A note on “possibility multi-fuzzy soft set and its application in decision making”

Abstract

The aim of this paper is to correct the assertions (3) and (4) of Theorem 3.23 proposed by Zhang and Shu [Journal of Intelligent and Fuzzy Systems 27 (2014) 2115-2125]. A counterexample illustrates the flaw of the assertions. We introduce the notion of generalized possibility multi-fuzzy soft sets and use it to correct the flaw in those assertions. Finally we introduce two new definitions and five theorems to improve the work of Zhang and Shu, along with two examples.

Keywords

Generalized distributive law multi-fuzzy set multi-fuzzy soft set possibility multi-fuzzy soft set

1 Introduction

The soft set theory, proposed by Molodtsov [1], can be used as a general mathematical tool for dealing with uncertainty. Maji et al. [2] presented the concept of fuzzy soft set which is based on a combination of the fuzzy set and soft set models. Later on, Maji et al. [3] defined some operations on soft sets and showed that the distributive law of soft sets is varied. Ali et al. [4] pointed out that the distributive law of soft sets is not true in general. This implies that the distributive law of fuzzy soft sets is not true. In this paper, we show that assertions (3) and (4) of Theorem 3.23 proposed by Zhang and Shu [5] are incorrect by a counterexample and correct the Theorem 3.23 (3) and (4) using the generalized distributive law of generalized possibility multi-fuzzy soft sets.

Definition 1.1. (see [6]) Let U be an initial universe set, k be a positive integer and F (U) be the set of all fuzzy subsets of U . A multi-fuzzy set $\tilde{A}$ in U is a set of ordered sequences $\tilde{A} = {u / (μ_{1} (u), μ_{2} (u), . . ., μ_{k} (u)) : u \in U},$ where μ_i ∈ F (U), i = 1, 2, . . . , k .

The function $μ_{\tilde{A}} = (μ_{1}, μ_{2}, . . ., μ_{K})$ is called the multi-membership function of multi-fuzzy set $\tilde{A},$ k is called the dimension of $\tilde{A}$ . The set of all multi-fuzzy sets of dimension k in U is denoted by M^kFS (U) .

Definition 1.2. (see [6]) Let $\tilde{A} = {u / (μ_{1} (u), μ_{2} (u), . . ., μ_{k} (u)) : u \in U}$ and $\tilde{B} = {u / (γ_{1} (u), γ_{2} (u), . . ., γ_{k} (u)) : u \in U}$ be two multi-fuzzy sets of dimension k in U. The following relations and operations

$\tilde{A} ⊑ \tilde{B}$ iff μ_i (u) ≤ γ_i (u), u ∈ U, and 1 ≤ i ≤ k,

$\tilde{A} = \tilde{B}$ iff μ_i (u) = γ_i (u), u ∈ U, and 1 ≤ i ≤ k,

$\tilde{A} ⊔ \tilde{B}$ iff {u/(μ₁ (u) ∨ γ₁ (u), μ₂ (u) ∨ γ₂ (u), . . . , μ_k (u) ∨ γ_k (u)) : u ∈ U} ,

$\tilde{A} ⊓ \tilde{B}$ iff {u/(μ₁ (u) ∧ γ₁ (u), μ₂ (u) ∧ γ₂ (u), . . . , μ_k (u) ∧ γ_k (u)) : u ∈ U} .

Definition 1.3. (see [1]) Suppose that U is an initial universe set, E be a set of parameters, P (U) be the power set of U and A ⊂ E . A pair (F, A) is called soft set over U, where F is a mapping given by F : A → P (U) .

Definition 1.4. (see [2]) A pair $(F, A)$ is called a fuzzy soft set over U, if A ⊆ E and $F : A \to F (U) .$

Definition 1.5. (see [7]) A pair $(\tilde{F}, A)$ is called a multi-fuzzy soft set of dimension k over U, where $\tilde{F}$ is a mapping given by $\tilde{F} : A \to M^{k} FS (U) .$

Definition 1.6. (see [5]) Let U be an initial universe and let E be a set of parameters. The pair (U, E) is called a soft universe. Suppose that $\tilde{F} : E \to M^{k} FS (U),$ and $\tilde{f}$ is a multi-fuzzy subset of E ; that is, $\tilde{f} : E \to M^{k} FS (U) .$ We say that ${\tilde{F}}_{\tilde{f}}$ is is a possibility multi-fuzzy soft set of dimension k (PM^kFSS, in short) over the soft universe (U, E) if and only if ${\tilde{F}}_{\tilde{f}}$ is a mapping given by ${\tilde{F}}_{\tilde{f}} : E \to M^{k} FS (U) \times M^{k} FS (U),$ where ${\tilde{F}}_{\tilde{f}} (e) = (\tilde{F} (e) (u), \tilde{f} (e) (u)),$ ∀ u ∈ U .

Definition 1.7. (see [5]) Let ${\tilde{F}}_{\tilde{f}}$ and ${\tilde{G}}_{\tilde{g}}$ be two PM^kFSSs over (U, E). Now ${\tilde{F}}_{\tilde{f}}$ is said to be a possibility multi-fuzzy soft subset of ${\tilde{G}}_{\tilde{g}}$ if and only if

$\tilde{f} (e)$ is a multi-fuzzy subset of $\tilde{g} (e);$

$\tilde{F} (e)$ is also a multi-fuzzy subset of $\tilde{G} (e), \forall$ e ∈ E .

