Soft covering based rough graphs and corresponding decision making

Definition 1

[2] A graph G^∗ is a pair (V, E) of sets, where V is a finite non-empty set whose members are called vertices (also called points or nodes) and E is a set of unordered pairs of distinct vertices called edges (also called lines or arcs). A graph is usually denoted as G^∗ = (V, E). Let G^∗ be a graph and {u, v} be an edge of G^∗. It is often more convenient to represent this edge by uv or vu. The vertex set is usually denoted by V(G^∗) and the edge set by E(G^∗). An edge of a graph that joins a node to itself is called loop or self loop. In a multigraph no loops are allowed but more than one edge can join two vertices and these edges are called multiple edges or parallel edges. A graph G^∗ is called simple if it has no loops or multiple edges.

Definition 2

[2] A directed graph or digraph G^∗ containing a vertex set V(G^∗), and an edge set E(G^∗) whose elements are ordered pairs of elements of V(G^∗) called the directed edges. The first element of the ordered pair is called the tail of the edge and the second is called the head, together, they are the endpoints.

Definition 3

[20] Let T be the set of parameters. A pair (k, T) is called a soft set over the set U of universe, where k : T → P(U) is a set valued mapping and P(U) is the power set of U.

Definition 4

[47] A quadruple 𝓖 = (G^∗, λ, μ, T) is called a soft graph, where

1. G^∗ = (V, E) is a simple graph,

2. (λ, T) is a soft set over V,

3. (μ, T) is a soft set over E,

4. (λ(a), μ(a)) is a subgraph of G^∗ for all a ∈ T.

Definition 5

[25] Let S = (k, T) be a soft set over U. Then the pair 𝓟 = (U, S) is called soft approximation space. Based on the soft approximation space 𝓟, we define

assigning to any set X ⊆ U, the sets appr_𝓟(X) and appr_𝓟(X) and are called soft 𝓟-lower approximation and soft 𝓟-upper approximation of X, respectively.

The sets

are called the soft 𝓟-positive region, the soft 𝓟 -negative region, and the soft 𝓟-boundary region of X, respectively. If appr_𝓟(X) = appr_𝓟(X) then X is said to be soft 𝓟-definable, otherwise X is called a soft 𝓟-rough set.

Definition 6

Let 𝓖 = (G^∗, λ, μ, T) be a soft graph over the simple graph G^∗ = (V, E). Then 𝓖 is called

1. full soft vertex graph if

   ⋃α∈Tλα=V.

2. full soft edge graph if

   ⋃α∈Tμα=E.

3. full soft graph if

   G∗=⋃α∈Tλα,⋃α∈Tμα.

4. covering soft vertex graph if λ(α) ≠ ∅ for all α ∈ T. In this case (λ, T) is called covering soft set over V, denoted by 𝓒_𝓥.

   Denote by 𝓢 = (V, 𝓒_𝓥) and call it soft vertex covering approximation space.

Definition 7

Let 𝓢 = (V, 𝓒_𝓥) be a soft vertex covering approximation space and v ∈ V. Then the set

is called soft minimal vertex description of v ∈ V.

Definition 8

Let 𝓢 = (V, 𝓒_𝓥) be a soft vertex covering approximation space. Based on 𝓢 = (V, 𝓒_𝓥), the sets defined by

and

for the subset X ⊆ V, are called the lower 𝓢-soft vertex covering and upper 𝓢-soft vertex covering approximations of X, respectively.

Furthermore

are called 𝓢-soft positive vertex covering region, 𝓢-soft negative vertex covering region and 𝓢-soft boundary vertex covering region respectively. If appr_𝓢(X) = appr_𝓢(X), for X ⊆ V, then X is called 𝓢-soft vertex covering definable set and G_𝓢 = (V, E) is called 𝓢-soft vertex covering definable graph. On the other hand if appr_𝓢(X) ≠ appr_𝓢(X) then X is called 𝓢-soft vertex covering based rough set and G_𝓢 is called an 𝓢-soft rough vertex covering graph. Define and denote the lower and upper 𝓢-soft vertex covering approximations of G_𝓢 by

and

for any X ⊆ V.

