Z -soft rough fuzzy graphs: A new approach to decision making

1 Introduction

Graph theory was developed with Leonhard Euler’s paper [13] in which he settled a famous unsolved problem known as Kōnigsberg Bridges Problem. The theory has greatly contributed to many developing fields, due to its applications in programming, civil engineering, communications theory, network theory, coding theory, switching circuits, psychology, computer science involving computations and algorithms, operation research(scheduling) and economics. Many extensions and applications of the subject of graph theory can be seen in [4, 51]. A graph can also be used to discuss the relationship between the objects where each object can be represented by a vertex and the relationship between the objects can be represented by an edge. Network problems, like sensor network problems and automatic channel allocation networks can also be tackled by using graphs and the related structures.

In real life, most of the problems involve complexity which arises from uncertainty in the form of ambiguity. Usually the data collected from real life situations consists of uncertain, imprecise and incomplete information in which the classical mathematics is not absolutely prosperous and not successful. Most of our conventional mathematical tools of modeling, reasoning and computing are crisp, deterministic and precise in character. Different approaches have been developed to solve the problems of uncertainty and vagueness. For example probability theory, fuzzy set theory, rough set theory, soft set theory and blend of these approaches have been used. The theory of probability has been an efficient tool to handle the problems of uncertainty, but this theory can be applied only to the situations whose characteristics are based only on random processes.

In 1965, Lotfi. A. Zadeh [55] originated the theory of fuzzy sets in his seminal paper which is another excellent effective tool to handle the problems involving uncertainty. Fuzzy set theory can be used to get, simulate and even reason the fuzziness in many practical material of information by mainly examining the fuzzy information granularity. Due to the wide applications of fuzzy sets, almost all the areas of mathematics, medicine, engineering etc., notions have been redefined using fuzzy sets. Fuzzy sets are described approximately through the membership degree of sets with respect to objects.

Rough sets was originally proposed by Z. Pawlak [33] as a better mathematics tool for replacing the probability and fuzzy set theory, and can be used to model and process incomplete information in an information system. The rough set approach seems to be of fundamental significance to artificial intelligence and cognitive sciences, particularly in the field of knowledge acquisition, machine learning, decision analysis, pattern recognition, expert systems and knowledge discovery from databases. Many applications of rough sets including data mining, pattern recognition, machine learning and knowledge discovery can be seen in [9, 49].

Molodtsov [30] introduced the notion of soft sets as a new mathematics tool for solving the problems involving vagueness and uncertainty. Since a soft set involve many parameters, so it is free from those difficulties which the previous theories possesses. Soft set theory has been modified and extended in some aspects to tackle many problems [6, 38]. A number of applications have been established and applied regarding decision making and multi-attributes modeling [10, 46]. A possible fusion has been established by Feng et al. [15] by considering approximation of a fuzzy set in soft approximation space and initiated a new concept called soft rough fuzzy sets, which is the generalization of Dubois and Prade’s rough fuzzy sets [12]. In their soft rough fuzzy model, a soft set is used to granulate the universe of discourse instead of equivalence class taken by Pawlak [33]. Meng et al. [28] further discussed the combination of rough sets, fuzzy sets and soft sets, and employed fuzzy soft sets to granulate the universe of discourse. Furthermore, they established a more general model called soft fuzzy rough model than the previous. Zhan et al. [60] introduced the notion of Z-soft rough fuzzy sets and gave a corresponding decision making. A number of applications and utilizations has been presented by many researchers in problems of multi-attributes, problems of reduction of attributes, knowledge base system, data mining and data labeling problems [5, 67]

It can be seen that soft sets, fuzzy sets and rough sets are closely linked, and some extensions of these are available in [3, 65]. An amalgamation of rough sets with fuzzy sets and referring them to theory of graphs can be found in [11, 45]. Akram [1] introduced the notion soft graphs which is an approach to handle the problems of multi-attributes related with graph theory. A number of extensions and generalizations of soft graphs are available in [2, 48]. The present article aims at providing a new approach to combine graphs, soft sets and rough sets which initiates the notion of Z-soft rough fuzzy graphs. This approach will not only enhance the applicability of soft graphs but will also use to discuss the uncertainty in soft graphs. The present paper is organized asbreak follows:

In Section 2 some basic concepts are revised. Section 3 is about the basic definitions and characterization of Z- soft rough fuzzy graphs, where the notions of lower (upper) Z- soft vertex approximations, lower (upper) Z- soft edge approximations and Z-soft rough fuzzy graph are defined to develop the basic theory with the help of examples. Section 4 is developed for the applications of Z- soft rough fuzzy graphs. An algorithm is proposed to compute the effectiveness of some diseases amongst some animals kept in a farm house in which the vertices are denoted by 20 animals and the edges are used to describe the interaction between the animals. This interaction may cause the spreadness of diseases amongst the animals. This is the motivation of the proposed theory. Conclusion of the article is presented in Section 5.

