Table Work

It looks like WordPress recently changed the way it handles LaTeX tables, perhaps deprecating the Tabular format in favor of ramping up the Array and Matrix formats. At any rate, I’ll have to play around with reworking my old tables on this page.

New Versions

Fourier Transforms of Boolean Functions : 1

$\begin{array}{|c||*{4}{c}|} \multicolumn{5}{c}{\text{Table 2.1. Values of}~ \boldsymbol{\chi}_\mathcal{S}(x)} \\[4pt] \hline \mathcal{S} & (1, 1) & (1, 0) & (0, 1) & (0, 0) \\ \hline\hline \varnothing & +1 & +1 & +1 & +1 \\ \{ u \} & -1 & -1 & +1 & +1 \\ \{ v \} & -1 & +1 & -1 & +1 \\ \{ u, v \} & +1 & -1 & -1 & +1 \\ \hline \end{array}$

$\begin{array}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{\text{Table 2.2. Fourier Coefficients of Bivariate Boolean Functions}} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ L_1 & L_2 && L_3 & L_4 & \hat{f}(\varnothing) & \hat{f}(\{u\}) & \hat{f}(\{v\}) & \hat{f}(\{u,v\}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1/4 & 1/4 & 1/4 & 1/4 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1/4 & 1/4 &-1/4 &-1/4 \\ f_{3} & f_{0011} && 0~0~1~1 & (u) & 1/2 & 1/2 & 0 & 0 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1/4 &-1/4 & 1/4 &-1/4 \\ f_{5} & f_{0101} && 0~1~0~1 & (v) & 1/2 & 0 & 1/2 & 0 \\ f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 1/2 & 0 & 0 &-1/2 \\ f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 3/4 & 1/4 & 1/4 &-1/4 \\ \hline f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 1/4 &-1/4 &-1/4 & 1/4 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1/2 & 0 & 0 & 1/2 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1/2 & 0 &-1/2 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 3/4 & 1/4 &-1/4 & 1/4 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1/2 &-1/2 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 3/4 &-1/4 & 1/4 & 1/4 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 3/4 &-1/4 &-1/4 &-1/4 \\ f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 0 & 0 & 0 \\ \hline \end{array}$

$\begin{array}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{\text{Table 2.3. Fourier Coefficients of Bivariate Boolean Functions}} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ L_1 & L_2 && L_3 & L_4 & \hat{f}(\varnothing) & \hat{f}(\{u\}) & \hat{f}(\{v\}) & \hat{f}(\{u,v\}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ \hline f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1/4 & 1/4 & 1/4 & 1/4 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1/4 & 1/4 &-1/4 &-1/4 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1/4 &-1/4 & 1/4 &-1/4 \\ f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 1/4 &-1/4 &-1/4 & 1/4 \\ \hline f_{3} & f_{0011} && 0~0~1~1 & (u) & 1/2 & 1/2 & 0 & 0 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1/2 &-1/2 & 0 & 0 \\ \hline f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 1/2 & 0 & 0 &-1/2 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1/2 & 0 & 0 & 1/2 \\ \hline f_{5} & f_{0101} && 0~1~0~1 & (v) & 1/2 & 0 & 1/2 & 0 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1/2 & 0 &-1/2 & 0 \\ \hline f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 3/4 & 1/4 & 1/4 &-1/4 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 3/4 & 1/4 &-1/4 & 1/4 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 3/4 &-1/4 & 1/4 & 1/4 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 3/4 &-1/4 &-1/4 &-1/4 \\ \hline f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 0 & 0 & 0 \\ \hline \end{array}$

Fourier Transforms of Boolean Functions : 2

$\begin{array}{|c||*{4}{c}|} \multicolumn{5}{c}{\text{Table 2.1. Values of}~ g(x)} \\[4pt] \hline g & f_{8} & f_{4} & f_{2} & f_{1} \\ & \texttt{ } u \texttt{ } v \texttt{ } & \texttt{ } u \texttt{ (} v \texttt{)} & \texttt{(} u \texttt{) } v \texttt{ } & \texttt{(} u \texttt{)(} v \texttt{)} \\ \hline\hline f_{7} & 0 & 1 & 1 & 1 \\ f_{11} & 1 & 0 & 1 & 1 \\ f_{13} & 1 & 1 & 0 & 1 \\ f_{14} & 1 & 1 & 1 & 0 \\ \hline \end{array}$

