
| δ1(., .) = δ(., .)   | A   | B   | C   | A ∪ B   |
|----------------------|-----|-----|-----|---------|
| A                    | 0   | 1   | 1   | 0       |
| B                    | 1   | 0   | 1   | 0       |
| C                    | 1   | 1   | 0   | 1       |
| A ∪ B                | 0   | 0   | 1   | 0       |
| TABLE VI             |     |     |     |         |

Whence, the first dissimilarity weights δ1(**., .**) defined by (29), *i.e.* δ1(X, Y ) = 1 − δ2(**X, Y** ) take the values:
VALUES FOR δ(**., .**)
The first similarity and dissimilarity weights δ2(**., .**) and δ1(**., .**) are not quite accurate, since for example: δ2(A, A∪B**) = 1**,
i.e. A and A ∪ B are 100% similar (which is not the case since A 6= A ∪ B) and δ1(A, A ∪ B) = 1 − δ2(A, A ∪ B) =
1 − **1 = 0**, *i.e.* A and A ∪ B are 100% dissimilar (which is not the case either since A ∩ (A ∪ B) 6= ∅).

| δ2(., .) = 1 − η(., .)   | A   | B   | C   | A ∪ B   |
|--------------------------|-----|-----|-----|---------|
| A                        | 1   | 0   | 0   | 1/2     |
| B                        | 0   | 1   | 0   | 1/2     |
| C                        | 0   | 0   | 1   | 0       |
| A ∪ B                    | 1/2 | 1/2 | 0   | 1       |
| TABLE VII                |     |     |     |         |

VALUES FOR 1 − η(**., .**)

$$\delta_{2}(A,A)=1-\eta(A,A)=\frac{\mathcal{C}(A\cap A)}{\operatorname*{max}\{\mathcal{C}(A),\mathcal{C}(A)\}}=\frac{\mathcal{C}(A)}{\mathcal{C}(A)}=1$$  $$\delta_{2}(A,B)=1-\eta(A,B)=\frac{\mathcal{C}(A\cap B)}{\operatorname*{max}\{\mathcal{C}(A),\mathcal{C}(B)\}}=0$$
because $A\cap B=$
because A ∩ B = ∅ and C(∅**) = 0**.

 $ \delta_2(A,A\cup B)=1-\eta(A,A\cup B)=\frac{\mathcal{C}(A\cap(A\cup B))}{\max\{\mathcal{C}(A),\mathcal{C}(A\cup B)\}}=\frac{\mathcal{C}(A)}{\mathcal{C}(A\cup B)}=\frac{1}{2}$ ,  $ \delta_2(A,A\cup B)=1-\delta(A,A\cup B)=1$. 
which is better than δ2(A, A ∪ B) = 1 − δ(A, A ∪ B**) = 1**.

etc.

Whence, the second dissimilarity weights η(**., .**) take the values:
η(A, A ∪ B**) = 1** −
1 2 =
1 2
, which is better than δ1(A, A ∪ B) = δ(A, A ∪ B**) = 0**.

b) The second similarity weights δ2(., .) = 1 − η(**., .**) given by the equation (30) overcomes this problem. We obtain on the previous example: since