2 Preliminaries

In this section, some basic notions are given. These will be very helpful in later sections.

Definition 1. [8, 51] A graph G^* = (V, E) consists of set of finite objects V = {v₁, v₂, v₃......., v _n } called vertices (also called points or nodes) and the set E = {e₁, e₂, e₃....... e _m } whose element are called edges (also called lines or arcs). Let {v₁, v₂} be an edge of G^*, then it is more convenient to represent this edge by v₁v₂. If e = v₁v₂ is an edges of a graph G^*, then we say that v₁ and v₂ are adjacent in G^* and that e joins v₁ and v₂. If every edge in a graph has direction then such a graph is called a digraph. An edge whose endpoints are equal, is called a loop and multiple edges are edges having the same pair of endpoints. A graph G^* is called simple if it has no loops or multiple edges.

Definition 2. [24] Let A be the set of parameters. A pair (k, A) is called a soft set over the set U of universe, where k: A → P (U) is a set valued mapping. A soft set can be represented by a table having values 0 and 1, where 1 represent that x ∈ k (ρ) and 0 represent that x ∉ k (ρ)  , for ρ ∈ A and x ∈ U.

Definition 3. [1] Let G^* = (V, E) be a given simple graph and A be the set of parameters. A quadruple G = ( G * , G , H , A ) is said to be a soft graph on G^* if

G = ( δ , A ) is a soft set over V ;

Definition 4. [16] Let S = (k, A) be a soft set over U. Then, the pair D = ( U , S ) is called soft approximation space. The following two sets, appr _ D ( X ) = { u ∈ U : ∃ ρ ∈ A , [ u ∈ k ( ρ ) ⊆ X ] } , appr ¯ D ( X ) = { u ∈ U : ∃ ρ ∈ A , [ u ∈ k ( ρ ) , k ( ρ ) ∩ X ≠ φ ] } assigning to any set X ⊆ U, are called soft D - lower approximation and soft D - upper approximation of X, respectively. Also, the sets Pos ( X ) = appr _ D ( X ) , Neg ( X ) = - appr _ D ( X ) , and Bnd ( X ) = appr ¯ D ( X ) - appr _ D ( X ) are called respectively, the soft D -positive region, the soft D - negative region, and the soft D -boundary region of X. If appr ¯ D ( X ) = appr _ D ( X ) , then X is said to be soft D -definable; otherwise X is called a soft D -rough set.

Definition 5. [15] Let S = (k, A) be a full soft set over U i.e., ⋃_ρ∈Rk (ρ) = U and D = ( U , S ) is a soft approximation space. Let F ( U ) be the set of all fuzzy subsets of U. For a fuzzy set μ∈ F ( U ) , the lower and upper soft rough approximation space of μ with respect to D are denoted and defined as, s a p _ D ( μ ) ( x ) = ∧ { μ ( y ) : ∃ ρ ∈ A [ { x , y } ⊆ k ( ρ ) % ] } , s a p ¯ D ( μ ) ( x ) = ∨ { μ ( y ) : ∃ ρ ∈ A [ { x , y } ⊆ k ( ρ ) ] } for all x ∈ U, the operators sap _ D and sap ¯ D , are called the lower and upper soft rough approximation operators on fuzzy sets. If sap _ D ( μ ) = sap ¯ D ( μ ) then μ is said to be soft definable; otherwise μ is called a soft rough fuzzy set.

3 Z-soft rough fuzzy graphs

In this section, Z-soft rough fuzzy graphs are defined. The notion of lower/upper Z-soft rough vertex approximation, lower/upper Z-soft rough edge approximation, Z-soft rough fuzzy vertex graph, Z-soft rough fuzzy edge graph is introduced. Basic theory of soft rough fuzzy graphs is evolved and some fundamental properties are interrogated.

