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

Authors: Ahmed Mostafa Khalil; Sheng-Gang Li; Hong-Xia Li; Hai-Dong Zhang
Journal: Mathematical Problems in Engineering
 (2019)

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## Body

## 1. Introduction and Preliminaries

The aim of this article is to correct assertions (3) and (4) ofTheorem17 proposed by Khalil et al. [1] and assertions (3) and (4) of Theorem29 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 Theorem17 in [1] and assertions (3) and (4) of Theorem29 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 beL-sets).)First, we present hesitant fuzzy set, soft set, fuzzy soft set, and interval-valued hesitant fuzzy soft set. We will useYX to denote the set of all mappings from X to Y, use 2X to denote the set of all subsets of X (it is a completely distributive complete lattice with the order ⊆), and write ↓E=p∈P∣∃e∈E,p≤e for each subset E of a poset (P,≤). For a mapping Φ:X→L (called L-subset or L-set and write also Φ∈LX, where L is a lattice with the least element 0), we call suppΦ=x∈X∣Φx≠0 the support of Φ; sometimes we identify Φ with its restriction to suppΦ for convenience.Definition 1.
(1) (cf. [6]) Each closed interval a=[aL,aU]=x∣aL≤x≤aU of R (the set of all real numbers) is called an interval number (we will identify a with the corresponding real number a if aL=aU=a). The set of all interval numbers (resp., all interval numbers included in [0,1]) is denoted by R (resp., by I). Obviously, I is a completely distributive complete lattice with the point-wise (We can identify a closed interval [a,b] of R with a point a,b in R2.) order ≤ which is defined by(1)aL,aU≤bL,bU⇔aL≤bLandaU≤bU.
(2) (cf. [6]) For two interval numbers a=[aL,aU] and b=[bL,bU], let la=aU-aL and lb=bU-bL, and define the degree p(a≥b) (i.e., p(b≤a)) of possibility of a≥b (i.e., b≤a) by (2)pa≥b=0,la+lb=0,a<b,12,la+lb=0,a=b,1,la+lb=0,a>b,max1-maxb+-a-la+lb,0,0,la+lb≠0.
The conclusions of Theorems2 and 3 (proofs of which will be given in our other paper) will be used in the sequel.Theorem 2.
The relation⩽ on I, defined by(3)a⩽b⇔pa≤b≥pa≥b,is a total preorder on I whose restriction to R is the same as the restriction of the point-wise order ≤ to R.Theorem 3.
For any two elementsA,B∈2(I) (the set of all nonempty finite subsets of I), where A={[a1L,a1U],[a2L,a2U],⋯, [amL,amU]} is an m-element set satisfying [a1L,a1U]⩽[a2L,a2U]⩽⋯⩽[amL,amU], and B=[b1L,b1U],[b2L,b2U],…,[bnL,bnU] is an n-element set satisfying [b1L,b1U]⩽[b2L,b2U]⩽⋯⩽[bnL,bnU] (m and n are nature numbers). Define A⪯B iff we have the following.
(1)[akL,akU]≤[bkL,bkU] (for k=1,2,…,m-1) and [amL,amU]≤[bkL,bkU] (for k=m,m+1,…,n) hold in (I,≤) when m≤n.
(2)[akL,akU]≤[bkL,bkU] (for k=1,2,…,n-1) and [akL,akU]≤[bnL,bnU] (for k=n,n+1,…,m) hold in (I,≤) when m≤n.
Then2I,⪯ is a completely distributive complete lattice. Moreover, the following hold (without loss of generality we assume m≤n):(4)A∨B=a1L∨b1L,a1U∨b1U,a2L∨b2L,a2U∨b2U,…,amL∨bmL,amU∨bmU,amL∨bnL,amU∨bnU,A∧B=a1L∧b1L,a1U∧b1U,a2L∧b2L,a2U∧b2U,…,amL∧bmL,amU∧bmU,amL∧bnL,amU∧bnU.Definition 4 (cf. [7, 8]).
Each element (Write alsoΓ=Γ(x1)/x1,Γ(x2)/x2,…,Γ(xn)/xn if X={x1,x2,…,xn} is a finite set.) Γ∈2(I)X (i.e., a family {Γ(x)}x∈X consisting of nonempty finite subsets of I) is called an interval-valued hesitant fuzzy set on X, where Γ(x) denotes all possible interval valued membership degrees of the element x∈X to Γ, and 2(I) has the order ⪯ as defined in Theorem 3.Definition 5 (cf. [3]).
