The proposed codification is more practical for computing union and intersection operations and the DSm cardinality, because only one integer represent

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

one of the distinct parts of the Venn diagram. With the Smarandache's codification computing union and intersection operations and the DSm cardinality could be very similar than with the practical codification, but adding a routine in order to treat the code of one part of the Venn diagram.

Hence, we propose to use the proposed codification to compute union, intersection and DSm cardinality, and the Smarandache's codification, easier to read, to present the results in order to safe eventually a scan of DΘ
r
.

## 3.2 Adding Constraints

With this codification, adding constraints is very simple and can reduce rapidly the number of integers. *E.g.* assume that in a given application we know θ1 ∩ θ3 ≡ ∅ (*i.e.* θ1 ∩ θ3 ∈/ DΘ
r
), that means that the integers [1 3] for |Θ| = 3 and [1 2 4 7] for |Θ| = 4 do not exist Θ. Hence, the codification of Θ with the reduced discernment space, noted Θr, is given respectively for |Θ| = 3 and |Θ| = 4 by:
Θr = {[2 5], [2 4 6], [4 7]}

## And Θr = {[3 6 9 12], [3 5 6 8 10 13], [5 8 11 14], [3 5 9 10 11 15]}.

Generally we have |Θ| = |Θr|, but it is not necessary if a constraint gives θi ≡ ∅, with θi ∈ Θ. This can happen in dynamic fusion, if one element of the discernment space can disappear.

Thereby, the introduction of the simple constraint θ1 ∩θ3 ≡ ∅ in Θ, includes all the other constraints that follow from it such as the intersection of all the elements of Θ is empty. In [15] all the constraints must be given by the user.