2}) = {b}, & 2 ξ_{2} (e_{1}) = {c, d}, & 2 ξ_{2} (e_{2}) = {b}, \\ 3 ξ_{1} (e_{1}) = {d}, & 3 ξ_{1} (e_{2}) = {d}, & 3 ξ_{2} (e_{1}) = \emptyset, & 3 ξ_{2} (e_{2}) = {c, d}, \\ 4 ξ_{1} (e_{1}) = {b, d}, & 4 ξ_{1} (e_{2}) = {c}, & 4 ξ_{2} (e_{1}) = {c}, & 4 ξ_{2} (e_{2}) = {a} . \end{matrix} \end{matrix}$ Now N₁ ∩ N₂ = {c} , N₁ ∪ N₂ = {a, c, d} . $\begin{matrix} {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e_{1}) = {4}, {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e_{1}) = {1, 2, 3, 4}, \\ {\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e_{1}) = \emptyset, \\ {\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e_{1}) = \emptyset, {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e_{1}) = {4}, \\ {\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e_{1}) = {1, 2, 3, 4}, \\ {\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e_{1}) \cap {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e_{1}) = {4}, \\ {\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e_{1}) \cup {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e_{1}) = {4}, \end{matrix}$ Thus ${\underline{ξ_{1} + ξ_{2}}}^{N_{1}} (e_{1}) \cap {\underline{ξ_{1} + ξ_{2}}}^{N_{2}} (e_{1}) = {4} ⊈ \emptyset = {\underline{ξ_{1} + ξ_{2}}}^{N_{1} \cap N_{2}} (e_{1})$

and

${\bar{ξ_{1} + ξ_{2}}}^{N_{1}} (e_{1}) \cup {\bar{ξ_{1} + ξ_{2}}}^{N_{2}} (e_{1}) = {4} ⊉ {1, 2, 3, 4} = {\bar{ξ_{1} + ξ_{2}}}^{N_{1} \cup N_{2}} (e_{1})$ .

Proposition 3.4. Let V≠ ∅ , W ≠ ∅ be two universal sets, N₁ ⊆ V, N₂ ⊆ V and (ξ₁, A) , (ξ₁, A) be two soft relations from V to W . Then the following properties hold, for all e ∈ A .

$^{N_{1} \cap N_{2}} \underline{ξ_{1} + ξ_{2}} (e) = (^{N_{1}} \underline{ξ_{1}} (e) \cap^{N_{2}} \underline{ξ_{1}} (e)) \cup (^{N_{1}} \underline{ξ_{2}} (e) \cap^{N_{2}} \underline{ξ_{2}} (e))$ ,

$^{N_{1} \cup N_{2}} \bar{ξ_{1} + ξ_{2}} (e) = (^{N_{1}} \bar{ξ_{1}} (e) \cup^{N_{2}} \bar{ξ_{1}} (e)) \cap (^{N_{1}} \bar{ξ_{2}} (e) \cup^{N_{2}} \bar{ξ_{2}} (e))$ ,

$N_{1} \subseteq N_{2} \Rightarrow^{N_{1}} \underline{ξ_{1} + ξ_{2}} (e) \subseteq^{N_{2}} \underline{ξ_{1} + ξ_{2}} (e)$ ,

$N_{1} \subseteq N_{2} \Rightarrow^{N_{1}} \bar{ξ_{1} + ξ_{2}} (e) \subseteq^{N_{2}} \bar{ξ_{1} + ξ_{2}} (e)$ ,

$^{N_{1} \cap N_{2}} \underline{ξ_{1} + ξ_{2}} (e) \subseteq^{N_{1}} \underline{ξ_{1} + ξ_{2}} (e) \cap^{N_{2}} \underline{ξ_{1} + ξ_{2}} (e)$ ,

$^{N_{1} \cup N_{2}} \bar{ξ_{1} + ξ_{2}} (e) \supseteq^{N_{1}} \bar{ξ_{1} + ξ_{2}} (e) \cup^{N_{2}} \bar{ξ_{1} + ξ_{2}} (e)$ ,

$^{N_{1} \cup N_{2}} \underline{ξ_{1} + ξ_{2}} (e) =^{N_{1}} \underline{ξ_{1} + ξ_{2}} (e) \cup^{N_{2}} \underline{ξ_{1} + ξ_{2}} (e)$ ,

$^{N_{1} \cap N_{2}} \bar{ξ_{1} + ξ_{2}} (e) \subseteq^{N_{1}} \bar{ξ_{1} + ξ_{2}} (e) \cap^{N_{2}} \bar{ξ_{1} + ξ_{2}} (e) .$

Proof. The proof is similar to that of Proposition 3.3.

If the SB-relations (ξ₁, A) , (ξ₂, A) are over V instead from V to W then all properties proved above are also true. Now we consider some special cases.

If (ξ, A) is a soft reflexive relation over V then each vξ (e) and ξ (e) v are non-empty and v belongs to vξ (e) and ξ (e) v .

If (ξ, A) is a soft symmetric relation over V and v′ ∈ vξ (e) then v ∈ v′ξ (e) and v′ ∈ ξ (e) v implies v ∈ ξ (e) v′ .

In general, if (ξ₁, A) , (ξ₂, A) are soft relations over V then ${\underline{ξ_{1} + ξ_{2}}}^{N} ⊈ {\bar{ξ_{1} + ξ_{2}}}^{N} .$ But if (ξ₁, A) , (ξ₂, A) are soft reflexive relations over V then ${\underline{ξ_{1} + ξ_{2}}}^{N} \subseteq N \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N} .$

Proposition 3.5. Let V be a non empty universal set and (ξ₁, A) , (ξ₂, A) be two soft reflexive relations over V and N ⊆ V . Then ${\underline{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq N \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N} (e)$ for all e ∈ A .

Proof.

Let $v \in {\underline{ξ_{1} + ξ_{2}}}^{N} (e) .$ Then vξ₁ (e) ⊆ N or vξ₂ (e) ⊆ N . Since (ξ₁, A) , (ξ₂, A) are soft reflexive relations so v ∈ vξ₁ (e) , v ∈ vξ₂ (e) , this implies that v ∈ N that is ${\underline{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq N .$ (1)

Now let v ∈ N . Since (ξ₁, A) , (ξ₂, A) are soft reflexive relations, so v ∈ vξ₁ (e) and v ∈ vξ₂ (e) ⇒vξ₁ (e)∩ N ≠ ∅ and vξ₂ (e)∩ N ≠ ∅ this implies that $v \in {\bar{ξ_{1} + ξ_{2}}}^{N} (e)$ that is $N \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N} (e) .$ (2) Combining Equations (1) and (2) we get ${\underline{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq N \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N} (e) .$

Proposition 3.6. Let V be a non empty universal set and (ξ₁, A) , (ξ₂, A) be two soft reflexive relations over V and N ⊆ V . Then $^{N} \underline{ξ_{1} + ξ_{2}} (e) \subseteq N \subseteq^{N} \bar{ξ_{1} + ξ_{2}} (e)$ for all e ∈ A .

Proof.

The proof is similar to that of Proposition 3.5.

Proposition 3.7. If (ξ₁, A) , (ξ₂, A) are two soft symmetric relations over a non-empty universe V and N ⊆ V, then the following hold. ${\bar{ξ_{1} + ξ_{2}}}^{{\underline{ξ_{1} + ξ_{2}}}^{N} (e)} (e) \subseteq N$

Proof.

