A rough set approximates a subset using lower and upper bounds determined by equivalence classes, thereby capturing both certainty and uncertainty in membership [1–3]. The following definitions formalize these concepts.

Definition 1 (Set). [4] A set is a well-defined collection of distinct elements or objects. If is an element of a set , we write ∈ ; otherwise, we write ∉ .

Definition 2 (Subset). [4] Let and be sets. is called a subset of , denoted ⊆ , if every element of is also an element of . If ⊆ but ≠ , then is called a proper subset of , denoted ⊂ .

Definition 3 (Empty Set). [4] The empty set, denoted by ∅, is the unique set containing no elements. Formally, for any set , ∅ ⊆ .

Definition 4 (Universal Set). A universal set, denoted by , is the set that contains all elements under consideration in a particular context. Every set discussed is assumed to be a subset of .

Definition 5 (Rough Set Approximation). [5] Let be a nonempty universe of discourse, and let ⊆ × be an equivalence relation (also called an indiscernibility relation) on . The relation partitions into disjoint equivalence classes, denoted by [ ] for each ∈ , where [ ] = { ∈ | (, ) ∈ }.

For any subset ⊆ , the lower approximation and the upper approximation are defined by:

(1) Lower Approximation:

= { ∈ | [ ] ⊆ }.

This set contains all elements whose entire equivalence class is contained within ; these elements definitely belong to .

(2) Upper Approximation:

= { ∈ | [ ] ∩ ≠ ∅}.

This set contains all elements whose equivalence class has a nonempty intersection with ; these elements possibly belong to .

Thus, the pair ( , ) forms the rough set representation of , satisfying ⊆ ⊆ .

Example 1 (Weather Forecasting using Rough Sets). Weather Forecasting is the science of predicting atmospheric conditions—like temperature, precipitation, and wind—based on data from satellites, radars, and models ( [6–9]). Consider a meteorological dataset collected over 100 days for a specific region. Each day is characterized by four attributes:

• Temperature (in °C), • Humidity (in %), • Wind speed (in km/h), • Cloud cover (in %).

For illustration, assume we have data for five specific days:

Day Temperature Humidity Wind Speed Cloud Cover 1 20 70 15 80 2 22 75 10 90 3 18 65 20 50 4 21 80 12 85 5 19 68 18 55 Define an equivalence relation on days as follows: two days are equivalent if their temperatures differ by no more than 2°C and their humidities differ by no more than 5%. Suppose this rule yields the equivalence classes [1] = {1, 2, 4} and [3] = {3, 5}.

Assume historical records indicate that days 1, 2, and 4 were rainy, so we set the target concept = {1, 2, 4}. Then the lower approximation is defined as = { ∈ | [ ] ⊆ }.

127 Fujita |Uncert. Disc. Appl. 2(2) (2025) 124-138

Since the class [1] is entirely contained in , days 1, 2, and 4 are in ; however, the class [3] is not, so days 3 and 5 fall into the boundary region. This analysis helps forecasters handle uncertainties by grouping similar weather days.

The HyperRough Set extends rough set theory by incorporating multiple attributes. Its formal definition is given below [10–12].

Definition 6 (HyperRough Set). [10,13] Let be a nonempty finite universe, and let 1, 2, . . . , be distinct attributes with corresponding domains 1, 2, . . . , . Define the Cartesian product = 1 × 2 × · · · × .

Let ⊆ × be an equivalence relation on , with [ ] denoting the equivalence class of . A HyperRough Set over is a pair (, ), where:

• : → P( ) is a mapping that assigns to each attribute value combination = ( 1, 2, . . . , ) ∈ a subset ( ) ⊆ .

• For each ∈ , the rough set approximations of ( ) are defined as ( ) = { ∈ | [ ] ⊆ ( )}, ( ) = { ∈ | [ ] ∩ ( ) ≠ ∅}.

Here, ( ) comprises all elements whose equivalence classes are completely contained within ( ), while ( ) contains elements whose equivalence classes intersect ( ). Additionally, the following properties hold for all ∈ :

• ( ) ⊆ ( ).

• If ( ) = ∅, then ( ) = ( ) = ∅.

• If ( ) = , then ( ) = ( ) = .