Each elementΓ∈2XI (i.e., a family {Γ(i)}i∈I of subsets of X) is called a soft set on X, where X is called an initial universe set and I is called a set of parameters.Definition 6 (cf. [9, 10]).
Each elementΓ∈[0,1]XI (i.e., a family {Γ(i)}i∈I of fuzzy sets on X) is called a fuzzy soft set on X, and each element Φ∈2(I)XI is called an interval-valued hesitant fuzzy soft set on X; we will use γi,x(k) to denote the k-th largest element (it exists by Theorem 3) in (Φix,⩽) (i∈I, x∈X, k≤Φix, and Φix denotes the cardinality of Φ(i)(x)).Definition 7 (cf. [1]).
LetΦ,Ψ∈2(I)XI with suppΦ=U and suppΨ=V. If, for each i∈U, there exists j∈V such that Φ(i)⪯Ψ(j) (i.e., Φ(i)(x)⪯Ψ(j)(x) for each x∈X), then we say that Φ is a generalized interval-valued hesitant fuzzy soft subset of Ψ (written as Φ⪯Ψ or Φ⋐Ψ). If both Φ⪯Ψ and Ψ⪯Φ hold, then we write Φ≅Ψ.Proposition8 holds.Proposition 8.
Φ ≅ Ψ  ⇔  ↓Φ=↓Ψ.Definition 9 (cf. [10]).
For a family{Φ1,Φ2}⊆2(I)XI, we call Φ1⊗Φ2∈2(I)XI×I, defined by(5)Φ1⊗Φ2i1,i2=Φ1i1∨Φ2i2∀i1,i2∈I×I,the hyper-product of {Φ1,Φ2} in 2(I)XI and call Φ1×Φ2∈2(I)XI×I, defined by(6)Φ1×Φ2i1,i2=Φ1i1∧Φ2i2∀i1,i2∈I×I,the product of {Φ1,Φ2} in 2(I)XI.Next, we present trapezoidal fuzzy set, trapezoidal fuzzy soft set, and generalized trapezoidal fuzzy soft set.Definition 10 (cf. [2, 11, 12]).
An elementn∈[0,1]R, defined by(7)nx=n1,n2,n3,n4x=x-n1n2-n1,n1≤x<n2,1,n2≤x<n3,x-n4n3-n4,n3≤x≤n4,0,x∈R-n1,n4,is called trapezoidal fuzzy number. The set of all trapezoidal fuzzy numbers satisfying 0≤n1≤n2≤n3≤n4≤1 is denoted by J; it is a Hutton algebra (i.e., a completely distributive complete lattice with an order-reversion involution) with the point-wise order ≤. Each element ξ∈JX is called a trapezoidal fuzzy set on X, each element Γ∈JXI (i.e., a family Γii∈I of trapezoidal fuzzy sets on X) is called a trapezoidal fuzzy soft set on X, and each element Φ∈JX×JI is called a generalized trapezoidal fuzzy soft set on X.Definition 11 (cf. [2]).
LetA={Φk}k∈K⊆JX×JI.
(1) The supremum (Each element (as a family) of2(I)XI is just a poset with the point-wise order, but JX×JI is a Hutton algebra with respect to the point-wise order (see Theorem 17).) ⋁k∈KΦk (write as also ⋓·k∈K⁡Φk) and the infimum ⋀k∈KΦk (write as also ⋒·k∈K⁡Φk) of Φkk∈K in JX×JI is called the generalized trapezoidal point-wise union and the generalized trapezoidal point-wise intersection of A in JX×JI, respectively.
(2) We call⨂A=⨂k∈KΦk∈JX×J(IK), defined by(8)⨂k∈K⁡Φkikk∈K=⋁k∈KΦkik∀ikk∈K∈IK,the hyper-product of A and ∏A=∏k∈KΦk∈JX×J(IK), defined by(9)∏k∈KΦkikk∈K=⋀k∈KΦkik∀ikk∈K∈IK,the product of A.


## 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{Φ,Ψ,Γ}⊆2(I)XI with suppΦ=U, suppΨ=V, and suppΓ=W. Then(3)
Φ×(Ψ⊗Γ)≅(Φ×Ψ)⊗(Φ×Γ).(4)
Φ⊗(Ψ×Γ)≅(Φ⊗Ψ)×(Φ⊗Γ).
However, neither (3) nor (4) is correct (see Example13).Example 13.