$\begin{array}{|*{9}{c|}} \multicolumn{9}{c}{\text{Table 2.2. Fourier Coefficients of Bivariate Boolean Functions}} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ L_1 & L_2 && L_3 & L_4 & \hat{f}(f_{7}) & \hat{f}(f_{11}) & \hat{f}(f_{13}) & \hat{f}(f_{14}) \\ ~&~&~&~&~&~&~&~&~\\ \hline &&u = & 1~1~0~0 &&&&& \\ &&v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1 & 1 & 1 & 0 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1 & 1 & 0 & 1 \\ f_{3} & f_{0011} && 0~0~1~1 & (u) & 0 & 0 & 1 & 1 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1 & 0 & 1 & 1 \\ f_{5} & f_{0101} && 0~1~0~1 & (v) & 0 & 1 & 0 & 1 \\ f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 0 & 1 & 1 & 0 \\ f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 1 & 0 & 0 & 0 \\ \hline f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 0 & 1 & 1 & 1 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1 & 0 & 0 & 1 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1 & 0 & 1 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 0 & 1 & 0 & 0 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1 & 1 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 0 & 0 & 1 & 0 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 0 & 0 & 0 & 1 \\ f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 1 & 1 & 1 \\ \hline \end{array}$

$\begin{array}{|*{9}{c|}} \multicolumn{9}{c}{\text{Table 2.3. Fourier Coefficients of Bivariate Boolean Functions}} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ L_1 & L_2 && L_3 & L_4 & \hat{f}(f_{7}) & \hat{f}(f_{11}) & \hat{f}(f_{13}) & \hat{f}(f_{14}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ \hline f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1 & 1 & 1 & 0 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1 & 1 & 0 & 1 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1 & 0 & 1 & 1 \\ f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 0 & 1 & 1 & 1 \\ \hline f_{3} & f_{0011} && 0~0~1~1 & (u) & 0 & 0 & 1 & 1 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1 & 1 & 0 & 0 \\ \hline f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 0 & 1 & 1 & 0 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1 & 0 & 0 & 1 \\ \hline f_{5} & f_{0101} && 0~1~0~1 & (v) & 0 & 1 & 0 & 1 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1 & 0 & 1 & 0 \\ \hline f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 1 & 0 & 0 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 0 & 1 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 0 & 0 & 1 & 0 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 0 & 0 & 0 & 1 \\ \hline f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 1 & 1 & 1 \\ \hline \end{array}$

Old Versions

Fourier Transforms of Boolean Functions : 1

$\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{Table 2.1. Values of $\boldsymbol{\chi}_\mathcal{S}(x)$} \\[4pt] \hline $ \mathcal{S} $ & $ (1, 1) $ & $ (1, 0) $ & $ (0, 1) $ & $ (0, 0) $ \\ \hline\hline $ \varnothing $ & $ +1 $ & $ +1 $ & $ +1 $ & $ +1 $ \\ $ \{ u \} $ & $ -1 $ & $ -1 $ & $ +1 $ & $ +1 $ \\ $ \{ v \} $ & $ -1 $ & $ +1 $ & $ -1 $ & $ +1 $ \\ $ \{ u, v \} $ & $ +1 $ & $ -1 $ & $ -1 $ & $ +1 $ \\ \hline \end{tabular}$