Definition 6. Let G = ( G ∗ , ψ , ξ , A ) be a soft graph and λ: V → P (A) be a map such that λ (x) = {ρ ∈ A: x ∈ ψ (ρ)}. Denote by Z _λ = (V, λ) and call it Z-soft vertex approximation space. Let F ( V ) be the set of all fuzzy subsets of V. Based on Z _λ = (V, λ) and for a fuzzy subset h of V, the lower Z- soft rough vertex approximation and upper Z- soft rough vertex approximation of h are denoted by appr _ Z λ ( h ) ( x ) and appr ¯ Z λ ( h ) ( x ) respectively, and are defined by appr _ Z λ ( h ) ( x ) = ⋀ { h ( y ) : y ∈ V , λ ( x ) = λ ( y ) } , appr ¯ Z λ ( h ) ( x ) = ⋁ { h ( y ) : y ∈ V , λ ( x ) = λ ( y ) } for all x ∈ V. The sets appr _ Z λ ( h ) and appr ¯ Z λ ( h ) are fuzzy sets in V. If appr ¯ Z λ ( h ) = appr _ Z λ ( h ) , then G _{Z λ} : = (V, E) is called Z- soft vertex definable graph, otherwise G _{Z λ} is called Z- soft rough fuzzy vertex graph. Denote and define the lower and upper Z- soft vertex approximations of G _{Z λ} by G Z λ _ = ( appr _ Z λ ( h ) , E ) and G Z λ ¯ = ( appr ¯ Z λ ( h ) , E ) for h ∈ F ( V ) .

Example 1. Let a simple graph be G^∗ = (V, E), where V ={ x₁, x₂, x₃, x₄, x₅, x₆, x₇ } is the set of vertices, E ={ e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀ } is the edge set as shown in Figure 1, given in Appendix. Let be the set of parameters and G = ( G ∗ , ψ , ξ , A ) be a soft graph. A soft set (ψ, A) over V is given in Table 1. and Z _λ = (V, λ) is a Z-soft vertex approximation space, a map λ: V → P (A) is defined as λ (x) = {ρ ∈ A: x ∈ ψ (ρ)} such that λ (x₁) = {ρ₁,ρ₂,ρ₄} = λ (x₃)  , λ (x₂) = {ρ₃}, λ (x₄) = {ρ₂,ρ₃, ρ₄} = λ (x₆)  , λ (x₅) = {ρ₁}.

Then for h ∈ F ( V ) , where h = { x 1 0.2 , x 2 0.4 , x 3 0.1 , x 4 0.5 , x 5 0.6 , x 6 0.9 , x 7 0.8 } , and one can calculate appr _ Z λ ( h ) ( x ) = { x 1 0.1 , x 2 0.4 , x 3 0.1 , x 4 0.5 , x 5 0.6 , x 6 0.5 , x 7 0.8 } and appr ¯ Z ( h ) ( x ) = { x 1 0.2 , x 2 0.4 , x 3 0.2 , x 4 0.9 , x 5 0.6 , x 6 0.9 , x 7 0.8 } . Since appr ¯ Z ( h ) ≠ appr _ Z ( h ) , So G _{Z λ} : = (V, E) is Z- soft rough fuzzy vertex graph, where G Z λ _ = ( appr _ Z λ ( h ) , E ) = ( { x 1 0.1 , x 2 0.4 , x 3 0.1 , x 4 0.5 , x 5 0.6 , x 6 0.5 , x 7 0.8 } , { e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 , e 8 , e 9 , e 10 } ) , G Z λ ¯ = ( appr ¯ Z λ ( h ) , E ) = ( { x 1 0.2 , x 2 0.4 , x 3 0.2 , x 4 0.9 , x 5 0.6 , x 6 0.9 , x 7 0.8 } , { e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 , e 8 , e 9 , e 10 } ) .

Definition 7. Let G = ( G ∗ , ψ , ξ , A ) be a soft graph and η: E → P (A) be a map such that η (e) = {ρ ∈ A: e ∈ ξ (ρ)}. Denote by Z _η = (E, η) and call it Z-soft edge approximation space. Let F ( E ) be the set of all fuzzy subsets of E. Based on Z _η = (E, η) and for a fuzzy subset k of E, the lower Z- soft rough edge approximation and upper Z-soft rough edge approximation of k are denoted by appr _ Z η ( k ) ( e ) and appr ¯ Z η ( k ) ( e ) respectively, and are defined by appr _ Z η ( k ) ( e ) = ⋀ { k ( f ) : f ∈ E , η ( e ) = η ( f ) } , appr ¯ Z η ( k ) ( e ) = ⋁ { k ( f ) : f ∈ E , η ( e ) = η ( f ) } . for all e ∈ E. The sets appr _ Z η ( k ) and appr ¯ Z η ( k ) are fuzzy sets in E. If appr ¯ Z η ( k ) = appr _ Z η ( k ) , then G _{Z η} : = (V, E) is called Z- soft edge definable graph, otherwise G _{Z η} is called Z- soft rough fuzzy edge graph. Denote and define the lower and upper Z- soft edge approximations of G _{Z η} by G Z η _ = ( appr _ Z η ( h ) , E ) and G Z η ¯ = ( appr ¯ Z η ( h ) , E ) for k ∈ F ( E ) .

