
| subset    | approximation   | subset    | approximation    | subset    | approximation    | subset    | approximation   |
|-----------|-----------------|-----------|------------------|-----------|------------------|-----------|-----------------|
| ∅         | (∅, ∅)          | {1}       | (∅, {1, 2})      | {2}       | (∅, {1, 2})      | {3}       | (∅, {3, 4})     |
| {4}       | (∅, {3, 4})     | {1, 2}    | ({1, 2}, {1, 2}) | {1, 3}    | (∅, U)           | {1, 4}    | (∅, U)          |
| {2, 3}    | (∅, U)          | {2, 4}    | (∅, U)           | {3, 4}    | ({3, 4}, {3, 4}) | {1, 2, 3} | ({1, 2}, U)     |
| {1, 2, 4} | ({1, 2}, U)     | {1, 3, 4} | ({3, 4}, U)      | {2, 3, 4} | ({3, 4}, U)      | U         | (U, U)          |

| Table 2: The rough set approximations and corresponding subsets in Example 3.1.   |                      |               |                                |                  |          |
|-----------------------------------------------------------------------------------|----------------------|---------------|--------------------------------|------------------|----------|
| approximation                                                                     | subsets              | approximation | subsets                        | approximation    | subsets  |
| (∅, ∅)                                                                            | ∅                    | (∅, {1, 2})   | {1}, {2}                       | (∅, {3, 4})      | {3}, {4} |
| ({1, 2}, {1, 2})                                                                  | {1, 2}               | (∅, U)        | {1, 3}, {1, 4}, {2, 3}, {2, 4} | ({3, 4}, {3, 4}) | {3, 4}   |
| ({1, 2}, U)                                                                       | {1, 2, 3}, {1, 2, 4} | ({3, 4}, U)   | {1, 3, 4}, {2, 3, 4}           | (U, U)           | U        |

Because we are concerned with the partition granulation of hU, πi with respect to the approximation operators app and app, we may assume that every subset of U appears with the same probability 1/2 n. As a result, the rough set approximation (Ai, A
′ i
) appears with the accumulative probability ri/2 nand we thus obtain a probability distribution

$$P(\pi)=\left({\frac{r_{1}}{2^{n}}},{\frac{r_{2}}{2^{n}}},\ldots,{\frac{r_{m}}{2^{n}}}\right).$$
$$\left(T\right)$$
. (6)
According to Shannon's information theory [22], the Shannon entropy function of the probability distribution P(π)
is defined as follows.

Definition 3.1. *Keep the notations as above. The* information entropy H(π) of hU, πi (with respect to the approximation operators app and *app) is defined by*

$${\cal H}(\pi)={\cal H}(P(\pi))=-\sum_{i=1}^{m}\frac{r_{i}}{2^{n}}\log\frac{r_{i}}{2^{n}}.\tag{1}$$

In the above definition, for simplicity we have used the notation H(π) instead of H(U, π). Following the explanation of Shannon entropy in information theory, the quantity H(π) measures the uncertainty associated to the partition π with respect to the approximation operators app and app. For instance, the probability distribution corresponding to the partition π = {{1, 2}, {3, 4}} in Example 3.1 is

$$P(\pi)=\left(\frac{1}{2^{4}},\frac{2}{2^{4}},\frac{2}{2^{4}},\frac{1}{2^{4}},\frac{4}{2^{4}},\frac{1}{2^{4}},\frac{2}{2^{4}},\frac{2}{2^{4}},\frac{1}{2^{4}}\right).$$

It follows from Definition 3.1 that

H(π) = − X 9 i=1 ri 2 4 log ri 2 4 = − "1 2 4 log 1 2 4 + 2 2 4 log 2 2 4 + 2 2 4 log 2 2 4 + 1 2 4 log 1 2 4 + 4 2 4 log 4 2 4 + 1 2 4 log 1 2 4 + 2 2 4 log 2 2 4 + 2 2 4 log 2 2 4 + 1 2 4 log 1 2 4 # = 3. 5