In this case, we write ${\tilde{F}}_{\tilde{f}} ⊑ {\tilde{G}}_{\tilde{g}} .$

Definition 1.8. (see [5]) Let ${\tilde{F}}_{\tilde{f}}$ and ${\tilde{G}}_{\tilde{g}}$ be two PM^kFSSs over (U, E). Now ${\tilde{F}}_{\tilde{f}}$ is said to be a possibility multi-fuzzy soft equal of ${\tilde{G}}_{\tilde{g}}$ if and only if

${\tilde{F}}_{\tilde{f}}$ is a possibility multi-fuzzy soft subset of ${\tilde{G}}_{\tilde{g}};$

${\tilde{G}}_{\tilde{g}}$ is a possibility multi-fuzzy soft subset of ${\tilde{F}}_{\tilde{f}};$

In this case, we write ${\tilde{F}}_{\tilde{f}} = {\tilde{G}}_{\tilde{g}} .$

Definition 1.9. (see [5]) The union operation on the two PM^kFSSs ${\tilde{F}}_{\tilde{f}}$ and ${\tilde{G}}_{\tilde{g}},$ denoted by ${\tilde{F}}_{\tilde{f}} ⊔ {\tilde{G}}_{\tilde{g}},$ is defined by a mapping given by ${\tilde{H}}_{\tilde{h}} : E \to M^{k} FS (U) \times M^{k} FS (U),$ such that for all $u \in U, {\tilde{H}}_{\tilde{h}} (e) = (\tilde{H} (e) (u), \tilde{h} (e) (u)),$ where $\tilde{H} (e) = \tilde{F} (e) ⊔ \tilde{G} (e)$ and $\tilde{h} (e) = \tilde{f} (e) ⊔ \tilde{g} (e) .$

Definition 1.10. (see [5]) The intersection operation on the two PM^kFSSs ${\tilde{F}}_{\tilde{f}}$ and ${\tilde{G}}_{\tilde{g}},$ denoted by ${\tilde{F}}_{\tilde{f}} ⊓ {\tilde{G}}_{\tilde{g}},$ is defined by a mapping given by ${\tilde{H}}_{\tilde{h}} : E \to M^{k} FS (U) \times M^{k} FS (U),$ such that for all $u \in U, {\tilde{H}}_{\tilde{h}} (e) = (\tilde{H} (e) (u), \tilde{h} (e) (u)),$ where $\tilde{H} (e) = \tilde{F} (e) ⊓ \tilde{G} (e)$ and $\tilde{h} (e) = \tilde{f} (e) ⊓ \tilde{g} (e) .$

Definition 1.11. (see [5]) Let $({\tilde{F}}_{\tilde{f}}, A)$ and $({\tilde{G}}_{\tilde{g}}, B)$ be two PM^kFSSs over (U, E). The “ $({\tilde{F}}_{\tilde{f}}, A)$ AND $({\tilde{G}}_{\tilde{g}}, B),$ ” denoted by $({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B),$ is defined by $({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B) = ({\tilde{H}}_{\tilde{h}}, A \times B),$ where ${\tilde{H}}_{\tilde{h}} (α, β) = (\tilde{H} (α, β) (x), \tilde{h} (α, β) (x)),$ for all (α, β) ∈ A × B, such that $\tilde{H} (α, β) = \tilde{F} (α) ⊓ \tilde{G} (β)$ and $\tilde{h} (α, β) = \tilde{f} (α) ⊓ \tilde{g} (β) .$

Definition 1.12. (see [5]) Let $({\tilde{F}}_{\tilde{f}}, A)$ and $({\tilde{G}}_{\tilde{g}}, B)$ be two PM^kFSSs over (U, E). The “ $({\tilde{F}}_{\tilde{f}}, A)$ OR $({\tilde{G}}_{\tilde{g}}, B),$ ” denoted by $({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{G}}_{\tilde{g}}, B),$ is defined by $({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{G}}_{\tilde{g}}, B) = ({\tilde{H}}_{\tilde{h}}, A \times B),$ where ${\tilde{H}}_{\tilde{h}} (α, β) = (\tilde{H} (α, β) (x), \tilde{h} (α, β) (x)),$ for all (α, β) ∈ A × B, such that $\tilde{H} (α, β) = \tilde{F} (α) ⊔ \tilde{G} (β)$ and $\tilde{h} (α, β) = \tilde{f} (α) ⊔ \tilde{g} (β) .$

2 Counterexample

We begin this section with Theorem 2.1 below, originally proposed as Theorem 3.23 in Zhang and Shu [5] and provide a counterexample to show that assertions (3) and (4) are not true.