$\begin{tabular}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $L_1$&$L_2$&&$L_3$&$L_4$& $\hat{f}(\varnothing)$&$\hat{f}(\{u\})$&$\hat{f}(\{v\})$&$\hat{f}(\{u,v\})$ \\ ~&~&~&~&~&~&~&~&~\\ \hline && $u =$& 1 1 0 0&&&&& \\ && $v =$& 1 0 1 0&&&&& \\ \hline $f_{0}$& $f_{0000}$&& 0 0 0 0& $(~)$& $0$& $0$& $0$& $0$ \\ $f_{1}$& $f_{0001}$&& 0 0 0 1& $(u)(v)$& $1/4$& $1/4$& $1/4$& $1/4$ \\ $f_{2}$& $f_{0010}$&& 0 0 1 0& $(u)~v~$& $ 1/4$& $ 1/4$& $-1/4$& $-1/4$ \\ $f_{3}$& $f_{0011}$&& 0 0 1 1& $(u)$& $1/2$& $1/2$& $ 0 $& $ 0 $ \\ $f_{4}$& $f_{0100}$&& 0 1 0 0& $~u~(v)$& $ 1/4$& $-1/4$& $ 1/4$& $-1/4$ \\ $f_{5}$& $f_{0101}$&& 0 1 0 1& $(v)$& $1/2$& $ 0 $& $1/2$& $ 0 $ \\ $f_{6}$& $f_{0110}$&& 0 1 1 0& $(u,~v)$& $ 1/2$& $ 0 $& $ 0 $& $-1/2$ \\ $f_{7}$& $f_{0111}$&& 0 1 1 1& $(u~~v)$& $ 3/4$& $ 1/4$& $ 1/4$& $-1/4$ \\ \hline $f_{8}$& $f_{1000}$&& 1 0 0 0& $~u~~v~$& $ 1/4$& $-1/4$& $-1/4$& $ 1/4$ \\ $f_{9}$& $f_{1001}$&& 1 0 0 1& $((u,~v))$& $1/2$& $ 0 $& $ 0 $& $1/2$ \\ $f_{10}$& $f_{1010}$&& 1 0 1 0& $v$& $ 1/2$& $ 0 $& $-1/2$& $ 0 $ \\ $f_{11}$& $f_{1011}$&& 1 0 1 1& $(~u~(v))$& $ 3/4$& $ 1/4$& $-1/4$& $ 1/4$ \\ $f_{12}$& $f_{1100}$&& 1 1 0 0& $u$& $ 1/2$& $-1/2$& $ 0 $& $ 0 $ \\ $f_{13}$& $f_{1101}$&& 1 1 0 1& $((u)~v~)$& $ 3/4$& $-1/4$& $ 1/4$& $ 1/4$ \\ $f_{14}$& $f_{1110}$&& 1 1 1 0& $((u)(v))$& $ 3/4$& $-1/4$& $-1/4$& $-1/4$ \\ $f_{15}$& $f_{1111}$&& 1 1 1 1& $((~))$& $1$& $0$& $0$& $0$ \\ \hline \end{tabular}$