Definition 8. A soft graph G = ( G ∗ , ψ , ξ , A ) is called Z-soft definable if (1) h is Z- soft vertex definable i.e., appr ¯ Z λ ( h ) = appr _ Z λ ( h ) and (2) k is Z- soft edge definable i.e., app r Z η ¯ ( k ) = app r _ z η ( k ) .

Definition 9. A soft graph G = ( G * , ψ , ξ , A ) is called Z-soft rough fuzzy graph if (1) h is Z-soft rough fuzzy vertex set i.e., a p p r ¯ Z λ ( h ) ≠ a p p r _ Z λ ( h ) (2) k is Z-soft rough fuzzy edge set i.e., a p p r ¯ Z η ( k ) ≠ a p p r _ Z η ( k ) A Z-soft rough fuzzy graph is denoted by

Definition 10. Let be a Z-soft rough fuzzy graph. Then a p p r _ ( G ) = ( a p p r _ Z λ ( h ) a p p r _ Z η ( k ) ) and a p p r ¯ ( G ) = ( a p p r ¯ Z λ ( h ) a p p r ¯ Z η ( k ) ) (k)) respectively, are called the lower and upper Z-soft approximations of G = ( G Z λ , G Z η ) , for h ∈ F ( V ) and k ∈ F ( E )

Definition 11. Let G = ( G ∗ , ψ , ξ , A ) be a soft graph over the simple graph G^∗ = (V, E). Then G is called

Full soft vertex graph if ⋃_ρ∈Aψ (ρ) = V and

A soft graph G = ( G ∗ , ψ , ξ , A ) is called a full soft graph if G^∗ = (⋃ _ρ∈Aψ (ρ)  , ⋃ _ρ∈Aξ (ρ).

Proposition 1. Let G = ( G ∗ , ψ , ξ , A ) be a full soft graph then for h ∈ F ( V ) and k ∈ F ( E ) , (1)

appr _ ( G ) = ( ⋀ x ∈ ψ ( ρ ) ⋀ y ∈ ψ ( ρ ) h ( y ) , ⋀ e ∈ ξ ( ρ ) ⋀ f ∈ ξ ( ρ ) k ( f ) ) and (2) appr ¯ ( G ) = ( ⋁ x ∈ ψ ( ρ ) ⋁ y ∈ ψ ( ρ ) h ( y ) , ⋁ e ∈ ξ ( ρ ) ⋁ f ∈ ξ ( ρ ) k ( f ) ) .

Proof. (1) Since a p p r _ ( G ) = ( a p p r _ Z λ ( h ) , a p p r _ Z η ( k ) ) (k)) so it is necessary to show only