Theorem 2.1. (see [5]) Let $({\tilde{F}}_{\tilde{f}}, A), ({\tilde{G}}_{\tilde{g}}, B)$ and $({\tilde{H}}_{\tilde{h}}, C)$ be any three PM^kFSSs over (U, E) . Then we have

$({\tilde{F}}_{\tilde{f}}, A) \land (({\tilde{G}}_{\tilde{g}}, B) \land ({\tilde{H}}_{\tilde{h}}, C)) = (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B)) \land ({\tilde{H}}_{\tilde{h}}, C),$

$({\tilde{F}}_{\tilde{f}}, A) \lor (({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C)) = (({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{G}}_{\tilde{g}}, B)) \lor ({\tilde{H}}_{\tilde{h}}, C),$

$({\tilde{F}}_{\tilde{f}}, A) \land (({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C)) = (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B)) \lor (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{H}}_{\tilde{h}}, C)),$

$({\tilde{F}}_{\tilde{f}}, A) \lor (({\tilde{G}}_{\tilde{g}}, B) \land ({\tilde{H}}_{\tilde{h}}, C)) = (({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{G}}_{\tilde{g}}, B)) \land (({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{H}}_{\tilde{h}}, C)) .$

The following example shows that assertions (3) and (4) of Theorem 2.1 above are not true in general.

Example 2.2. Let U = {u₁, u₂} be a set of two houses and E = {e₁, e₂, e₃, e₄, e₅, e₆} be a set of parameters, which stand for expensive, beautiful, cheap, size, location and in the green surroundings respectively. Consider A, B and C to be subsets of E, where A = {e₁, e₂} = {expensive, beautiful} , B = {e₃, e₄} = {cheap, size} and C = {e₅, e₆} = {location, in the green surroundings} . Suppose that $({\tilde{F}}_{\tilde{f}}, A), ({\tilde{G}}_{\tilde{g}}, B)$ and $({\tilde{H}}_{\tilde{h}}, C)$ be three PM^kFSSs defined by $\begin{matrix} {\tilde{F}}_{\tilde{f}} (e_{1}) & = & {(u_{1} / (0.2, 0.5, 0.6), (0.4, 0.5, 0.7)), \\ (u_{2} / (0.5, 0.4, 0.6), (0.1, 0.5, 0.7))}, \\ {\tilde{F}}_{\tilde{f}} (e_{2}) & = & {(u_{1} / (0.1, 0.8, 0.1), (0.2, 0.5, 0.9)), \\ (u_{2} / (0.3, 0.1, 0.6), (0.4, 0.5, 0.2))}, \\ {\tilde{G}}_{\tilde{g}} (e_{3}) & = & {(u_{1} / (0.4, 0.2, 0.1), (0.1, 0.1, 0.9)), \\ (u_{2} / (0.5, 0.1, 0.6), (0.3, 0.5, 0.2))}, \\ {\tilde{G}}_{\tilde{g}} (e_{4}) & = & {(u_{1} / (0.1, 0.2, 0.7), (0.1, 0.7, 0.9)), \\ (u_{2} / (0.2, 0.1, 0.4), (0.1, 0.5, 0.2))}, \\ {\tilde{H}}_{\tilde{h}} (e_{5}) & = & {(u_{1} / (0.6, 0.2, 0.7), (0.1, 0.3, 0.9)), \\ (u_{2} / (0.2, 0.3, 0.1), (0.2, 0.5, 0.4))}, \\ {\tilde{H}}_{\tilde{h}} (e_{6}) & = & {(u_{1} / (0.3, 0.3, 0.8), (0.1, 0.2, 0.4)), \\ (u_{2} / (0.5, 0.3, 0.2), (0.5, 0.5, 0.1))} . \end{matrix}$

By Definitions 1.11 and 1.12, the possibility multi-fuzzy soft set $({\tilde{F}}_{\tilde{f}}, A) \land (({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C))$ has the parameter set A × (B × C) and possibility multi-fuzzy soft set $(({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B)) \lor (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{H}}_{\tilde{h}}, C))$ has a set of parameters as (A × B) × (A × C) . But we can not find any notion which ensure A × (B × C) = (A × B) × (A × C) . Hence Theorem 2.1 above is not true.

3 Main results

In this section we introduce the concept of a generalized possibility multi-fuzzy soft subset and fuzzy soft equal and give some theorems and examples.

We begin by proposing the definition of a generalized possibility multi-fuzzy soft subset followed by an example.

Definition 3.1. Let U be an initial universe and E be a set of parameters. For subsets A and B of E, let $({\tilde{F}}_{\tilde{f}}, A)$ and $({\tilde{G}}_{\tilde{g}}, B)$ be two PM^kFSSs over (U, E) . $({\tilde{F}}_{\tilde{f}}, A)$ is generalized possibility multi-fuzzy soft subset of $({\tilde{G}}_{\tilde{g}}, B)$ , denoted by $({\tilde{F}}_{\tilde{f}}, A) {\tilde{\subset}}_{g} ({\tilde{G}}_{\tilde{g}}, B),$ if, for every α ∈ A there exists β ∈ B such that ${\tilde{F}}_{\tilde{f}} (α) = {\tilde{G}}_{\tilde{g}} (β),$ where for all $α \in A, β \in B, \tilde{f} (α) = \tilde{g} (β)$ in PM^kFSSs ${\tilde{F}}_{\tilde{f}} (α)$ and ${\tilde{G}}_{\tilde{g}} (β),$ respectively.

