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exactly definable sets RX and RX in U , which are obtained as

=
[
{Y∈U/R : Y⊆X}

RX

(3.1)

=
[
{Y∈U/R : Y∩X 6=∅}
(3.2)

RX

In the above, RX and RX are respectively called the R-lower approximation and the R-

upper approximation of X. In essence, the pair of sets < RX, RX > is the representation

of any arbitrary set X⊆U in the approximation space < U, R >, where X can not be

defined. As given in (Pawlak, 1991), an inexactness measure of the set X can be defined as

ρ
R
(X) = 1−
|RX|

|RX|

(3.3)

where|RX|and|RX|are respectively the cardinalities of the sets RX and RX in U . The

inexactness measure ρ
R
(X) is called the R-roughness measure of X and it takes a value in

the interval [0, 1].

3.2.2 The Lower and Upper Approximations of a Set

The expressions for the lower and upper approximations of the set X depends on the type

of relation R and whether X is a crisp (Klir and Yuan, 2005) or a fuzzy (Klir and Yuan,

2005) set. Here we shall consider the upper and lower approximations of the set X when

R denotes an equivalence, a fuzzy equivalence, a tolerance or a fuzzy tolerance relation and

X is a crisp or a fuzzy set.

When X is a crisp or a fuzzy set and the relation R is a crisp or a fuzzy equivalence

relation, we consider the expressions for the lower and the upper approximations of the set

X as

RX

= {(u, M (u))|u∈U}

RX

= {(u, M (u))|u∈U}

(3.4)

where

=
X

Y∈U/R

M (u)

m
Y
(u)×inf

ϕ∈U

max(1−m
Y
(ϕ), µ
X
(ϕ))

=
X

Y∈U/R

M (u)

m
Y
(u)×sup

ϕ∈U

min(m
Y
(ϕ), µ
X
(ϕ))

(3.5)

where the membership function m
Y
represents the belongingness of every element (u) in

the universe (U ) to a granule Y∈U/R and it takes values in the interval [0, 1] such that

P
Y
m
Y
(u) = 1, and µ
X
, which takes values in the interval [0, 1], is the membership function

associated with X. When X is a crisp set, µ
X
would take values only from the set{0, 1}.

Similarly, when R is a crisp equivalence relation m
Y
would take values only from the set

{0, 1}. In the above, the symbols
P
(sum) and×(product) respectively represent specific

fuzzy union and intersection operations (Klir and Yuan, 2005), which are chosen judging

their suitability with respect to the underlying application of measuring ambiguity.

In the above, we have considered the indiscernibility relation R⊆U×U to be an

equivalence relation, that is, R satisfies crisp or fuzzy reflexivity, symmetry and transitivity

properties (Klir and Yuan, 2005). We shall also consider here the case when the transitivity

property is not satisfied.

Such a relation R is said to be a tolerance relation (Skowron

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