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\begin{center}\begin{tabular}{|l|l|l|l|}
\begin{center}\begin{tabular}{|l|l|l|l|}
\multicolumn{4}{c}{\textbf{Table 4. Propositional Calculus : Basic Notation}} \\
\multicolumn{4}{c}{\textbf{Table 4. Propositional Calculus : Basic Notation}} \\[8pt]
\hline
\hline
\textbf{Symbol} & \textbf{Notation} & \textbf{Description} & \textbf{Type} \\
\textbf{Symbol} & \textbf{Notation} & \textbf{Description} & \textbf{Type} \\[4pt]
\hline
\hline
$\mathcal{A}$ & $\{ a_1, \ldots, a_n \}$ & Alphabet & $[n] = \mathbf{n}$ \\
$\mathcal{A}$ & $\{ a_1, \ldots, a_n \}$ & Alphabet & $[n] = \mathbf{n}$ \\[4pt]
\hline
\hline
$A_i$ & $\{ \overline{a_i}, a_i \}$ & Dimension $i$ & $\mathbb{B}$ \\
$A_i$ & $\{ \overline{a_i}, a_i \}$ & Dimension $i$ & $\mathbb{B}$ \\[4pt]
\hline
\hline
$A$ & $\langle \mathcal{A} \rangle$ & Set of cells, & $\mathbb{B}^n$ \\
$A$ & $\langle \mathcal{A} \rangle$ & Set of cells, & $\mathbb{B}^n$ \\[4pt]
& $\langle a_1, \ldots, a_n \rangle$ & coordinate tuples, & \\
& $\langle a_1, \ldots, a_n \rangle$ & coordinate tuples, & \\[4pt]
& $\{ (a_1, \ldots, a_n) \}$ & points, or vectors & \\
& $\{ (a_1, \ldots, a_n) \}$ & points, or vectors & \\[4pt]
& $A_1 \times \ldots \times A_n$ & in the universe & \\
& $A_1 \times \ldots \times A_n$ & in the universe & \\[4pt]
& $\textstyle \prod_{i=1}^n A_i$ & of discourse & \\
& $\textstyle \prod_{i=1}^n A_i$ & of discourse & \\[4pt]
\hline
\hline
$A^*$ & $(\operatorname{hom} : A \to \mathbb{B})$ & Linear functions &
$A^*$ & $(\operatorname{hom} : A \to \mathbb{B})$ & Linear functions &
$(\mathbb{B}^n)^* \cong \mathbb{B}^n$ \\
$(\mathbb{B}^n)^* \cong \mathbb{B}^n$ \\[4pt]
\hline
\hline
$A^\uparrow$ & $(A \to \mathbb{B})$ & Boolean functions &
$A^\uparrow$ & $(A \to \mathbb{B})$ & Boolean functions &
$\mathbb{B}^n \to \mathbb{B}$ \\
$\mathbb{B}^n \to \mathbb{B}$ \\[4pt]
\hline
\hline
$A^\circ$ & $[ \mathcal{A} ]$ & Universe of discourse &
$A^\circ$ & $[ \mathcal{A} ]$ & Universe of discourse &
$(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))$ \\
$(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))$ \\[4pt]
& $(A, A^\uparrow)$ & based on the features &
& $(A, A^\uparrow)$ & based on the features &
$(\mathbb{B}^n\ +\!\to \mathbb{B})$ \\
$(\mathbb{B}^n\ +\!\to \mathbb{B})$ \\[4pt]
& $(A\ +\!\to \mathbb{B})$ & $\{ a_1, \ldots, a_n \}$ &
& $(A\ +\!\to \mathbb{B})$ & $\{ a_1, \ldots, a_n \}$ &
$[\mathbb{B}^n]$ \\
$[\mathbb{B}^n]$ \\[4pt]
& $(A, (A \to \mathbb{B}))$ & & \\
& $(A, (A \to \mathbb{B}))$ & & \\[4pt]
& $[ a_1, \ldots, a_n ]$ & & \\
& $[ a_1, \ldots, a_n ]$ & & \\[4pt]
\hline
\hline
\end{tabular}\end{center}
\end{tabular}\end{center}
\begin{center}\begin{tabular}{|l|l|l|l|}
\begin{center}\begin{tabular}{|l|l|l|l|}
\multicolumn{4}{c}{\textbf{Table 5. Differential Extension : Basic Notation}} \\
\multicolumn{4}{c}{\textbf{Table 5. Differential Extension : Basic Notation}} \\[8pt]
\hline
\hline
\textbf{Notation} &
\textbf{Notation} &
\textbf{Description} &