Example 2 (Project Management using Hyperrough Sets). Project Management is the systematic process of planning, organizing, executing, and monitoring tasks to achieve specific goals within defined time, budget, and scope constraints (cf. [14,15]). Imagine a project portfolio with 6 projects characterized by two attributes:

• Duration (in months), • Risk Level (on a scale from 1 to 5).

Suppose the following data is available:

Project Duration Risk Level 1 12 2 2 18 4 3 15 3 4 20 5 5 10 1 6 16 3 Define an equivalence relation where two projects are equivalent if their durations differ by at most 3 months and their risk levels by at most 1. Assume the following equivalence classes result:

[ 1] = { 1, 3, 5} and [ 2] = { 2, 4, 6}.

Let the target concept be the set of projects completed on schedule, say = { 1, 3, 5}. The Hyperrough Set model uses the Cartesian product of the attribute domains (e.g., Duration × Risk Level) to form refined groups. For example, consider the attribute combination (12, 2) with ((12, 2)) = { 1, 5}.

Then the lower approximation of with respect to this combination includes projects whose equivalence class is completely contained in . This multi-attribute analysis allows project managers to evaluate projects more comprehensively by considering both duration and risk factors simultaneously.

Probabilistic hyperrough set and coveringhyper rough set 128

An n-SuperHyperRough Set generalizes rough sets by using power sets of attribute values to produce nuanced approximations under uncertainty. The definition of -SuperHyperRough Sets is described as follows.

Definition 7 ( -SuperHyperRough Set). [10, 16] Let be a nonempty finite universe, and let 1, 2, . . . , be distinct attributes with respective domains 1, 2, . . . , . For each attribute , let P( ) denote its power set. Define the set of all possible attribute value combinations as = P( 1) × P( 2) × · · · × P( ).

Let ⊆ × be an equivalence relation on . An -SuperHyperRough Set over is a pair (, ), where:

• : → P( ) is a mapping that assigns to each attribute value combination = ( 1, 2, . . . , ) ∈ (with ⊆ for all ) a subset ( ) ⊆ .

• For each ∈ , the lower and upper approximations are defined as ( ) = { ∈ | [ ] ⊆ ( )}, ( ) = { ∈ | [ ] ∩ ( ) ≠ ∅}.

Thus, ( ) consists of all elements whose equivalence classes are entirely contained in ( ), and ( ) includes those elements whose equivalence classes intersect ( ). The following properties hold for all ∈ :

• ( ) ⊆ ( ).

• If ( ) = ∅, then ( ) = ( ) = ∅.

• If ( ) = , then ( ) = ( ) = .

• For any , ∈ , ( ∩ ) ⊆ ( ) ∩ ( ), ( ∪ ) ⊇ ( ) ∪ ( ).

Example 3 (Project Management using Superhyperrough Sets). Consider a scenario where a company evaluates 4 projects based on cost and quality. The attributes are defined as follows:

• Cost with domain 1 = {Low, Medium, High}, • Quality with domain 2 = {Poor, Fair, Good}.

For a more flexible analysis, the Superhyperrough Set model considers the power sets P( 1) and P( 2). For instance, an element of P( 1) could be {Low} or {Low, Medium}, and similarly for P( 2). Define the super attribute space as = P( 1) × P( 2).

Suppose the following project data is given:

Project Cost Quality 1 Low Good 2 Medium Fair 3 High Poor 4 Low Fair Let the target concept be projects that are considered "acceptable" based on performance, say = { 1, 2, 4}. Using the super attribute space, we can form attribute combinations such as ({Low}, {Good, Fair}), and define ({Low}, {Good, Fair}) = { 1, 4}.

Then, the lower approximation of with respect to this combination consists of projects for which every project in the corresponding equivalence class is in . This finer granularity in handling attribute subsets helps decision makers identify projects with acceptable performance more precisely, even when cost and quality assessments are subject to uncertainty. 129 Fujita | | Uncert. Disc. Appl. 2(2) (2025)124-138

Probabilistic Rough Sets provide a framework for estimating uncertainty by utilizing membership probabilities, allowing for the definition of lower and upper approximations based on specified threshold values [17,18]. These sets offer a flexible and intuitive approach to handling imprecise data. We formally define Probabilistic Rough Sets below. Due to their simplicity and effectiveness, they have been extensively studied in numerous research works [19–23].

