Comment on “Corrigendum to ‘On Interval-Valued Hesitant Fuzzy Soft Sets’” and “Generalized Trapezoidal Fuzzy Soft Set and Its Application in Medical Diagnosis”

1. Introduction and Preliminaries

The aim of this article is to correct assertions (3) and (4) of proposed by Khalil et al. [1] and assertions (3) and (4) of proposed by Zhang et al. [2]. We first generalize (in an unambiguous manner) the fitting or remedy of the involved preliminary notions concerning interval-valued hesitant fuzzy soft set and generalized trapezoidal fuzzy soft sets and then use these notions to remedy the flaw in those assertions.

The concept of soft set was introduced by Molodtsov [3] in 1999. Maji et al. [4] defined some operations on soft sets and showed that the distributive law on soft sets is varied. Later on, Ali et al. [5] pointed out that the distributive laws of soft sets are not true in general. It is necessary to present the theory in a mathematical (at least errorless) way. So, in this paper, we will show how to remedy flows in soft set theory by correcting assertions (3) and (4) of in [1] and assertions (3) and (4) of in [2] (including also remedy preliminaries they involved).

In this section we present (in a clear, rigorous, and nonburdensome manner) some notions concerning interval-valued hesitant fuzzy soft set and generalized trapezoidal fuzzy soft sets, most of which are generalizations of the fitting or remedy of the original notions in the references. (All key notions involved in this note are defined for arbitrary sets; this is convenient for subsequent study (including categorical approach to soft sets, study process relating to soft sets since soft sets can be used in decision-making and study approximation and control relating to soft sets since soft sets can be looked to be L-sets).)

First, we present hesitant fuzzy set, soft set, fuzzy soft set, and interval-valued hesitant fuzzy soft set. We will use to denote the set of all mappings from to , use to denote the set of all subsets of (it is a completely distributive complete lattice with the order ), and write for each subset of a poset . For a mapping (called -subset or -set and write also , where is a lattice with the least element ), we call the support of ; sometimes we identify with its restriction to for convenience.

Definition 1. (1) (cf. [6]) Each closed interval of (the set of all real numbers) is called an interval number (we will identify with the corresponding real number if ). The set of all interval numbers (resp., all interval numbers included in ) is denoted by (resp., by ). Obviously, is a completely distributive complete lattice with the point-wise (We can identify a closed interval of with a point in .) order which is defined by(2) (cf. [6]) For two interval numbers and , let and , and define the degree (i.e., ) of possibility of (i.e., ) by The conclusions of Theorems 2 and 3 (proofs of which will be given in our other paper) will be used in the sequel.

Theorem 2. The relation on , defined byis a total preorder on whose restriction to is the same as the restriction of the point-wise order to .

Theorem 3. For any two elements (the set of all nonempty finite subsets of ), where , is an -element set satisfying , and is an -element set satisfying ( and are nature numbers). Define iff we have the following.
(1) (for ) and (for ) hold in when .
(2) (for ) and (for ) hold in () when .
Then is a completely distributive complete lattice. Moreover, the following hold (without loss of generality we assume ):

Definition 4 (cf. [7, 8]). Each element (Write also if is a finite set.) (i.e., a family consisting of nonempty finite subsets of ) is called an interval-valued hesitant fuzzy set on , where denotes all possible interval valued membership degrees of the element to , and has the order as defined in Theorem 3.

Definition 5 (cf. [3]). Each element (i.e., a family of subsets of ) is called a soft set on , where is called an initial universe set and is called a set of parameters.

Definition 6 (cf. [9, 10]). Each element (i.e., a family of fuzzy sets on ) is called a fuzzy soft set on , and each element is called an interval-valued hesitant fuzzy soft set on ; we will use to denote the -th largest element (it exists by Theorem 3) in () (, , , and denotes the cardinality of ).

Definition 7 (cf. [1]). Let with and . If, for each , there exists such that (i.e., for each , then we say that is a generalized interval-valued hesitant fuzzy soft subset of (written as or ). If both and hold, then we write .

Proposition 8 holds.

Proposition 8.     .

