Definition 1. (see [2].)Let Π be a universal set (space of points or objects). A fuzzy set (class) Ϝ over Π is a set described by a function η_Ϝ : Π⟶[0, 1]. η_Ϝ is said to be the characteristic, indicator, or membership function of the fuzzy set Ϝ, and η_Ϝ(u) is said to be the membership grade value of π ∈ Π in Ϝ. A fuzzy set Ϝ over a universal set Π can be written as Ϝ = {(η_Ϝ(π)/π): π ∈ Π, η_Ϝ(π) ∈ [0, 1]} or Ϝ = {(π, η_Ϝ(π)): π ∈ Π, η_Ϝ(π) ∈ [0, 1]}.

Definition 2. (see [32].)A bipolar-valued fuzzy set $$ on Π can be represented as the following formula: $$, taking into account that $$ expresses the positive characteristic function describing the satisfaction amount of π to the feature corresponding to $$, and $$ expresses the negative characteristic function describing the satisfaction amount of π to the counter-feature of $$.

Definition 3. (see [63].)A multifuzzy set $$ of dimension n over Π is represented by a set of ordered sequences as follows: $$, where for i = 1, 2, ⋯, n, we have $$ representing the membership functions. The function $$ is said to be a fuzzy multimembership function of a multifuzzy set $$ of dimension n.

Definition 4. (see [65].)A bipolar-valued multifuzzy set $$ of dimension n over Π is given by the following structure: $$, where for i = 1, 2, ⋯, n, we have $$ representing the positive membership degrees denoting the satisfaction degrees of π to some properties corresponding to $$ and $$ representing the negative membership degrees denoting the satisfaction degrees of π to some implicit counter-properties of $$.

Definition 5. (see [4].)Given the universal set Π = {π₁, π₂, ⋯, π_n}, a vague set V over Π is a set determined by a truth membership function τ_V and a false membership function η_V. The exact grade of membership of π ∈ Π ( μ_V(π)) belongs to an interval [τ_V(π), 1 − η_V(π)]⊆[0, 1], i.e., μ_V(π) may be unknown, but it is bounded by τ_V(π) ≤ μ_V(π) ≤ 1 − η_V(π), where τ_V(π) + η_V(π) ≤ 1 and τ_V, η_V : Π⟶[0, 1]. V can be written as V = {([τ_V(π), 1 − η_V(π)]/π): π ∈ Π, τ_V(π), η_V(π) ∈ [0, 1]}, or V = {(π, [τ_V(π), 1 − η_V(π)]): π ∈ Π, τ_V(π), η_V(π) ∈ [0, 1]}.

Definition 6. (see [68].)A bipolar-valued vague set $$ over the universal set Π is defined by $$, where $$ are the positive truth and false membership functions denoting the satisfaction degree of an element π to the property corresponding to $$, such that $$ and $$ are the negative truth and false membership functions denoting the satisfaction degree of π to some implicit counter-property of $$, such that $$, i.e., the intervals $$ and $$ denote the satisfaction region of π to the property corresponding to $$ and to some implicit counter-property of $$, respectively.

Definition 7. (see [64].)A multivague set $$ of dimension n over Π is represented by a set of ordered sequences as follows: $$, where for i = 1, 2, ⋯, n, we have $$ and $$, the truth and false membership functions, respectively, such that $$, i.e., the intervals $$ denote the satisfaction intervals of π to some properties corresponding to $$. The function $$ is said to be a vague multimembership function of a multivague set $$ of dimension n.

Definition 8. (see [5].)Suppose that Π is a universal set, Y is a set of attributes (or parameters) and Λ⊆Y. The power set of Π is given by P(Π) = 2^Π. A pair (Γ, Λ) or Γ_Λ is said to be a soft set over Π, where Γ is a mapping characterized by Γ : Λ⟶P(Π). Also, Γ_Λ may be written as a set of ordered pairs Γ_Λ = {(λ, Γ_Λ(λ)): λ ∈ Λ, Γ_Λ(λ) ∈ P(Π)}. Λ is called the support of Γ_Λ, and we have Γ_Λ(λ) ≠ ϕ for all λ ∈ Λ and Γ_Λ(λ) = ϕ for all λ ∉ Λ. In other words, a soft set (Γ, Λ) over Π is a parameterized family of subsets of the set Π.

Definition 9. (see [64].)A multivague soft set of dimension n over the universal set Π is a parameterized family of multivague sets of the universal set Π, stated above in Definition 7.

Definition 10. A bipolar-valued multivague set $$ of dimension n over the universal set Π is defined by $$, where for i = 1, 2, ⋯, n, we have $$, the positive truth and false membership functions denoting the satisfaction degrees of an element π to some properties corresponding to $$, such that $$ and $$ are the negative truth and false membership functions denoting the satisfaction degrees of π to some implicit counter-properties of $$, such that $$, i.e., the intervals $$ and $$ denote the satisfaction regions of π to some properties corresponding to $$ and to some implicit counter-properties of $$, respectively.

