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manually assign a membership to an element. This can avoid any possible
individual bias. Moreover, uncertainty is considered to be relative to membership
relation instead of the set itself (as defined in fuzzy set). To describe uncertainty
clearly, we have the following definition of membership relation:
Definition 11.4
Suppose X
⊆
U and x
∈
U, the membership function of set X (or
called as rough membership function) is defined as
card X
%
R x
% &&
k
R
X
(11.3)
% &
x
;
card R x
% % &&
where
y
∈
U)
∧
(
y R x
)}.
In this chapter, we use card to represent the cardinal number of a set.
According to above definition, some properties can be got as follows:
R
is an indiscernibility relation, and
R
(
x
)=[
x
]
R
={
y
:(
R
X
m
(1)
(
x
)=1 iff [
x
]
R
⊆
X
;
R
X
m
]
R
∩
X
≠
∅
;
(2)
(
x
)>0 iff [
x
R
X
m
]
R
∩X=
∅
.
(3)
(
x
)=0 iff [
x
m
)∈[0,1]. Note that the membership relation here is
calculated based on classification of knowledge which is already known, thus it
can be explained as some kind of conditional probability which can be calculated
for individuals on the universe instead of predefined manually.
Obviously, we have
(
x
X
11.1.3
The View of Concept's Boundary
The granularity of knowledge is the reason that known knowledge can not
explicitly express some concepts. This is the origin of the idea of “boundary” to
describe inaccuracy. Frege concludes that “concepts must have an explicit
boundary. The concept without explicit boundary will result in an area without
clear borders.” The fuzziness in rough set theory is a concept based on boundary,
that is, an inaccurate concept has an ambiguous boundary that can't be explicitly
classified. To describe fuzziness, every inaccurate concept is expressed by a pair
of accurate concepts namely the upper approximation and lower approximation.
They can be defined by membership function as follows:
