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H ∗ p (Ah ) p (Ah ) ˜ p∗ (Bz1 ), . . , p∗ (Bz ) + H k ˜ ∗ (Bz1 ), . . , p∗ (Bz )) by H(B ˜ \ A). , Bz1 , . . , Bzk ∈ π(B) such that Bz1 , . . , Bzk ∈ / A. Thus we have ∗ 1 ∗ m ˜ ˜ ˜ p (Bh ) , . . , p (Bh ) H(B) = H(A) + p∗ (Ah ) · H ∗ ∗ p (Ah ) p (Ah ) ˜ \ A) + H(B ˜ ˜ This result means that we always have H(B) ≥ H(A), and thus we can write A⊆B implies ˜ ˜ H(A) ≤ H(B) (40) ˜ \ A) > 0, and from this In particular, if |XB | > |XA | then we have that H(B ˜ ˜ trivially H(A) < H(B) follows.

Cattaneo Example 6. Making reference to example 5 we have that although π1 ≺ π2 , for (P ) the subset Y0 we get ρπ1 (Y0 ) = ρπ2 (Y0 ) (for both roughness measures ρπ (Y0 ) (C) and ρπ (Y0 )). , Eπ : Y ∈ P(X) → Eπ (Y ) ∈ [0, log2 m(X) ] can be considered as a (local) rough granularity mapping. Example 7. Let us consider the universe X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, its subset Y = {2, 3, 5, 8, 9, 10, 11}, and the following three diﬀerent partitions of the universe X by granules: π1 = {{2, 3, 5, 8, 9}, {1, 4}, {6, 7, 10, 11}}, π2 = {{2, 3}, {5, 8, 9}, {1, 4, }, {6, 7, 10, 11}}, π3 = {{2, 3}, {5, 8, 9}, {1, 4, }, {7, 10}, {6, 11}} with π3 ≺ π2 ≺ π1 .

16]. From the fact that π2 is a partition we have that Al = Al ∩(∪k Bk ) = ∪k (Al ∩Bk ), with these latter pairwise disjoints, and so we get that p(Al ) = k p(Al ∩ Bk ), leading to the result p(Al ) = p(Al |Bk ) p(Bk ) (17) k which can be expressed in the matrix form: ⎛ ⎞ ⎛ ⎞ p(A1 ) p(A1 |B1 ) . . p(A1 |Bk ) . . p(A1 |BN ) ⎜ .. ⎟ ⎜ ⎟ .. .. .. ⎜ . ⎟ ⎜ ⎟ . . . ⎜ ⎟ ⎜ ⎟ ⎜ p(Al ) ⎟ = ⎜ p(Al |B1 ) . . p(Al |Bk ) . . p(Al |BN ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ .. .. .. ⎝ .. ⎠ ⎝ ⎠ . . . p(AM ) p(AM |B1 ) .