Definition 11. Assume that Π is a universal set, Y is a parameter set and Λ⊆Y. Then, a pair (ℶ, Λ) given by $$ is said to be a bipolar-valued multivague soft set of dimension n over the universal set Π, taking into account that ℶ is a mapping defined as $$, where $$ is the power set of bipolar-valued multivague sets of dimension n on Π (i.e., the family of all bipolar-valued multivague subsets of dimension n of Π) and the bipolar-valued multivague subset of dimension n of Π is as stated above in Definition 10.

Definition 12. Any bipolar-valued multivague soft set (ℶ, Λ) of dimension n on the universal set Π is said to be complete (or absolute), which stands for $$, if for all λ ∈ Λ, we have $$. That is to say that, for i = 1, 2, ⋯, n, we have $$, $$ (i.e., $$), $$ (i.e., $$), and $$, for all λ ∈ Λ and for all π ∈ Π. According to those assumptions, we have $$.

Definition 13. Any bipolar-valued multivague soft set (ℶ, Λ) of dimension n on the universal set Π is said to be null (or empty), which stands for $$, if for all λ ∈ Λ, we have ℶ_Λ(λ) = ϕ. That is to say that, for i = 1, 2, ⋯, n, we have $$, $$ (i.e., $$), $$ (i.e., $$), and $$, for all λ ∈ Λ and for all π ∈ Π. According to those assumptions, we have $$.

Definition 14. The complement of a bipolar-valued multivague soft set (ℶ, Λ) of dimension n is defined by (ℶ, Λ)^c = (ℶ^c, Λ), where $$ is a mapping given, for i = 1, 2, ⋯, n, by the following: $$, $$, $$, and $$, for all λ ∈ Λ and for all π ∈ Π. Then, we have

Definition 15. Assume that (Γ, Λ) and (Ψ, Θ) are two bipolar-valued multivague soft sets of dimension n on a universal set Π. Then, (Γ, Λ) is called a bipolar-valued multivague soft subset of (Ψ, Θ) if Λ⊆Θ, and Γ(λ)⊆Ψ(λ) for all λ ∈ Λ, that is to say that, for i = 1, 2, ⋯, n, we have $$, $$, $$, and $$, i.e., for i = 1, 2, ⋯, n, we have $$, $$, $$, and $$, for all λ ∈ Λ and for all π ∈ Π. One can write $$. In this case, (Ψ, Θ) is called a bipolar-valued multivague soft superset of (Γ, Λ), denoted by $$.

Definition 16. Two bipolar-valued multivague soft sets (Γ, Λ) and (Ψ, Θ) of dimension n on a common universal set Π are called bipolar-valued multivague soft equal if they are bipolar-valued multivague soft subsets of each other, i.e., $$ and $$.

Definition 17. The union of two bipolar-valued multivague soft sets (Γ, Λ) and (Ψ, Θ) of dimension n on a common universal set Π is a bipolar-valued multivague soft set (ℶ, Δ) of dimension n, written as $$, where Δ = Λ ∪ Θ and for all δ ∈ Δ:

Definition 18. The union of a family {(Γ_i, Λ_i): i ∈ I} of bipolar-valued multivague soft sets of dimension n over a universal set Π is a bipolar-valued multivague soft set (ℶ, Δ) of dimension n, written as $$, where Δ = ∪Λ_i, for all i ∈ I defined as follows, for all δ ∈ Δ:

Definition 19. The restricted union of two bipolar-valued multivague soft sets (Γ, Λ) and (Ψ, Θ) of dimension n on a common universal set Π is a bipolar-valued multivague soft set (ℶ, Δ) of dimension n, written as $$, where Δ = Λ∩Θ ≠ ϕ and for all δ ∈ Δ:

Definition 20. The restricted union of a family {(Γ_i, Λ_i): i ∈ I} of bipolar-valued multivague soft sets of dimension n over a universal set Π is a bipolar-valued multivague soft set (ℶ, Δ) of dimension n, written as $$, where Δ = ∩Λ_i, for all i ∈ I defined by ℶ_Δ(δ) = ∪Γ_i(δ), δ ∈ Δ = ∩Λ_i.

