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complement of the set X in the universe. The various entropy measures of this class are

obtained by calculating the roughness values ρ
R
(X) and ρ
R
(X
{
) considering the different

ways of obtaining the lower and upper approximations of the vaguely definable set X. Note

that, the 'gain in incompleteness' term is taken as−log
β
ρ
β
in (3.7) and for β > 1

it takes a value in the interval [1,∞]. The other class of entropy measures proposed is

obtained by considering an exponential function (Pal and Pal, 1991) to measure the 'gain

in incompleteness'. This second proposed class of entropy measures for quantifying the

incompleteness of knowledge about U with respect to the definability of a set X⊆U is

given as

ρ
R
(X)β
1−ρ
R
(X)
+ ρ
R
(X
{
)β
1−ρ
R
(X
{

)

1

2

H
R
(X) =

(3.8)

where β denotes the base of the exponential function used. The authors in (Pal and Pal,

1991) had considered only the case when β equaled e. Similar to the class of entropy

measures H
R
, the various entropy measures of this class are obtained by using the different

ways of obtaining the lower and upper approximations of X in order to calculate ρ
R
(X)

and ρ
R
(X
{
). The 'gain in incompleteness' term is taken as β
1−ρ
R
in (3.8) and for β > 1

it takes a value in the finite interval [1, β]. Note that, an analysis on the appropriate values

that β in H
R
and H
R
can take is given later in Section 3.2.5.

We shall name a proposed entropy measure using attributes that represent the class

(logarithmic or exponential) it belongs to, and the type of the pair of sets < RX, RX >

considered. For example, if < RX, RX > represents a tolerance rough-fuzzy set and the

expression of the proposed entropy in (3.8) is considered, then we call such an entropy as the

exponential tolerance rough-fuzzy entropy. Some other examples of names for the proposed

entropy measures are, the logarithmic rough entropy, the exponential fuzzy rough entropy

and the logarithmic tolerance fuzzy rough-fuzzy entropy.

Let us first consider a brief discussion on fuzzy set theory based uncertainty measures.

Assume that a set F S is fuzzy in nature and it is associated with a membership function

µ
F S
. As mentioned in (Pal and Bezdek, 1994), most of the appropriate fuzzy set theory

based uncertainty measures can be grouped into two classes, namely, the multiplicative class

and the additive class. It should be noted from (Pal and Bezdek, 1994) that the measures

belonging to these classes are functions of µ
F S
and µ
F S
{
where µ
F S
= 1−µ
F S
{
.

Now, as mentioned in (Jumarie, 1990) and pointed out in (Pal and Bezdek, 1994), the

existence of an exact relation between µ
F S
and µ
F S
{
suggests that they 'theoretically'

convey the same. However, sometimes such unnecessary terms should to be retained as

dropping them would cause the corresponding measures to fail certain important properties

(Pal and Bezdek, 1994).

We shall now analyze the relation between ρ
R
(X) and ρ
R
(X
{
), and show that there exist

no unnecessary terms in the classes of entropy measures (see (3.7) and (3.8)) proposed using

rough set theory and its certain generalizations. As it is known that ρ
R
(X) takes a value

in the interval [0, 1], let us consider

1

C
, 1≤C≤∞
(3.9)

ρ
R
(X) =

Let us now find the range of values that ρ
R
(X
{
) can take when the value of ρ
R
(X) is given.

Let the total number of elements in the universe U under consideration be n. As we have

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