LetX={u,v} be a set of two cars and I={i,j,k,l} be a set of parameters, where i (resp., j, k, l) stands for the parameter “cheap" (resp., “equipment", “fuel consumption", and expensive). Again let {Φ,Ψ,Γ}⊆2(I)XI, defined by(10)Φi=0.3,0.5,0.4,0.8u,0.1,0.2,0.4,0.5v,Φj=0.2,0.3,0.4,0.5u,0.7,0.9,0.5,0.6v,Ψk=0.2,0.6,0.6,0.8u,0.4,0.5,0.7,0.8v,Γl=0.1,0.4,0.2,0.7u,0.3,0.7,0.4,0.9v,where suppΦ=U={i,j}, suppΨ=V={k}, and suppΓ=W={l}.
(1) LetΘ=Φ×(Ψ⊗Γ) and Ξ=(Φ×Ψ)⊗(Φ×Γ). As(11)Ψk∨Γl=0.2,0.6,0.6,0.8u,0.4,0.7,0.7,0.9v,Θi,k,l=Φi∧Ψk∨Γl=0.2,0.5,0.4,0.8u,0.1,0.2,0.4,0.5vand(12)Θj,k,l=Φj∧Ψk∨Γl=0.2,0.3,0.4,0.5u,0.4,0.7,0.5,0.6v, similarly,(13)Φi∧Ψk=0.2,0.5,0.4,0.8u,0.1,0.2,0.4,0.5v,Φj∧Ψk=0.2,0.3,0.4,0.5u,0.4,0.5,0.7,0.8v,Φi∧Γl=0.1,0.4,0.2,0.7u,0.1,0.2,0.4,0.5v,and(14)Φj∧Γl=0.1,0.3,0.2,0.5u,0.3,0.6,0.4,0.9v.Thus(15)Ξi,k,i,l=Φi∧Ψk∨Φi∧Γl=0.2,0.5,0.4,0.8u,0.1,0.2,0.4,0.5v,Ξj,k,j,l=Φj∧Ψk∨Φj∧Γl=0.2,0.3,0.4,0.5u,0.4,0.6,0.7,0.9v,Ξi,k,j,l=Φi∧Ψk∨Φj∧Γl=0.2,0.5,0.4,0.8u,0.3,0.6,0.4,0.9v,Ξj,k,i,l=Φj∧Ψk∨Φi∧Γl=0.2,0.4,0.4,0.7u,0.4,0.5,0.7,0.8v.As Ξi,k,j,l⋠Θ(i,k,l) and Ξ((i,k),(j,l))⋠Θ(j,k,l), (Φ×Ψ)⊗(Φ×Γ)⋠Φ×(Ψ⊗Γ). Therefore, Φ×(Ψ⊗Γ)≅(Φ×Ψ)⊗(Φ×Γ) does not hold.
(2) Now we showΦ⊗(Ψ×Γ)≅(Φ⊗Ψ)×(Φ⊗Γ) does not hold. Let Δ=Φ⊗(Ψ×Γ) and Ω=(Φ⊗Ψ)×(Φ⊗Γ). As(16)Ψk∧Γl=0.1,0.4,0.2,0.7u,0.3,0.5,0.4,0.8v,Δi,k,l=Φi∨Ψk∧Γl=0.3,0.5,0.4,0.8u,0.3,0.5,0.4,0.8vand(17)Δj,k,l=Φj∨Ψk∧Γl=0.2,0.4,0.4,0.7u,0.5,0.6,0.7,0.9v,similarly,(18)Φi∨Ψk=0.3,0.6,0.6,0.8u,0.4,0.5,0.7,0.8v,Φj∨Ψk=0.2,0.6,0.6,0.8u,0.5,0.6,0.7,0.9v,Φi∨Γl=0.3,0.5,0.4,0.8u,0.3,0.7,0.4,0.9v,and(19)Φj∨Γl=0.2,0.4,0.4,0.7u,0.5,0.7,0.7,0.9v.Thus(20)Ωi,k,i,l=Φi∨Ψk∧Φi∨Γl=0.3,0.5,0.4,0.8u,0.3,0.5,0.4,0.8v,Ωj,k,j,l=Φj∨Ψk∧Φj∨Γl=0.2,0.4,0.4,0.7u,0.5,0.6,0.7,0.9v,Ωi,k,j,l=Φi∨Ψk∧Φj∨Γl=0.2,0.4,0.4,0.7u,0.4,0.7,0.5,0.8v,Ωj,k,i,l=Φj∨Ψk∧Φi∨Γl=0.2,0.5,0.4,0.8u,0.3,0.6,0.4,0.9v.As Ωi,k,j,l⋠Δ(i,k,l) and Ω((i,k),(j,l))⋠Δ(j,k,l), (Φ⊗Ψ)×(Φ⊗Γ)⋠Φ⊗(Ψ×Γ). Therefore, Φ⊗(Ψ×Γ)≅(Φ⊗Ψ)×(Φ⊗Γ) does not hold.Correction 1 to Theorem12. Let {Φ,Ψ,Γ}⊆2(I)XI with suppΦ=U, suppΨ=V, and suppΓ=W. Then(3){[Φ(i)∧Ψ(j)]∨[Φ(i)∧Γ(k)]}((i,j),(i,k))∈(U×V)×(U×W) = {Φ(i)∧[Ψ(j)∨Γ(k)]}(i,j,k)∈U×V×W, and thus Φ×(Ψ⊗Γ)⪯(Φ×Ψ)⊗(Φ×Γ).(4){[Φ(i)∨Ψ(j)]∧[Φ(i)∨Γ(k)]}((i,j),(i,k))∈(U×V)×(U×W) = {Φ(i)∨[Ψ(j)∧Γ(k)]}(i,j,k)∈U×V×W, and thus Φ⊗(Ψ×Γ)⪯(Φ⊗Ψ)×(Φ⊗Γ).Proof.