Definition 9

Let G_𝓢 be an 𝓢-soft rough vertex covering graph. Then the 𝓢-roughness membership function of X ⊆ V is given by

Thus if appr_𝓢(X) = appr_𝓢(X) then ξ_G𝓢(X) = 0, and the graph G_𝓢 is soft vertex covering definable, i.e., no roughness.

Figure 1

Example 1

Consider a simple graph G^∗ = (V, E), where V = {v₁, v₂, v₃, v₄, v₅, v₆, v₇} is set of vertices and E = { e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀} is the set of edges as shown in figure below. Let T = {α₁, α₂, α₃, α₄, α₅, α₆} be the set of parameters and 𝓖 = (G^∗, λ, μ, T) be a soft covering vertex graph over the simple graph G^∗. The covering soft set (λ, T) over V is given in Table 1 such that λ(α₁) = {v₃, v₄, v₅, v₆}, λ(α₂) = {v₁, v₂}, λ(α₃) = {v₃, v₅, v₆}, λ(α₄) = {v₃, v₄}, λ (α₅) = {v₁, v₂, v₃, v₇}, λ(α₆) = {v₁, v₂, v₅, v₆}

Then 𝓢 = (V, 𝓒_𝓥) is a soft vertex covering approximation space. Let X = {v₁, v₂, v₄} ⊆ V. Then

Since appr_𝓢(X) ≠ appr_𝓢(X) . So X is a soft vertex covering based rough set and G_𝓢 : = (V, E) is an 𝓢-soft rough vertex covering graph, where

Note if X = {v₁, v₂} ⊆ V, then X is 𝓢-soft vertex covering based definable because appr_𝓢(X) = appr_𝓢(X) = {v₁, v₂} and G_𝓢 : = (V, E) is an 𝓢-soft vertex covering definable graph, where

Thus there is no roughness.

Definition 10

Let 𝓖 = (G^∗, λ, μ, T) be a full soft edge graph. Then 𝓖 is called covering soft edge graph if μ(α) ≠ ∅ for all α ∈ T. In this case (μ, T) is called covering soft edge set over E, denoted by 𝓒_𝓔. Denote by 𝓠 = (E, 𝓒_𝓔) and call it a soft edge covering approximation space.

Definition 11

Let 𝓠 = (E, 𝓒_𝓔) be a soft edge covering approximation space and e ∈ E. Then the set

is called soft minimal edge description of e ∈ E.

Definition 12

Let 𝓠 = (E, 𝓒_𝓔) be a soft edge covering approximation space. Based on 𝓠 = (E, 𝓒_𝓔), the sets defined by

and

for the subset Y ⊆ E, are called the lower 𝓠-soft edge covering and upper 𝓠-soft edge covering approximations of Y, respectively.

Also,

are called 𝓠-soft positive edge covering region, 𝓠-soft negative edge covering region and 𝓠-soft boundary edge covering region respectively. If appr_𝓠(Y) = appr_𝓠(Y), for Y ⊆ E, then Y is called 𝓠-soft edge covering definable set and G_𝓠 = (V, E) is called 𝓠-soft edge covering definable graph. On the other hand if appr_𝓠(Y) ≠ appr_𝓠(Y) then Y is called 𝓠-soft edge covering based rough set and G_𝓠 is called 𝓠-soft rough edge covering graph. Define and denote the lower and upper Q-soft edge covering approximations of G_𝓠 by

and

for any Y ⊆ E.

Definition 13

Let G_𝓠 be a 𝓠-soft rough edge covering graph. Then the 𝓠-roughness membership function of Y ⊆ E is given by

Thus if appr_𝓠(Y) = appr_𝓠(Y) then ξ_G𝓠(Y) = 0, and so the graph G_𝓠 is soft edge covering definable, i.e., no roughness.

