Letter to the Editor | Open Access
Comment on “Rough Multisets and Information Multisystems”
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.
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 [1–3]. 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.
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 ). An mset drawn from the set is represented by a count function defined as : , where represents the set of nonnegative integers.
Here is the number of occurrences of the element in the mset . The mset is drawn from the set and it is written as , where is the number of occurrences of the element , in the mset .
Definition 2 (see ). A domain is defined as a set of elements from which msets are constructed. The mset space is the set of all msets whose elements are in such that no element in the mset occurs more than times.
The mset space is the set of all msets over a domain such that there is no limit on the number of occurrences of an element in an mset. If , then , .
Definition 3 (see ). Let and be two msets drawn from a set . Then, (1) if for all ,(2) if for all ,(3) if for all ,(4) if for all ,(5) if for all ,(6) if for all , where and represent mset addition and mset subtraction, respectively.
Let be an mset drawn from a set . The support set of denoted by is a subset of and ; that is, is an ordinary set.
Definition 4 (see ). Let be an mset drawn from the set . If for all , then is called an empty mset and denoted by ; that is, for all .
Definition 5 (see ). Let be an mset drawn from the set and be the mset space defined over . Then, for any mset , the complement of in is an element of such that for every .
Definition 6 (see ). Let and be two msets drawn from a set . Then, the Cartesian product of and is defined as .
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 of the nonempty msets is the mset of all ordered -tuples , where , , and with , where and .
Definition 7 (see ). A submset of is said to be an mset relation on if every member of has a count . Then, related to is denoted by .
Definition 8 (see ). The domain and range of the mset relation on are defined as follows, respectively.
such that , where .
such that , where .
Definition 9 (see ). An -equivalence class in containing an element is denoted by . The pair is called an mset approximation space. For any , the lower mset approximation and upper mset approximation of are defined, respectively, byThe pair is referred to as the rough mset of .
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 , p. 12], the authors introduced the fact that, for any submsets and of ,(1),(2).
The following example shows that(1),(2).
Example 1. Let and , . Then, and . If such that(1) and , then , , and . Thus, . Hence, ,(2) and , then , , and . Thus, . Hence, .
The following theorem is the correction form of [Theorem , p. 12] in .
Theorem 2. For any submsets and of ,(1),(2).
Proof. (1) (2)
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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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.