Definition 8 (Probabilistic Rough Set). (cf. [17, 18]) Let be a finite universe and let ⊆ × be an equivalence relation on . This relation partitions into disjoint equivalence classes, denoted by [ ] for each ∈ . For any subset ⊆ , define the rough membership function : → [0, 1] by | ∩ [ ] | ( ) = , for every ∈ , | [ ] | where | | denotes the cardinality of set . Given two thresholds , ∈ [0, 1] satisfying 0 ≤ ≤ ≤ 1, the -level probabilistic lower approximation of is defined as = { ∈ | ( ) ≥ }, and the -level probabilistic upper approximation of is defined as = { ∈ | ( ) > }.

These definitions yield a probabilistic characterization of the set : comprises those elements that are almost certainly in (with membership at least ), while contains those elements that possibly belong to (with membership exceeding

Example 4 (Example of a Probabilistic Rough Set). Consider the finite universe = {1, 2, 3, 4, 5, 6}, and define an equivalence relation on such that the equivalence classes are:

[1] = {1, 2}, [3] = {3, 4}, [5] = {5, 6}.

Let ⊆ be given by = {1, 2, 3, 5}.

We compute the rough membership function ( ) for each ∈ :

• For ∈ [1] = {1, 2}:

|{1, 2} ∩ | 2 (1) = = = 1, (2) = 1.

• For ∈ [3] = {3, 4}:

|{3, 4} ∩ | 1 (3) = = = 0.5, (4) = 0.5.

• For ∈ [5] = {5, 6}:

|{5, 6} ∩ | 1 (5) = = = 0.5, (6) = 0.5.

Now, select thresholds = 0.8 and = 0.3. Then the -level probabilistic lower approximation of is = { ∈ | ( ) ≥ 0.8} = {1, 2}, since only the elements in the equivalence class {1, 2} have a membership value of 1 (which is ≥ 0.8). In contrast, the -level probabilistic upper approximation of is 0.3 = { ∈ | ( ) > 0.3} = , because for every ∈ , ( ) is either 0.5 or 1, both of which exceed 0.3.

Thus, the probabilistic rough set approximations of are:

= {1, 2} and 0.3 = {1, 2, 3, 4, 5, 6}.

This example concretely illustrates how the rough membership function and the threshold parameters and are used to derive probabilistic approximations of a set.

In classical rough set theory, an equivalence relation (or partition) is used to approximate a subset of a universe. In many real-world applications, however, the information available does not naturally yield a partition but rather a cover – a family of overlapping subsets whose union is the entire universe. Covering rough sets extend the classical model by using such covers to define approximations [24–28].

Definition 9 (Cover). (cf. [25]) Let be a finite, nonempty set (called the universe). A collection C of nonempty subsets of is called a cover of if Ø = .

Unlike a partition, the subsets in C may overlap; that is, for some ∈C , ∈ C with ≠ , we may have ∩ ≠ ∅.

1 2 1 2 1 2

Definition 10 (Covering Rough Set). [25] Let be a finite universe and let C be a cover of . For any subset ⊆ , we define:

(1) The covering lower approximation of with respect to C as = { ∈ C | ⊆ }.

This is the union of all covering sets that are completely contained in . Intuitively, these are the elements that definitely belong to as their entire covering set is a subset of .

(2) The covering upper approximation of with respect to C as C = { ∈ C | ∩ ≠ ∅}.

This is the union of all covering sets that have a nonempty intersection with . Intuitively, these are the elements that possibly belong to because at least part of their covering set intersects .

The pair , C is called the covering rough set of with respect to the cover C.

A special case occurs when C is a partition. In that case, the covering lower and upper approximations coincide with the classical rough set approximations.

Example 5 (Example of a Covering Rough Set). Consider the universe = {1, 2, 3, 4, 5, 6}.

Define a cover C of by C = { 1, 2, 3, 4}, where 1 = {1, 2, 3}, 2 = {2, 3, 4}, 3 = {4, 5}, 4 = {5, 6}.

Note that = {1, 2, 3, 4, 5, 6} = .

Now, let = {2, 3, 4, 5} ⊆ .

We compute the approximations as follows:

(1) Covering Lower Approximation: We include every ∈ C that is entirely contained in :

• 1 = {1, 2, 3} is not included because 1 ∉ .

• 2 = {2, 3, 4} ⊆ is included.