Definition 14

A full soft graph 𝓖 = (G^∗, λ, μ, T) is called covering soft graph if λ(α) ≠ ∅ and μ(α) ≠ ∅ for all α ∈ T.

Example 2

(Continued from Example 1) Let (μ, T) be a covering soft set over E and 𝓠 = (E, 𝓒_𝓔) be a soft edge covering approximation space such that μ(α₁) = {e₁, e₃, e₅, e₁₀}, μ(α₂) = {e₄}, μ(α₃) = {e₄, e₅, e₈}, μ(α₄) = {e₁, e₂, e₅, e₆, e₈, e₉}, μ(α₅) = {e₁, e₃} and μ(α₆) = {e₆, e₇, e₉} as shown in the Table 2

Let Y = {e₁, e₃, e₄, e₁₀} ⊆ E. Then

As appr_𝓠(Y) ≠ appr_𝓠(Y), so Y is a soft edge covering based rough set and G_𝓠 := (V, E) is 𝓠-soft rough edge covering graph, where

Definition 15

The G^∗-roughness membership function of any subgraph graph G^∗∗ = (X, Y) of G^∗ is given by

Definition 16

A soft graph 𝓖 = (G^∗, λ, μ, T) is called soft covering based definable if

1. X is 𝓢-soft vertex covering definable i.e., appr_𝓢(X) = appr_𝓢(X) for X ⊆ V and

2. Y is 𝓠-soft edge covering definable i.e., appr_𝓠(Y) = appr_𝓠(Y) for Y ⊆ E.

Definition 17

A soft graph 𝓖 = (G^∗, λ, μ, T) is called soft covering based rough graph if

1. X is 𝓢-soft vertex covering based rough set i.e., appr_𝓢(X) ≠ appr_𝓢(X)

2. Y is 𝓠-soft edge covering based rough set i.e., appr_𝓠(Y) ≠ appr_𝓠(Y).

   A soft covering based rough graph is denoted by G = (G_𝓢, G_𝓠).

Definition 18

The lower and upper approximations of the soft covering based rough graph G = (G_𝓢, G_𝓠), is denoted and defined as appr(G) = (appr_𝓢(X), appr_𝓠(Y) ) and appr(G) = (appr_𝓢(X), appr_𝓠(Y) ) respectively, for any X ⊆ V and Y ⊆ E.

Proposition 1

Let 𝓖 = (G^∗, λ, μ, T) be a covering soft graph and 𝓢 = (V, 𝓒_𝓥) and 𝓠 = (E, 𝓒_𝓔) be soft vertex covering approximation space and soft edge covering approximation space, respectively. Then

1. appr(G) = ⋃α∈T {(λ(α), μ(α)) : λ(α) ⊆ X, μ(α) ⊆ Y}

2. appr(G) = ⋃ {(Mdes_𝓢(v), Mdes_𝓠(e)) : v ∈ X ⊆ V and e ∈ Y ⊆ E}

3. appr(∅) = appr (∅) = ∅

4. appr_G∗=appr¯G∗=G∗=⋃α∈Tλα,⋃α∈Tμα

5. appr(G₁ ∩ G₂) ⊆ appr(G₁) ∩ appr(G₂)

6. appr(G₁ ∪ G₂) ⊇ appr(G₁) ∪ appr(G₂)

7. appr(G₁ ∪ G₂) = appr(G₁) ∪ appr(G₂)

8. appr(G₁ ∩ G₂) ⊆ appr(G₁) ∩ appr(G₂)

9. G₁ ⊆ G₂ implies appr(G₁) ⊆ appr(G₂) and appr(G₁) ⊆ appr(G₂) where G₁ and G₂ are subgraphs of G^∗.

Proof

Straightforward. □

Proposition 2

Let 𝓖 = (G^∗, λ, μ, T) be a covering soft graph and let 𝓢 = (V, 𝓒_𝓥) and 𝓠 = (E, 𝓒_𝓔) be soft vertex covering approximation space and soft edge covering approximation space, respectively. Then

