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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 17:43, 21 February 2013→6.36. Irreducibly Triadic Relations: markup
Most likely, any triadic relation <math>L \subseteq X \times Y \times Z\!</math> imposed on arbitrary domains <math>X, Y, Z\!</math> could find use as a sign relation, provided it embodies any constraint at all, in other words, so long as it forms a proper subset of its total space, a relationship symbolized by writing <math>L \subset X \times Y \times Z.\!</math> However, triadic relations of this sort are not guaranteed to form the most natural examples of sign relations.
Most likely, any triadic relation <math>L \subseteq X \times Y \times Z\!</math> imposed on arbitrary domains <math>X, Y, Z\!</math> could find use as a sign relation, provided it embodies any constraint at all, in other words, so long as it forms a proper subset of its total space, a relationship symbolized by writing <math>L \subset X \times Y \times Z.\!</math> However, triadic relations of this sort are not guaranteed to form the most natural examples of sign relations.
In order to show what irreducibly triadic relations look 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.
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 74.1 and 75.1 show a pair of irreducibly triadic relations <math>L_0\!</math> and <math>L_1,\!</math> respectively. Tables 74.2 to 74.4 and Tables 75.2 to 75.4 show the dyadic relations comprising <math>\operatorname{Proj}^{(2)} L_0\!</math> and <math>\operatorname{Proj}^{(2)} L_1,\!</math> respectively.
Tables 74.1 and 75.1 show a pair of irreducibly triadic relations <math>L_0\!</math> and <math>L_1,\!</math> respectively. Tables 74.2 to 74.4 and Tables 75.2 to 75.4 show the dyadic relations comprising <math>\operatorname{Proj}^{(2)} L_0\!</math> and <math>\operatorname{Proj}^{(2)} L_1,\!</math> respectively.
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The relations <math>L_0, L_1 \subseteq \mathbb{B}^3\!</math> are defined by the following equations, with algebraic operations taking place as in <math>\text{GF}(2),\!</math> that is, with <math>1 + 1 = 0.\!</math>
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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.
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.