Definition 21. The intersection of two bipolar-valued multivague soft sets (Γ, Λ) and (Ψ, Θ) of dimension n over a common universal set Π is a bipolar-valued multivague soft set (ℶ, Δ) of dimension n, written as $$, where Δ = Λ ∪ Θ and for all δ ∈ Δ:

Definition 22. The intersection of a family {(Γ_i, Λ_i): i ∈ I} of bipolar-valued multivague soft sets of dimension n over a universal set Π is a bipolar-valued multivague soft set (ℶ, Δ) of dimension n, written as $$, where Δ = ∪Λ_i, for all i ∈ I defined as follows, for all δ ∈ Δ:

Definition 23. The restricted intersection of two bipolar-valued multivague soft sets (Γ, Λ) and (Ψ, Θ) of dimension n on a common universal set Π is a bipolar-valued multivague soft set (ℶ, Δ) of dimension n, written as $$, where Δ = Λ∩Θ ≠ ϕ and for all δ ∈ Δ:

Definition 24. The restricted intersection of a family {(Γ_i, Λ_i): i ∈ I} of bipolar-valued multivague soft sets of dimension n over a universal set Π is a bipolar-valued multivague soft set (ℶ, Δ) of dimension n, written as $$, where Δ = ∩Λ_i, for all i ∈ I defined by ℶ_Δ(δ) = ∩Γ_i(δ), δ ∈ Δ = ∩Λ_i.

Example 1. Suppose that the bipolar-valued multivague soft set (ℶ, Y) of dimension (order) 3 describes some features of five employees nominated for a position in a certain organization, denoted by π₁, π₂, π₃, π₄, π₅ in the universal set Π, i.e., Π = {π₁, π₂, π₃, π₄, π₅}. Let the two major sets of attributes be Y = {λ₁, λ₂, λ₃}, λ_i(i = 1, 2, 3) and its opposite (counter) set −Y = {−λ₁, −λ₂, −λ₃}, −λ_i(i = 1, 2, 3) stand for the properties and the counter-properties, respectively. These attributes are represented in the following main parameters: mental abilities, leadership or social skills, and professional or organizational talents, respectively. Mental abilities and their counter-abilities are as follows: “intelligence” and “stupidity”; “memory” and “forgetfulness”; and “problem-solving” and “puzzlement.” Leadership skills and their counter-skills are as follows: “patience” and “boredom”; “fairness” and “injustice”; and “cooperation” and “competition.” Professional talents and their counter-talents are as follows: “flexibility” and “strictness”; “precision” and “negligence”; and “experience” and “rawness.” After doing several oral and written tests for them as well as subjecting them to some personal interviews and tests for intelligence and psychological aspects by a selection committee, the following bipolar-valued multivague soft set of dimension 3, namely (ℶ, Y) is obtained:

(ℶ, Y) = {(λ₁, {(π₁, [0.8,0.9], [0.2,0.3], [0.4,0.5], [−0.2, −0.1], [−0.8, −0.6], [−0.3, −0.2]), (π₂, [0.5,0.7], [0.4,0.6], [0.9,1], [−0.3, −0.1], [−0.5, −0.4], [−1, −0.8]), (π₃, [0.4,0.6], [0.7,0.8], [0.2,0.3], [−0.5, −0.4], [−0.8, −0.7], [−0.9, −0.6]), (π₄, [0.9,1], [0.1,0.3], [0.5,0.6], [−0.2, −0.1], [−0.6, −0.5], [−0.8, −0.7]), (π₅, [0.5,0.6], [0,0.1], [0.6,0.9], [−1, −0.8], [−0.5, −0.4], [−1, −0.9])}), (λ₂, {(π₁, [0.1,0.2], [0.7,0.8], [0.5,0.6], [−0.3, −0.2], [−0.9, −0.8], [−0.2, −0.1]), (π₂, [0.8,0.9], [0.6,0.7], [0.5,0.6], [−0.4, −0.3], [−0.8, −0.7], [−0.9, −0.8]), (π₃, [0.3,0.4], [0.9,1], [0.2,0.3], [−1, −0.9], [−0.2, −0.1], [−0.5, −0.4]), (π₄, [0.8,0.9], [0.2,0.4], [0.7,0.8], [−0.6, −0.5], [−0.7, −0.6], [−0.2, −0.1]), (π₅, [0.7,0.8], [0.5,0.6], [0.8,0.9], [−0.9, −0.8], [−0.5, −0.4], [−0.7, −0.6])}), (λ₃, {(π₁, [0.2,0.3], [0.8,0.9], [0.8,1], [−0.8, −0.7], [−0.9, −0.8], [−0.6, −0.5]), (π₂, [0.8,1], [0.5,0.7], [0.8,0.9], [−0.2, −0.1], [−0.4, −0.3], [−0.9, −0.7]), (π₃, [0.6,0.7], [0.2,0.4], [0.9,1], [−1, −0.9], [−0.2, −0.1], [−0.4, −0.3]), (π₄, [0.4,0.5], [0.6,0.8], [0.3,0.4], [−0.7, −0.6], [−0.3, −0.2], [−0.8, −0.7]), (π₅, [0.3,0.4], [0.5,0.6], [0.8,0.9], [−1, −0.9], [−0.6, −0.5], [−0.3, −0.2])})}Thus, the complement of (ℶ, Y), which is again of dimension 3, is defined as following:

In addition, we can define two bipolar-valued multivague soft sets (Γ, Λ) and (Ψ, Θ) of dimension 3 on Π, respectively, as follows:

It is clear that $$.