1. For any X ⊆ V, X is 𝓢 − soft vertex covering definable iffappr_𝓢(X) ⊆ X.

2. For any Y ⊆ E, Y is 𝓠 − soft edge covering definable iffappr_𝓠(Y) ⊆ Y.

Proof

1. Suppose X is an 𝓢-soft vertex covering definable. Then appr_𝓢(X) = appr_𝓢(X) and so appr_𝓢(X) = appr_𝓢(X) ⊆ X. Conversely suppose that appr_𝓢(X) ⊆ X for X ⊆ V. To prove X is an 𝓢-soft vertex covering definable, we have to prove only appr_𝓢(X) ⊆ appr_𝓢(X) . Let v ∈ appr_𝓢(X) = ⋃ {Mdes_𝓢(v) : v ∈ X} . Then v ∈ Mdes_𝓢(v) and Mdes_𝓢(v) ∩ X ≠ ∅, showing that v ∈ Mdes_𝓢(v) ⊆ appr_𝓢(X) ⊆ X. So v ∈ λ(α) for some α ∈ T. Hence v ∈ appr_𝓢(X) = ⋃α∈T {λ(α) : λ(α) ⊆ X} and so appr_𝓢(X) ⊆ appr_𝓢(X).

2. Can be proved in similar way. □

The above proposition is illustrated in the following example:

Example 3

Let 𝓖 = (G^∗, λ, μ, T) be a covering soft graph over the simple graph G^∗ = (V, E), where V = {v₁, v₂, v₃, v₄, v₅, v₆} is a set of vertices, E = {e₁, e₂, e₃, e₄, e₅, e₆} is the set of edges and T = {α₁, α₂, α₃, α₄, α₅} is the set of parameters. Let (λ, T) be a covering soft set over V such that λ (α₁) = {v₂, v₃, v₅, v₆}, λ(α₂) = {v₂, v₆}, λ (α₃) = {v₁, v₂, v₃, v₆}, λ(α₄) = {v₁, v₅}, λ (α₅) = {v₁, v₃, v₄, v₅} . Then 𝓢 = (V, 𝓒_𝓥) is soft vertex covering approximation space. For X = {v₂, v₆} ⊆ V, we have appr_𝓥(X) = {v₂, v₆} = appr_𝓥(X) . Hence appr_𝓢(X) ⊆ X and X is 𝓢-soft vertex covering definable.

(2) Let (μ, T) is called covering soft set over E such that μ(α₁) = {e₁, e₃, e₆}, μ (α₂) = {e₂, e₃, e₄, e₅}, μ(α₃) = {e₁, e₂, e₄, e₆}, μ (α₄) = {e₂, e₄}, μ(α₅) = {e₁, e₂, e₄, e₅, e₆} . Then 𝓠 = (E, 𝓒_𝓔) is a soft edge covering approximation space. Let Y = {e₂, e₄} ⊆ E. Then we can see that appr_𝓠(Y) = {e₂, e₄} = appr_𝓠(Y) showing that Y is 𝓠-soft edge covering definable. Clearly, 𝓖 = (G^∗, λ, μ, T) is soft covering based definable graph.

Proposition 3

Let G = (G_𝓢, G_𝓠) be a soft covering based rough graph. Then

1. appr(appr(G)) = appr(G)

2. appr(appr(G)) ⊇ appr(G)

3. appr(appr(G)) = appr(G)

4. appr(appr(G) ) ⊇ appr(G).

Proof

Let G = (G_𝓢, G_𝓠) be a soft covering based rough graph . Let 𝓢 = (V, 𝓒_𝓥) be a soft vertex covering approximation space and 𝓠 = (E, 𝓒_𝓔) be a soft edge covering approximation space.

