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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 { 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.
3.2.4 Relation between ρ R (X) and ρ R (X { )
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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