Example 3.2. Let U = {u₁, u₂, u₃} be an initial universe and E = {e₁, e₂, e₃, e₄} be a set of parameters. With A = {e₁, e₃} and B = {e₂, e₄} , let $({\tilde{F}}_{\tilde{f}}, A)$ and $({\tilde{G}}_{\tilde{g}}, B)$ be two PM^kFSSs over (U, E) defined by $\begin{matrix} {\tilde{F}}_{\tilde{f}} (e_{1}) & = & {\tilde{G}}_{\tilde{g}} (e_{2}) \\ = & {(u_{1} / (0.2, 0.5, 0.6), (0.4, 0.5, 0.7)), \\ (u_{2} / (0.5, 0.4, 0.6), (0.1, 0.5, 0.7)), \\ (u_{3} / (0.1, 0.8, 0.1), (0.3, 0.1, 0.6))}, \\ {\tilde{F}}_{\tilde{f}} (e_{3}) & = & {\tilde{G}}_{\tilde{g}} (e_{4}) \\ = & {(u_{1} / (0.1, 0.4, 0.6), (0.2, 0.1, 0.3)), \\ (u_{2} / (0.4, 0.4, 0.6), (0.1, 0.5, 0.7)), \\ (u_{3} / (0.3, 0.5, 0.2), (0.1, 0.3, 0.6))} . \end{matrix}$

Then $({\tilde{F}}_{\tilde{f}}, A) {\tilde{\subset}}_{L} ({\tilde{G}}_{\tilde{g}}, B) .$

We establish the definition of generalized possibility multi-fuzzy soft set equal below followed by a theorem on its equality.

Definition 3.3. Let $({\tilde{F}}_{\tilde{f}}, A)$ and $({\tilde{G}}_{\tilde{g}}, B)$ be two PM^kFSSs over (U, E). $({\tilde{F}}_{\tilde{f}}, A)$ and $({\tilde{G}}_{\tilde{g}}, B)$ are generalized possibility multi-fuzzy soft set equal, denoted by $({\tilde{F}}_{\tilde{f}}, A) =_{g} ({\tilde{G}}_{\tilde{g}}, B),$ if $({\tilde{F}}_{\tilde{f}}, A) {\tilde{\subset}}_{g} ({\tilde{G}}_{\tilde{g}}, B)$ and $({\tilde{G}}_{\tilde{g}}, B) {\tilde{\subset}}_{g} ({\tilde{F}}_{\tilde{f}}, A) .$

Theorem 3.4. Let $({\tilde{F}}_{\tilde{f}}, A)$ and $({\tilde{G}}_{\tilde{g}}, B)$ be generalized possibility multi-fuzzy soft sets. Then $({\tilde{F}}_{\tilde{f}}, A) =_{g} ({\tilde{G}}_{\tilde{g}}, B)$ if and only if ${{\tilde{F}}_{\tilde{f}} (α) : α \in A} = {{\tilde{G}}_{\tilde{g}} (β) : β \in B} .$

Proof. If $({\tilde{F}}_{\tilde{f}}, A) =_{g} ({\tilde{G}}_{\tilde{g}}, B),$ then $({\tilde{F}}_{\tilde{f}}, A) {\tilde{\subset}}_{g} ({\tilde{G}}_{\tilde{g}}, B)$ and $({\tilde{G}}_{\tilde{g}}, B) {\tilde{\subset}}_{g} ({\tilde{F}}_{\tilde{f}}, A) .$ For any ${\tilde{F}}_{\tilde{f}} (α) \in {{\tilde{F}}_{\tilde{f}} (α) : α \in A},$ there exists β_α ∈ B such that ${\tilde{F}}_{\tilde{f}} (α) = {\tilde{G}}_{\tilde{g}} (β_{α}) \in {{\tilde{G}}_{\tilde{g}} (β) : β \in B}$ since $({\tilde{F}}_{\tilde{f}}, A) {\tilde{\subset}}_{g} ({\tilde{G}}_{\tilde{g}}, B) .$ Hence we obtain ${{\tilde{F}}_{\tilde{f}} (α) : α \in A} \subseteq {{\tilde{G}}_{\tilde{g}} (β) : β \in B} .$ In a similar fashion, we can show that ${{\tilde{G}}_{\tilde{g}} (β) : β \in B} \subseteq {{\tilde{F}}_{\tilde{f}} (α) : α \in A}$ since $({\tilde{G}}_{\tilde{g}}, B) {\tilde{\subset}}_{g} ({\tilde{F}}_{\tilde{f}}, A) .$ Conversely, assume that ${{\tilde{F}}_{\tilde{f}} (α) : α \in A} = {{\tilde{G}}_{\tilde{g}} (β) : β \in B} .$ Then for every $α \in A, {\tilde{F}}_{\tilde{f}} (α) \in {{\tilde{F}}_{\tilde{f}} (α) : α \in A} = {{\tilde{G}}_{\tilde{g}} (β) : β \in B} .$ Hence there exists β_α ∈ B such that ${\tilde{G}}_{\tilde{g}} (β_{α}) = {\tilde{F}}_{\tilde{f}} (α) .$ This implies that $({\tilde{F}}_{\tilde{f}}, A) {\tilde{\subset}}_{g} ({\tilde{G}}_{\tilde{g}}, B) .$ Similarly, we can show that $({\tilde{G}}_{\tilde{g}}, B) {\tilde{\subset}}_{g} ({\tilde{F}}_{\tilde{f}}, A),$ and so we deduce that $({\tilde{F}}_{\tilde{f}}, A) =_{g} ({\tilde{G}}_{\tilde{g}}, B)$ as required.

