It follows immediately from definition that H(π) + G(π) = log n.

Contrary to the uncertainty measure H, the co-entropy function G achieves the maximum value log n when π = πˇ and the minimum value 0 when π = πˆ; moreover, it has been known [11] that for any two partitions π and σ of U, if σ ≺ π, then G(σ) < G(π).

As argued in [2, 3], the entropy H(π) can be interpreted as the uncertainty measure of the partition π, while the co-entropy G(π) can be regarded as the granularity measure of π. In [21], Sen and Pal introduced two other entropy measures for crisp sets and fuzzy sets with (crisp or fuzzy) equivalence relations or (crisp or fuzzy) tolerance relations, which are based upon the roughness measures of X and of the complement of X in the universe and have been used to analyze the grayness and spatial ambiguities in images. Under the same name, there are some different concepts of entropy in the literature of rough set theory (see, for example, [9, 20]).

## 3. A Novel Pair Of Entropy And Co-Entropy

In this section, we first introduce a novel entropy and the corresponding co-entropy and then explore their properties.

Let us begin with some notations. Throughout this section, we write hU, πi for an approximation space and assume that |U| = n. Given a hU, πi, we use A(U, π) to denote the set of rough set approximations of all subsets of U. More formally, we set A(U, π) =
appπ X, appπX X ⊆ U
. (4)
It follows from Eq. (4) that A(U, π) has at least two elements: (∅, ∅) and (U, U). If n = 1, then A(U, π) exactly consists of the two elements; if n > 1 and π = πˇ, then A(U, π) contains one more element (∅, U); for any n ≥ 1, if π = πˆ, then we see that A(U, π) = {(X, X) | X ⊆ U}, which consists of 2nelements. Note that the set A(U, π) is not a multiset, that is, the same element cannot appear more than once in A(U, π). In general, we have that |A(U, π)| ≤ 2 n since the subset X of U in Eq. (4) has only 2nalternatives.

For simplicity, we use m to stand for |A(U, π)|. For any (Ai, A
′ i
) ∈ A(U, π), 1 ≤ i ≤ m, we set

$${\mathcal{A}}_{i}=\left\{X\subseteq U\left|\left({\underline{{a p p}}},X,{\overline{{a p p}}}_{\pi}X\right)=(A_{i},A_{i}^{\prime})\right\}\,{\mathrm{~and~}}|{\mathcal{A}}_{i}|=r_{i}.\right.$$
and |Ai| = ri. (5)
In other words, riis the number of subsets of U that have the rough set approximation (Ai, A
′
i
). It turns out that
{A1, A2, . . . , Am} gives rise to a partition of P(U). Therefore, we get by Eq. (4) that

$$\sum_{i=1}^{m}r_{i}=2^{n}.$$

To illustrate the above concepts, let us see an example.

Example 3.1. Consider U = {1, 2, 3, 4} and π = {{1, 2}, {3, 4}}. In this case, U has 16 subsets. For each subset X of U, we compute the rough set approximation of X; the results are listed in Table 1.

Hence, we see that A(U, π) = {(∅, ∅), (∅, {1, 2}), (∅, {3, 4}), ({1, 2}, {1, 2}), (∅, U), ({3, 4}, {3, 4}), ({1, 2}, U), ({3, 4}, U), (U, U)} .

As an example, let us calculate r2*. By definition,*

$$\begin{array}{l l l}{{r_{2}}}&{{=}}&{{\left|\left\{X\subseteq U\left|\left(\underline{{{a p p}}}_{\pi}X,\overline{{{a p p}}}_{\pi}X\right)=\left(0,\left\{1,2\right\}\right)\right\}\right|\right.}}\\ {{}}&{{=}}&{{\left|\left\{\left\{1\right\},\left\{2\right\}\right\}\right|}}\\ {{}}&{{=}}&{{2.}}\end{array}$$

This is exactly the number of subsets of U that have the rough set approximation (∅, {1, 2})*, which can be counted* from the table. In light of this, we may get Table 2 by rearranging Table 1. It follows immediately from Table 2 that r1 = r4 = r6 = r9 = 1, r2 = r3 = r7 = r8 = 2, and r5 = 4.

4