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Revision as of 03:32, 14 January 2009
, 03:32, 14 January 2009→1.3.10.3. Propositions and Sentences: mathematical markup
The ''play of images'' determined by <math>\underline{f}</math> and <math>x,\!</math> more specifically, the play of the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> that has to do with the element <math>x \in X,</math> is the <math>k\!</math>-tuple <math>\underline{y} = (y_1, \ldots, y_k)</math> of values in <math>\underline\mathbb{B}</math> that satisfies the equations <math>y_j = f_j (x),\!</math> for <math>j = 1 ~\text{to}~ k.</math>
The ''play of images'' determined by <math>\underline{f}</math> and <math>x,\!</math> more specifically, the play of the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> that has to do with the element <math>x \in X,</math> is the <math>k\!</math>-tuple <math>\underline{y} = (y_1, \ldots, y_k)</math> of values in <math>\underline\mathbb{B}</math> that satisfies the equations <math>y_j = f_j (x),\!</math> for <math>j = 1 ~\text{to}~ k.</math>
A ''projection'' of <math>\underline\mathbb{B}^k,</math> written <math>\pi_j\!</math> or <math>\operatorname{pr}_j,\!</math> is one of the maps <math>\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> that is defined as follows:
A "projection" of Bk, typically denoted by "pj" or "prj", is one of the maps pj : Bk �> B, for j = 1 to k, that is defined as follows:
If v = <v1, ..., vk> C Bk,
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{cccccc}
\text{If} & \underline{y} & = & (y_1, \ldots, y_k) & \in & \underline\mathbb{B}^k, \\
\\
\text{then} & \pi_j (\underline{y}) & = & \pi_j (y_1, \ldots, y_k) & = & y_j. \\
\end{array}</math>
|}
The ''projective imagination'' of <math>\underline\mathbb{B}^k</math> is the imagination <math>(\pi_1, \ldots, \pi_k).</math>
The "projective imagination" of Bk is the imagination <p1, ..., pk>.
<pre>
A "sentence about things in the universe", for short, a "sentence", is a sign that denotes a proposition. In other words, a sentence is any sign that denotes an indicator function, any sign whose object is a function of the form f : U �> B.
A "sentence about things in the universe", for short, a "sentence", is a sign that denotes a proposition. In other words, a sentence is any sign that denotes an indicator function, any sign whose object is a function of the form f : U �> B.