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and
A = \
B∈ ˚ B
c B (A) = \
B∈B
c B (A).
(4.10)
The following equations also relate the interior and closure to erosion and dilation, (Serra,
1983):
A = [
B∈ ˚ B
e B (A) = [
B∈B
e B (A).
(4.11)
and
A = \
B∈ ˚ B
d B (A) = \
B∈B
d B (A).
(4.12)
In words, the interior of a set A is the union of the opening or erosions of the set A with
open or closed balls of different sizes. The closure of a set A is the union of the closing or
dilations of A with open/closed balls of different sizes.
4.4
Rough Sets
In rough set theory, objects in a universe X, are perceived by means of their attributes(features).
Let ϕ i denote a real-valued function that represents an object feature. Each element
x∈A⊆X is defined by its feature vector, ϕ(x) = (ϕ 1 (x), ϕ 2 (x), . . . , ϕ n (x)) Peters and
Wasilewski (2009). An equivalence (called an indiscernibility relation (Pawlak and Skowron,
2007c)) can be defined on X. Let∼be an equivalence relation defined on X, i.e.,∼is re-
flexive, symmetric and transitive. An equivalent relation(∼) on X classifies objects (x∈X)
into classes called equivalence classes. Objects in each class have the same feature-value
vectors and are treated as one generalized item. The indiscernibility(equivalence) relation
X,ϕ is defined in (4.13).
X,ϕ ={(x, y)∈X×X : ϕ(x) = ϕ(y)}.
(4.13)
X,ϕ partitions universe X into non-overlapping equivalence classes denoted by X /∼ ϕ
or
simply X /∼ . Let x /∼ denote a class containing an element x as in (4.14).
x /∼ ={y∈X|x∼ ϕ y}.
(4.14)
Let X /∼ denote the quotient set as defined in (4.15).
X /∼ ={x /∼ |x∈X}.
(4.15)
The relation∼holds for all members of each class x /∼ in a partition.
In a rough set model, elements of the universe are described based on the available infor-
mation about them. For each subset A⊆X, rough set theory defines two approximations
based on equivalence classes, lower approximation A and upper approximation, A :
A ={x∈X : x /∼ ⊆A}
(4.16)
A ={x∈X : x /∼ ∩A 6= φ}
(4.17)
The set A⊂X is called a rough set if A 6= A , otherwise it is called exact set, (Pawlak,
1991). Figure 4.1 shows the set A and lower and upper approximation of it in terms of the
partitioned space. The space X is partitioned in to squares of the form (j, j + 1] 2 .

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