# (2112-7123) Generalization of rough fuzzy sets based on a fuzzy ideal

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

2 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

10.22111/ijfs.2022.7127

Abstract

Since Pawlak defined the notion of rough sets in 1982, many authors made wide research studying rough sets in the ordinary case and the fuzzy case. This paper introduced a new style of rough fuzzy sets based on %arbitrary fuzzy relation $R$ and a fuzzy ideal $\ell$ on a universal finite set $X$. New lower and new upper fuzzy sets are introduced, and consequently, fuzzy interior and  fuzzy closure of a rough fuzzy set are discussed. These definitions, if $\ell$ is restricted to $\ell^{\circ} = \{\overline{0}\}$, imply the fuzzification of previous definitions given in the ordinary case, and moreover in the crisp case, we get exactly these previous definitions. The new style gives us a better accuracy value of roughness than the previous styles. Rough fuzzy connectedness is introduced as a sample of applications on the recent style of roughness.

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