appr _ Z λ ( h ) ( x ) = ⋀ x ∈ ψ ( ρ ) ⋀ y ∈ ψ ( ρ ) h ( y ) and

a p p r _ Z λ ( h ) ( x ) = ∧ x ∈ ψ ( ρ ) ∧ y ∈ ψ ( ρ ) h ( y ) a n d a p p r _ Z η ( k ) ( e ) = ∧ e ∈ ξ ( ρ ) ∧ f ∈ ξ ( ρ ) k ( f ) As G = ( G * , ψ , ξ , A ) be a full soft graph so for x ∈ V, e ∈ E there exists ρ ∈ A such that x ∈ ψ (ρ) and e ∈ ξ (ρ). But a p p r _ Z λ ( h ) ( x ) = ∧ { h ( y ) : y ∈ V , λ ( x ) = λ ( y ) } and a p p r _ Z η ( k ) ( e ) = ∧ { k ( f ) : f ∈ E , η ( e ) = η ( f ) } So h ( y ) ≥ a p p r _ Z λ ( h ) ( x ) and k ( f ) ≥ a p p r _ Z η ( k ) ( e ) Which implies ∧ y ∈ ψ ( ρ ) h ( y ) ≥ a p p r _ Z λ ( h ) ( x ) and ∧ f ∈ ξ ( ρ ) k ( f ) ≥ a p p r _ Z η ( k ) ( e ) or ∧ x ∈ ψ ( ρ ) ∧ y ∈ ψ ( ρ ) h ( y ) ≥ a p p r _ Z λ ( h ) ( x ) and ∧ e ∈ ξ ( ρ ) ∧ f ∈ ξ ( ρ ) k ( f ) ≥ a p p r _ Z η ( k ) ( e ) Let λ (x) = λ (q) and η (e) = η (l)  , so ⋀{ h (q): q ∈ V, λ (x) = λ (q) } ≥ ⋀ _x∈ψ(ρ) ⋀ _y∈ψ(ρ)h (y) and ⋀ { k (l): l ∈ E, η (e) = η (l)} ≥ ⋀ _e∈ξ(ρ) ⋀ _f∈ξ(ρ)k (f) showing appr _ Z λ ( h ) ( x ) ≥ ⋀ x ∈ ψ ( ρ ) ⋀ y ∈ ψ ( ρ ) h ( y ) and appr _ Z η ( k ) ( e ) ≥ ⋀ e ∈ ξ ( ρ ) ⋀ f ∈ ξ ( ρ ) k ( f ) . Since ⋀ x ∈ ψ ( ρ ) ⋀ y ∈ ψ ( ρ ) h ( y ) ≥ appr _ Z λ ( h ) ( x ) , appr _{Z λ} (h) (x) ≥ ⋀ _x∈ψ(ρ) ⋀ _y∈ψ(ρ)h (y) and ⋀ e ∈ ξ ( ρ ) ⋀ f ∈ ξ ( ρ ) k ( f ) ≥ appr _ Z η ( k ) ( e ) , appr _ Z η ( k ) ( e ) ≥ ⋀ e ∈ ξ ( ρ ) ⋀ f ∈ ξ ( ρ ) k ( f ) . Therefore appr _ Z λ ( h ) ( x ) = ⋀ x ∈ ψ ( ρ ) ⋀ y ∈ ψ ( ρ ) h ( y ) and appr _ Z η ( k ) ( e ) = ⋀ e ∈ ξ ( ρ ) ⋀ f ∈ ξ ( ρ ) k ( f ) . Hence (1) is proved. (2) As appr ¯ ( G ) = ( appr ¯ Z λ ( h ) , appr ¯ Z η ( h ) ) , so it is only to show appr ¯ Z λ ( h ) ( x ) = ⋁ x ∈ ψ ( ρ ) ⋁ y ∈ ψ ( ρ ) h ( y ) and appr ¯ Z η ( k ) ( e ) = ⋁ e ∈ ξ ( ρ ) ⋁ f ∈ ξ ( ρ ) k ( f ) . Given G = ( G ∗ , ψ, ξ, A) is a full soft graph so for x ∈ V, e ∈ E there exists ρ ∈ A such that x ∈ ψ (ρ) and e ∈ ξ (ρ). But appr ¯ Z λ ( h ) ( x ) = ⋁ { h ( y ) : y ∈ V , λ ( x ) = λ ( y ) } and appr ¯ Z η ( h ) ( e ) = ⋁ { k ( f ) : f ∈ E , η ( e ) = η ( f ) } .

Therefore h ( y ) ≤ appr ¯ Z λ ( h ) ( x ) and k ( f ) ≤ appr ¯ Z η ( h ) ( e ) , which gives ⋁ y ∈ ψ ( ρ ) h ( y ) ≤ appr ¯ Z λ ( h ) ( x ) and ⋁_f∈ξ(ρ) k ( f ) ≤ appr ¯ Z η ( h ) ( e ) . So ⋁ x ∈ ψ ( ρ ) ⋁ y ∈ ψ ( ρ ) h ( y ) ≤ appr ¯ Z λ ( h ) ( x ) and ⋁ e ∈ ξ ( ρ ) ⋁ f ∈ ξ ( ρ ) k ( f ) ≤ appr ¯ Z η ( h ) ( e ) . Now let λ (x) = λ (q) and η (e) = η (l)  , so h (q) ≤ ⋁ _x∈ψ(ρ) ⋁ _y∈ψ(ρ)h (y) and k (l) ≤ ⋁ _e∈ξ(ρ) ⋁ _f∈ξ(ρ) k (f)  , ⋁ { k (l): l ∈ E, η (e) = η (l) } ≤ ⋁ _e∈ξ(ρ) ⋁ _f∈ξ(ρ) k (f) showing that appr ¯ Z λ ( h ) ( x ) ≤ ⋁ x ∈ ψ ( ρ ) ⋁ y ∈ ψ ( ρ ) h ( y ) and appr ¯ Z η ( k ) ( e ) ≤ ⋁ e ∈ ξ ( ρ ) ⋁ f ∈ ξ ( ρ ) k ( f ) . As ⋁ x ∈ ψ ( ρ ) ⋁ y ∈ ψ ( ρ ) h ( y ) ≥ appr ¯ Z λ ( h ) ( x ) , appr ¯ Z λ ( h ) ( x ) ≤ ⋁ x ∈ ψ ( ρ ) ⋁ y ∈ ψ ( ρ ) h ( y )