Let $v \in {\bar{ξ_{1} + ξ_{2}}}^{{\underline{ξ_{1} + ξ_{2}}}^{N} (e)} (e) .$ Then $v ξ_{1} (e) \cap {\underline{ξ_{1} + ξ_{2}}}^{N} (e) \neq \emptyset$ and $v ξ_{2} (e) \cap {\underline{ξ_{1} + ξ_{2}}}^{N} (e) \neq \emptyset,$ so there exist $v_{1} \in v ξ_{1} (e) \cap {\underline{ξ_{1} + ξ_{2}}}^{N} (e)$ and $v_{2} \in v ξ_{2} (e) \cap {\underline{ξ_{1} + ξ_{2}}}^{N} (e) .$ These imply that v₁ ∈ vξ₁ (e) and $v_{1} \in {\underline{ξ_{1} + ξ_{2}}}^{N} (e)$ and v₂ ∈ vξ₂ (e) and $v_{2} \in {\underline{ξ_{1} + ξ_{2}}}^{N} (e),$ that is v₁ξ₁ (e) ⊆ N or v₁ξ₂ (e) ⊆ N, v₂ξ₁ (e) ⊆ N or v₂ξ₂ (e) ⊆ N and (v, v₁) ∈ ξ₁ (e) , (v, v₂) ∈ ξ₂ (e) since (ξ₁, A) , (ξ₂, A) are symmetric therefore (v₁, v) ∈ ξ₁ (e) , (v₂, v) ∈ ξ₂ (e) , that is v ∈ v₁ξ₁ (e) ⊆ N or v ∈ v₂ξ₂ (e) ⊆ N . These imply that v ∈ N .

Thus ${\bar{ξ_{1} + ξ_{2}}}^{{\underline{ξ_{1} + ξ_{2}}}^{N} (e)} (e) \subseteq N$

Proposition 3.8. If (ξ₁, A) , (ξ₂, A) are two soft symmetric relation over a non-empty universe V and N ⊆ V, then the following hold. ${^{^{N}}}^{\underline{ξ_{1} + ξ_{2}}} \bar{ξ_{1} + ξ_{2}} (e) \subseteq N$

Proof. The proof is similar to that of Proposition 3.7.

Proposition 3.9. If the SB-relation (ξ₁, A) and (ξ₂, A) are soft transitive relations over V, then ${\bar{ξ_{1} + ξ_{2}}}^{{\bar{ξ_{1} + ξ_{2}}}^{N} (e)} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N} (e)$ .

Proof.

Let $v \in {\bar{ξ_{1} + ξ_{2}}}^{{\bar{ξ_{1} + ξ_{2}}}^{N} (e)} (e),$ that is $v ξ_{1} (e) \cap {\bar{ξ_{1} + ξ_{2}}}^{N} (e) \neq \emptyset$ and $v ξ_{2} (e) \cap {\bar{ξ_{1} + ξ_{2}}}^{N} (e) \neq \emptyset .$ Then there exist $v_{1} \in v ξ_{1} (e) \cap {\bar{ξ_{1} + ξ_{2}}}^{N} (e)$ and $v_{1}^{'} \in v ξ_{2} \cap {\bar{ξ_{1} + ξ_{2}}}^{N} (e)$ that is $v_{1} \in v ξ_{1} (e), v_{1}^{'} \in v ξ_{2} (e)$ and $v_{1} \in {\bar{ξ_{1} + ξ_{2}}}^{N} (e), v_{1}^{'} \in {\bar{ξ_{1} + ξ_{2}}}^{N} (e) .$ Now $v_{1} \in {\bar{ξ_{1} + ξ_{2}}}^{N} (e)$ and $v_{1}^{'} \in {\bar{ξ_{1} + ξ_{2}}}^{N} (e) .$ These imply that v₁ξ₁ (e)∩ N ≠ ∅ , v₁ξ₂ (e) ∩ N ≠ ∅ and $v_{1}^{'} ξ_{1} (e) \cap N \neq \emptyset, v_{1}^{'} ξ_{2} (e) \cap N \neq \emptyset .$ Then there exist v₂ ∈ v₁ξ₁ (e) ∩ N, v₂ ∈ v₁ξ₂ (e) ∩ N and $v_{2}^{'} \in v_{1}^{'} ξ_{1} (e) \cap N, v_{2}^{'} \in v_{1}^{'} ξ_{2} (e) \cap N,$ that is v₂ ∈ ξ₁ (e) , v₂ ∈ ξ₂ (e) , v₂ ∈ N and $v_{2} \in ξ_{1} (e), v_{2}^{'} \in ξ_{2} (e), v_{2}^{'} \in N .$ But v₁ ∈ vξ₁ (e) , v₁ ∈ vξ₂ (e) and $v_{1}^{'} \in v ξ_{1} (e), v_{1}^{'} \in v ξ_{2} (e),$ that is $(v, v_{1}) \in ξ_{1} (e), (v, v_{1}) \in ξ_{2} (e), (v, v_{1}^{1}) \in ξ_{1} (e), (v, v_{1}^{'}) \in ξ_{2} (e)$ and $v_{2} \in v_{1} ξ_{1} (e), v_{2} \in v_{1} ξ_{2} (e), v_{2}^{'} \in v_{1}^{'} ξ_{1} (e), v_{2}^{'} \in v_{1}^{'} ξ_{2} (e),$ that is $(v_{1}, v_{2}) \in ξ_{1} (e), (v_{1}, v_{2}) \in ξ_{2} (e), (v_{1}^{'}, v_{2}^{'}) \in ξ_{1} (e), (v_{1}^{'}, v_{2}^{'}) \in ξ_{2} (e) .$ Since ξ₁ and ξ₂ are soft transitive therefore $(v, v_{2}) \in ξ_{1} (e), (v, v_{2}^{'}) \in ξ_{2} (e)$ and $(v, v_{2}^{'}) \in ξ_{1} (e), (v, v_{2}) \in ξ_{2} (e),$ these imply that $v_{2} \in v ξ_{1} (e), v_{2}^{'} \in v ξ_{2} (e),$ that is vξ₁ (e)∩ N ≠ ∅ and vξ₂ (e)∩ N ≠ ∅ that is $v \in {\bar{ξ_{1} + ξ_{2}}}^{N} (e) .$ Thus ${\bar{ξ_{1} + ξ_{2}}}^{{\bar{ξ_{1} + ξ_{2}}}^{N} (e)} (e) \subseteq {\bar{ξ_{1} + ξ_{2}}}^{N} (e) .$

Proposition 3.10. If the SB-relation (ξ₁, A) and (ξ₂, A) are two soft transitive relations over V, then ${^{^{N}}}^{\underline{ξ_{1} + ξ_{2}} (e)} \underline{ξ_{1} + ξ_{2}} (e) \subseteq^{N} \underline{ξ_{1} + ξ_{2}} (e) .$ .

Proof. The proof is similar to that of Proposition 3.9.

Proposition 3.11. If the SB-relation (ξ₁, A) and (ξ₂, A) are soft reflexive and soft transitive relations over V, then ${\underline{ξ_{1} + ξ_{2}}}^{{\underline{ξ_{1} + ξ_{2}}}^{N} (e)} (e) = {\underline{ξ_{1} + ξ_{2}}}^{N} (e)$ .

Proof. Since the SB-relation is soft transitive from Proposition 3.9 we have ${\underline{ξ_{1} + ξ_{2}}}^{{\underline{ξ_{1} + ξ_{2}}}^{N} (e)} (e) \subseteq {\underline{ξ_{1} + ξ_{2}}}^{N} (e)$ and sine SB-relations are soft reflexive, we have from Proposition 3.5, ${\underline{ξ_{1} + ξ_{2}}}^{N} (e) \subseteq {\underline{ξ_{1} + ξ_{2}}}^{{\underline{ξ_{1} + ξ_{2}}}^{N} (e)} (e) .$ Thus ${\underline{ξ_{1} + ξ_{2}}}^{{\underline{ξ_{1} + ξ_{2}}}^{N} (e)} (e) = {\underline{ξ_{1} + ξ_{2}}}^{N} (e) .$

Proposition 3.12. If the SB-relation (ξ₁, A) and (ξ₂, A) are two soft reflexive and soft transitive relations over V, then $^{^{N} \underline{ξ_{1} + ξ_{2}} (e)} \underline{ξ_{1} + ξ_{2}} (e) =^{N} \underline{ξ_{1} + ξ_{2}} (e)$ .