• 3 = {4, 5} ⊆ is included.

• 4 = {5, 6} is not included because 6 ∉ .

Therefore, = 2 ∪ 3 = {2, 3, 4} ∪ {4, 5} = {2, 3, 4, 5} = .

131 Fujita| |Uncert. Disc. Appl. 2(2) (2025) 124-138

(2) Covering Upper Approximation: We include every ∈ C that has a nonempty intersection with :

• 1 ∩ = {2, 3} ≠ ∅, so 1 is included.

• 2 ∩ = {2, 3, 4} ≠ ∅, so 2 is included.

• 3 ∩ = {4, 5} ≠ ∅, so 3 is included.

• 4 ∩ = {5} ≠ ∅, so 4 is included.

Therefore, C = 1 ∪ 2 ∪ 3 ∪ 4 = {1, 2, 3} ∪ {2, 3, 4} ∪ {4, 5} ∪ {5, 6} = {1, 2, 3, 4, 5, 6} = .

Thus, the covering rough set approximation of is , C = , .

The boundary region of is given by BND( ) = C \ = \ = {1, 6}.

This example illustrates a scenario where is exactly defined by the cover in the sense that the lower approximation equals , yet the upper approximation spans the entire universe due to overlapping cover sets.

We define Probabilistic hyperrough Sets as follows.

Definition 11 (Probabilistic Hyperrough Set). Let be a finite universe and let ⊆ × be an equivalence relation on that partitions into disjoint equivalence classes [ ] for each ∈ . Let 1, 2, . . . , be distinct attributes with corresponding domains 1, 2, . . . , . Define the Cartesian product = 1 × 2 × · · · × .

A mapping : → P( ) assigns to each attribute combination = ( 1, 2, . . . , ) ∈ a subset ( ) ⊆ . For each ∈ , the rough membership function of ( ) is defined by | ( ) ∩ [ ] | ( ) ( ) = , ∀ ∈ .

| [ ] | Given thresholds , ∈ [0, 1] satisfying 0 ≤ ≤ ≤ 1, we define the -level probabilistic lower approximation of ( ) by ( ) = { ∈ | ( ) ( ) ≥ }, and the -level probabilistic upper approximation of ( ) by ( ) = { ∈ | ( ) ( ) > }.

The pair (, ), together with the family of approximation operators { ( ) , ( ) | ∈ }, is called a Probabilistic Hyperrough Set over .

Theorem 1. A Probabilistic Hyperrough Set generalizes both the classical Probabilistic Rough Set and the Hyperrough Set.

Proof: We consider two limiting cases:

(1) Singleton Attribute Space: If the attribute space is a singleton (i.e., = { 0}), then the mapping reduces to a single subset ( 0) ⊆ . In this case, the rough membership function ( ) ( ) is identical to that in the standard Probabilistic Rough Set model. Consequently, the approximations ( 0) and ( 0) recover the classical probabilistic rough set approximations.

(2) Extreme Thresholds: If we set the thresholds to = 1 and = 0, then for every ∈ , ( ) = { ∈ | ( ) ( ) = 1} and ( )0 = { ∈ | ( ) ( ) > 0}.

These definitions coincide with those of the Hyperrough Set model, where the lower approximation comprises all elements whose entire equivalence classes are contained in ( ), and the upper approximation includes all elements whose equivalence classes intersect ( ).

Thus, the Probabilistic Hyperrough Set model encompasses both the Probabilistic Rough Set (singleton case) and the Hyperrough Set (extreme threshold case) models. □

Example 6 (Probabilistic Hyperrough Set). Consider the finite universe = {1, 2, 3, 4, 5, 6}, with an equivalence relation defined by the following equivalence classes:

[1] = {1, 2}, [3] = {3, 4}, [5] = {5, 6}.

Let there be a single attribute 1 with domain 1 = {red, blue}. Then, = 1 = {red, blue}.

Define the mapping : → P( ) by:

(red) = {1, 2, 3} and (blue) = {4, 5, 6}.

For = red, compute the rough membership function:

• For ∈ [1] = {1, 2}:

|{1, 2} ∩ {1, 2, 3}| 2 (red) ( ) = = = 1.

• For ∈ [3] = {3, 4}:

|{3, 4} ∩ {1, 2, 3}| 1 (red) ( ) = = = 0.5.