1. Let 𝓛 = appr(G) = (appr_𝓢(X), appr_𝓠(Y)) for every X ⊆ V, Y ⊆ E. Let (l₁, l₂) ∈ 𝓛 be such that l₁ ∈ appr_𝓢(X) and l₂ ∈ appr_𝓠(Y) . As l₁ ∈ appr_𝓢(X), so l₁ ∈ λ(α) and Mdes_𝓢(v) ∩ X ≠ ∅ for some α ∈ T. Since l₂ ∈ appr_𝓠(Y) so l₂ ∈ μ(α) and Mdes_𝓠(e) ∩ Y ≠ ∅ for some α ∈ T. But

   appr¯G=⋃MdesSv,MdesQ e:v∈X⊆V and e∈Y⊆E

   so there exists α ∈ T such that (l₁, l₂) ∈ (λ(α), μ(α)) ⊆ 𝓛. Hence (l₁, l₂) ∈ appr(𝓛), and so 𝓛 ⊆ appr(𝓛) . But appr(𝓛) ⊆ 𝓛 which shows 𝓛 = appr(𝓛) . Therefore appr(appr (G)) = appr(G).

2. Let 𝓛 = appr(G) = (appr_𝓢(X), appr_𝓠(Y)) for every X ⊆ V, Y ⊆ E. Let (l₁, l₂) ∈ 𝓛 such that l₁ ∈ appr_𝓢(X) and l₂ ∈ appr_𝓠(Y) . As l₁ ∈ appr_𝓢(X), so l₁ ∈ λ(α) and Mdes_𝓢(l₁) ∩ X ≠ ∅ for some α} ∈ T. Since l₂ ∈ appr_𝓠(Y), so l₂ ∈ μ(α) and Mdes_𝓠(l₂) ∩ Y ≠ ∅ for some α ∈ T. But

   appr_G=⋃α∈Tλα,μα:λα⊆X,μα⊆Y,

   Mdes_𝓢(l₁) ∩ appr_𝓢(X) = Mdes_𝓢(l₁) ≠ ∅ and Mdes_𝓠(l₂) ∩ appr_𝓠(Y) = Mdes_𝓠(l₂) ≠ ∅ . Then (Mdes_𝓢(l₁), Mdes_𝓠(l₂)) ∩ 𝓛 = (Mdes_𝓢(l₁), Mdes_𝓠(l₂)), α ∈ T, which shows (l₁, l₂) ∈ appr(𝓛) . So 𝓛 ⊆ appr(𝓛) or appr(appr(G)) ⊇ appr(G).

3. Let 𝓛 = appr(G) = (appr_𝓢(X), appr_𝓠(Y)) for every X ⊆ V, Y ⊆ E. Let (l₁, l₂) ∈ 𝓛 be such that l₁ ∈ appr_𝓢(X) and l₂ ∈ appr_𝓠(Y) . Since l₁ ∈ appr_𝓢(X), so l₁ ∈ λ(α) and λ(α) ⊆ X for some α ∈ T. Since l₂ ∈ appr_𝓠(Y), so l₂ ∈ μ(α) and μ(α) ⊆ Y for some α ∈ T.

   Since

   appr_G=⋃α∈Tλα,μα:λα⊆X,μα⊆Y.

   So there exists α ∈ T such that (l₁, l₂) ∈ (λ(α), μ(α)) ⊆ appr(𝓛) . Thus 𝓛 ⊆ appr(𝓛) . But appr(𝓛) ⊆ (𝓛) . So 𝓛 = appr(𝓛) . Hence appr(G) = appr(appr(G)).

4. To prove appr(appr(G)) ⊇ appr(G) . Let 𝓛 = appr(G) = (appr_𝓢(X), appr_𝓠(Y)) for every X ⊆ V, Y ⊆ E. Let (l₁, l₂) ∈ 𝓛 such that l₁ ∈ appr_𝓢(X) and l₂ ∈ appr_𝓠(Y) . Since l₁ ∈ appr_𝓢(X), so l₁ ∈ λ(α) and Mdes_𝓢(l₁) ∩ X ≠ ∅ for some α ∈ T. Since l₂ ∈ appr_𝓠(Y) so l₂ ∈ μ(α) and Mdes_𝓠(l₂) ∩ Y ≠ ∅ for some α ∈ T. Since

   appr¯G=⋃MdesSl1,MdesQl2:l1∈X⊆V and l2∈Y⊆E,appr¯SX=⋃MdesSl1:l1∈X⊆V andappr¯QY=⋃ MdesQl2:l2∈Y⊆E.

