(2) It follows from (1) that H reaches the maximum value when π = πˆ. In this case, we get by definition that

$${\mathcal{H}}({\hat{\pi}})=-\sum_{i=1}^{2^{n}}{\frac{1}{2^{n}}}\log{\frac{1}{2^{n}}}=n.$$

This proves (2).

(3) By (1), we see that H reaches the minimum value when π = πˇ. In this case, the empty subset ∅ of U has the rough set approximation (∅, ∅) and U itself has the rough set approximation (U, U). For any proper subset of U, if any, it has the rough set approximation (∅, U). Hence, r1 = r2 = 1 and r3 = 2 n − 2. We thus obtain by definition that

$$\begin{array}{r c l}{{{\mathcal H}({\tilde{\pi}})}}&{{=}}&{{-{\frac{1}{2^{n}}}\log{\frac{1}{2^{n}}}-{\frac{1}{2^{n}}}\log{\frac{1}{2^{n}}}-{\frac{2^{n}-2}{2^{n}}}\log{\frac{2^{n}-2}{2^{n}}}}}\\ {{}}&{{=}}&{{n-{\frac{2^{n}-2}{2^{n}}}\log(2^{n}-2).}}\end{array}$$
$\text{\rm{ti}}\alpha\text{\rm{m}}$. 
Whence, (3) holds, finishing the proof of the proposition.

Note that in the clause (3) of Theorem 3.1, if n = 1, the value of the corresponding summand 0 log 0 is taken to be 0, which is consistent with the limit:
lim x→0+
x log x = 0.

For later need, let us recall the following definition from [31].

Definition 3.2. Let hU, πi and hV, σi be two approximation spaces, and suppose that f : U −→ *V is a mapping.* (1) *The mapping f is called a* homomorphism *from* hU, πi to hV, σi if for any C ∈ π, there exists D ∈ σ such that f(C) ⊆ *D, where f*(C) = {f(u) | u ∈ C}.

(2) *A homomorphism f is called a* monomorphism *if f is an injective mapping.*
(3) *A monomorphism f is called* strictly monomorphic if either there exist C ∈ π and D ∈ σ such that f(C) ( D,
namely, f(C) ⊆ D and f(C) , *D, or* |V| > |U|.

(4) *The mapping f is called an* isomorphism if the mapping f : U −→ *V is bijective, and moreover, both f and its* inverse mapping f −1 *are homomorphisms.*
We can now state the following facts.

Proposition 3.1. Let hU, πi and hV, σi be two approximation spaces with |U| = |V|, and let f : U −→ *V be a mapping.* (1) *If f is a monomorphism from* hU, πi to hV, σi, in particular, π  σ*, then* H(π) ≥ H(σ).

(2) *If f is a strict monomorphism from* hU, πi to hV, σi, in particular, π ≺ σ*, then* H(π) > H(σ).

(3) *If f is an isomorphism from* hU, πi to hV, σi*, then* H(π) = H(σ).

Proof. It follows immediately from Definition 3.1 and Theorem 3.1.

To measure the granularity with respect to the approximation operators app and app carried by the partition π, we introduce the concept of co-entropy, which corresponds to the information entropy in Definition 3.1.

Definition 3.3. *Keep the notations as in Definition 3.1. The* co-entropy G(π) of hU, πi (with respect to the approximation operators app and *app) is defined by*

$$\mathcal{G}(\pi)=\mathcal{G}(P(\pi))=\sum_{t=1}^{m}\frac{r_{t}}{2^{n}}\log r_{t}.\tag{8}$$  The quantity $\mathcal{G}(\pi)$ furnishes a measure of the average granularity carried by the partition $\pi$ as a whole. It follows
immediately from definition that H(π) + G(π) = n. (9)
It means that the two measures complement each other with respect to the constant quantity n = |U|, which is invariant with respect to the choice of the partition π of U.

The co-entropy function G is of the following properties.

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