(3) For each(i,j,k)∈U×V×W and each x∈X,(21)Φix∧Ψjx∨Φix∧Γkx=Φix∧Ψjx∨Γkx=Φ×Ψ⊗Γi,j,kxby Theorem 3, which means(22)Φi∧Ψj∨Φi∧Γki,j,i,k∈U×V×U×W=Φi∧Ψj∨Γki,j,k∈U×V×W,and thus Φ×(Ψ⊗Γ)⪯(Φ×Ψ)⊗(Φ×Γ).
(4) For each(i,j,k)∈U×V×W and each x∈X,(23)Φix∨Ψjx∧Φix∨Γkx=Φix∨Ψjx∧Γkx=Φ⊗Ψ×Γi,j,kxby Theorem 3, which means(24)Φi∨Ψj∧Φi∨Γki,j,i,k∈U×V×U×W=Φi∨Ψj∧Γki,j,k∈U×V×W,and thus Φ⊗(Ψ×Γ)⪯(Φ⊗Ψ)×(Φ⊗Γ).Correction 2 to Theorem12. Let {Φ,Ψ,Γ}⊆2(I)XI with suppΦ=U, suppΨ=V, and suppΓ=W. Then(1)
Φ∧(Ψ∨Γ)=(Φ∧Ψ)∨(Φ∧Γ).(2)
Φ∨(Ψ∧Γ)=(Φ∨Ψ)∧(Φ∨Γ).Proof.
It follows from the fact that2(I),⪯ is a completely distributive complete lattice.Analogously, the following (on community and associativity) hold.Theorem 14.
Let{Φ,Ψ,Γ}⊆2(I)XI with suppΦ=U, suppΨ=V, and suppΓ=W. Then
(1){Φ(i)∧Ψ(j)}(i,j)∈(U×V) = {Ψ(j)∧Φ(i)}(i,j)∈(U×V), and thus Φ×Ψ=Ψ×Φ.
(2){Φ(i)∨Ψ(j)}(i,j)∈(U×V) = {Ψ(j)∨Φ(i)}(i,j)∈(U×V), and thus Φ⊗Ψ=Ψ⊗Φ.
(3){Φ(i)∧[Ψ(j)∧Γ(k)]}(i,j,k)∈U×V×W = {[Φ(i)∧Ψ(j)]∧Γ(k)}(i,j,k)∈U×V×W, and thus Φ×(Ψ×Γ)=(Φ×Ψ)×Γ.
(4){Φ(i)∨[Ψ(j)∨Γ(k)]}(i,j,k)∈U×V×W = {[Φ(i)∨Ψ(j)]∨Γ(k)}(i,j,k)∈U×V×W, and thus Φ⊗(Ψ⊗Γ)=(Φ⊗Ψ)⊗Γ.


## 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{Φ,Ψ,Γ}⊆JX×JI with suppΦ=I1, suppΨ=I2, and suppΓ=I3. Then(3)
Φ×(Ψ⊗Γ) = (Φ×Ψ)⊗(Φ×Γ).(4)
Φ⊗(Ψ×Γ) = (Φ⊗Ψ)×(Φ⊗Γ).