   Therefore (Mdes_𝓢(l₁), Mdes_𝓠(l₂)) ∩ 𝓛 = (Mdes_𝓢(l₁), Mdes_𝓠(l₂)) with Mdes_𝓢(l₁) ≠ ∅ and Mdes_𝓠(l₂) ≠ ∅ . So (l₁, l₂) ∈ (Mdes_𝓢(l₁), Mdes_𝓠(l₂)) ⊆ appr(𝓛) showing (l₁, l₂) ∈ appr(𝓛) . Thus 𝓛 ⊆ appr(𝓛) . Hence appr(appr(G)) ⊇ appr(G) . □

Corollary 1

Let 𝓖 = (G^∗, λ, μ, T) be a soft graph and 𝓢 = (V, 𝓒_𝓥) and 𝓠 = (E, 𝓒_𝓔) be soft vertex covering approximation space and soft edge covering approximation space, respectively. Then for every X ⊆ V and Y ⊆ E, the following hold.

1. appr_𝓢(appr_𝓢(X)) = appr_𝓢(X).

2. appr_𝓢(appr_𝓢(X)) ⊇ appr_𝓢(X)

3. appr_𝓢(appr_𝓢(X)) = appr_𝓥(X)

4. appr_𝓢(appr_𝓢(X)) ⊇ appr_𝓢(X) for every X ⊆ V.

5. appr_𝓠(appr_𝓠(Y)) = appr_𝓠(Y).

6. appr_𝓠(appr_𝓠(Y)) ⊇ appr_𝓠(Y)

7. appr_𝓠(appr_𝓠(Y)) = appr_𝓠(Y)

8. appr_𝓠(appr_𝓠(Y)) ⊇ appr_𝓠(Y) for every Y ⊆ E.

Example 4

Let 𝓖 = (G^∗, λ, μ, T) be a soft graph as in Example 1 with μ(α₁) = {v₁, v₅, v₆, v₇}, μ(α₂) = {v₃, v₅, v₆}, μ(α₃) = {v₁, v₃, v₄}, μ (α₄) = {v₅, v₆}, μ(α₅) = {v₂, v₄, v₆, v₇} and μ(α₆) = {v₁, v₂, v₃, v₆, v₇} . Let X = {v₂, v₃, v₅, v₆} ⊆ V. Then

and

1. Let P = appr_𝓢(X) = {v₁, v₂, v₃, v₄, v₅, v₆, v₇} . Then appr_𝓢(P) = appr_𝓢(appr_𝓢(X)) = P.

2. Let R = appr_𝓢(X) = {v₃, v₅, v₆} . Then appr_𝓢(R) = appr_𝓢(appr_𝓢(X)) = {v₁, v₂, v₃, v₄, v₅, v₆, v₇} = appr_𝓢(X) ⊇ R.

3. appr_𝓢(appr_𝓢(X)) = appr_𝓢(R) = {v₃, v₅, v₆} = R = appr_𝓢(X) . Similarly rest of the above results can be verified.

Proposition 4

Let 𝓖 = (G^∗, λ, μ, T) be a soft graph. Then 𝓖 = (G^∗, λ, μ, T) is a full soft graph if and only if appr(G^∗) = G^∗ = appr(G^∗).