Proof. The proof is similar to that of Proposition 3.11.

4 Roughness of a set by multi soft relations

Generalizing the concept of roughness by two soft relations we define the roughness by "n" soft relations.

Definition 4.1. Let V, W be two non-empty universal sets, (ξ₁, A) , (ξ₂, A) , (ξ₃, A) …… (ξ_n, A) be soft relations from V to W and N ⊆ W. Then we define two soft sets over V by $\begin{matrix} {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e) = {v \in V ∣ \emptyset \neq v ξ_{l} (e) \subseteq N, \\ for some l = 1, 2, \dots n} \\ {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e) = {v \in V ∣ v ξ_{l} (e) ⋂ N \neq \emptyset, \\ for all l = 1, 2, \dots, n} \end{matrix}$ called the lower and upper approximations of N with respect to aftersets, we denote these soft sets by $({\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e), A), ({\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e), A)$ respectively.

If N ⊆ V, then we define two soft sets over W by $^{N} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) = {w \in W ∣ \emptyset \neq ξ_{l} (e) w \subseteq N,$ $for some l = 1, 2, \dots n}$ $^{N} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) = {w \in W ∣ ξ_{l} (e) w \cap N \neq \emptyset,$ $for all l = 1, 2, \dots, n}$ called the lower and upper approximations of N with respect to foresets, we denote these soft sets by $({\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e), A), ({\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e), A)$ respectively.

Proposition 4.1. Let V≠ ∅ , W ≠ ∅ be two sets, (ξ₁, A) , (ξ₂, A) , (ξ₃, A) …… (ξ_n, A) be soft relations from V to W and N ⊆ W. Then the following properties hold.

${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e) = ⋃_{l = 1}^{n} {\underline{ξ_{l}}}^{N} (e),$

${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e) = ⋂_{l = 1}^{n} {\bar{ξ_{l}}}^{N} (e),$

${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N^{c}} (e) \subseteq ({\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e))^{c},$

${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N^{c}} (e) \subseteq ({\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e))^{c} .$

Proof. The proof of 1, 2, is similar to that of Proposition 3.1 and the proof of 3, 4, is similar to that of Lemma 3.1.

Remark 4.1. If vξ_l (e)≠ ∅ for all e ∈ A, v ∈ V and for all l, then ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N^{c}} (e) = ({\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e))^{c}$ and ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N^{c}} (e) = ({\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e))^{c} .$

Proof. The proof is similar to that of Remark 3.2.

Proposition 4.2. Let V≠ ∅ , W ≠ ∅ be two sets, (ξ₁, A) , (ξ₂, A) , (ξ₃, A) …… (ξ_n, A) be soft relations from V to W and N ⊆ V. Then the following properties hold.

$^{N} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) = ⋃_{l = 1}^{n}^{N} \underline{ξ_{l}} (e),$

$^{N} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) = ⋂_{l = 1}^{n}^{N} \bar{ξ_{l}} (e),$

$^{N^{c}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) \subseteq (^{N} \bar{\sum_{l = 1}^{n} ξ_{l}} (e))^{c},$

$^{N^{c}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) \subseteq (^{N} \underline{\sum_{l = 1}^{n} ξ_{l}} (e))^{c} .$

Proof. The proof is similar to that of Proposition 4.1.

Remark 4.2. If ξ_l (e) w≠ ∅ for all e ∈ A, w ∈ W and for all l, then $^{N^{c}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) = (^{N} \bar{\sum_{l = 1}^{n} ξ_{l}} (e))^{c}$ and $^{N^{c}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) = {(^{N} \underline{\sum_{l = 1}^{n} ξ_{l}} (e))}^{c} .$

Proposition 4.3. Let V≠ ∅ , W ≠ ∅ be two sets, N₁ ⊆ W, N₂ ⊆ W, …, N_m ⊆ W and (ξ₁, A) , (ξ₂, A) , …, (ξ_n, A) be soft relations from V to W. Then the following properties hold.

${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) = ⋃_{l = 1}^{n} (⋂_{j = 1}^{m} {\underline{ξ_{l}}}^{N_{j}} (e)),$

${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) = ⋂_{l = 1}^{n} (⋃_{j = 1}^{m} {\bar{ξ_{l}}}^{N_{j}} (e)),$

${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) \subseteq ⋂_{j = 1}^{m} ({\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)),$

${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) \supseteq ⋃_{j = 1}^{m} ({\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)),$

${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) = ⋃_{j = 1}^{m} ({\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)),$

${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) = ⋂_{j = 1}^{m} ({\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)) .$

Proof.

${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) = ⋃_{l = 1}^{n} {\underline{ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) = ⋃_{l = 1}^{n} ({\underline{ξ_{l}}}^{N_{1}} (e) ⋂ {\underline{ξ_{l}}}^{N_{2}} (e) ⋂ \dots ⋂ {\underline{ξ_{l}}}^{N_{m}} (e)),$ that is ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) = ⋃_{l = 1}^{n} (⋂_{j = 1}^{m} {\underline{ξ_{l}}}^{N_{j}} (e)) .$

${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) = ⋂_{l = 1}^{n} {\bar{ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) = ⋂_{l = 1}^{n} ({\bar{ξ_{l}}}^{N_{1}} (e) ⋃ {\bar{ξ_{l}}}^{N_{2}} (e) ⋃ \dots ⋃ {\bar{ξ_{l}}}^{N_{m}} (e)),$ that is ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) = ⋂_{l = 1}^{n} (⋃_{j = 1}^{m} {\bar{ξ_{l}}}^{N_{j}} (e)) .$

We know that $⋂_{j = 1}^{m} N_{j} \subseteq N_{j}$ for all 1 ≤ j ≤ m. Then by (3) of Proposition 3.3, we have ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ for all 1 ≤ j ≤ m, this implies that ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) \subseteq ⋂_{j = 1}^{m} {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e) .$

We know that $⋃_{j = 1}^{m} N_{j} \supseteq N_{j}$ for all 1 ≤ j ≤ m. Then by (4) of Proposition 3.3, we have ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) \supseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ for all 1 ≤ j ≤ m, this implies that ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) \supseteq ⋃_{j = 1}^{m} {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e) .$

We know that $⋃_{j = 1}^{m} N_{j} \supseteq N_{j}$ for all 1 ≤ j ≤ m. Then by (3) of Proposition 3.3, we have ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) \supseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e) \forall 1 \leq j \leq m,$ this implies that ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{n} N_{j}} (e) \supseteq ⋃_{j = 1}^{m} {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ .

Conversely, let $v \in {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{m} N_{j}} (e) .$ Then ∅ ≠ vξ_l (e) ⊆ N₁ or ∅ ≠ vξ_l (e) ⊆ N₂ or … or ∅ ≠ vξ_l (e) ⊆ N_m, this implies that $v \in {\underline{\sum_{l = 1}^{n} ξ_{1}}}^{N_{1}} (e)$ or $v \in {\underline{\sum_{l = 1}^{n} ξ_{1}}}^{N_{2}} (e)$ or … $v \in {\underline{\sum_{l = 1}^{n} ξ_{1}}}^{N_{m}} (e),$ that is $v \in ⋃_{j = 1}^{m} {\underline{\sum_{l = 1}^{n} ξ_{1}}}^{N_{j}} (e) .$ So ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{n} N_{j}} (e) \subseteq ⋃_{j = 1}^{m} {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ .

