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Directory:Jon Awbrey/Papers/Inquiry Driven Systems (view source)
Revision as of 19:27, 17 January 2009
, 19:27, 17 January 2009→1.3.10.6. Stretching Principles: mathematical markup
From these materials, it is required to construct a proposition <math>p : X \to \underline\mathbb{B}</math> such that <math>p(x) = F(f_1 (x), \ldots, f_k (x)),</math> for all <math>x \in X.</math> The desired construction is determined as follows:
From these materials, it is required to construct a proposition <math>p : X \to \underline\mathbb{B}</math> such that <math>p(x) = F(f_1 (x), \ldots, f_k (x)),</math> for all <math>x \in X.</math> The desired construction is determined as follows:
The cartesian power <math>\underline\mathbb{B}^k,</math> as a cartesian product, is characterized by the possession of a projective imagination <math>\pi = (\pi_1, \ldots, \pi_k)</math> of degree <math>k\!</math> on <math>\underline\mathbb{B}^k,</math> along with the property that any imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> of degree <math>k\!</math> on an arbitrary set <math>W\!</math> determines a unique map <math>f! : W \to \underline\mathbb{B}^k,</math> the play of whose projective images <math>(\pi_1 (f!(w), \ldots, \pi_k (f!(w))</math> on the functional image <math>f!(w)</math> matches the play of images <math>(f_1 (w), \ldots, f_k (w))</math> under <math>\underline{f},</math> term for term and at every element <math>w\!</math> in <math>W.\!</math>
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Just to be on the safe side, I state this again in more standard terms. The cartesian power Bk, as a cartesian product, is characterized by the possession of k projection maps pj : Bk -> B, for j = 1 to k, along with the property that any k maps fj : W -> B, from an arbitrary set W to B, determine a unique map f! : W -> Bk such that pj(f!(w)) = fj(w), for all j = 1 to k, and for all w C W.
Just to be on the safe side, I state this again in more standard terms. The cartesian power Bk, as a cartesian product, is characterized by the possession of k projection maps pj : Bk -> B, for j = 1 to k, along with the property that any k maps fj : W -> B, from an arbitrary set W to B, determine a unique map f! : W -> Bk such that pj(f!(w)) = fj(w), for all j = 1 to k, and for all w C W.