Moreover, we have the union and intersection of bipolar-valued multivague soft sets of dimension 3, namely $$, and $$, which are again of dimension 3, where Δ = Λ ∪ Θ = {λ₁, λ₂, λ₃}, respectively, as follows:

Finally, we have the restricted union and the restricted intersection of bipolar-valued multivague soft sets of dimension 3, namely $$ and $$, which are again of dimension 3, where Ω = Λ∩Θ = {λ₁, λ₂}, respectively, as follows:

Assume the following throughout the remainder of the paper, except otherwise indicated:

Theorem 25. Let (Γ, Λ) be any bipolar-valued multivague soft set of dimension n, $$ be the complete bipolar-valued multivague soft set of dimension n, and $$ be the null bipolar-valued multivague soft set of dimension n on a common universal set Π, then:

Proof. We prove (8), and from (1) to (12) follow similarly. From Definitions 12 and 13, $$ and $$, respectively. Assume, for Δ = Λ ∪ Λ = Λ, that

This result is true for the third case (when δ ∈ Λ∩Λ = Λ) of Definition 21 of intersection of two bipolar-valued multivague soft sets of dimension n. For the first and second case, we have no parameters, since δ ∈ Λ − Λ = ϕ.

Theorem 26. Assume that (Γ, Λ) and (Ψ, Θ) are two bipolar-valued multivague soft sets of dimension n on a universal set Π; then we have the absorption properties satisfied for them as follows:

Proof. We just prove (1). The same steps can be followed to prove (5). Let

According to Definition 17, we have to show that (1) holds for all following three cases:

• (i)

  If δ ∈ Λ − Θ, then we obtain from Definition 23 that:

Then, by using (6) from Theorem 25, we have:

• (ii)

  If δ ∈ Θ − Λ, then we obtain from Definition 23 that:

Then, by using (6) from Theorem 25, we have:

• (iii)

  If δ ∈ Λ∩Θ, then we obtain from Definition 23 that:

Therefore, we have from Definition 17 that:

Corollary 27. For two bipolar-valued multivague soft sets of dimension n on a common universal set (Γ, Λ) and (Ψ, Θ):

Proof. One can prove this result directly with the help of the above Theorem 26.

Theorem 28. Suppose that (Γ, Λ) and (Ψ, Θ) are two bipolar-valued multivague soft sets of dimension n on a universal set Π; then, we have the commutative (Abelian) property satisfied for them as follows:

Proof. By applying similar techniques used to prove the above theorems with the help of Definitions 21 and 17 of the intersection and union of two bipolar-valued multivague soft sets of dimension n on a universal set Π, respectively, we can show that (1) and (5) hold.

Proposition 29. If (Γ, Λ) and (Ψ, Θ) are two bipolar-valued multivague soft sets of dimension n, $$, then

Proof. Apply similar methods but by using Definitions 23 and 19 of restricted intersection and restricted union of two bipolar-valued multivague soft sets of dimension n, respectively, to prove (1) and (5).

Theorem 30. Suppose that (Γ, Λ), (Ψ, Θ), and (Σ, Ω) are bipolar-valued multivague soft sets of dimension n on a common universal set Π; then associative and distributive laws, respectively, hold for them as follows:

Proof. Using Definitions 21 and 17 of the intersection and union of two bipolar-valued multivague soft sets of dimension n on a universal set Π, respectively, one can prove the theorem, similarly as the above theorems.

Theorem 31. De Morgan’s laws are valid for any two bipolar-valued multivague soft sets (Γ, Λ) and (Ψ, Θ) of dimension n on a common universal set Π as follows:

Proof. To prove (1), let δ ∈ Δ = Λ ∪ Θ. We must show that (1) is true for all three cases according to Definitions 17 and 21. Assume that δ ∈ Λ∩Θ; therefore, we obtain from Definitions 14, 17, and 21 that:

This proves the third case. The first and second cases are trivial and follow similarly as above. In addition, (5) can be proved similarly as (1).

Definition 32. (comparison table). The comparison table is a square table; its rows and columns are labeled by the object name of the universe such as π₁, π₂, π₃, ⋯, π_n, and the entries d_ij, where d_ij is the number of parameters for which, the value of d_i (the membership value of π_i) exceeds or equal to (≥) the value of d_j (the membership value of π_j).