Thus ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{⋃_{j = 1}^{n} N_{j}} (e) = ⋃_{j = 1}^{m} {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ .

We know that $⋂_{j = 1}^{m} N_{j} \subseteq N_{j}$ for all 1 ≤ j ≤ m. Then by (4) of Proposition 3.3, we have ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) \subseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ for all 1 ≤ j ≤ m, this implies that ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) \subseteq ⋂_{j = 1}^{m} {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ .

Conversely, let $v \in ⋂_{j = 1}^{m} ({\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)) \Rightarrow v \in {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ for all 1 ≤ j ≤ m ⇒ vξ_i ⋂ N_j ≠ ∅ , for all l = 1, 2, …, n and $1 \leq j \leq m \Rightarrow v ξ_{l} (e) ⋂ (⋂_{j = 1}^{m} N_{j}) \neq \emptyset$ for all $l = 1, 2, \dots, n \Rightarrow v \in {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) \Rightarrow {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) \supseteq ⋂_{j = 1}^{m} {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ .

So ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{⋂_{j = 1}^{m} N_{j}} (e) = ⋂_{j = 1}^{m} {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} (e)$ .

Proposition 4.4. Let V≠ ∅ , W ≠ ∅ be two sets N₁ ⊆ V, N₂ ⊆ V, …, N_m ⊆ V and (ξ₁, A) , (ξ₂, A) , …, (ξ_n, A) be soft relations from V to W. Then the following properties hold.

$^{⋂_{j = 1}^{m} N_{j}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) = ⋃_{l = 1}^{n} (⋂_{j = 1}^{m}^{N_{j}} \underline{ξ_{l}} (e)),$

$^{⋃_{j = 1}^{m} N_{j}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) = ⋂_{l = 1}^{n} (⋃_{j = 1}^{m}^{N_{j}} \bar{ξ_{l}} (e)),$

$^{⋂_{j = 1}^{m} N_{j}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) \subseteq ⋂_{j = 1}^{m} (^{N_{j}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e)),$

$^{⋃_{j = 1}^{m} N_{j}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) \supseteq ⋃_{j = 1}^{m} (^{N_{j}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e)),$

$^{⋃_{j = 1}^{m} N_{j}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) = ⋃_{j = 1}^{m} (^{N_{j}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e)),$

$^{⋂_{j = 1}^{m} N_{j}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) = ⋂_{j = 1}^{m} (^{N_{j}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e)) .$

Proof. The Proof is similar to that of Proposition 4.3.

Proposition 4.5. Let V≠ ∅ , W ≠ ∅ be two sets, N₁, N₂, N₃, …, N_n ⊆ W with N₁ ⊆ N₂ ⊆ N₃ ⊆ … ⊆ N_m and (ξ₁, A) , (ξ₂, A) , (ξ₃, A) , …, (ξ_n, A) be soft relations from V to W. Then the following properties.

${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{1}} (e) \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{2}} (e) \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{3}} (e) \subseteq \dots \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{m}} (e),$

${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{1}} (e) \subseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{2}} (e) \subseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{3}} (e) \subseteq \dots \subseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{m}} (e) .$

Proof. Suppose 1 ≤ k ≤ j ≤ n then N_k ⊆ N_j hold

Clearly N_k ⋂ N_j = N_k that is ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k}} = {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k} ⋂ N_{j}} .$ Then from part (5) of Proposition 3.3 we have ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k}} = {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k} ⋂ N_{j}} \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k}} ⋂ {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}},$ this implies that ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k}} \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k}} ⋂ {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}},$ that is ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k}} \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} .$ This implies. ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{1}} \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{2}} \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{3}} \subseteq \dots \subseteq {\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N_{m}} .$

Clearly N_k ⋃ N_j = N_j that is ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} = {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k} ⋃ N_{j}} .$ Then from part (6) of Proposition 3.3 we have ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} = {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k} ⋃ N_{j}} \supseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k}} ⋃ {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}},$ this implies that $\Rightarrow {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} \supseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k}} ⋃ {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}},$ that is ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{j}} \supseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{k}} .$ Thus ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{1}} \subseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{2}} \subseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{3}} \subseteq \dots \subseteq {\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N_{m}} .$

Proposition 4.6. Let V≠ ∅ , W ≠ ∅ be two sets, N₁, N₂, N₃, …, N_n ⊆ V with N₁ ⊆ N₂ ⊆ N₃ ⊆ … ⊆ N_m and (ξ₁, A) , (ξ₂, A) , (ξ₃, A) , …, (ξ_n, A) be soft relations from V to W. Then the following properties hold.

$^{N_{1}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) \subseteq^{N_{2}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) \subseteq^{N_{3}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) \subseteq \dots \subseteq^{N_{m}} \underline{\sum_{l = 1}^{n} ξ_{l}} (e),$

$^{N_{1}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) \subseteq^{N_{2}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) \subseteq^{N_{3}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) \subseteq \dots \subseteq^{N_{m}} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) .$

Proof. The proof is similar to that of Proposition 4.5.

Proposition 4.7. Let V≠ ∅ , W ≠ ∅ be two universal sets, (ξ₁, A) , (ξ₂, A) , (ξ₃, A) , …, (ξ_n, A) be soft relations from V to W, with (ξ₁, A) ⊆ (ξ₂, A) ⊆ (ξ₃, A) ⊆ … ⊆ (ξ_n, A) and N ⊆ W . Then

${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e) = {\underline{ξ_{1}}}^{N} (e),$

${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e) = {\bar{ξ_{1}}}^{N} (e) .$

Proof. Suppose 1 ≤ j ≤ k ≤ n and (ξ_j, A) ⊆ (ξ_k, A) , for any v ∈ V, vξ_j (e) ⊆ vξ_k (e)

for all e ∈ A therefore, we have that ${\underline{ξ_{k}}}^{N} (e) \subseteq {\underline{ξ_{j}}}^{N} (e)$ (3) ${\bar{ξ_{j}}}^{N} (e) \subseteq {\bar{ξ_{k}}}^{N} (e)$ (4) Thus

${\underline{ξ_{j} + ξ_{k}}}^{N} (e) = {\underline{ξ_{j}}}^{N} (e) ⋃ {\underline{ξ_{k}}}^{N} (e),$ that is ${\underline{ξ_{j} + ξ_{k}}}^{N} (e) = {\underline{ξ_{j}}}^{N} (e)$ by Eq. 3

Since (ξ₁, A) ⊆ (ξ₂, A) ⊆ (ξ₃, A) ⊆ … ⊆ (ξ_n, A) therefore, we have ${\underline{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e) = {\underline{ξ_{1}}}^{N} (e) .$

${\bar{ξ_{j} + ξ_{k}}}^{N} (e) = {\bar{ξ_{j}}}^{N} (e) ⋂ {\bar{ξ_{k}}}^{N} (e),$ that is ${\bar{ξ_{j} + ξ_{k}}}^{N} (e) = {\bar{ξ_{j}}}^{N} (e)$ by Eq. 4

Since (ξ₁, A) ⊆ (ξ₂, A) therefore, we have ${\bar{\sum_{l = 1}^{n} ξ_{l}}}^{N} (e) = {\bar{ξ_{1}}}^{N} (e) .$

Proposition 4.8. Let V≠ ∅ , W ≠ ∅ be two universal sets, (ξ₁, A) , (ξ₂, A) , (ξ₃, A) , …, (ξ_n, A) be soft relations from V to W, with (ξ₁, A) ⊆ (ξ₂, A) ⊆ (ξ₃, A) ⊆ … ⊆ (ξ_n, A) and N ⊆ V . Then

$^{N} \underline{\sum_{l = 1}^{n} ξ_{l}} (e) =^{N} \underline{ξ_{1}} (e),$

$^{N} \bar{\sum_{l = 1}^{n} ξ_{l}} (e) =^{N} \bar{ξ_{1}} (e) .$

Proof. The proof is similar to that of Proposition 4.7.

