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and P
≠
, the intersection set of all
equivalence relation on P is called as indiscernibility relation on P, denoted as
IND(P). That is,
Definition 11.2
Suppose P
R
⊆
X Z
x
X Z
x
;
<
(11.2)
IN D
% &
P
R
R
¦
P
Note, IND(
) is an equivalence relation, and it is unique.
Definition 11.3
P
Given approximate space K=(U, R), subset X
⊆
U is called as a
concept on U. Formally, empty set can also be viewed as a concept. The
indiscernibility relation IND(P) generated from non-empty family subset P
⊆
R,
denoted as U/IND(P), is called as basic knowledge, and its corresponding
equivalence class is called as basic concept. Specially, if relation Q
∈
R, then Q
is called as elementary knowledge, and its corresponding equivalence classes are
called as elementary concepts.
Capitalized letters such as
P
,
Q
and
R
usually denote relations whereas bolded
P, Q
and
R
denote the family set of relations respectively; [
x
]
R
or
R
(
x
) denotes
that relation
R
contains the concept or equivalent class of element
x
which
satisfies
(
P
).
According to above definition, concept is the set of objects whereas family of
concepts is called knowledge on
x
∈
U
. For simplicity, sometimes
P
is used to replace
IND
U
. Classified family sets on
U
can be viewed as
knowledge base on
, or to put it differently, knowledge base is the assembly of
classification methods.
U
11.1.2
A New Type of Membership Relations
The rough set theory has some similarity compared with traditional set theory,
whereas they have totally different motivations. An element must either belong to
or be excluded from a set, that is, the membership function is
X
(
x
)
∈
{0,1}.
Fuzzy set theory extended this definition by giving a membership to elements in
fuzzy set, i.e.
µ
X
(
x
)
∈
[0,1], which enables the fuzzy set ability to handle some
fuzzy and uncertain data. However, usually the membership functions are
manually defined which hinders its application in real life. Moreover, traditional
set theory and fuzzy set theory treat membership relations as basic concepts, and
the union and intersection of sets are based on min/max operations on the
membership of their elements. This is because the memberships are predefined,
and for a traditional set the default membership is 1 or 0. In rough set theory, the
membership relation is no longer a basic concept thus there's no need to
