We show that some results introduced in Girish and John (2011) are incorrect. Moreover, a counterexample is given to confirm our claim. Furthermore, the correction form of the incorrect results in Girish and John (2011) is presented.

1. Introduction

In addition to Girish and John (2011) [1], many authors were recently interested in studying the extensions of results and properties of rough set to rough multiset [1–3]. There exist many of the applications on rough multisets in several fields such as the medicine field in [3]. Additionally, the concept of rough multisets and the basic definitions of relations in multiset context are introduced by Girish and John [4, 5]. Therefore, the notion of multisets (briefly, msets) was introduced by Yager [6], and Blizard [7, 8] and Jena et al. [9] have mentioned them as well.

2. Preliminaries

The aim of this section is to present the basic concepts and properties of msets. At the end of this section, rough msets and the definitions and notions of relations in msets are introduced.

Definition 1 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

An mset M drawn from the set X is represented by a count function CM defined as CM: X→N, where N represents the set of nonnegative integers.

Here CM(x) is the number of occurrences of the element x in the mset M. The mset M is drawn from the set X={x1,x2,x3,…,xn} and it is written as M={m1/x1,m2/x2,m3/x3,…,mn/xn}, where mi is the number of occurrences of the element xi,i=1,2,3,…,n, in the mset M.

Definition 2 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

A domain X is defined as a set of elements from which msets are constructed. The mset space [X]w is the set of all msets whose elements are in X such that no element in the mset occurs more than w times.

The mset space [X]∞ is the set of all msets over a domain X such that there is no limit on the number of occurrences of an element in an mset. If X={x1,x2,…,xk}, then [X]w={{m1/x1,m2/x2,…,mk/xk}:mi∈{0,1,2,…,w}, i=1,2,…,k}.

Definition 3 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

Let M and N be two msets drawn from a set X. Then,

M=N if CM(x)=CN(x) for all x∈X,

M⊆N if CM(x)≤CN(x) for all x∈X,

P=M∪N if CP(x)=maxCMx,CNx for all x∈X,

P=M∩N if CP(x)=minCMx,CNx for all x∈X,

P=M⊕N if CP(x)=minCMx+CNx,w for all x∈X,

P=M⊖N if CP(x)=maxCMx-CNx,0 for all x∈X, where ⊕ and ⊖ represent mset addition and mset subtraction, respectively.

Let M be an mset drawn from a set X. The support set of M denoted by M∗ is a subset of X and M∗={x∈X:CM(x)>0}; that is, M∗ is an ordinary set.

Definition 4 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

Let M be an mset drawn from the set X. If CM(x)=0 for all x∈X, then M is called an empty mset and denoted by ϕ; that is, ϕ(x)=0 for all x∈X.

Definition 5 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

Let M be an mset drawn from the set X and [X]w be the mset space defined over X. Then, for any mset M∈[X]w, the complement Mc of M in [X]w is an element of [X]w such that CMc(x)=w-CM(x) for every x∈X.

Definition 6 (see [<xref ref-type="bibr" rid="B6">4</xref>]).

Let M1 and M2 be two msets drawn from a set X. Then, the Cartesian product of M1 and M2 is defined as M1×M2={(m/x,n/y)/mn:x∈mM1,y∈nM2}.

The Cartesian product of three or more nonempty msets can be defined by generalizing the definition of the Cartesian product of two msets. Thus, the Cartesian product M1×M2×⋯×Mn of the nonempty msets M1,M2,…,Mn is the mset of all ordered n-tuples (m1,m2,…,mn), where mi∈riMi, i=1,2,…,n, and (m1,m2,…,mn)∈pM1×M2×⋯×Mn with p=∏ri, where ri=CMi(mi) and i=1,2,…,n.

Definition 7 (see [<xref ref-type="bibr" rid="B6">4</xref>]).

A submset R of M×M is said to be an mset relation on M if every member (m/x,n/y) of R has a count (mn). Then, m/x related to n/y is denoted by m/xRn/y.

Definition 8 (see [<xref ref-type="bibr" rid="B6">4</xref>]).

The domain and range of the mset relation R on M are defined as follows, respectively.

DomR={x∈rM:∃y∈sM such that (r/x)R(s/y)}, where CDomR(x)=sup{C1(x,y):x∈rM}.

RanR={y∈sM:∃x∈rM such that (r/x)R(s/y)}, where CRanR(y)=sup{C2(x,y):y∈sM}.

Definition 9 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

An m-equivalence class in R containing an element x∈mM is denoted by [m/x]. The pair (M,R) is called an mset approximation space. For any N⊆M, the lower mset approximation and upper mset approximation of N are defined, respectively, by(1)RLN=x∈mM:mx⊆N,(2)RUN=x∈mM:mx∩N≠ϕ.

The pair (RL(N),RU(N)) is referred to as the rough mset of N.

3. Counterexample

In this section, we point out where the errors occur in [1] and then give counterexamples to confirm our claim. Finally, the correction form of these errors is presented.

In [[1], Theorem 4.5, p. 12], the authors introduced the fact that, for any submsets M1 and M2 of M,

RL[(M1⊖M2)c]=RL(M1c)⊕RL(M2),

RL[(M1⊕M2)c]=RL(M1c)⊖RL(M2).

The following example shows that

RL[(M1⊖M2)c]⊈RL(M1c)⊕RL(M2),

RL[(M1⊕M2)c]⊉RL(M1c)⊖RL(M2).

Example 1.

Let M={3/x,2/y,4/z,8/r} and R={(3/x,3/x)/9,(2/y,2/y)/4,(4/z,4/z)/16,(8/r,8/r)/64, (3/x,2/y)/6,(2/y,3/x)/6,(3/x,4/z)/12,(4/z,3/x)/12,(2/y,4/z)/8,(4/z,2/y)/8}. Then, [3/x]=[2/y]=[4/z]={3/x,2/y,4/z} and [8/r]={8/r}. If M1,M2⊆M such that

M1={3/x,4/z,8/r} and M2={3/x,4/z}, then RL[(M1⊖M2)c]={3/x,2/y,4/z}, RL(M1c)=ϕ, and RL(M2)=ϕ. Thus, RL(M1c)⊕RL(M2)=ϕ. Hence, RL[(M1⊖M2)c]⊈RL(M1c)⊕RL(M2),

M1=ϕ and M2={2/x,2/y,8/r}, then RL[(M1⊕M2)c]=ϕ, RL(M1c)=M, and RL(M2)={8/r}. Thus, RL(M1c)⊖RL(M2)={3/x,2/y,4/z}. Hence, RL[(M1⊕M2)c]⊉RL(M1c)⊖RL(M2).

The following theorem is the correction form of [Theorem 4.5, p. 12] in [1].

The authors declare that they have no conflicts of interest.

GirishK. P.JohnS. J.Rough multisets and information multisystemsGirishK.JohnS. J.On rough multiset relationsHosnyM.RaafatM.On generalization of rough multiset via multiset idealsGirishK. P.JohnS. J.Relations and functions in multiset contextGirishK. P.JohnS. J.Multiset topologies induced by multiset relationsYagerR. R.On the theory of bagsBlizardW. D.Multiset theoryBlizardW. D.Real-valued multisets and fuzzy setsJenaS. P.GhoshS. K.TripathyB. K.On the theory of bags and lists