5 Examples

Decision making is a major area of study in almost all types of data analysis. Many researchers and experts developed many methods to find a wise decision. Soft set theory introduced by Molodrsov [30] and Rough set theory developed by Pawlak [31] are the theories which are mostly used in the decision making problems. Pawlak [31] rough set theory is based on a single relation. Qian et al. [32], generalized the concept and used finite number of equivalence relations and defined multigranulation rough sets. On the other-hand Shabir et al. [36] used soft binary relation to approximate a set. In this paper we used finite number of soft sets to approximate a set. We apply the theory developed above in data classification in the following examples:

Example 5.1. Suppose there are forty students in class 10, which we have divided in two sections represented by V = {s_i : 1 ≤ i ≤ 20} and $W = {s_{i}^{'} : 1 \leq i \leq 20} .$ We compare them by their results of two tests which are taken on the basis of Mathematics and physics, set of parameters A = {p = Physic, m = Mathematics}. The comparison of students are given in Table 1 from which we get two soft relations ξ₁ : A ↦ P (V × W) and ξ₂ : A ↦ P (V × W) , whose aftersets are given in Table 2. If the students $N = {s_{5}^{'}, s_{8}^{'}, s_{11}^{'}, s_{14}^{'}, s_{15}^{'}, s_{19}^{'}, s_{20}^{'}} \subset W,$ of section W get more than 70% marks in their test, then who will get more than 70% marks in section V ?

Table 1
Aftersets of ξ₁ and ξ₂

s₁ξ₁ (p) = ${s_{3}^{'}, s_{10}^{'}, s_{15}^{'},}$ s₁ξ₁ (m) = ∅, s₁ξ₂ (p) = ∅, s₁ξ₂ (m) = ∅,

s₂ξ₁ (p) = ∅, s₂ξ₁ (m) = ${s_{10}^{'}, s_{15}^{'}},$ s₂ξ₂ (p) = ${s_{18}^{'}},$ s₂ξ₂ (m) = ∅,

s₃ξ₁ (p) = ${s_{5}^{'}, s_{7}^{'}}$ s₃ξ₁ (m) = ${s_{11}^{'}},$ s₃ξ₂ (p) = ${s_{15}^{'}, s_{17}^{'}},$ s₃ξ₂ (m) = ∅,

s₄ξ₁ (p) = ${s_{2}^{'}},$ s₄ξ₁ (m) = ${s_{13}^{'}, s_{16}^{'}},$ s₄ξ₂ (p) = ${s_{6}^{'}},$ s₄ξ₂ (m) = ${s_{12}^{'}, s_{16}^{'}, s_{19}^{'}}$

s₅ξ₁ (p) = ${s_{1}^{'}, s_{20}^{'}},$ s₅ξ₁ (m) = ∅, s₅ξ₂ (p) = ∅, s₅ξ₂ (m) = ∅,

s₆ξ₁ (p) = ${s_{17}^{'}, s_{18}^{'}},$ s₆ξ₁ (m) = ${s_{17}^{'}},$ s₆ξ₂ (p) = ∅, s₆ξ₂ (m) = ∅,

s₇ξ₁ (p) = ∅, s₇ξ₁ (m) = ${s_{14}^{'}},$ s₇ξ₂ (p) = ${s_{16}^{'}},$ s₇ξ₂ (m) = ${s_{7}^{'}, s_{11}^{'}, s_{17}^{'}},$

s₈ξ₁ (p) = ${s_{6}^{'}},$ s₈ξ₁ (m) = ∅, s₈ξ₂ (p) = ∅, s₈ξ₂ (m) = ∅,

s₉ξ₁ (p) = ${s_{11}^{'}, s_{13}^{'}},$ s₉ξ₁ (m) = ${s_{19}^{'}},$ s₉ξ₂ (p) = ${s_{19}^{'}},$ s₉ξ₂ (m) = ∅,

s₁₀ξ₁ (p) = ∅, s₁₀ξ₁ (m) = ${s_{5}^{'}},$ s₁₀ξ₂ (p) = ∅, s₁₀ξ₂ (m) = ${s_{20}^{'}},$

s₁₁ξ₁ (p) = ${s_{8}^{'}},$ s₁₁ξ₁ (m) = ∅, s₁₁ξ₂ (p) = ∅, s₁₁ξ₂ (m) = ∅,

s₁₂ξ₁ (p) = ∅, s₁₂ξ₁ (m) = ∅, s₁₂ξ₂ (p) = ${s_{1}^{'}, s_{5}^{'}, s_{9}^{'}},$ s₁₂ξ₂ (m) = ∅,

s₁₃ξ₁ (p) = ∅, s₁₃ξ₁ (m) = ∅, s₁₃ξ₂ (p) = ${s_{14}^{'}},$ s₁₃ξ₂ (m) = ∅,

s₁₄ξ₁ (p) = ∅, s₁₄ξ₁ (m) = ∅, s₁₄ξ₂ (p) = ∅, s₁₄ξ₂ (m) = ${s_{18}^{'}},$

s₁₅ξ₁ (p) = ${s_{9}^{'}},$ s₁₅ξ₁ (m) = ${s_{2}^{'}},$ s₁₅ξ₂ (p) = ∅, s₁₅ξ₂ (m) = ∅,

s₁₆ξ₁ (p) = ∅, s₁₆ξ₁ (m) = ∅, s₁₆ξ₂ (p) = ${s_{10}^{'}},$ s₁₆ξ₂ (m) = ${s_{2}^{'}},$

s₁₇ξ₁ (p) = ∅ , s₁₇ξ₁ (m) = ${s_{7}^{'}},$ s₁₇ξ₂ (p) = ∅, s₁₇ξ₂ (m) = ∅,

s₁₈ξ₁ (p) = ∅, s₁₈ξ₁ (m) = ${s_{3}^{'}},$ s₁₈ξ₂ (p) = ∅, s₁₈ξ₂ (m) = ${s_{1}^{'}, s_{5}^{'}},$

s₁₉ξ₁ (p) = ∅, s₁₉ξ₁ (m) = ${s_{8}^{'}},$ s₁₉ξ₂ (p) = ${s_{2}^{'}},$ s₁₉ξ₂ (m) = ${s_{3}^{'}},$

s₂₀ξ₁ (p) = ${s_{14}^{'}},$ s₂₀ξ₁ (m) = ${s_{4}^{'}, s_{9}^{'}},$ s₂₀ξ₂ (p) = ${s_{3}^{'}, s_{11}^{'}},$ s₂₀ξ₂ (m) = ${s_{6}^{'}}$

