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, 13:25, 17 August 2011→6.36. Irreducibly Triadic Relations: + raw text
===6.36. Irreducibly Triadic Relations===
===6.36. Irreducibly Triadic Relations===
<pre>
Most likely, any triadic relation R c XxYxZ that is imposed on arbitrary domains X, Y, Z could find use as a sign relation, provided that it embodies any constraint at all, in other words, so long as it forms a proper subset of its total space, R c XxYxZ. However, these sorts of uses of triadic relations are not guaranteed to capture the most natural examples of sign relations.
In order to show what an irreducibly triadic relation looks like, this section presents a pair of triadic relations that have the same dyadic projections, and thus cannot be distinguished from each other on this basis alone. As it happens, these examples of triadic relations can be discussed independently of sign relational concerns, but structures of their general ilk are frequently found arising in signal theoretic applications, and they are undoubtedly closely associated with problems of reliable coding and communication.
Tables 71.1 and 72.1 show a pair of irreducibly triadic relations R0 and R1, respectively. Tables 71.2 to 71.4 and Tables 72.2 to 72.4 show the dyadic relations comprising Proj (R0) and Proj (R1), respectively.
Table 71.1 Relation R0 = {<x, y, z> C B3 : x + y + z = 0}
x y z
0 0 0
0 1 1
1 0 1
1 1 0
Table 71.2 Dyadic Projection R012
x y
0 0
0 1
1 0
1 1
Table 71.3 Dyadic Projection R013
x z
0 0
0 1
1 1
1 0
Table 71.4 Dyadic Projection R023
y z
0 0
1 1
0 1
1 0
Table 72.1 Relation R1 = {<x, y, z> C B3 : x + y + z = 1}
x y z
0 0 1
0 1 0
1 0 0
1 1 1
Table 72.2 Dyadic Projection R112
x y
0 0
0 1
1 0
1 1
Table 72.3 Dyadic Projection R113
x z
0 1
0 0
1 0
1 1
Table 72.4 Dyadic Projection R123
y z
0 1
1 0
0 0
1 1
The relations R0, R1 c B3 are defined by the following equations, with algebraic operations taking place as in GF(2), that is, with 1 + 1 = 0.
1. The triple <x, y, z> in B3 belongs to R0 iff x + y + z = 0. Thus, R0 is the set of even parity bit vectors, with x + y = z.
2. The triple <x, y, z> in B3 belongs to R1 iff x + y + z = 1. Thus, R1 is the set of odd parity bit vectors, with x + y = z + 1.
The corresponding projections of Proj (R0) and Proj (R1) are identical. In fact, all six projections, taken at the level of logical abstraction, constitute precisely the same dyadic relation, isomorphic to the whole of BxB and expressed by the universal constant proposition 1 : BxB > B. In summary:
R012 = R112 = 112 = B2,
R013 = R113 = 113 = B2,
R023 = R123 = 123 = B2.
Thus, R0 and R1 are both examples of irreducibly triadic relations.
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===6.37. Propositional Types===
===6.37. Propositional Types===
