ONT Re: Relations And Their Divisitudes
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RATD. Note 28
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Relational Composition as Logical Matrix Multiplication (cont.)
With an eye to extracting a general formula, let us now examine
what we did in multiplying the 2-adic relations G and H together
to obtain their relational composite G o H.
Given the space X = {1, 2, 3, 4, 5, 6, 7}, whose cardinality |X| is 7,
we naturally observe that there are |X x X| = |X| x |X| = 7 x 7 = 49
elementary relations of the form i:j, where i and j range over the
space X. Although they might be organized in many different ways,
it is convenient to regard the collection of elementary relations
as being arranged in a lexicographic block of the following form:
1:1 1:2 1:3 1:4 1:5 1:6 1:7
2:1 2:2 2:3 2:4 2:5 2:6 2:7
3:1 3:2 3:3 3:4 3:5 3:6 3:7
4:1 4:2 4:3 4:4 4:5 4:6 4:7
5:1 5:2 5:3 5:4 5:5 5:6 5:7
6:1 6:2 6:3 6:4 6:5 6:6 6:7
7:1 7:2 7:3 7:4 7:5 7:6 7:7
We may think of G and H as being logical sums of the following forms:
G = Sum_ij G_ij (i:j)
H = Sum_ij H_ij (i:j)
The notation "Sum_ij" indicates a logical sum over the collection of
elementary relations i:j, while the factors G_ij, H_ij are values in
the boolean domain B = {0, 1} that are known as the "coefficients"
of the relations G, H, respectively, with regard to each of the
elementary relations i:j in turn.
In general, for a 2-adic relation L, the coefficient L_ij of
the elementary relation i:j in the relation L will be 0 or 1,
respectively, as i:j is excluded from or included in L.
Given all this, we may write out the expansions of G and H as follows:
G = 4:3 + 4:4 + 4:5 =
0(1:1) + 0(1:2) + 0(1:3) + 0(1:4) + 0(1:5) + 0(1:6) + 0(1:7) +
0(2:1) + 0(2:2) + 0(2:3) + 0(2:4) + 0(2:5) + 0(2:6) + 0(2:7) +
0(3:1) + 0(3:2) + 0(3:3) + 0(3:4) + 0(3:5) + 0(3:6) + 0(3:7) +
0(4:1) + 0(4:2) + 1(4:3) + 1(4:4) + 1(4:5) + 0(4:6) + 0(4:7) +
0(5:1) + 0(5:2) + 0(5:3) + 0(5:4) + 0(5:5) + 0(5:6) + 0(5:7) +
0(6:1) + 0(6:2) + 0(6:3) + 0(6:4) + 0(6:5) + 0(6:6) + 0(6:7) +
0(7:1) + 0(7:2) + 0(7:3) + 0(7:4) + 0(7:5) + 0(7:6) + 0(7:7)
H = 3:4 + 4:4 + 5:4 =
0(1:1) + 0(1:2) + 0(1:3) + 0(1:4) + 0(1:5) + 0(1:6) + 0(1:7) +
0(2:1) + 0(2:2) + 0(2:3) + 0(2:4) + 0(2:5) + 0(2:6) + 0(2:7) +
0(3:1) + 0(3:2) + 0(3:3) + 1(3:4) + 0(3:5) + 0(3:6) + 0(3:7) +
0(4:1) + 0(4:2) + 0(4:3) + 1(4:4) + 0(4:5) + 0(4:6) + 0(4:7) +
0(5:1) + 0(5:2) + 0(5:3) + 1(5:4) + 0(5:5) + 0(5:6) + 0(5:7) +
0(6:1) + 0(6:2) + 0(6:3) + 0(6:4) + 0(6:5) + 0(6:6) + 0(6:7) +
0(7:1) + 0(7:2) + 0(7:3) + 0(7:4) + 0(7:5) + 0(7:6) + 0(7:7)
Presenting just the coefficients of G and H on the above plan:
G =
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 1 1 1 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
H =
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 1 0 0 0
0 0 0 1 0 0 0
0 0 0 1 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
These are the logical matrix representations of the 2-adic relations G and H.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o