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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 22:30, 22 February 2013
, 22:30, 22 February 2013→6.36. Irreducibly Triadic Relations: markup
# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_0\!</math> if and only if <math>x + y + z = 0.\!</math> Thus, <math>L_0\!</math> is the set of even-parity bit vectors, with <math>x + y = z.\!</math>
# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_0\!</math> if and only if <math>x + y + z = 0.\!</math> Thus, <math>L_0\!</math> is the set of even-parity bit vectors, with <math>x + y = z.\!</math>
# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_1\!</math> if and only if <math>x + y + z = 1.\!</math> Thus, <math>L_1\!</math> is the set of odd-parity bit vectors, with <math>x + y = z + 1.\!</math>
# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_1\!</math> if and only if <math>x + y + z = 1.\!</math> Thus, <math>L_1\!</math> is the set of odd-parity bit vectors, with <math>x + y = z + 1.\!</math>
The corresponding projections of <math>\operatorname{Proj}^{(2)} L_0\!</math> and <math>\operatorname{Proj}^{(2)} L_1\!</math> 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 <math>\mathbb{B} \times \mathbb{B}\!</math> and expressed by the universal constant proposition <math>1 : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math> In summary:
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R012 = R112 = 112 = B2,
R012 = R112 = 112 = B2,
R013 = R113 = 113 = B2,
R013 = R113 = 113 = B2,