and ⋁ e ∈ ξ ( ρ ) ⋁ f ∈ ξ ( ρ ) k ( f ) ≥ appr ¯ Z η ( k ) ( e ) ,

appr ¯ Z η ( k ) ( e ) ≤ ⋁ e ∈ ξ ( ρ ) ⋁ f ∈ ξ ( ρ ) k ( f ) . So appr ¯ Z λ ( h ) ( x ) = ⋁ x ∈ ψ ( ρ ) ⋁ y ∈ ψ ( ρ ) h ( y ) and

appr ¯ Z η ( k ) ( e ) = ⋁ e ∈ ξ ( ρ ) ⋁ f ∈ ξ ( ρ ) k ( f ) . Which completes the proof.□

Proposition 2. Let G = ( G ∗ , ψ , ξ , A ) be a full soft graph then for h ∈ F ( V ) and k ∈ F ( E ) , we have ( 1 ) appr _ ( G ) ⊆ appr ¯ ( G ) ( 2 ) appr _ ( φ ) = appr ¯ ( φ ) = φ ( 3 ) appr _ ( G ∗ ) = appr ¯ ( G ∗ ) = G ∗ (4) ( appr ¯ ( G ) ) c = appr _ ( G c ) , ( appr _ ( G ) ) c = appr ¯ ( G c ) ( 5 ) appr _ ( G 1 ∩ G 2 ) = appr _ ( G 1 ) ∩ appr _ ( G 2 ) (6) appr _ ( G 1 ∪ G 2 ) ⊇ appr _ ( G 1 ) ∪ appr _ ( G 2 ) (7) appr ¯ ( G 1 ∪ G 2 ) = appr ¯ ( G 1 ) ∪ appr ¯ ( G 2 ) (8) appr ¯ ( G 1 ∩ G 2 ) ⊆ appr ¯ ( G 1 ) ∩ appr ¯ ( G 2 ) .

Proof. (1), (2) and (3) are straightforward. (4) Since appr ¯ ( G ) = ( appr ¯ Z λ ( h ) , appr ¯ Z η ( k ) ) , so we will prove only ( appr ¯ Z λ ( h ) ) c = appr _ Z λ ( h c ) and ( appr ¯ Z η ( k ) ) c = appr _ Z η ( k c ) .

Since appr ¯ Z λ ( h ) ( x ) = ⋁ { h ( y ) : y ∈ V , λ ( x ) = λ ( y ) }

and appr ¯ Z η ( k ) ( e ) = ⋁ { k ( f ) : f ∈ E , η ( e ) = η (f) }   . So h ( y ) ≤ appr ¯ Z λ ( h ) ( x ) and k ( f ) ≤ appr ¯ Z η ( k ) ( e ) . 1 - h ( y ) ≥ 1 - appr ¯ Z λ ( h ) ( x ) and 1 - k ( f ) ≥ 1 - appr ¯ Z η ( k ) ( e ) . h c ( y ) ≥ ( appr ¯ Z λ ( h ) ) c ( x ) and k c ( f ) ≥ ( appr ¯ Z η ( k ) ) c ( e ) . Or ⋀ { h c ( y ) : y ∈ V , λ ( x ) = λ ( y ) } ⊇ appr ¯ Z λ ( h ) c ( x ) and ⋀ { k c ( f ) : f ∈ E , η ( e ) = η ( f ) } ⊇ ( appr ¯ Z η ( k ) ) c ( e ) . Which shows

appr _ Z λ ( h c ) ( x ) ≥ ( appr ¯ Z λ ( h ) ) c ( x )

and appr _ Z η ( k c ) ( e ) ≥ ( appr ¯ Z η ( k ) ) c ( e ) .