Definition 9 (cf. [10]). For a family , we call , defined bythe hyper-product of in and call , defined bythe product of in .

Next, we present trapezoidal fuzzy set, trapezoidal fuzzy soft set, and generalized trapezoidal fuzzy soft set.

Definition 10 (cf. [2, 11, 12]). An element , defined byis called trapezoidal fuzzy number. The set of all trapezoidal fuzzy numbers satisfying is denoted by ; it is a Hutton algebra (i.e., a completely distributive complete lattice with an order-reversion involution) with the point-wise order . Each element is called a trapezoidal fuzzy set on , each element (i.e., a family of trapezoidal fuzzy sets on ) is called a trapezoidal fuzzy soft set on , and each element is called a generalized trapezoidal fuzzy soft set on .

Definition 11 (cf. [2]). Let .
(1) The supremum (Each element (as a family) of is just a poset with the point-wise order, but is a Hutton algebra with respect to the point-wise order (see Theorem 17).) (write as also ) and the infimum (write as also ) of in is called the generalized trapezoidal point-wise union and the generalized trapezoidal point-wise intersection of in , respectively.
(2) We call , defined bythe hyper-product of and , defined bythe product of .

2. Correction to Paper [1]

Khalil et al. [1] proposed the following Theorem 12 (and hoped it to be a corrected version of assertions (3) and (4) of Theorem 36 in Zhang et al. [10]):

Theorem 12 (see [1, Theorem 17, p.3]). Let with , , and . Then(3).(4) However, neither (3) nor (4) is correct (see Example 13).

Example 13. Let be a set of two cars and be a set of parameters, where (resp., , , ) stands for the parameter “cheap" (resp., “equipment", “fuel consumption", and expensive). Again let , defined bywhere , , and .
(1) Let and . Asand similarly,andThusAs and , . Therefore, does not hold.
(2) Now we show does not hold. Let and . Asandsimilarly,andThusAs and , . Therefore, does not hold.

Correction 1 to Theorem 12. Let with , , and . Then

(3) = , and thus .

(4) = , and thus

Proof. (3) For each and each ,by Theorem 3, which meansand thus .
(4) For each and each ,by Theorem 3, which meansand thus .

Correction 2 to Theorem 12. Let with , , and . Then(1).(2).

Proof. It follows from the fact that is a completely distributive complete lattice.

Analogously, the following (on community and associativity) hold.

Theorem 14. Let with , , and . Then
(1) = , and thus .
(2) = , and thus .
(3) = , and thus .
(4) = , and thus .

3. Correction to Paper [2]

For the finite set case, Zhang et al. [2] defined the concept of generalized trapezoidal fuzzy soft set and showed some examples of its operations, along with an application in decision-making. In this section we first give an example to show the following Theorem 15 is incorrect. Then we consider some possible corrections or replacements of the theorem.

Theorem 15 (see [2, Theorem 29, p.7]). Let with , , and . Then(3) = .(4) = However, neither (3) nor (4) is correct mainly because the cardinality of (the support of the left) is less than the cardinality of (the support of the right) in general.

Example 16. Let be a two-element set, a four-element set, a two-element set, , and . Again let (with supports , , and , respectively), defined byThen has the following two members: has the following four members:Apparently, . However,    (because and ) and . Therefore,   , especially, (whether we look and to be sets or look them to be mappings).

Now we consider some possible corrections or replacements of the theorem.

A correction to Theorem 15. Let with , , and . Then(3).(4)

Proof. We only show (3). For each ,since is a distributive lattice. Therefore,which implies .

Theorem 17. is a Hutton algebra (with the point-wise order), and thus the following hold for a family :where is the set of all mappings (disjoint union) satisfying .

Proof. It follows from the fact that is a Hutton algebra.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundations of China (Grants nos. 11771263 and 11501496), the Key Research and Development Project of Shaanxi Province of China (Grant no. 2018KW-050), the Fundamental Research Funds For the Central Universities (Grant no. 2018CBLY001), and the Fundamental for Graduate students to participate in international academic conference (Grant no. 2018CBLY001).