However, neither (3) nor (4) is correct mainly because the cardinalityI1×I2×I3 of I1×(I2×I3) (the support of the left) is less than the cardinality I1×I2×I1×I3 of (I1×I2)×(I1×I3) (the support of the right) in general.Example 16.
LetX={u,v} be a two-element set, I={i,j,k,m} a four-element set, I1={i,j} a two-element set, I2={k}, and I3={m}. Again let {Φ,Ψ,Γ}⊆JX×JI (with supports I1, I2, and I3, respectively), defined by(25)Φi=0.3,0.4,0.6,0.8u,0.5,0.7,0.7,0.9v;0.1,0.4,0.7,0.8,Φj=0.2,0.3,0.4,0.5u,0.7,0.8,0.8,0.9v;0.5,0.6,0.7,0.8,Ψk=0.1,0.2,0.2,0.3u,0.2,0.3,0.4,0.5v;0.4,0.4,0.7,0.9,Γm=0.4,0.5,0.5,0.8u,0.1,0.1,0.4,0.5v;0.6,0.8,0.9,0.9.Then Υ1=Φ×(Ψ⊗Γ) has the following two members:(26)Υ1i,k,m=0.3,0.4,0.5,0.8u,0.2,0.3,0.4,0.5v;0.1,0.4,0.7,0.8,Υ1j,k,m=0.2,0.3,0.4,0.5u,0.2,0.3,0.4,0.5v;0.5,0.6,0.7,0.8.Υ2=(Φ×Ψ)⊗(Φ×Γ) has the following four members:(27)Υ2i,k,i,m=0.3,0.4,0.5,0.8u,0.2,0.3,0.4,0.5v;0.1,0.4,0.7,0.8,Υ2j,k,i,m=0.3,0.4,0.5,0.8u,0.2,0.3,0.4,0.5v;0.4,0.4,0.7,0.8,Υ2i,k,j,m=0.2,0.3,0.4,0.5u,0.2,0.3,0.4,0.5v;0.5,0.6,0.7,0.8,Υ2j,k,j,m=0.2,0.3,0.4,0.5u,0.2,0.3,0.4,0.5v;0.5,0.6,0.7,0.8.
Apparently,Υ1⊆Υ2. However, ↓Υ1⊉  ↓Υ2 (because Υ2(j,k,i,m)≰Υ1(i,k,m) and Υ2(j,k,i,m)≰Υ1(j,k,m)) and I1×I2×I3=2≠4=I1×I2×I1×I3. Therefore, ↓Υ1≠  ↓Υ2, especially, Υ1≠Υ2 (whether we look Υ1 and Υ2 to be sets or look them to be mappings).Now we consider some possible corrections or replacements of the theorem.A correction to Theorem15. Let Φ,Ψ,Γ∈JX×JI with suppΦ=I1, suppΨ=I2, and suppΓ=I3. Then(3)
Φ×(Ψ⊗Γ)⊆(Φ×Ψ)⊗(Φ×Γ).(4)
Φ⊗(Ψ×Γ)⊆(Φ⊗Ψ)×(Φ⊗Γ).Proof.
We only show (3). For each(i1,(i2,i3))∈I1×(I2×I3),(28)Φ×Ψ⊗Γi1,i2,i3=Φi1∧Ψi2∨Γi3=Φi1∧Ψi2∨Ψi2∧Γi3=Φ×Ψ⊗Φ×Γi1,i2i2,i3since JX×J is a distributive lattice. Therefore,(29)Φ×Ψ⊗Γi1,i2,i3∣i1,i2,i3∈I1×I2×I3⊆Φ×Ψ⊗Φ×Γi1,i2i2,i3∣i1,i2i2,i3∈I1×I2×I2×I3,which implies Φ×(Ψ⊗Γ)⊆(Φ×Ψ)⊗(Φ×Γ).Theorem 17.
J X × J I is a Hutton algebra (with the point-wise order), and thus the following hold for a family Φk,j∣j∈Jk,k∈K⊆JX×JK:(30)⋁j∈Jj,k∈KΦk,j′=⋀j∈Jk,k∈KΦk,j′,⋀j∈Jk,k∈IΦk,j′=⋁j∈Jk,k∈KΦk,j′,⋁k∈K⋀j∈JkΦk,j=⋀f∈F⋁k∈KΦk,fk,⋀k∈K⋁j∈JkΦk,j=⋁f∈F⋀j∈KΦk,fk,where F is the set of all mappings f:K→∐k∈KJk (disjoint union) satisfying f(k)∈Jk(∀k∈K).Proof.
It follows from the fact thatJ is a Hutton algebra.



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