In the following Theorem 3.5 is the corrected version of assertions (3) and (4) of Theorem 2.1, originally written as Theorem 3.23 of Zhang and Shu [5].

Theorem 3.5. Let $({\tilde{F}}_{\tilde{f}}, A), ({\tilde{G}}_{\tilde{g}}, B)$ and $({\tilde{H}}_{\tilde{h}}, C)$ be any three PM^kFSSs over (U, E) . Then we have

$({\tilde{F}}_{\tilde{f}}, A) \land (({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C)) {\tilde{\subset}}_{g} (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B)) \lor (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{H}}_{\tilde{h}}, C)),$

$({\tilde{F}}_{\tilde{f}}, A) \lor (({\tilde{G}}_{\tilde{g}}, B) \land ({\tilde{H}}_{\tilde{h}}, C)) {\tilde{\subset}}_{g} (({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{G}}_{\tilde{g}}, B)) \land (({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{H}}_{\tilde{h}}, C)) .$

Proof. We only show the validity of (1); then the proof of (2) can be obtained using similar techniques. Now, let us write $({\tilde{T}}_{\tilde{t}}, B \times C)$ for $({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C),$ where ${\tilde{T}}_{\tilde{t}} (β, γ) = {\tilde{G}}_{\tilde{g}} (β) ⊔ {\tilde{H}}_{\tilde{h}} (γ)$ for all (β, γ) ∈ B × C . Then let $({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{T}}_{\tilde{t}}, B \times C) = ({\tilde{L}}_{\tilde{l}}, A \times (B \times C)),$ where ${\tilde{L}}_{\tilde{l}} (α, (β, γ)) = {\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{T}}_{\tilde{t}} (β, γ) = {\tilde{F}}_{\tilde{f}} (α) ⊓ ({\tilde{G}}_{\tilde{g}} (β) ⊔ {\tilde{H}}_{\tilde{h}} (γ))$ for all (α, (β, γ)) ∈ A × (B × C) . In a similar fashion, we can show that ${\tilde{R}}_{\tilde{r}} ((α, β), (α, γ)) = ({\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{G}}_{\tilde{g}} (β)) ⊔ ({\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{H}}_{\tilde{h}} (γ))$ for all ((α, β), (α, γ)) ∈ (A × B) × (A × C) . Hence for every (α, (β, γ)) ∈ A × (B × C), there exists ((α, β), (α, γ)) ∈ (A × B) × (A × C) such that ${\tilde{L}}_{\tilde{l}} (α, (β, γ)) = {\tilde{F}}_{\tilde{f}} (α) ⊓ ({\tilde{G}}_{\tilde{g}} (β) ⊔ {\tilde{H}}_{\tilde{h}} (γ)) = ({\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{G}}_{\tilde{g}} (β)) ⊔ ({\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{H}}_{\tilde{h}} (γ)) = {\tilde{R}}_{\tilde{r}} ((α, β), (α, γ)) .$ By Definition 3.1, we obtain $({\tilde{F}}_{\tilde{f}}, A) \land (({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C)) {\tilde{\subset}}_{g} (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B)) \lor (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{H}}_{\tilde{h}}, C)) .$

The equalities in Theorem 3.5 are not true in general as illustrated by the following example.

Example 3.6. Let U = {u₁, u₂} be a set of two houses and E = {e₁, e₂, e₃, e₄} be a set of parameters, which stand for expensive, beautiful, cheap and size, respectively. Consider A, B and C to be subsets of E, where A = {e₁, e₂} = {expensive, beautiful} , B = {e₃} = {cheap} and C = {e₄} = { size} . Suppose that $({\tilde{F}}_{\tilde{f}}, A), ({\tilde{G}}_{\tilde{g}}, B)$ and $({\tilde{H}}_{\tilde{h}}, C)$ be three PM^kFSSs defined by $\begin{matrix} {\tilde{F}}_{\tilde{f}} (e_{1}) & = & {(u_{1} / (0.4, 0.5, 0.6), (0.1, 0.5, 0.3)), \\ (u_{2} / (0.3, 0.4, 0.9), (0.2, 0.4, 0.6))}, \\ {\tilde{F}}_{\tilde{f}} (e_{2}) & = & {(u_{1} / (0.3, 0.5, 0.6), (0.3, 0.1, 0.5)), \\ (u_{2} / (0.7, 0.8, 0.9), (0.4, 0.5, 0.6))}, \\ {\tilde{G}}_{\tilde{g}} (e_{3}) & = & {(u_{1} / (0.2, 0.4, 0.7), (0.1, 0.7, 0.1)), \\ (u_{2} / (0.5, 0.7, 0.7), (0.1, 0.3, 0.5))}, \\ {\tilde{H}}_{\tilde{h}} (e_{4}) & = & {(u_{1} / (0.6, 0.6, 0.8), (0.3, 0.2, 0.5)), \\ (u_{2} / (0.5, 0.6, 0.9), (0.3, 0.5, 0.7))} . \end{matrix}$