• For ∈ [5] = {5, 6}:

|{5, 6} ∩ {1, 2, 3}| (red) ( ) = = 0.

If we choose thresholds = 0.8 and = 0.3, then:

(red) = { ∈ | (red) ( ) ≥ 0.8} = {1, 2}, and (red)0.3 = { ∈ | (red) ( ) > 0.3} = {1, 2, 3, 4}.

Thus, the probabilistic hyperrough approximations for the attribute value "red" are (red) = {1, 2} and (red)0.3 = {1, 2, 3, 4}.

Similarly, for = blue, with (blue) = {4, 5, 6}:

• For ∈ [1] = {1, 2}:

|{1, 2} ∩ {4, 5, 6}| (blue) ( ) = = 0.

• For ∈ [3] = {3, 4}:

|{3, 4} ∩ {4, 5, 6}| 1 (blue) ( ) = = = 0.5.

133 Fujita| |Uncert. Disc. Appl. 2(2) (2025) 124-138

• For ∈ [5] = {5, 6}:

|{5, 6} ∩ {4, 5, 6}| 2 (blue) ( ) = = = 1.

With the same thresholds, we obtain:

(blue) = { ∈ | (blue) ( ) ≥ 0.8} = {5, 6}, and (blue)0.3 = { ∈ | (blue) ( ) > 0.3} = {3, 4, 5, 6}.

This example clearly illustrates how the probabilistic hyperrough set model works.

We define Probabilistic -Superhyperrough Sets as follows.

Definition 12 (Probabilistic -Superhyperrough Set). Let be a finite universe and ⊆ × be an equivalence relation on . Let 1, 2, . . . , be distinct attributes with corresponding domains 1, 2, . . . , . For each attribute , denote by P( ) its power set, and define the Cartesian product = P( 1) × P( 2) × · · · × P( ).

A mapping : → P( ) assigns to each combination = ( 1, 2, . . . , ) ∈ (with ⊆ ) a subset ( ) ⊆ . For each ∈ , define the rough membership function ( ) : → [0, 1] by | ( ) ∩ [ ] | ( ) ( ) = , ∀ ∈ .

| [ ] | Given thresholds , ∈ [0, 1] with 0 ≤ ≤ ≤ 1, the -level probabilistic lower approximation of ( ) is defined as ( ) = { ∈ | ( ) ( ) ≥ }, and the -level probabilistic upper approximation is defined as ( ) = { ∈ | ( ) ( ) > }.

Then, the pair (, ) is called a Probabilistic -Superhyperrough Set over .

Theorem 2. A Probabilistic -Superhyperrough Set generalizes both the Probabilistic Hyperrough Set and the classical -Superhyperrough Set.

Proof: (i) If, for each attribute , the power set P ( ) is restricted to singletons (i.e., is a singleton for every ), then reduces to the Cartesian product 1 × 2 × · · · × . In this case, the mapping becomes identical to that of the Hyperrough Set model, and the approximations ( ) and ( ) are the same as those in the Probabilistic Hyperrough Set model. (ii) Conversely, if the thresholds are set to their extreme values, = 1 and = 0, then for each ∈ , ( ) = { ∈ | ( ) ( ) = 1} and ( )0 = { ∈ | ( ) ( ) > 0}, which coincides with the classical -Superhyperrough Set definitions.

Thus, the Probabilistic -Superhyperrough Set model encompasses both the Probabilistic Hyperrough Set and the -Superhyperrough Set. □

Example 7 (Probabilistic -Superhyperrough Set Example). Let = {1, 2, 3, 4, 5}, with an equivalence relation defined by:

[1] = {1, 2}, [3] = {3, 4}, [5] = {5}.

Assume a single attribute 1 with domain 1 = {A, B}. Then, its power set is P ( 1) = {∅, {A}, {B}, {A, B}}, and we set = P ( 1). Define the mapping : → P ( ) as follows:

(∅) = ∅, ({A}) = {1, 2, 3}, ({B}) = {4, 5}, ({A, B}) = .

For = {A}, we have ({A}) = {1, 2, 3}. Then:

• For ∈ [1] = {1, 2}:

|{1, 2} ∩ {1, 2, 3}| ( {A} ) ( ) = = 1.