So appr _ Z λ ( h c ) ⊇ ( appr ¯ Z λ ( h ) ) c ( x )

and appr _ Z η ( k c ) ( e ) ⊇ ( appr ¯ Z η ( k ) ) c ( e ) .

Now let r ∈ F ( V ) and m ∈ F ( E )

such that appr _ Z λ ( r ) ( x ) = ⋀ { r ( y ) : y ∈ V , λ (x) = λ (y)} and appr _ Z η ( m ) ( e ) = ⋀ { m ( f ) : f ∈ E , η ( e ) = η (f) }   .

Then r ( y ) ≥ appr _ Z λ ( r ) ( x ) and m ( f ) ≥ appr _ Z η (m) (e)  ,

or 1 − r ( y ) ≤ 1 − a p p r _ Z λ ( r ) ( x ) and 1 - m ( f ) ≤ 1 - appr _ Z η ( m ) ( e ) . r c ( y ) ≤ ( appr _ Z λ ( r ) ) c ( x ) and m c ( f ) ≤ ( appr _ Z η (m)) ^c (e) ⋁{ r ^c (y): y ∈ V, λ (x) = λ (y) }

r c ( y ) ≤ ( a p p r _ Z λ ( r ) ) c ( x ) and ⋁{ m ^c (f): f ∈ E, η (e) = η (f) }

So ∨ { r c ( y ) : y ∈ V , λ ( x ) = λ ( y ) } ≤ ( a p p r _ Z λ ( r ) ) c ( x ) a n d ∨ { m c ( f ) : f ∈ E , η ( e ) = η ( f ) } ≤ ( a p p r ¯ Z η ( m ) ) c ( e ) Replacing r by h ^c and m by n ^c , we get a p p r ¯ Z λ ( h c ) c ( x ) ≤ ( a p p r _ Z λ ( h c ) ) c ( x ) and a p p r ¯ Z η ( n c ) c ( e ) ≤ ( a p p r _ Z η ( n c ) ) c ( e ) a p p r ¯ Z η ( n c ) c ( e ) ≤ ( a p p r _ Z η ( n c ) ) c ( e ) and a p p r ¯ Z λ ( h c ) c ( x ) ⊆ ( a p p r _ Z λ ( h c ) ) c ( x ) So a p p r ¯ Z η ( n ) ( e ) ⊆ ( a p p r _ Z η ( n c ) ) c ( e ) and a p p r _ Z λ ( h c ) ⊆ ( a p p r ¯ Z η ( n ) ) c Since a p p r _ Z η ( n c ) ⊆ ( a p p r ¯ Z η ( n ) ) c , a p p r _ Z η ( n c ) ⊆ ( a p p r ¯ Z η ( n ) ) c and a p p r _ Z η ( n c ) ⊆ ( a p p r ¯ Z η ( n ) ) c , a p p r _ Z λ ( h c ) ⊆ ( a p p r ¯ Z λ ( h ) ) c . Therefore a p p r _ Z η ( n c ) ⊆ ( a p p r ¯ Z η ( n ) ) c and ( a p p r ¯ Z λ ( h ) ) c = a p p r _ Z λ ( h c ) . Hence ( a p p r ¯ Z η ( n ) ) c = a p p r _ Z η ( n c ) Similarly one can show that ( a p p r ¯ ( G ) ) c = a p p r _ ( G c ) (5) Suppose ( a p p r _ ( G ) ) c = a p p r ¯ ( G c ) and a p p r _ ( G 1 ∩ G 2 ) = ( a p p r _ Z λ ( h 1 ∩ h 2 ) , a p p r _ Z η ( K 1 ∩ k 2 ) ) where h _i ∈ a p p r _ ( G i ) = ( a p p r _ Z λ ( h i ) , a p p r _ Z η ( k i ) ) and h i ∈ F ( V ) for i = 1, 2. Using appr _ Z λ ( h 1 ∩ h 2 ) ( x ) =⋀ { h₁ (y) ∧ h₂ (y): y ∈ V, λ (x) = λ (y) } and appr _ Z η ( k 1 ∩ k 2 ) ( e ) = ⋀ { k₁ ((f)) ∧ k₂ (f): f ∈ E, η (e) = η (f) }   , we have, appr _ Z λ ( h 1 ∩ h 2 ) ( x ) ≤ h 1 ( y ) ∧ h 2 ( y ) ≤ h 1 ( y ) and appr _ Z η ( k 1 ∩ k 2 ) ( e i ) ≤ k 1 ( ( f ) ) ∧ k 2 ( f ) ≤ k 1 ( f ) . Which shows a p p r _ Z λ ( h 1 ∩ ​ h 2 ) ( x )

≤⋀ { h₁ (y): y ∈ V, λ (x) = λ (y) }

k i ∈ F ( E ) appr _ Z η ( k 1 ∩ k 2 ) ( e )

≤⋀ { m₁ (f): f ∈ E, η (e) = η (f) }

= appr _ Z η ( m 1 ) ( e ) .