Table 2

Comparison of students

Test 1, ξ₁ : A ↦ P (V × W)		Test 2, ξ₂ : A ↦ P (V × W)
Physic	Mathematics	Physic	Mathematics
$(s_{1}, s_{3}^{'})$	$(s_{2}, s_{10}^{'})$	$(s_{2}, s_{18}^{'})$	$(s_{4}, s_{12}^{'})$
$(s_{1}, s_{10}^{'})$	$(s_{2}, s_{15}^{'})$	$(s_{3}, s_{15}^{'})$	$(s_{4}, s_{16}^{'})$
$(s_{1}, s_{15}^{'})$	$(s_{3}, s_{11}^{'})$	$(s_{3}, s_{17}^{'})$	$(s_{4}, s_{19}^{'})$
$(s_{3}, s_{5}^{'})$	$(s_{4}, s_{16}^{'})$	$(s_{5}, s_{6}^{'})$	$(s_{7}, s_{7}^{'})$
$(s_{3}, s_{7}^{'})$	$(s_{4}, s_{13}^{'})$	$(s_{7}, s_{16}^{'})$	$(s_{7}, s_{11}^{'})$
$(s_{4}, s_{2}^{'})$	$(s_{6}, s_{17}^{'})$	$(s_{9}, s_{13}^{'})$	$(s_{7}, s_{17}^{'})$
$(s_{5}, s_{1}^{'})$	$(s_{7}, s_{14}^{'})$	$(s_{9}, s_{19}^{'})$	$(s_{10}, s_{20}^{'})$
$(s_{5}, s_{20}^{'})$	$(s_{9}, s_{19}^{'})$	$(s_{12}, s_{1}^{'})$	$(s_{14}, s_{18}^{'})$
$(s_{6}, s_{18}^{'})$	$(s_{10}, s_{5}^{'})$	$(s_{12}, s_{5}^{'})$	$(s_{16}, s_{2}^{'})$
$(s_{6}, s_{17}^{'})$	$(s_{15}, s_{20}^{'})$	$(s_{12}, s_{9}^{'})$	$(s_{18}, s_{1}^{'})$
$(s_{8}, s_{6}^{'})$	$(s_{15}, s_{2}^{'})$	$(s_{13}, s_{14}^{'})$	$(s_{18}, s_{5}^{'})$
$(s_{9}, s_{11}^{'})$	$(s_{17}, s_{7}^{'})$	$(s_{16}, s_{10}^{'})$	$(s_{19}, s_{3}^{'})$
$(s_{9}, s_{13}^{'})$	$(s_{18}, s_{3}^{'})$	$(s_{19}, s_{12}^{'})$	$(s_{20}, s_{6}^{'})$
$(s_{11}, s_{8}^{'})$	$(s_{19}, s_{8}^{'})$	$(s_{20}, s_{2}^{'})$
$(s_{15}, s_{9}^{'})$	$(s_{20}, s_{4}^{'})$	$(s_{20}, s_{3}^{'})$
$(s_{20}, s_{14}^{'})$	$(s_{20}, s_{9}^{'})$	$(s_{20}, s_{11}^{'})$

Now we categorise the students of section V who got more than 70% marks in their tests.

${\underline{ξ_{1} + ξ_{2}}}^{N} (p) = {s_{9}, s_{11}, s_{20}}$ , ${\underline{ξ_{1} + ξ_{2}}}^{N} (m) = {s_{3}, s_{7}, s_{9}, s_{10}, s_{19}}$ ${\bar{ξ_{1} + ξ_{2}}}^{N} (p) = {s_{9}, s_{20}}$ , ${\bar{ξ_{1} + ξ_{2}}}^{N} (m) = {s_{7}, s_{10}} .$ In this case lower approximation contains all those students who get more than 70% in one test and the upper approximation contains all those students who get more than 70% in both tests.

Example 5.2. Suppose eight students apply for Ph.D. admission in the department of Mathematics. If department take their test and interview on the basis of pure and applied mathematics to choose the students for Ph.D. admission. Suppose the students are represented by V = {v₁, v₂, v₃, v₄, v₅, v₆, v₇, v₈} and the marks represented by W = {50, 55, 60, 65, 70, 75, 80, 85} and the set of parameters given by A = {a = applied math, p=pure math} . If passing marks are 70 then the applicant who have marks in N = {70, 75, 80, 85} ⊂ W secure their admission. The result of test and interview are given in Table 3. From which we get two soft relations ξ₁ : A ⟼ P (V × W) and ξ₂ : A ⟼ P (V × W) , based on test and interview, respectively, whose afterset are given by v₁ξ₁ (a) = {60} , v₂ξ₁ (a) = {70} , v₃ξ₁ (a) = {80} , v₄ξ₁ (a) = {65} , v₅ξ₁ (a) = {55} , v₆ξ₁ (a) = {55} , v₇ξ₁ (a) = {50} , v₈ξ₁ (a) = {70} , v₁ξ₁ (p) = {55} , v₂ξ₁ (p) = {55} , v₃ξ₁ (p) = {85} , v₄ξ₁ (p) = {50} , v₅ξ₁ (p) = {60} , v₆ξ₁ (p) = {65} , v₇ξ₁ (p) = {70} , v₈ξ₁ (p) = {50}

Table 3

Test and interview results

Test result			Interview result
ξ₁ : A → P (V × W)			ξ₂ : A → P (V × W)
Applicant	a	p	Applicant	a	p
v ₁	60	55	v ₁	50	70
v ₂	70	55	v ₂	65	85
v ₃	80	85	v ₃	70	60
v ₄	65	50	v ₄	85	50
v ₅	55	60	v ₅	60	65
v ₆	55	65	v ₆	75	70
v ₇	65	70	v ₇	55	80
v ₈	70	50	v ₈	50	55

And

v₁ξ₂ (a) = {50} , v₂ξ₂ (a) = {65} , v₃ξ₂ (a) = {70} , v₄ξ₂ (a) = {85} , v₅ξ₂ (a) = {60} , v₆ξ₂ (a) = {75} , v₇ξ₂ (a) = {55} , v₈ξ₂ (a) = {50} , v₁ξ₂ (p) = {70} , v₂ξ₂ (p) = {85} , v₃ξ₂ (p) = {60} , v₄ξ₂ (p) = {50} , v₅ξ₂ (p) = {65} , v₆ξ₂ (p) = {70} , v₇ξ₂ (p) = {80} , v₈ξ₂ (p) = {65}

Now, we categorise the applicants who secure their admission in the field of mathematics by $\underline{ξ_{1} + ξ_{2}} (a) = {v_{2}, v_{3}, v_{4}, v_{6}, v_{8}}, \underline{ξ_{1} + ξ_{2}} (p) = {v_{1}, v_{2}, v_{3}, v_{6}, v_{7}},$ $\bar{ξ_{1} + ξ_{2}} (a) = {v_{3}}, \bar{ξ_{1} + ξ_{2}} (p) = {v_{7}} .$

In this case lower approximation contains all those applicants who passed one from their test or interview and in the upper approximation contains all those applicant who passed both test and interview.

6 Conclusion

In the present article we studied multigranulation rough set based on soft binary relations. Initially, we defined the roughness of a set with respect to aftersets and foresets of two soft binary relations from which we got two new soft sets. Also we discussed some fundamental properties of granulation roughness of a set. Then we generalized these definitions to multigranulation roughness of a set based on soft binary relations. Moreover, we presented some examples to applications in decision making problems. The main advantage of this approach is that we can approximate a subset of universal set through parameters in some other universal set. For future work, we have the following ideas.

1) We will study the multigranulation roughness of fuzzy set by the soft binary relations and will study ( , S)- fuzzy rough set models as developed by Zhang et al. (2020).

2) We will study the multigranulation roughness of Pythagorean fuzzy sets by the soft binary relations and will develop PF - TOPSIS method as developed by Zhan et al. (2020).

3) We will study the multigranulation roughness of Intuitionistic fuzzy sets by the soft binary relations and will develop TOPSIS method as developed by Zhang et al. (2020).

References

Acar

, Koyuncu

and Tanay

, Soft sets and soft rings, Computers and Mathematics with Applications 59(11) (2010), 3458–3463.

Aktaş

and Çağman

, Soft sets and soft groups, Information Sciences 177(13) (2007), 2726–2735.

Alcantud

J.C.R.

, Feng

and Yager

R.R.

, An N-Soft Set Approach to Rough Sets, IEEE Transactions on Fuzzy Systems 28(11) (2019), 2996–3007.

Ali

M.I.

