

![0_image_0.png](0_image_0.png)

with n = |Θ| and C
p n the number of p-uplets with n numbers. The number 1 will be still by convention the intersection of all the elements of Θ. The codification of θ1 ∩ θ3 is given by [1 3] for |Θ| = 3 and [1 2 4 7] for |Θ| = 4. And the codification of θ1 ∪ θ3 is given by [1 2 3 4 5 7] for |Θ| = 3 and [1 2 3 4 6 7 9 12] for |Θ| = 4.

In order to reduce the complexity, especially using more hardware language than Matlab, we could use binary numbers instead of the integer numbers.

The Smarandache's codification [11], was introduce for the enumeration of distinct parts of a Venn diagram. If |Θ| = n, **< i >** denotes the part of θi with no covering with other θj , i 6= j. **< ij >** denotes the part of θi ∩ θj with no covering with other parts of the Venn diagram. So if n = 2, θ1 ∩ θ2 = {< 12 >} and if n = 3, θ1 ∩ θ2 = {< 12 **>, <** 123 >}, see the figure 3 for an illustration for n = 3. The authors note a problem for n ≥ 10, but if we introduce space in the codification we can conserve integers instead of other symbols and we write < 1 2 3 > instead of < 123 >.

On the contrary of the Smarandache's codification, the proposed codification gives only one integer number to each part of the Venn diagram. This codification is more complex for the reader then the Smarandache's codification. Indeed, the reader can understand directly the Smarandache's codification thanks to the mining of the numbers knowing the n: each disjoint part of the Venn diagram is seen as an intersection of the elements of Θ. More exactly, this is a part of the intersections. For example, θ1 ∩θ2 is given with the Smarandache's codification by {< 12 >} if n = 2 and by {< 12 **>, <** 123 >} if n = 3. With the codification practical codification the same element has also different codification according to the number n. For the previous example θ1 ∩ θ2 is given by [1] if n = 2, and by [1 2] if n = 3.