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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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