Commentary on “Type-2 soft sets” [Journal of Intelligent and Fuzzy Systems 29 (2015) 885-898]

Abstract

In this paper, we point out that the expression of the assertions proposed by Chatterjee et al. [Journal of Intelligent and Fuzzy Systems 29 (2015) 885-898] is not true by a counterexample, and give a revised version.

Keywords

Soft sets type-2 soft sets counterexample

1 Introduction

The concept of type-2 soft sets has been introduced by Chatterjee et al. [1]. It opens up a new field of soft sets, and it also shows its superiority in decision making problems. However, the expressions in Proposition 4.15 [1] are defective. In the following, we will give a counterexample to illustrate the defects of these assertions. To do this, we firstly review the basic knowledge what will be used in subsequence.

Definition 1. [1] Let (X, E) be a soft universe and S (X) be the collection of all type-1 soft sets (T1SS, in short) over (X, E). Then a mapping $F : A \to S (X)$ , A ⊂ E is called a type-2 soft set (T2SS, in short) over (X, E) and it is denoted by [ $F, A$ ].

In this case, corresponding to each parametere ∈ A, $F (e)$ is a T1SS. Thus, for each e ∈ A, this exists a T1SS (Fe, Se) such that $F (e) = (Fe, Se)$ where Fe : Se → P (X) and Se ⊂ E. In this case, we refer to the parameter set A as the “primary set of parameters”, while the set of parameters ∪Se is known as the “underlying set of parameters”.

Definition 2. [1] A T2SS [ $F, A$ ] is said to be an absolute T2SS if and only if for each parameter e ∈ A, the T1SS corresponding to $F (e)$ is an absolute T1SS. An absolute T2SS is denoted by $\tilde{𝔸}$ .

Definition 3. [1] A T2SS [ $F, A$ ] is said to be a null T2SS if and only if for each parameter e ∈ A, the T1SS corresponding to $F (e)$ is a null T1SS. A null T2SS is denoted by $\tilde{Φ}$ .

Definition 4. [1] The union of two T2SSs [ $F$ , A] and [ $G$ , B] is a T2SS, denoted by [ $F$ , A] ⊔ [ $G$ , B] = [ $H, C$ ], where C = A ∪ B, is defined as, ∀ e ∈ C, $H (e) = {\begin{matrix} F (e), & if e \in A - B \\ G (e), & if e \in B - A \\ F (e) \tilde{\cup} G (e), & if e \in A \cap B \end{matrix}$

It may be mentioned here that $F$ (e) $\tilde{\cup}$ $G$ (e) for all e ∈ A ∩ B, refers to the usual soft union between the respective T1SS corresponding to $F$ (e) and $G$ (e) respectively.

Definition 5. [1] The intersection of two T2SSs [ $F$ , A] and [ $G$ , B] is a T2SS, denoted by [ $F$ , A] ⊓ [ $G$ , B] = [ $K$ , C],where C = A ∩ B, is defined as, ∀ e ∈ C, $K (e) = F (e) \tilde{\cap} G (e),$ where $F$ (e) $\tilde{\cap}$ $G$ (e) refers to the usual soft intersection between the respective T1SS corresponding to $F$ (e) and $G$ (e) respectively.

Definition 6. [1] The complement of a T2SS [ $F$ , A] is denoted as [ $F$ , A]^c and is defined by [ $F, A]^{c} = [F^{c}, A]$ , where

$F^{c} (α) = (F_{α}, S_{α})^{c}$ , ∀α ∈ A such that

$F_{α}^{c} (β) = X - F_{α} (β)$ , ∀β ∈ S _α.

2 Counterexample

Here we begin this section with Proposition 4.15 in [1] below.

Proposition 1. [1] Let [ $F$ , A] be a T2SS over the soft universe (X, E) and $\tilde{𝔸}$ and $\tilde{Φ}$ are the absolute and null T2SS respectively. Then,

${\tilde{Φ}}^{c}$ = $\tilde{𝔸}$ ;

${\tilde{𝔸}}^{c}$ = $\tilde{Φ}$ ;

[ $F$ , A] ⊔ $\tilde{Φ}$ = [ $F$ , A];

[ $F$ , A] ⊓ $\tilde{Φ}$ = $\tilde{Φ}$ ;

[ $F$ , A] ⊔ $\tilde{𝔸}$ = $\tilde{𝔸}$ ;

[ $F$ , A] ⊓ $\tilde{𝔸}$ = [ $F$ , A].

In what follows, we will give a counterexample to illustrate the defects of Proposition 1.

Example 1. Let X = {x ₁, x ₂, x ₃, x ₄, x ₅} be a universe set, E = {e ₁, e ₂, e ₃, e ₄, e ₅, e ₆, e ₇} be a parameter set, and $\tilde{𝔸}$ and $\tilde{Φ}$ be the absolute and null T2SS respectively.

Suppose that $[F_{1}, A_{1}]$ and $[G_{1}, B_{1}]$ are two T2SSs defined as follows, where A ₁ = {e ₁} , B ₁ = {e ₃}. Now, $\begin{matrix} [F_{1}, A_{1}] & = & {F (e_{1}) = {\frac{e_{4}}{Φ} \frac{e_{5}}{Φ}}}, \\ [G_{1}, B_{1}] & = & {G (e_{3}) = {\frac{e_{6}}{U}, \frac{e_{7}}{U}}} . \end{matrix}$

Clearly, $[F_{1}, A_{1}]$ is a $\tilde{Φ}$ , $[G_{1}, B_{1}]$ is a $\tilde{𝔸}$ . According to Definition 6, we can get $\begin{matrix} [F_{1}, A_{1}]^{c} & = & {F (e_{1}) = {\frac{e_{4}}{U}, \frac{e_{5}}{U}}}, \\ [G_{1}, B_{1}]^{c} & = & {G (e_{3}) = {\frac{e_{6}}{Φ}, \frac{e_{7}}{Φ}}}, \end{matrix}$ where $[F_{1}, A_{1}]^{c}$ is a $\tilde{𝔸}$ , $[G_{1}, B_{1}]^{c}$ is a $\tilde{Φ}$ . Actually, $[F_{1}, A_{1}]^{c} \neq [G_{1}, B_{1}]$ , $[F_{1}, A_{1}] \neq [G_{1}, B_{1}]^{c}$ . Therefore, the assertions (i) and (ii) of the above Proposition 1 are not true.

