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These definitions are easily extended to fuzzy sets for dealing with uncertainty. The dein-
ition of rough fuzzy sets we propose to adopt here takes inspiration, as firstly made in Ref.
[ 29 ] , from the notion of composite sets [ 21 , 11 ] . Let a fuzzy set
on T defined by adding to each element of T the degree of its membership to the set through a
mapping μ F : T → [0, 1]. The operations on fuzzy sets are extensions of those used for conven-
tional sets (intersection, union, comparison, etc.). The basic operations are the intersection and
union as defined as follows:
The previous are only a restricted set of operations applicable among fuzzy sets, but they
are the most significant for our aim. A composite set or C -set is a triple C = ( Γ , m , M ) (where
Γ = { T 1 , …, T p } is a partition of T in p disjoint subsets T 1 , …, T p , while m and M are mappings of
kind T → [0, 1] such that
for each choice of function f : T →[0, 1]. Γ and f uniquely define a composite set. Based on these
assumptions, we may formulate the following definition of rough fuzzy set:
1. If f is the membership function μ F and the partition Γ is made with respect to a relation
, i.e., Γ = T / , a fuzzy set F gets two approximations RS ( F ) and RS ( F ), which are
again fuzzy sets with membership functions defined as Equation (1) , i.e.,
. The couple of sets ( RS ( F ), RS ( F )) is a rough fuzzy set .
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