Let us write $({\tilde{T}}_{\tilde{t}}, B \times C)$ for $({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C),$ where ${\tilde{T}}_{\tilde{t}} (β, γ) = {\tilde{G}}_{\tilde{g}} (β) ⊔ {\tilde{H}}_{\tilde{h}} (γ)$ for all (β, γ) ∈ B × C . Then $({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{T}}_{\tilde{t}}, B \times C) = ({\tilde{L}}_{\tilde{l}}, A \times (B \times C)),$ where ${\tilde{L}}_{\tilde{l}} (α, (β, γ)) = {\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{T}}_{\tilde{t}} (β, γ) = {\tilde{F}}_{\tilde{f}} (α) ⊓ (\tilde{G} (β) ⊔ \tilde{H} (γ))$ for all (α, (β, γ)) ∈ A × (B × C) . It is easily seen that A × (B × C) = {(e₁, (e₃, e₄)), (e₂, (e₃, e₄))} .

By calculation we obtain $\begin{matrix} {\tilde{L}}_{\tilde{l}} ((e_{1}, (e_{3}, e_{4}))) \\ = \tilde{F} (e_{1}) ⊓ ({\tilde{G}}_{\tilde{g}} (e_{3}) ⊔ {\tilde{H}}_{\tilde{h}} (e_{4})) \\ = {(u_{1} / (0.4, 0.5, 0.6), (0.1, 0.5, 0.3)), \\ (u_{2} / (0.3, 0.4, 0.7), (0.2, 0.4, 0.6))}, \end{matrix}$ and $\begin{matrix} {\tilde{L}}_{\tilde{l}} ((e_{2}, (e_{3}, e_{4}))) \\ = \tilde{F} (e_{2}) ⊓ ({\tilde{G}}_{\tilde{g}} (e_{3}) ⊔ {\tilde{H}}_{\tilde{h}} (e_{4})) \\ = {(u_{1} / (0.3, 0.5, 0.6), (0.3, 0.1, 0.5)), \\ (u_{2} / (0.5, 0.7, 0.7), (0.3, 0.5, 0.6))} . \end{matrix}$

Next, we write ${\tilde{R}}_{\tilde{r}} ((α, β), (α, γ)) = ({\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{G}}_{\tilde{g}} (β)) ⊔ ({\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{H}}_{\tilde{h}} (γ))$ for all ((α, β), (α, γ)) ∈ (A × B) × (A × C) . Hence A × B = {(e₁, e₃), (e₂, e₃)} and A × C = {(e₁, e₄), (e₂, e₄)} . Thus we have $\begin{matrix} (A \times B) \times (A \times C) \\ = {((e_{1}, e_{3}), (e_{1}, e_{4})), ((e_{1}, e_{3}), \\ (e_{2}, e_{4})), ((e_{2}, e_{3}), (e_{1}, e_{4})), ((e_{2}, e_{3}), (e_{2}, e_{4}))} . \end{matrix}$

By calculation we obtain $\begin{matrix} {\tilde{R}}_{\tilde{r}} ((e_{1}, e_{3}), (e_{1}, e_{4})) \\ = ({\tilde{F}}_{\tilde{f}} (e_{1}) ⊓ {\tilde{G}}_{\tilde{g}} (e_{3})) ⊔ ({\tilde{F}}_{\tilde{f}} (e_{1}) ⊓ {\tilde{H}}_{\tilde{h}} (e_{4})) \\ = {(u_{1} / (0.4, 0.5, 0.6), (0.1, 0.5, 0.3)), \\ (u_{2} / (0.3, 0.4, 0.9), (0.2, 0.4, 0.6))}, \\ {\tilde{R}}_{\tilde{r}} ((e_{1}, e_{3}), (e_{2}, e_{4})) \\ = ({\tilde{F}}_{\tilde{f}} (e_{1}) ⊓ {\tilde{G}}_{\tilde{g}} (e_{3})) ⊔ ({\tilde{F}}_{\tilde{f}} (e_{2}) ⊓ {\tilde{H}}_{\tilde{h}} (e_{4})) \\ = {(u_{1} / (0.3, 0.5, 0.6), (0.3, 0.5, 0.5)), \\ (u_{2} / (0.5, 0.6, 0.9), (0.3, 0.5, 0.7))}, \\ {\tilde{R}}_{\tilde{r}} ((e_{2}, e_{3}), (e_{1}, e_{4})) \\ = ({\tilde{F}}_{\tilde{f}} (e_{2}) ⊓ {\tilde{G}}_{\tilde{g}} (e_{3})) ⊔ ({\tilde{F}}_{\tilde{f}} (e_{1}) ⊓ {\tilde{H}}_{\tilde{h}} (e_{4})) \\ = {(u_{1} / (0.4, 0.5, 0.6), (0.1, 0.2, 0.3)), \\ (u_{2} / (0.5, 0.7, 0.9), (0.2, 0.4, 0.6))}, \end{matrix}$ and $\begin{matrix} {\tilde{R}}_{\tilde{r}} ((e_{2}, e_{3}), (e_{2}, e_{4})) \\ = ({\tilde{F}}_{\tilde{f}} (e_{2}) ⊓ {\tilde{G}}_{\tilde{g}} (e_{3})) ⊔ ({\tilde{F}}_{\tilde{f}} (e_{2}) ⊓ {\tilde{H}}_{\tilde{h}} (e_{4})) \\ = {(u_{1} / (0.3, 0.5, 0.6), (0.3, 0.1, 0.5)), \\ (u_{2} / (0.5, 0.7, 0.9), (0.3, 0.5, 0.7))} . \end{matrix}$