Proof

Suppose 𝓖 = (G^∗, λ, μ, T) is a full soft graph. Then G∗=⋃α∈Tλα,⋃α∈Tμα. Since

so appr_𝓢(V) = ⋃α∈T {λ(α) : λ (α) ⊆ V} = ⋃α∈T λ(α) = V. Hence appr_𝓢(V) = V. Now appr_𝓠(Y) = ⋃α∈T {μ(α) : μ(α) ⊆ Y}, so appr_𝓠(E) = ⋃α∈T {μ(α) : μ(α) ⊆ E} = ⋃α∈T μ(α) = E. Therefore appr (G^∗) = (appr_𝓢(V), appr_𝓠(E)) = (V, E) = G^∗. Also appr_𝓢(X) = ⋃ {Mdes_𝓢(v) : v ∈ X }, so appr_𝓢(V) = ⋃ {Mdes_𝓢(v) : v ∈ V} = ⋃ Mdes_𝓢(v) = V

Similarly appr_𝓠(Y) = ⋃ {Mdes_𝓠(e) : e ∈ Y }, so appr_𝓠(E) = ⋃ {Mdes_𝓠(e) : e ∈ E} = ⋃ Mdes_𝓠(e) = E. Hence G^∗ = appr(G^∗).

Conversely suppose thatappr(G^∗) = G^∗ = appr(G^∗) . To show 𝓖 = (G^∗, λ, μ, T) is a full soft graph. Since appr(G^∗) = (appr_𝓢(V), appr_𝓠(E)) = G^∗ so appr_𝓢(V) = V and appr_𝓠(E) = E.

So appr_𝓢(V) = ⋃α∈T {λ(α) : λ (α) ⊆ V} = V and appr_𝓠(E) = ⋃α∈T {μ(α) : μ(α) ⊆ E} = E, which shows that ⋃α∈T λ(α) = V and ⋃α∈T μ(α) = E. Hence G∗=V,E=⋃α∈Tλα,⋃α∈Tμα showing that 𝓖 = (G^∗, λ, μ, T) is a full soft graph. □

Proposition 5

Let 𝓖 = (G^∗, λ, μ, T) be a covering soft graph and 𝓢 = (V, 𝓒_𝓥) be a soft vertex covering approximation space. Then the following are equivalent:

1. (λ, T) is a full soft set.

2. appr_𝓢(V) = V

3. appr_𝓢(V) = V

4. X ⊆ appr_𝓢(X) for all X ⊆ V

5. appr_𝓢({v}) ≠ ∅ for all v ∈ V.

Proof

Using Proposition 4, it is easy to show that the conditions (1) and (2) are equivalent. Similarly the conditions (1) and (3) are equivalent.

Now to prove conditions (4), (5) and (1) are equivalent. Suppose condition (4) holds. To prove that (5) is true. For v ∈ V, by condition (4), {v} ⊆ appr_𝓢({v}) . Thus appr_𝓢({v}) ≠ ∅ because v ∈ appr_𝓢({v}) . Hence (4) implies(5) . Now to show (5) implies (1) . Suppose appr_𝓢({v}) ≠ ∅ for all v ∈ V. Let l ∈ appr_𝓢({v}) . Then by definition of appr_𝓢({v}), there exists some α in T with l ∈ λ (α) and λ(α) ∩ {v} ≠ ∅ . It follows that v = l ∈ λ(α) and so v ∈ ⋃α∈T λ(α) . Hence (λ, T) is a full soft set showing that (5) implies (1) . Now to complete the proposition, it is left to show (1) implies (4) . Suppose (λ, T) is a full soft set with X ⊆ V. For any p ∈ X, since (λ, T) is a full soft set so there exists some α ∈ T such that p ∈ λ(α) . Also X∩Mdes_𝓢(p) ≠ ∅ because p ∈ X ∩ Mdes_𝓢(p) . Hence p ∈ appr_𝓢(X) . Therefore X ⊆ appr_𝓢(X) for all X ⊆ V. □

Proposition 6

Let 𝓖 = (G^∗, λ, μ, T) be a covering soft graph and 𝓠 = (E, 𝓒_𝓔) be a soft edge covering approximation space. Then the following are equivalent:

1. (μ, T) is a full soft set.

2. appr_𝓠(E) = E.

3. appr_𝓠(E) = E.

4. Y ⊆ appr_𝓠(Y) for all Y ⊆ E.

5. appr_𝓠({e}) ≠ ∅ for all e ∈ E.

Proof

Similar to the proof of the Proposition 5. □