and upper approximation operators. For example, in [6, 13, 23, 26] the information entropy H(π) of the partition π = {U1, U2, . . . , Uk} is defined as

$$H(\pi)=-\sum_{i=1}^{k}{\frac{n_{i}}{n}}\log{\frac{n_{i}}{n}},$$

where niis the cardinality of Ui and n =Pk i=1 ni. As a result, it often yields that some partitions like {{1}, {2}} and
{{1, 2}, {3, 4}} have the same entropy (or co-entropy). This seems somewhat unreasonable since the basic idea of rough set theory aims at describing vague concepts by the lower and upper approximations. In other words, the result of this description relies on both the partition and the approximations. In light of this, we should pay more attention to the lower and upper approximation operators.

The previous observation motivates us to propose another information-theoretic entropy function to measure the uncertainty associated to the partition and the approximation operators in this paper. More concretely, given a universe U with n elements and a partition π of U, we take count of the subsets of U described by every pair of lower and upper approximations. Assume that ri, 1 ≤ i ≤ m, is the number of subsets described by the rough set approximation (Ai, A
′ i
) and every subset of U appears with the same probability. It follows that the rough set approximation (Ai, A
′ i
)
appears with the accumulative probability ri/2 nsince the amount of all subsets of U is precisely 2n. In this way, we obtain a probability distribution

## P(Π) = R1 2 N , R2 2 N , . . . , Rm 2 N

.

It gives rise to an information entropy, say H(π), according to Shannon's information theory [22]. On the other hand, we can get by the probability distribution a co-entropy G(π). It turns out that H(π) + G(π) = n. After exploring some properties of the entropy and co-entropy, we discuss the relationships of co-entropies between different universes.

Roughly speaking, the co-entropy monotonically increases when the partition becomes coarser. For example, the co-entropy of {{1, 2}, {3, 4}} is greater than that of {{1}, {2}}.

The remainder of the paper is structured as follows. In Section 2, we briefly review some basics of Pawlak's rough set theory and the information-theoretic measures of uncertainty and granularity for rough sets in the literature. Section 3 is devoted to our novel notions of entropy and co-entropy and their properties. We address the relationship of co-entropies between different universes in Section 4 and conclude the paper in Section 5 with a brief discussion on the future research.

## 2. Preliminaries

This section consists of two subsections. We briefly recall the definition of Pawlak's rough sets in the first subsection and then review two information-theoretic measures of uncertainty and granularity in rough set theory in the second subsection.

## 2.1. Rough Sets

We start by recalling some basic notions in Pawlak's rough set theory [14, 15].

Let U be a finite and nonempty universal set, and let R ⊆ U × U be an equivalence relation on U. Denote by U/R
the set of all equivalence classes induced by R. Such equivalence classes are also called *elementary sets*; every union (not necessarily nonempty) of elementary sets is called a *definable set*.

For any X ⊆ U, one can characterize X by a pair of lower and upper approximations. The lower approximation appR
X of X is defined as the greatest definable set contained in X, while the upper approximation appRX of X is defined as the least definable set containing X. Formally,

## Appr X = ∪{C ∈ U/R |C ⊆ X} And Apprx = ∪{C ∈ U/R |C ∩ X , ∅}.

The pair appR
X, appRXis referred to as the *rough set approximation* of X. It follows immediately from definition

## That Appr X ⊆ X ⊆ Apprx For Any X ⊆ U.

The ordered pair hU, Ri is said to be an *approximation space*. A *rough set* in hU, Ri is the family of all subsets of U having the same lower and upper approximations. Thus, the general notion of rough set can be simply identified with the rough approximation of any given set.

2