• For ∈ [3] = {3, 4}:

|{3, 4} ∩ {1, 2, 3}| 1 ( {A} ) ( ) = = = 0.5.

• For ∈ [5] = {5}:

|{5} ∩ {1, 2, 3}| ( {A} ) ( ) = = 0.

Choosing thresholds = 0.8 and = 0.3, we obtain:

({A}) = { ∈ | ( {A} ) ( ) ≥ 0.8} = {1, 2}, and ({A})0.3 = { ∈ | ( {A} ) ( ) > 0.3} = {1, 2, 3, 4}.

For = {B}, with ({B}) = {4, 5}:

• For ∈ [3] = {3, 4}:

|{3, 4} ∩ {4, 5}| 1 ( {B} ) ( ) = = = 0.5.

• For ∈ [5] = {5}:

|{5} ∩ {4, 5}| ( {B} ) ( ) = = 1.

• For ∈ [1] = {1, 2}:

( {B} ) ( ) = 0.

Thus, ({B}) = {5} and ({B})0.3 = {3, 4, 5}.

This example clearly demonstrates how the Probabilistic -Superhyperrough Set model refines the approximations by incorporating power set combinations of attribute values.

We define Covering Hyperrough Set as follows.

Definition 13 (Covering Hyperrough Set). Let be a finite universe, C a cover of , and let 1, 2, . . . , be distinct attributes with corresponding domains 1, 2, . . . , . Define the Cartesian product = 1 × 2 × · · · × .

A mapping : → P( ) 135 Fujita|Uncert. Disc. Appl. 2(2) (2025) 124-138

assigns to each attribute combination = ( 1, 2, . . . , ) ∈ a subset ( ) ⊆ . Then, for each ∈ , the covering hyperrough approximations of ( ) with respect to the cover C are defined as:

( ) = { ∈ C | ⊆ ( )},

( ) C = { ∈ C | ∩ ( ) ≠ ∅}.

The collection (, ) together with the family {( ( ) , ( ) C ) | ∈ } is called a Covering Hyperrough Set over . Theorem 3. A Covering Hyperrough Set generalizes both the classical Covering Rough Set and the Hyperrough Set.

Proof: Case 1 (Reduction to Covering Rough Set): If the attribute space is a singleton (i.e., = { 0}) and ( 0) = for some ⊆ , then the definitions reduce to:

( 0) = and ( 0) C = C, which is exactly the classical covering rough set approximation of .

Case 2 (Reduction to Hyperrough Set): If the cover C is a partition of (i.e., each ∈ C is an equivalence class of an equivalence relation ) and is constant (so that ( ) = for all ∈ ), then the approximations become ( ) = { ∈ | [ ] ⊆ } and ( ) C = { ∈ | [ ] ∩ ≠ ∅}, which are precisely the hyperrough set approximations of .

Thus, the Covering Hyperrough Set model indeed generalizes both frameworks. □

Example 8 (Covering Hyperrough Set). Let = {1, 2, 3, 4, 5, 6}, and consider the cover C = { 1, 2, 3}, with 1 = {1, 2}, 2 = {2, 3, 4}, 3 = {5, 6}.

Let there be one attribute 1 with domain 1 = {red, blue} so that = {red, blue}.

Define the mapping : → P ( ) by:

(red) = {1, 2, 3, 4}, (blue) = {3, 4, 5, 6}.

For = red, we have (red) = {1, 2, 3, 4}:

• 1 = {1, 2} ⊆ (red) ⇒ included in the lower approximation.

• 2 = {2, 3, 4} ⊆ (red) ⇒ included in the lower approximation.

• 3 = {5, 6} does not intersect (red) ⇒ excluded.

Thus, (red) = 1 ∪ 2 = {1, 2, 3, 4}.

For the upper approximation, all cover elements that intersect (red) are included. Here, 1 ∩ (red) ≠ ∅, 2 ∩ (red) ≠ ∅, so (red) C = 1 ∪ 2 = {1, 2, 3, 4}.

For = blue, with (blue) = {3, 4, 5, 6}:

• 1 = {1, 2} does not intersect (blue) ⇒ excluded.

• 2 = {2, 3, 4} has nonempty intersection ({3, 4}) ⇒ included in the upper approximation.

• 3 = {5, 6} ⊆ (blue) ⇒ included in both the lower and upper approximations.

