Suppose the truth is in the set F∪E. First we combine m1(.) with m2(E) = 1 using the conjunctive rule, and its result m1+ m2 is in the fourth row in Table 1. All below conditioning rules are referred to the result of this conjunctive rule, and they differ through the way the conflicting mass, i.e. mass of empty intersections, is transferred to the other elements. 

In DCR, since N∩ (F∪E) = ϕ the conflicting mass m1(N)· m2 (F∪E) = 0.3·1 = 0.3, is transferred to the non-empty sets F, E, and F∪E proportionally with respect to their masses acquired after applying the conjunctive rule (m1+ m2), i.e. with respect to 0.2, 0.1, and respectively 0.4. Thus, we get mDST(X|F∪E) as in the fifth row of Table 1, where X∈{ ϕ, F, E, N, F∪E, F∪E∪N, N∩ (F∪E)}. 

In Smets' TBM (Transferable Believe Model), the conflicting mass, 0.3, is transferred to the empty set, since TBM considers an open world (non-exhaustive hypotheses). See row \# 6.

With DSm first conditioning rule (row \# 7) the conflicting mass 0.3 is transferred to the whole set that the truth belongs to, F∪E. So, mDSmT1(F∪E |F∪E) = (m1+ m2)( F∪E) + 0.3 = 0.4+0.3 
= 0.7. 

In DSm second conditioning rule (row \# 8) the conflicting mass 0.3 is uniformly transferred to the non-empty sets F, E, and F∪E, therefore each such set receives 0.3/3 = 0.1. 

## Example 5.2.

Let m1(.) be defined on the frame {A = Airplane, T = tank, S = ship, M = submarine}, where the hypotheses A, T, S, M are mutually exclusive, in the following way (see the second row): 

| ϕ             | A   | T   | S   | M   | A∪S   | T∪M   | A∩ (T∪M)   | S∩ (T∪M)   | (A∪S)∩ (T∪M)   |     |
|---------------|-----|-----|-----|-----|-------|-------|------------|------------|----------------|-----|
| m1            | 0   | 0.4 | 0   | 0.5 | 0     | 0.1   | 0          |            |                |     |
| m2            | 0   | 0   | 0   | 0   | 0     | 0     | 1          |            |                |     |
| m1+ m2        | 0   | 0   | 0   | 0   | 0     | 0     | 0          | 0.4        | 0.5            | 0.1 |
| mDCR(X|T∪M)   | 0   | N/A | N/A | N/A | N/A   | N/A   | N/A        |            |                |     |
| mTBM(X|T∪M)   | 1   | 0   | 0   | 0   | 0     | 0     | 0          | 0          | 0              | 0   |
| mDSmT1(X|T∪M) | 0   | 0   | 0   | 0   | 0     | 0     | 1          | 0          | 0              | 0   |
| mDSmT2(X|T∪M) | 0   | 0   | 1/3 | 0   | 1/3   | 0     | 1/3        | 0          | 0              | 0   |
| Table 2       |     |     |     |     |       |       |            |            |                |     |

Suppose the truth is in T∪M. Since the sets A∩ (T∪M), S∩ (T∪M), and (A∪S)∩ (T∪M) 
are empty, their masses 0.4, 0.5, and respectively 0.1 have to be transferred to non-empty sets belonging to P(T∪M), where P(T∪M) means the set of all subsets of T∪M. 

7 This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June and July 2009.