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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 19:40, 18 May 2008
, 19:40, 18 May 2008→Current Version @ PlanetMath : TeX Format: update
\begin{center}\begin{tabular}{ccccccc}
\begin{center}\begin{tabular}{ccccccc}
\multicolumn{7}{c}{Table 3. Differential Inference Rules} \\[12pt]
\multicolumn{7}{c}{\textbf{Table 3. Differential Inference Rules}} \\[12pt]
From & $\overline{q}$ & and & $\overline{\operatorname{d}q}$ & infer & $\overline{q}$ & next. \\[6pt]
From & $\overline{q}$ & and & $\overline{\operatorname{d}q}$ & infer & $\overline{q}$ & next. \\[6pt]
From & $\overline{q}$ & and & $\operatorname{d}q$ & infer & $q$ & next. \\[6pt]
From & $\overline{q}$ & and & $\operatorname{d}q$ & infer & $q$ & next. \\[6pt]
\section{Formal development}
\section{Formal development}
\begin{center}\begin{tabular}{|l|l|l|l|}
\multicolumn{4}{c}{\textbf{Table 4. Propositional Calculus : Basic Notation}} \\
\hline
\textbf{Symbol} & \textbf{Notation} & \textbf{Description} & \textbf{Type} \\
\hline
$\mathcal{A}$ & $\{ a_1, \ldots, a_n \}$ & Alphabet & $[n] = \mathbf{n}$ \\
\hline
$A_i$ & $\{ (a_i), a_i \}$ & Dimension $i$ & $\mathbb{B}$ \\
\hline
$A$ & $\langle \mathcal{A} \rangle$ & Set of cells, & $\mathbb{B}^n$ \\
& $\langle a_1, \ldots, a_n \rangle$ & coordinate tuples, & \\
& $\{ (a_1, \ldots, a_n) \}$ & points, or vectors & \\
& $A_1 \times \ldots \times A_n$ & in the universe & \\
& $\textstyle \prod_{i=1}^n A_i$ & of discourse & \\
\hline
$A^*$ & $(\operatorname{hom} : A \to \mathbb{B})$ & Linear functions &
$(\mathbb{B}^n)^* \cong \mathbb{B}^n$ \\
\hline
$A^\uparrow$ & $(A \to \mathbb{B})$ & Boolean functions &
$\mathbb{B}^n \to \mathbb{B}$ \\
\hline
$A^\circ$ & $[ \mathcal{A} ]$ & Universe of discourse &
$(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))$ \\
& $(A, A^\uparrow)$ & based on the features &
$(\mathbb{B}^n\ +\!\to \mathbb{B})$ \\
& $(A\ +\!\to \mathbb{B})$ & $\{ a_1, \ldots, a_n \}$ &
$[\mathbb{B}^n]$ \\
& $(A, (A \to \mathbb{B}))$ & & \\
& $[ a_1, \ldots, a_n ]$ & & \\
\hline
\end{tabular}\end{center}
$\ldots$
$\ldots$