is the result of performing a logical "AND" operation on all IDM's which

occur in incidences of the FVIM which are assigned as outputs. For

example, refer to Fig. 3-6. If y is assigned to f and x to g, either

ii or i1+i can be the index output. If, however, x is assigned to f

and y to g, only ii can be the index output. The logical "AND" then

provides a means for discovering which index outputs can be chosen inde-

pendent of function type. When only structural considerations are being

made, this can save time. This is because the logical "AND" can be

performed faster than an output set can be assigned and because only one

output set assignment is required in this case.

 An important result is that index ordering does not affect the

number of tears in the problem. Consider the two different index

orderings in Fig. 3-7. Both incidence matrices can be solved with four

tears.

Theorem 3-4: If two function indices are adjacent in ordering, the

number of tears for the expanded incidence matrix is independent of the

ordering of the indices.

Proof: Let the IDM's for the two indices be called a and b. Let the

number of tears for a be NT and for b be N ,. Let Nca and Ncb be the

number of columns (and rows) in the IDM's a and b respectively. Assume

b is nested inside a. Then the total number of tears is

 N = NTa x No. of columns for each occurence of a

 + NTb x No. of occurences of b not already torn

 -= Ta x Nb + NTb(ca NTa
 = NTa x N b + N Tb x Nca NTbNTa 3-

which is symmetric in NTa, N Tb NTc and Ncb. Thus, the number of tears

is independent of the index ordering. The theorem is proved.