It can also be seen that a p p r _ Z λ ( h 1 ∩ ​ h 2 ) ( x ) ≤ a p p r _ Z λ ( h 2 ) ( x )

and appr _ Z η ( k 1 ∩ k 2 ) ( e ) ≤ appr _ Z η ( k 2 ) ( e ) .

As appr _ Z λ ( h 1 ∩ h 2 ) ( x ) ≤ appr _ Z λ ( h 1 ) ( x )

and appr _ Z λ ( h 1 ∩ h 2 ) ( x ) ≤ appr _ Z λ ( h 2 ) ( x )

so appr _ Z λ ( h 1 ∩ h 2 ) ( x ) ≤ appr _ Z λ ( h 1 ) ( x ) ∧ appr _ Z λ ( h 2 ) ( x ) . Or appr _ Z λ ( h 1 ∩ h 2 ) ⊆ appr _ Z λ ( h 1 ) ∩ appr _ Z λ ( h 2 ) . Also appr _ Z η ( k 1 ∩ k 2 ) ( e ) ≤ appr _ Z η ( k 1 ) ( e )

and appr _ Z η ( k 1 ∩ k 2 ) ( e ) ≤ appr _ Z η ( k 2 ) ( e ) so appr _ Z η ( k 1 ∩ k 2 ) ( e ) ≤ appr _ Z η ( k 1 ) ( e ) ∧ appr _ Z η (k₂) (e) or appr _ Z η ( k 1 ∩ k 2 ) ⊆ appr _ Z η ( k 1 ) ∩ appr _ Z η ( k 2 ) . Hence appr _ Z λ ( h 1 ∩ h 2 ) ⊆ appr _ Z λ ( h 1 ) ∩ appr _ Z λ ( h 2 ) and appr _ Z η ( k 1 ∩ k 2 ) ⊆ appr _ Z η ( k 1 ) ∩ appr _ Z η ( k 2 ) . Now we show the reverse inclusion. It can see seen that ( appr _ Z λ ( h 1 ) ∩ appr _ Z λ ( h 2 ) ) ( x )

= appr _ Z λ ( h 1 ) ( x ) ∧ appr _ Z λ ( h 2 ) ( x ) ≤ appr _ Z λ ( h 1 ) ( x ) ≤ ( h 1 ) ( y ) and

( appr _ Z λ ( h 1 ) ∩ appr _ Z λ ( h 2 ) ) ( x ) = appr _ Z λ ( h 1 ) ( x ) ∧ appr _ Z λ ( h 2 ) ( x )

≤ appr _ Z λ ( h 2 ) ( x ) ≤ ( h 2 ) ( y ) so ( appr _ Z λ ( h 1 ) ∩ appr _ Z λ ( h 2 ) ) ( x ) ≤ ( h 1 ) ( y ) ∧ ( h 2 ) ( y ) Which shows ( appr _ Z λ ( h 1 ) ∩ appr _ Z λ ( h 2 ) ) ( x ) ≤ ⋀ { ( h 1 ) ( y ) ∧ ( h 2 ) ( y ) : y ∈ V , λ ( x ) = λ ( y ) } , = appr _ Z λ ( h 1 ∩ h 2 ) ( x ) , so ( appr _ Z λ ( h 1 ) ∩ appr _ Z λ ( h 2 ) ) ⊆ appr _ Z λ (h₁ ∩ h₂). Similarly it can be seen that ( appr _ Z η ( k 1 ) ∩ appr _ Z η ( k 2 ) ) ⊆ appr _ Z η ( k 1 ∩ k 2 ) . As ( appr _ Z λ ( h 1 ) ∩ appr _ Z λ ( h 2 ) ) ⊆ appr _ Z λ ( h 1 ∩ h 2 ) and ( appr _ Z λ ( h 1 ) ∩ appr _ Z λ ( h 2 ) ) ⊇ appr _ Z λ ( h 1 ∩ h 2 ) ,