\textbf{Description} &
\textbf{Type} \\
\textbf{Type} \\[4pt]
\hline
\hline
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
Alphabet of differential features &
Alphabet of differential features &
$[n] = \mathbf{n}$ \\
$[n] = \mathbf{n}$ \\[4pt]
\hline
\hline
$\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}$ &
$\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}$ &
Differential dimension $i$ &
Differential dimension $i$ &
$\mathbb{D}$ \\
$\mathbb{D}$ \\[4pt]
\hline
\hline
$\langle \operatorname{d}\mathcal{A} \rangle$ &
$\langle \operatorname{d}\mathcal{A} \rangle$ &
Tangent space at a point: &
Tangent space at a point: &
$\mathbb{D}^n$
$\mathbb{D}^n$ \\[4pt]
\\
&
&
$\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle$ &
$\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle$ &
Set of changes, &
Set of changes, &
\\
\\[4pt]
&
&
$\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}$ &
$\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}$ &
motions, steps, &
motions, steps, &
\\
\\[4pt]
&
&
$\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n$ &
$\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n$ &
tangent vectors &
tangent vectors &
\\
\\[4pt]
&
&
$\textstyle \prod_{i=1}^n \operatorname{d}A_i$ &
$\textstyle \prod_{i=1}^n \operatorname{d}A_i$ &
at a point &
at a point &
\\
\\[4pt]
\hline
\hline
$(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})$ &
$(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})$ &
Linear functions on $\operatorname{d}A$ &
Linear functions on $\operatorname{d}A$ &
$(\mathbb{D}^n)^* \cong \mathbb{D}^n$ \\
$(\mathbb{D}^n)^* \cong \mathbb{D}^n$ \\[4pt]
\hline
\hline
$(\operatorname{d}A \to \mathbb{B})$ &
$(\operatorname{d}A \to \mathbb{B})$ &
Boolean functions on $\operatorname{d}A$ &
Boolean functions on $\operatorname{d}A$ &
$\mathbb{D}^n \to \mathbb{B}$ \\
$\mathbb{D}^n \to \mathbb{B}$ \\[4pt]
\hline
\hline
$[ \operatorname{d}\mathcal{A} ]$ &
$[ \operatorname{d}\mathcal{A} ]$ &
Tangent universe &
Tangent universe &
$(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))$
$(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))$ \\[4pt]
\\
&
&
$(\operatorname{d}A, \operatorname{d}A^\uparrow)$ &
$(\operatorname{d}A, \operatorname{d}A^\uparrow)$ &
at a point of $A^\circ,$ &
at a point of $A^\circ,$ &
$(\mathbb{D}^n\ +\!\to \mathbb{B})$
$(\mathbb{D}^n\ +\!\to \mathbb{B})$ \\[4pt]
\\
&
&
$(\operatorname{d}A\ +\!\to \mathbb{B})$ &
$(\operatorname{d}A\ +\!\to \mathbb{B})$ &
based on the &
based on the &
$[\mathbb{D}^n]$
$[\mathbb{D}^n]$ \\[4pt]
\\
&
&
$(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))$ &
$(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))$ &
tangent features &
tangent features &
\\
\\[4pt]
&
&
$[ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ]$ &
$[ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ]$ &
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
\\
\\[4pt]
\hline
\hline
\end{tabular}\end{center}
\end{tabular}\end{center}
