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A comparison of three types of rough fuzzy sets based on two universal sets

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Abstract

The extension of rough set model is an important research direction in rough set theory. This paper presents two new extensions of the rough set model over two different universes. By means of a binary relation between two universes of discourse, two pairs of rough fuzzy approximation operators are proposed. These models guarantee that the approximating sets and the approximated sets are on the same universes of discourse. Furthermore, some interesting properties are investigated, the connections between relations and rough fuzzy approximation operators are examined. Finally, the connections of these approximation operators are made, and conditions under which these approximation operators made equivalent are obtained.

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Abbreviations

\({\mathbf{\mathcal{P}}}(U)\) :

Power set of the universe set U

\({\mathbf{\mathcal{F}}}({\text{U}})\) :

Fuzzy power set of the universe set U

\({\text{F}}\left( {\text{x}} \right)\) :

Successor neighborhood of x

\(G\left( y \right)\) :

Predecessor neighborhood of y

\({\underline{R}}_{s}\) :

Generalized rough fuzzy lower approximation operator with respect to the successor neighborhood

\(\bar{R}_{s}\) :

Generalized rough fuzzy upper approximation operator with respect to the successor neighborhood

\({\underline{R}}_{p}\) :

Generalized rough fuzzy lower approximation operator with respect to the predecessor neighborhood

\(\bar{R}_{p}\) :

Generalized rough fuzzy upper approximation operator with respect to the predecessor neighborhood

\({\underline{R}}^{*}\) :

Revised rough fuzzy lower approximation operator

\(\bar{R}^{*}\) :

Revised rough fuzzy upper approximation operator

\({\underline{R}}^{{\prime }}\) :

Weak rough fuzzy lower approximation operator

\(\bar{R}^{\prime }\) :

Weak rough fuzzy upper approximation operator

\({\underline{R}}^{\prime \prime }\) :

Strong rough fuzzy lower approximation operator

\(\bar{R}^{\prime \prime }\) :

Strong rough fuzzy upper approximation operator

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Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

    M. E. Abd El-Monsef & R. M. Aqeel

  2. Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt

    M. A. El-Gayar

Authors
  1. M. E. Abd El-Monsef
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  2. M. A. El-Gayar
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  3. R. M. Aqeel
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Correspondence to R. M. Aqeel.

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Abd El-Monsef, M.E., El-Gayar, M.A. & Aqeel, R.M. A comparison of three types of rough fuzzy sets based on two universal sets. Int. J. Mach. Learn. & Cyber. 8, 343–353 (2017). https://doi.org/10.1007/s13042-015-0327-8

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