, A note on soft sets, rough soft sets and fuzzy soft Sets, Applied Soft Computing 11(4) (2011), 3329–3332.

Ali

M.I.

, Feng

, Liu

, Min

W.K.

and Shabir

, On some new operations in soft set theory, Computers and Mathematics with Applications 57(9) (2009), 1547–1553.

Ali

, Ali

M.I.

and Rehman

, New types of dominance based multi-granulation rough sets and their applications in Conflict analysis problems, Journal of Intelligent and Fuzzy Systems 35(3) (2018), 3859–3871.

Atanassov

K.T.

, Intuitionistic fuzzy Sets, In Intuitionistic Fuzzy Sets, (1999), 1–137.

Çağman

and Enginoğlu

, Soft set theory and uni–int decision making, European Journal of Operational Research 207(2) (2010), 848–855.

Cagman

and Enginoglu

, FP-soft set theory and its applications, Ann Fuzzy Math Inform 2(2) (2011), 219–226.

10.

Cagman

and Enginoglu

, Soft matrix theory and its decision making, Comput Math Appl 59 (2010), 3308–3314.

11.

Chen

, Tsang

E.C.C.

, Yeung

D.S.

and Wang

, The parameterization reduction of soft sets and its applications, Computers Mathematics with Applications 49(5-6) (2005), 757–763.

12.

Degang

, Changzhong

and Qinghua

, A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough Sets, Information Sciences 177(17) (2007), 3500–3518.

13.

Feng

, Ali

M.I.

and Shabir

, Soft relations applied to semigroups, Filomat 27(7) (2013), 1183–1196.

14.

Feng

, Jun

Y.B.

and Zhao

, Soft semirings, Computers and Mathematics with Applications 56(10) (2008), 2621–2628.

15.

Feng

, Jun

Y.B.

, Liu

and Li

, An adjustable approach to fuzzy soft set based decision making, Journal of Computational and Applied Mathematics 234(1) (2010), 10–20.

16.

Feng

, Li

, Davvaz

and Ali

M.I.

, Soft sets combined with fuzzy sets and rough sets a tentative approach, Soft Computing 14(9) (2010), 899–911.

17.

Feng

, Liu

, Leoreanu-Fotea

and Jun

Y.B.

, Soft sets and soft rough Sets, Information Sciences 181(6) (2011), 1125–113.

18.

Gau

W.L.

and Buehrer

D.J.

, Vague Sets, IEEE Trans Syst Man Cybernet 23(2) (1993), 610–614.

19.

Gorzałczany

M.B.

, A method of inference in approximate reasoning based on interval-valued fuzzy Sets, Fuzzy Sets and Systems 21(1) (1987), 1–17.

20.

Greco

, Matarazzo

and Slowinski

, Rough approximation by dominance relations, International Journal of Intelligent Systems 17(2) (2002), 153–171.

21.

Jun

Y.B.

, Soft bck/bci-algebras, Computers and Mathematics with Applications 56(5) (2008), 1408–1413.

22.

Jun

Y.B.

and Park

C.H.

, Applications of soft sets in ideal theory of BCK/BCI-algebras, Information Sciences 178(11) (2008), 2466–2475.

23.

Kumar

and Inbarani

, Optimistic multi-granulation rough set based classification for medical diagnosis, Procedia Computer Science 47 (2015), 374–382.

24.

, Xie

and Gao

, Rough approximations based on soft binary relations and knowledge bases, Soft Computing 21(4) (2017), 839–852.

25.

T.J.

, Leung

and Zhang

W.X.

, Generalized fuzzy rough approximation operators based on fuzzy coverings, International Journal of Approximate Reasoning 48(3) (2008), 836–856.

26.

Liu

and Zhu

, The algebraic structures of generalized rough set theory, Information Sciences 178(21) (2008), 4105–4113.

27.

Liu

, Rough set theory based on two universal sets and its applications, Knowledge-Based Systems 23(2) (2010), 110–115.

28.

Maji

P.K.

, Roy

A.R.

and Biswas

, An application of soft sets in a decision making problem, Computers and Mathematics with Applications 44(8-9) (2002), 1077–1083.

29.

Maji

P.K.

, Biswas

and Roy

, Soft set theory, Computers and Mathematics with Applications 45(4-5) (2003), 555–562.

30.

Molodtsov

, Soft set theory–first results, Computers and Mathematics with Applications 37(4-5) (1999), 19–31.

31.

Pawlak

, Rough sets, International Journal of computer and Information Sciences 11(5) (1982), 341–356.

32.

Qian

, Liang

, Yao

and Dang

, A multi-granulation rough set, Information Sciences 180(6) (2010), 949–970.

33.

Qian

Y.H.

and Liang

J.Y.

, Rough set method based on multi-granulations, In 5th IEEE International Conference on Cognitive Informatics (Vol. 1, pp. 297–304) IEEE. (2006).

34.

Qian

Y.H.

, Cheng

H.H.

, Wang

J.T.

, Liang

J.Y.

, Pedrycz

and Dang

C.Y.

, Grouping granular structures in human granulation intelligence, Inf Sci 382–383 (2017), 150–169.

35.

Sezgin

and Atagün

A.O.

, On operations of soft Sets, Computers and Mathematics with Applications 61(5) (2011), 1457–1467.

36.

Shabir

, Kanwal

R.S.

and Ali

M.I.

, Reduction of an information system, Soft Computing (2019). doi.org/10.1007/s00500-019-04582-3

37.

Shabir

, Ali

M.I.

and Shaheen

, Another approach to soft rough Sets, Knowledge-Based Systems 40 (2013), 72–80.

38.

Shaheen

, Mian

, Shabir

and Feng

, A novel approach to decision analysis using dominance-based soft rough sets, International Journal of Fuzzy Systems 21(3) (2019), 954–962.

39.

Sharma

H.K.

, Kumari

and Kar

, A rough set approach for forecasting models, Decision Making: Applications in Management and Engineering 3(1) (2020), 1–21.

40.

Skowron

and Stepaniuk

, Tolerance approximation spaces, Fundamenta Informaticae 27(2, 3) (1996), 245–253.

41.

Slowinski

and Vanderpooten

, A generalized definition of rough approximations based on similarity, IEEE Transactions on Knowledge and Data Engineering 12(2) (2000), 331–336.

42.

Sun

and Ma

, Multigranulation rough set theory over two universes, Journal of Intelligent and Fuzzy Systems 28(3) (2015), 1251–1269.

43.

Sun

, Ma

and Qian

, Multigranulation fuzzy rough set over two universes and its application to decision making, Knowledge-Based Systems 123 (2017), 61–74.

44.

Tan

, Wu

W.Z.

, Shi

and Zhao

, Granulation selection and decision making with multigranulation rough set over two universes, International Journal of Machine Learning and Cybernetics 10(9) (2019), 2501–2513.

45.

W.Z.

and Zhang

W.X.

, Neighborhood operator systems and approximations, Information Sciences 144(1-4) (2002), 201–217.

46.

W.H.

and Zhang

W.X.

, Measuring roughness of generalized rough sets induced by a covering, Fuzzy Sets and Systems 158(22) (2007), 2443–2455.

47.

, Zhang

, Wang

and Sun

, On general binary relation based rough set, Journal of Information and Computing Science 7(1) (2012), 054–066.

48.

, Li

and Zhang

, Generalized multigranulation rough sets and optimal granularity selection, Granular Computing 2(4) (2017), 271–288.

49.

, Wang

and Luo

, Multi-granulation fuzzy rough Sets, Journal of Intelligent and Fuzzy Systems 26(3) (2014), 1323–1340.

50.

, Wang

and Zhang

, Multi-granulation Fuzzy Rough Sets in a Fuzzy Tolerance Approximation Space, International Journal of Fuzzy Systems 13(4) (2011).

