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objects definitely belonging to the vague concept, whereas the upper approximation is the
set of objects possibly belonging to the same.
An information system is a pair (U, A), where U represents a non-empty finite set called
the universe and A a non-empty finite set of attributes. Let B⊆A and X⊆U . Taking
into account these two sets, it is possible to approximate the set X making only the use
of the information contained in B by the process of construction of the lower and upper
approximations of X.
Let IN D(B)⊆U×U be defined by
(x, y)∈IN D(B) if and only if, for all a∈B a(x) = a(y)
An approximation space AS B = (U, IN D(B)).
For X⊆U , the sets
LOW (AS B , X) ={x∈U : [x] B ⊆X},
U P P (AS B , X) ={x∈U : [x] B ∩X 6=∅}.
where [x] B denotes the equivalence class of the object x relative to B (the equivalence
relation IN D(B)), are called the B-lower and B-upper approximations of X in U.
It is possible to express numerically the roughness R(AS B , X) of a set X with respect to
B (Pawlak, 1991) by assignment
R(AS B , X) = 1− card(LOW (AS B , X))
card(U P P (AS B , X)) .
In this way, the value of the roughness of the set X equal 0 means that X is crisp with
respect to B, and conversely if R(AS B , X) > 0 then X is rough (i.e., X is vague with respect
to B). Detailed information on rough set theory is provided in (Pawlak, 1991; Pawlak and
Skowron, 2007; Stepaniuk, 2008).
11.4.2 Generalized Rough Set Theory
Let U be a non-empty set of objects called the universal set and P (U ) be the power set of
U as described in previous subsection. Binary relation R on U is referred to as reflexive if
for all x∈U, xRx. Relation R is referred to as symmetric if for all x, y∈U, xRy implies
yRx. Relation R is referred to as transitive if for all x, y, z∈U, if xRy and yRz then xRz.
Relation R is an equivalence relation if it is reflexive, symmetric and transitive.
Parameterized approximation space
In (Skowron and Stepaniuk, 1996; Stepaniuk, 2008) parameterized approximation space
has been defined as a system AS #,$ = (U, I # , ν $ ) where
•U non empty set of objects
•I # : U→P (U ) uncertainty function
: P (U )×P (U )→[0, 1] rough inclusion function
and #, $ describe parameter indices ( #, $ may be omitted if the context is clear).
In this way, each predefined number of cluster centers defines for each data object different
uncertainty function I # : U→P (U ). Detailed description is given in (Stepaniuk, 2008).
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