Assume that $[F_{2}, A_{2}]$ be a T2SS, defined as follows, where A ₂ = {e ₁, e ₂}. Now, $\begin{matrix} [F_{2}, A_{2}] & = & {F (e_{1}) = {\frac{e_{3}}{{x_{1}}}, \frac{e_{5}}{{x_{3}}}}, \\ F (e_{2}) = {\frac{e_{3}}{{x_{1}, x_{2}}}, \frac{e_{5}}{{x_{4}, x_{5}}}}} . \end{matrix}$

According to Definition 4, we obtain $\begin{matrix} [F_{1}, A_{1}] ⊔ [F_{2}, A_{2}] \\ = {F (e_{1}) = {\frac{e_{4}}{Φ}, \frac{e_{5}}{Φ}}} \\ ⊔ {F (e_{1}) = {\frac{e_{3}}{{x_{1}}}, \frac{e_{5}}{{x_{3}}}}, \\ F (e_{2}) = {\frac{e_{3}}{{x_{1}, x_{2}}}, \frac{e_{5}}{{x_{4}, x_{5}}}}} \\ = {F (e_{1}) = {\frac{e_{3}}{{x_{1}}}, \frac{e_{4}}{Φ}, \frac{e_{5}}{{x_{3}}}}, \\ F (e_{2}) = {\frac{e_{3}}{{x_{1}, x_{2}}}, \frac{e_{5}}{{x_{4}, x_{5}}}}} \end{matrix}$ and according to Definition 5, we can get $\begin{matrix} [F_{1}, A_{1}] ⊓ [F_{2}, A_{2}] \\ = {F (e_{1}) = {\frac{e_{4}}{Φ}, \frac{e_{5}}{Φ}}} \\ ⊓ {F (e_{1}) = {\frac{e_{3}}{{x_{1}}}, \frac{e_{5}}{{x_{3}}}}, \\ F (e_{2}) = {\frac{e_{3}}{{x_{1}, x_{2}}}, \frac{e_{5}}{{x_{4}, x_{5}}}}} \\ = {F (e_{1}) = {\frac{e_{5}}{Φ}}} \end{matrix}$

Clearly, $[F_{1}, A_{1}] ⊔ [F_{2}, A_{2}] \neq [F_{2}, A_{2}], [F_{1}, A_{1}] ⊓ [F_{2}, A_{2}] \neq [F_{1}, A_{1}]$ . Hence, the assertions (iii) and (iv) of Proposition 1 are incorrect. In a similar way, we can show the assertions (v) and (vi) are also incorrect.

On the basis of the analysis above, obviously, we can find out that assertions in Proposition 1 are ambiguous. The reason is that $\tilde{Φ}$ and $\tilde{𝔸}$ are not unique in T2SS. For the purpose of solving this issue, in what follows, we introduce an improved proposition. Firstly, we develop two definitions.

Definition 7. Let [ $F$ , A] is a T2SS over (X, E), A be the “primary set of parameters”, ∪Se be the “underlying set of parameters”. If $F (e)$ is a null T1SS, i.e. $F (e) = Φ_{s_{e}}$ , for ∀e ∈ A, then [ $F, A$ ] is said to be the null T2SS with respect to A and ∪S _e, denoted by ${\tilde{Φ}}_{s_{e}}^{A}$ .

Definition 8. Let [ $F$ , A] is a T2SS over (X, E), A be the “primary set of parameters”, ∪Se be the “underlying set of parameters”. If $F (e)$ is an absolute T1SS, i.e. $F (e) = {\tilde{A}}_{s_{e}}$ , for ∀e ∈ A, then [ $F, A$ ] is said to be the absolute T2SS with respect to A and ∪S _e, denoted by ${\tilde{𝔸}}_{s_{e}}^{A}$ .

Proposition 2. Let [ $F$ , A] be T2SS over the soft universe (X, E). Then,

$({\tilde{Φ}}_{s_{e}}^{A})^{c} = {\tilde{𝔸}}_{s_{e}}^{A}$ ;

$({\tilde{𝔸}}_{s_{e}}^{A})^{c} = {\tilde{Φ}}_{s_{e}}^{A}$ ;

[ $F$ , A] ⊔ ${\tilde{Φ}}_{s_{e}}^{A} = [F, A$ ];

[ $F$ , A] ⊓ ${\tilde{Φ}}_{s_{e}}^{A}$ = ${\tilde{Φ}}_{s_{e}}^{A}$ ;

[ $F$ , A] ⊔ ${\tilde{𝔸}}_{s_{e}}^{A}$ = ${\tilde{𝔸}}_{s_{e}}^{A}$ ;

[ $F$ , A] ⊓ ${\tilde{𝔸}}_{s_{e}}^{A}$ = [ $F$ , A].

Proofs are straight-forward.

References

Chatterjee

, Majumdar

and Samanta

S.K.

, Type-2 soft sets, Journal of Intelligent and Fuzzy Systems 29 (2015), 885–898.