Clearly, we have $\begin{matrix} {\tilde{R}}_{\tilde{r}} ((e_{1}, e_{3}), (e_{1}, e_{4})) \\ ⊈ {\tilde{L}}_{\tilde{l}} ((e_{1}, (e_{3}, e_{4}))), {\tilde{R}}_{\tilde{r}} ((e_{1}, e_{3}), (e_{2}, e_{4})) \\ ⊈ {\tilde{L}}_{\tilde{l}} ((e_{1}, (e_{3}, e_{4}))), {\tilde{R}}_{\tilde{r}} ((e_{2}, e_{3}), (e_{1}, e_{4})) \\ ⊈ {\tilde{L}}_{\tilde{l}} ((e_{2}, (e_{3}, e_{4}))) \end{matrix}$ and $\begin{matrix} {\tilde{R}}_{\tilde{r}} ((e_{2}, e_{3}), (e_{2}, e_{4})) \\ ⊈ {\tilde{L}}_{\tilde{l}} ((e_{2}, (e_{3}, e_{4}))) . \end{matrix}$

This shows that $({\tilde{F}}_{\tilde{f}}, A) \land (({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C)) =_{g} (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B)) \lor (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{H}}_{\tilde{h}}, C))$ does not hold. Using similar techniques, we can also show that the second statement of Theorem 3.5 does not hold.

In order to give a deeper insight on associativity, intersection and union of generalized possibility multi-fuzzy soft sets equal, we propose the following Theorems 3.7, 3.8 and 3.9 along with their corresponding proofs.

Theorem 3.7. Let $({\tilde{F}}_{\tilde{f}}, A), ({\tilde{G}}_{\tilde{g}}, B)$ and $({\tilde{H}}_{\tilde{h}}, C)$ be any three PM^kFSSs over (U, E) . Then we have

$({\tilde{F}}_{\tilde{f}}, A) \land (({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C)) {\tilde{\subset}}_{g} (({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{G}}_{\tilde{g}}, B)) \land (({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{H}}_{\tilde{h}}, C)),$

$({\tilde{F}}_{\tilde{f}}, A) \lor (({\tilde{G}}_{\tilde{g}}, B) \land ({\tilde{H}}_{\tilde{h}}, C)) {\tilde{\subset}}_{g} (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B)) \lor (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{H}}_{\tilde{h}}, C)) .$

Proof. (1) and (2) are similar to the proof of Theorem 3.5 (1) and (2) respectively.

Theorem 3.8. Let $({\tilde{F}}_{\tilde{f}}, A)$ and $({\tilde{G}}_{\tilde{g}}, B)$ be two PM^kFSSs over (U, E) . Then

$({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B) =_{g} ({\tilde{G}}_{\tilde{g}}, B) \land ({\tilde{F}}_{\tilde{f}}, A);$

$({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{G}}_{\tilde{g}}, B) =_{g} ({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{F}}_{\tilde{f}}, A) .$

Proof. To prove the first assertion, Let us write $({\tilde{L}}_{\tilde{l}}, A \times B)$ for $({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B),$ where ${\tilde{L}}_{\tilde{l}} (α, β) = {\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{G}}_{\tilde{g}} (β)$ for all (α, β) ∈ A × B . We denote $({\tilde{G}}_{\tilde{g}}, B) \land ({\tilde{F}}_{\tilde{f}}, A)$ by $({\tilde{R}}_{\tilde{r}}, B \times A),$ where ${\tilde{R}}_{\tilde{r}} (β, α) = {\tilde{G}}_{\tilde{g}} (β) ⊓ {\tilde{F}}_{\tilde{f}} (α)$ for all (β, α) ∈ B × A . For any (α, β) ∈ A × B, there exists (β, α) ∈ B × A such that ${\tilde{L}}_{\tilde{l}} (α, β) = {\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{G}}_{\tilde{g}} (β) = {\tilde{G}}_{\tilde{g}} (β) ⊓ {\tilde{F}}_{\tilde{f}} (α) = {\tilde{R}}_{\tilde{g}} (β, α) .$ Hence, by Definition 3.1, we have $({\tilde{L}}_{\tilde{l}}, A \times B) {\tilde{\subset}}_{g} ({\tilde{R}}_{\tilde{r}}, B \times A) .$ Similarly, we can show that $(\tilde{R}, B \times A) {\tilde{\subset}}_{g} ({\tilde{L}}_{\tilde{l}}, A \times B) .$ Therefore, we have $({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B) =_{g} ({\tilde{G}}_{\tilde{g}}, B) \land ({\tilde{F}}_{\tilde{f}}, A) .$ The second assertion can be proven in a similar fashion and thus omitted.