51.

Yang

, Lin

T.Y.

, Yang

, Li

and Yu

, Combination of interval-valued fuzzy set and soft set, Computers and Mathematics with Applications 58(3) (2009), 521–527.

52.

Yao

Y.Y.

, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences 111(1-4) (1998), 239–259.

53.

Yao

and Yao

, Covering based rough set approximations, Information Sciences 200 (2012), 91–107.

54.

Yan

, Zheng

, Liu

and Zhai

, Research on the model of rough set over dual-universes, Knowledge-Based Systems 23(8) (2010), 817–822.

55.

Yang

H.L.

and Guo

Z.L.

, Multigranulation decision-theoretic rough sets in incomplete information systems, Int J Mach Learn Cybern 6 (2015), 1005–1018.

56.

Yang

, Song

, Dou

and Yang

, Multi-granulation rough set: from crisp to fuzzy case, Annals of Fuzzy Mathematics and Informatics 1(1) (2011), 55–70.

57.

You

, Li

and Wang

, Relative reduction of neighborhood-covering pessimistic multigranulation rough set based on evidence theory, Information 10(11) (2019), 334.

58.

Zadeh

L.A.

, Fuzzy Sets, Information and Control 8(3) (1965), 338–353.

59.

Zhan

and Wang

, Certain types of soft coverings based rough sets with applications, International Journal of Machine Learning and Cybernetics 10(5) (2019), 1065–1076.

60.

Zhan

, Sun

and Zhang

, PF-TOPSIS method based on CPFRS models An application to unconventional emergency events, Computers and Industrial Engineering 139 (2020), 106192.

61.

Zhang

, Zhan

and Wu

W.Z.

, Novel fuzzy rough set models and corresponding applications to multi-criteria decision-making, Fuzzy Sets and Systems 383 (2020), 92–126.

62.

Zhang

, Zhang

and Wu

W.Z.

, On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse, International Journal of Approximate Reasoning 51(1) (2009), 56–70.

63.

Zhang

, Zhan

and Yao

, Intuitionistic fuzzy TOPSIS method based on CVPIFRS models an application to biomedical problems, Information Sciences 517 (2020), 315–339.

64.

Zhu

, Generalized rough sets based on relations, Information Sciences 177(22) (2007), 4997–5011.

65.

Zhu

, Relationship between generalized rough sets based on binary relation and covering, Information Sciences 179(3) (2009), 210–225.

66.

Zou

and Xiao

, Data analysis approaches of soft sets under incomplete information, Knowledge-Based Systems 21(8) (2008), 941–945.

s₁ξ₁ (p) =	${s_{3}^{'}, s_{10}^{'}, s_{15}^{'},}$	s₁ξ₁ (m) =	∅,	s₁ξ₂ (p) =	∅,	s₁ξ₂ (m) =	∅,
s₂ξ₁ (p) =	∅,	s₂ξ₁ (m) =	${s_{10}^{'}, s_{15}^{'}},$	s₂ξ₂ (p) =	${s_{18}^{'}},$	s₂ξ₂ (m) =	∅,
s₃ξ₁ (p) =	${s_{5}^{'}, s_{7}^{'}}$	s₃ξ₁ (m) =	${s_{11}^{'}},$	s₃ξ₂ (p) =	${s_{15}^{'}, s_{17}^{'}},$	s₃ξ₂ (m) =	∅,
s₄ξ₁ (p) =	${s_{2}^{'}},$	s₄ξ₁ (m) =	${s_{13}^{'}, s_{16}^{'}},$	s₄ξ₂ (p) =	${s_{6}^{'}},$	s₄ξ₂ (m) =	${s_{12}^{'}, s_{16}^{'}, s_{19}^{'}}$
s₅ξ₁ (p) =	${s_{1}^{'}, s_{20}^{'}},$	s₅ξ₁ (m) =	∅,	s₅ξ₂ (p) =	∅,	s₅ξ₂ (m) =	∅,
s₆ξ₁ (p) =	${s_{17}^{'}, s_{18}^{'}},$	s₆ξ₁ (m) =	${s_{17}^{'}},$	s₆ξ₂ (p) =	∅,	s₆ξ₂ (m) =	∅,
s₇ξ₁ (p) =	∅,	s₇ξ₁ (m) =	${s_{14}^{'}},$	s₇ξ₂ (p) =	${s_{16}^{'}},$	s₇ξ₂ (m) =	${s_{7}^{'}, s_{11}^{'}, s_{17}^{'}},$
s₈ξ₁ (p) =	${s_{6}^{'}},$	s₈ξ₁ (m) =	∅,	s₈ξ₂ (p) =	∅,	s₈ξ₂ (m) =	∅,
s₉ξ₁ (p) =	${s_{11}^{'}, s_{13}^{'}},$	s₉ξ₁ (m) =	${s_{19}^{'}},$	s₉ξ₂ (p) =	${s_{19}^{'}},$	s₉ξ₂ (m) =	∅,
s₁₀ξ₁ (p) =	∅,	s₁₀ξ₁ (m) =	${s_{5}^{'}},$	s₁₀ξ₂ (p) =	∅,	s₁₀ξ₂ (m) =	${s_{20}^{'}},$
s₁₁ξ₁ (p) =	${s_{8}^{'}},$	s₁₁ξ₁ (m) =	∅,	s₁₁ξ₂ (p) =	∅,	s₁₁ξ₂ (m) =	∅,
s₁₂ξ₁ (p) =	∅,	s₁₂ξ₁ (m) =	∅,	s₁₂ξ₂ (p) =	${s_{1}^{'}, s_{5}^{'}, s_{9}^{'}},$	s₁₂ξ₂ (m) =	∅,
s₁₃ξ₁ (p) =	∅,	s₁₃ξ₁ (m) =	∅,	s₁₃ξ₂ (p) =	${s_{14}^{'}},$	s₁₃ξ₂ (m) =	∅,
s₁₄ξ₁ (p) =	∅,	s₁₄ξ₁ (m) =	∅,	s₁₄ξ₂ (p) =	∅,	s₁₄ξ₂ (m) =	${s_{18}^{'}},$
s₁₅ξ₁ (p) =	${s_{9}^{'}},$	s₁₅ξ₁ (m) =	${s_{2}^{'}},$	s₁₅ξ₂ (p) =	∅,	s₁₅ξ₂ (m) =	∅,
s₁₆ξ₁ (p) =	∅,	s₁₆ξ₁ (m) =	∅,	s₁₆ξ₂ (p) =	${s_{10}^{'}},$	s₁₆ξ₂ (m) =	${s_{2}^{'}},$
s₁₇ξ₁ (p)	= ∅ ,	s₁₇ξ₁ (m) =	${s_{7}^{'}},$	s₁₇ξ₂ (p) =	∅,	s₁₇ξ₂ (m) =	∅,
s₁₈ξ₁ (p) =	∅,	s₁₈ξ₁ (m) =	${s_{3}^{'}},$	s₁₈ξ₂ (p) =	∅,	s₁₈ξ₂ (m) =	${s_{1}^{'}, s_{5}^{'}},$
s₁₉ξ₁ (p) =	∅,	s₁₉ξ₁ (m) =	${s_{8}^{'}},$	s₁₉ξ₂ (p) =	${s_{2}^{'}},$	s₁₉ξ₂ (m) =	${s_{3}^{'}},$
s₂₀ξ₁ (p) =	${s_{14}^{'}},$	s₂₀ξ₁ (m) =	${s_{4}^{'}, s_{9}^{'}},$	s₂₀ξ₂ (p) =	${s_{3}^{'}, s_{11}^{'}},$	s₂₀ξ₂ (m) =	${s_{6}^{'}}$