ADS Advances in Decision Sciences 2090-3367 2090-3359 Hindawi 10.1155/2017/3436073 3436073 Letter to the Editor Comment on “Rough Multisets and Information Multisystems” El-Sheikh Sobhy Ahmed 1 Hosny Mona 1 http://orcid.org/0000-0002-8710-3875 Raafat Mahmoud 1 Peiris Shelton Department of Mathematics Faculty of Education Ain Shams University Cairo Egypt asu.edu.eg 2017 28122017 2017 26 07 2016 15 11 2017 28122017 2017 Copyright © 2017 Sobhy Ahmed El-Sheikh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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) , many authors were recently interested in studying the extensions of results and properties of rough set to rough multiset . There exist many of the applications on rough multisets in several fields such as the medicine field in . 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 , and Blizard [7, 8] and Jena et al.  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: XN, 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 xX,

MN if CM(x)CN(x) for all xX,

P=MN if CP(x)=maxCMx,CNx for all xX,

P=MN if CP(x)=minCMx,CNx for all xX,

P=MN if CP(x)=minCMx+CNx,w for all xX,

P=MN if CP(x)=maxCMx-CNx,0 for all xX, 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={xX: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 xX, then M is called an empty mset and denoted by ϕ; that is, ϕ(x)=0 for all xX.

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 xX.

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:xmM1,ynM2}.

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 miriMi, 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.

Dom R = { x r M : y s M such that (r/x)R(s/y)}, where CDomR(x)=sup{C1(x,y):xrM}.

Ran R = { y s M : x r M such that (r/x)R(s/y)}, where CRanR(y)=sup{C2(x,y):ysM}.

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

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

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  and then give counterexamples to confirm our claim. Finally, the correction form of these errors is presented.

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

RL[(M1M2)c]=RL(M1c)RL(M2),

RL[(M1M2)c]=RL(M1c)RL(M2).

The following example shows that

RL[(M1M2)c]RL(M1c)RL(M2),

RL[(M1M2)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,M2M such that

M1={3/x,4/z,8/r} and M2={3/x,4/z}, then RL[(M1M2)c]={3/x,2/y,4/z}, RL(M1c)=ϕ, and RL(M2)=ϕ. Thus, RL(M1c)RL(M2)=ϕ. Hence, RL[(M1M2)c]RL(M1c)RL(M2),

M1=ϕ and M2={2/x,2/y,8/r}, then RL[(M1M2)c]=ϕ, RL(M1c)=M, and RL(M2)={8/r}. Thus, RL(M1c)RL(M2)={3/x,2/y,4/z}. Hence, RL[(M1M2)c]RL(M1c)RL(M2).

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

Theorem 2.

For any submsets M1 and M2 of M,

RL[(M1M2)c]RL(M1c)RL(M2),

RL[(M1M2)c]RL(M1c)RL(M2).

Proof.

(3)RLM1M2c=xmM:mxM1M2c=xmM:mxM1cM2xmM:mxM1cxmM:mxM2=RLM1cRLM2.

(4)RLM1M2c=xmM:mxM1M2c=xmM:mxM1cM2xmM:mxM1cxmM:mxM2=RLM1cRLM2.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Girish K. P. John S. J. Rough multisets and information multisystems Advances in Decision Sciences 2011 2011 17 495392 10.1155/2011/495392 2-s2.0-84858315909 Girish K. John S. J. On rough multiset relations International Journal of Granular Computing, Rough Sets and Intelligent Systems 2014 3 4 306 326 10.1504/IJGCRSIS.2014.068036 Hosny M. Raafat M. On generalization of rough multiset via multiset ideals Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology 2017 33 2 1249 1261 10.3233/JIFS-17102 Girish K. P. John S. J. Relations and functions in multiset context Information Sciences 2009 179 6 758 768 MR2489175 10.1016/j.ins.2008.11.002 Zbl1162.03316 2-s2.0-58049098625 Girish K. P. John S. J. Multiset topologies induced by multiset relations Information Sciences 2012 188 298 313 MR2873678 10.1016/j.ins.2011.11.023 Zbl1305.54019 2-s2.0-84855471098 Yager R. R. On the theory of bags International Journal of General Systems 1986 13 1 23 37 2-s2.0-0002312129 10.1080/03081078608934952 Blizard W. D. Multiset theory Notre Dame Journal of Formal Logic 1989 30 1 36 66 MR990203 2-s2.0-84972519707 10.1305/ndjfl/1093634995 Blizard W. D. Real-valued multisets and fuzzy sets Fuzzy Sets and Systems 1989 33 1 77 97 2-s2.0-38249004569 10.1016/0165-0114(89)90218-2 Jena S. P. Ghosh S. K. Tripathy B. K. On the theory of bags and lists Information Sciences 2001 132 1-4 241 254 MR1822770 10.1016/S0020-0255(01)00066-4 Zbl0980.68041 2-s2.0-0035247387