Theorem 3.9. Let $({\tilde{F}}_{\tilde{f}}, A), ({\tilde{G}}_{\tilde{g}}, B)$ and $({\tilde{H}}_{\tilde{h}}, C)$ be any three PM^kFSSs over (U, E) . Then we have

$({\tilde{F}}_{\tilde{f}}, A) \land (({\tilde{G}}_{\tilde{g}}, B) \land ({\tilde{H}}_{\tilde{h}}, C)) =_{g} (({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B)) \land ({\tilde{H}}_{\tilde{h}}, C);$

$({\tilde{F}}_{\tilde{f}}, A) \lor (({\tilde{G}}_{\tilde{g}}, B) \lor ({\tilde{H}}_{\tilde{h}}, C)) =_{g} (({\tilde{F}}_{\tilde{f}}, A) \lor ({\tilde{G}}_{\tilde{g}}, B)) \lor ({\tilde{H}}_{\tilde{h}}, C) .$

Proof. We only show the validity of (1); then the proof of (2) can be obtained using similar techniques. Now, let us write $({\tilde{N}}_{\tilde{n}}, B \times C)$ for $({\tilde{G}}_{\tilde{g}}, B) \land ({\tilde{H}}_{\tilde{h}}, C),$ where ${\tilde{N}}_{\tilde{n}} (β, γ) = {\tilde{G}}_{\tilde{g}} (β) ⊓ {\tilde{H}}_{\tilde{h}} (γ)$ for all (β, γ) ∈ B × C . Then let $({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{N}}_{\tilde{n}}, B \times C) = ({\tilde{L}}_{\tilde{l}}, A \times (B \times C)),$ where ${\tilde{L}}_{\tilde{l}} (α, (β, γ)) = {\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{N}}_{\tilde{n}} (β, γ) = {\tilde{F}}_{\tilde{f}} (α) ⊓ ({\tilde{G}}_{\tilde{f}} (β) ⊓ {\tilde{H}}_{\tilde{h}} (γ))$ for all (α, (β, γ)) ∈ A × (B × C) . On the other hand, let us write $({\tilde{M}}_{\tilde{m}}, A \times B)$ for $({\tilde{F}}_{\tilde{f}}, A) \land ({\tilde{G}}_{\tilde{g}}, B),$ where ${\tilde{M}}_{\tilde{m}} (α, β) = {\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{G}}_{\tilde{g}} (β)$ for all (α, β) ∈ A × B . Then let $({\tilde{M}}_{\tilde{m}}, A \times B) \land ({\tilde{H}}_{\tilde{h}}, C) = ({\tilde{R}}_{\tilde{r}}, (A \times B) \times C)),$ where ${\tilde{R}}_{\tilde{r}} ((α, β), γ) = {\tilde{M}}_{\tilde{m}} (α, β) ⊓ {\tilde{H}}_{\tilde{h}} (γ) = ({\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{G}}_{\tilde{g}} (β)) ⊓ {\tilde{H}}_{\tilde{h}} (γ)$ for all ((α, β), γ) ∈ (A × B) × C . Since $\tilde{F} (α) ⊓ ({\tilde{G}}_{\tilde{g}} (β) ⊓ {\tilde{H}}_{\tilde{h}} (γ)) = ({\tilde{F}}_{\tilde{f}} (α) ⊓ {\tilde{G}}_{\tilde{g}} (β)) ⊓ {\tilde{H}}_{\tilde{h}} (γ),$ we deduce that ${{\tilde{L}}_{\tilde{l}} (α, (β, γ)) : (α, (β, γ)) \in A \times (B \times C)}$ and ${{\tilde{R}}_{\tilde{r}} ((α, β), γ) : ((α, β), γ) \in (A \times B) \times C}$ are indeed the same set. Hence by Theorem 3.4, we can conclude that $({\tilde{L}}_{\tilde{l}}, A \times (B \times C)) =_{L} ({\tilde{R}}_{\tilde{r}}, (A \times B) \times C)$ as required.

4 Conclusion

Zhang and Shu [5] introduced the concept of possibility multi-fuzzy soft sets based on a generalization parameter which itself is multi-fuzzy and proposed several theorems and some operations on a possibility multi-fuzzy soft set. However, we pointed out that assertion in Theorem 3.23 (3) and (4) of [5] is flawed. Using the notions of a generalized possibility multi-fuzzy soft subset and fuzzy soft equal, the Theorem 3.23 (3) and (4) of [5] is corrected, along with the introduction of a few new theorems and definitions.

Footnotes

Acknowledgments

We are indebted to Universiti Kebangsaan Malaysia for providing financial support and facilities for this research under the grant DPP-2015-FST.

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