Thus, (blue) = 3 = {5, 6}, (blue) C = 2 ∪ 3 = {2, 3, 4, 5, 6}.

This example demonstrates how the Covering Hyperrough Set model uses both the cover C and the attribute mapping to yield approximations.

We define Covering -Superhyperrough Set as follows.

Definition 14 (Covering -Superhyperrough Set). Let be a finite universe and let C be a cover of . Let 1, 2, . . . , be distinct attributes with corresponding domains 1, 2, . . . , . For each attribute , let P( ) denote its power set, and define the Cartesian product = P( 1) × P( 2) × · · · × P( ).

A mapping : → P( ) assigns to each attribute value combination = ( 1, 2, . . . , ) ∈ (with ⊆ for each ) a subset ( ) ⊆ . The covering -superhyperrough approximations of ( ) with respect to the cover C are defined as:

( ) = { ∈ C | ⊆ ( )},

( ) C = { ∈ C | ∩ ( ) ≠ ∅}.

The pair (, ) is then called a Covering -Superhyperrough Set over .

Theorem 4. A Covering -Superhyperrough Set generalizes both the Covering Hyperrough Set and the classical -Superhyperrough Set.

Proof: (i) Reduction to Covering Hyperrough Set: If for each attribute we restrict P ( ) to singleton sets (i.e., every is a singleton), then becomes = 1 × 2 × · · · × , and the mapping coincides with that used in the Covering Hyperrough Set model. Thus, the approximations ( ) and ( ) C are exactly those of a Covering Hyperrough Set.

(ii) Reduction to Classical -Superhyperrough Set: If, additionally, the cover C is a partition (i.e., it is induced by an equivalence relation on ), then the approximations reduce to:

( ) = { ∈ | [ ] ⊆ ( )}, ( ) C = { ∈ | [ ] ∩ ( ) ≠ ∅}, which are precisely the approximations used in the classical -Superhyperrough Set model.

Therefore, the Covering -Superhyperrough Set model unifies and generalizes both the Covering Hyperrough Set and the -Superhyperrough Set. □

Example 9 (Covering -Superhyperrough Set). Let = {1, 2, 3, 4, 5, 6, 7, 8}, and let the cover C of be given by C = { 1, 2, 3}, where 1 = {1, 2, 3}, 2 = {3, 4, 5}, 3 = {6, 7, 8}.

Assume there is one attribute 1 with domain 1 = {X, Y}. Then, the power set is P( 1) = {∅, {X}, {Y}, {X, Y}}, 137 Fujita|Uncert. Disc. Appl. 2(2) (2025) 124-138

and we set = P ( 1). Define the mapping : → P ( ) by:

(∅) = ∅, ({X}) = {1, 2, 3, 4}, ({Y}) = {4, 5, 6, 7}, ({X, Y}) = .

For = {X}, ({X}) = {1, 2, 3, 4}:

• 1 = {1, 2, 3} ⊆ {1, 2, 3, 4} is included in the lower approximation.

• 2 = {3, 4, 5} is not completely contained in {1, 2, 3, 4} (since 5 ∉ ({X})), but 2 ∩ ({X}) = {3, 4} ≠ ∅ so it is included in the upper approximation.

• 3 = {6, 7, 8} does not intersect {1, 2, 3, 4} and is excluded.

Thus, ({X}) = 1 = {1, 2, 3}, ({X}) C = 1 ∪ 2 = {1, 2, 3, 4, 5}.

For = {Y}, ({Y}) = {4, 5, 6, 7}:

• 1 does not intersect ({Y}) and is excluded.

• 2 = {3, 4, 5} has a nonempty intersection with ({Y}) ({4, 5}) and is included.

• 3 = {6, 7, 8} satisfies 3 ∩ ({Y}) = {6, 7} and is also included.

Thus, = 3 ∩ ({Y}) (if 3 ⊆ ({Y})), here 3 ⊈ ({Y}), ({Y}) so we take the union of those completely contained in ({Y}). Suppose only 3 qualifies partially, then the lower approximation might be empty; however, the upper approximation is given by ({Y})C = 2 ∪ 3 = {3, 4, 5, 6, 7, 8}.

This example shows that the approximations vary with the chosen attribute combination. The model is flexible enough to capture both disjunctive and conjunctive relationships among attribute values.