$\begin{tabular}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $L_1$&$L_2$&&$L_3$&$L_4$& $\hat{f}(\varnothing)$&$\hat{f}(\{u\})$&$\hat{f}(\{v\})$&$\hat{f}(\{u,v\})$ \\ ~&~&~&~&~&~&~&~&~\\ \hline && $u =$& 1 1 0 0&&&&& \\ && $v =$& 1 0 1 0&&&&& \\ \hline $f_{0}$& $f_{0000}$&& 0 0 0 0& $(~)$& $0$& $0$& $0$& $0$ \\ \hline $f_{1}$& $f_{0001}$&& 0 0 0 1& $(u)(v)$& $1/4$& $1/4$& $1/4$& $1/4$ \\ $f_{2}$& $f_{0010}$&& 0 0 1 0& $(u)~v~$& $ 1/4$& $ 1/4$& $-1/4$& $-1/4$ \\ $f_{4}$& $f_{0100}$&& 0 1 0 0& $~u~(v)$& $ 1/4$& $-1/4$& $ 1/4$& $-1/4$ \\ $f_{8}$& $f_{1000}$&& 1 0 0 0& $~u~~v~$& $ 1/4$& $-1/4$& $-1/4$& $ 1/4$ \\ \hline $f_{3}$& $f_{0011}$&& 0 0 1 1& $(u)$& $1/2$& $1/2$& $ 0 $& $ 0 $ \\ $f_{12}$& $f_{1100}$&& 1 1 0 0& $u$& $ 1/2$& $-1/2$& $ 0 $& $ 0 $ \\ \hline $f_{6}$& $f_{0110}$&& 0 1 1 0& $(u,~v)$& $ 1/2$& $ 0 $& $ 0 $& $-1/2$ \\ $f_{9}$& $f_{1001}$&& 1 0 0 1& $((u,~v))$& $1/2$& $ 0 $& $ 0 $& $1/2$ \\ \hline $f_{5}$& $f_{0101}$&& 0 1 0 1& $(v)$& $1/2$& $ 0 $& $1/2$& $ 0 $ \\ $f_{10}$& $f_{1010}$&& 1 0 1 0& $v$& $ 1/2$& $ 0 $& $-1/2$& $ 0 $ \\ \hline $f_{7}$& $f_{0111}$&& 0 1 1 1& $(u~~v)$& $ 3/4$& $ 1/4$& $ 1/4$& $-1/4$ \\ $f_{11}$& $f_{1011}$&& 1 0 1 1& $(~u~(v))$& $ 3/4$& $ 1/4$& $-1/4$& $ 1/4$ \\ $f_{13}$& $f_{1101}$&& 1 1 0 1& $((u)~v~)$& $ 3/4$& $-1/4$& $ 1/4$& $ 1/4$ \\ $f_{14}$& $f_{1110}$&& 1 1 1 0& $((u)(v))$& $ 3/4$& $-1/4$& $-1/4$& $-1/4$ \\ \hline $f_{15}$& $f_{1111}$&& 1 1 1 1& $((~))$& $1$& $0$& $0$& $0$ \\ \hline \end{tabular}$

Fourier Transforms of Boolean Functions : 2

$\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{Table 2.1. Values of $ g(x) $} \\[4pt] \hline $ g $ & $ f_{8} $ & $ f_{4} $ & $ f_{2} $ & $ f_{1} $ \\ & \texttt{ $u$ $v$ } & \texttt{ $u$ ($v$)} & \texttt{($u$) $v$ } & \texttt{($u$)($v$)} \\ \hline\hline $ f_{7} $ & $0$ & $1$ & $1$ & $1$ \\ $ f_{11} $ & $1$ & $0$ & $1$ & $1$ \\ $ f_{13} $ & $1$ & $1$ & $0$ & $1$ \\ $ f_{14} $ & $1$ & $1$ & $1$ & $0$ \\ \hline \end{tabular}$

$\begin{tabular}{|*{9}{c|}} \multicolumn{9}{c}{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $L_1$& $L_2$&& $L_3$& $L_4$& $\hat{f}(f_{7} )$& $\hat{f}(f_{11})$& $\hat{f}(f_{13})$& $\hat{f}(f_{14})$ \\ ~&~&~&~&~&~&~&~&~\\ \hline && $u =$& 1 1 0 0&&&&& \\ && $v =$& 1 0 1 0&&&&& \\ \hline $f_{0}$& $f_{0000}$&& 0 0 0 0& $(~)$& $0$& $0$& $0$& $0$ \\ $f_{1}$& $f_{0001}$&& 0 0 0 1& $(u)(v)$& $1$& $1$& $1$& $0$ \\ $f_{2}$& $f_{0010}$&& 0 0 1 0& $(u)~v~$& $1$& $1$& $0$& $1$ \\ $f_{3}$& $f_{0011}$&& 0 0 1 1& $(u)$& $0$& $0$& $1$& $1$ \\ $f_{4}$& $f_{0100}$&& 0 1 0 0& $~u~(v)$& $1$& $0$& $1$& $1$ \\ $f_{5}$& $f_{0101}$&& 0 1 0 1& $(v)$& $0$& $1$& $0$& $1$ \\ $f_{6}$& $f_{0110}$&& 0 1 1 0& $(u,~v)$& $0$& $1$& $1$& $0$ \\ $f_{7}$& $f_{0111}$&& 0 1 1 1& $(u~~v)$& $1$& $0$& $0$& $0$ \\ \hline $f_{8}$& $f_{1000}$&& 1 0 0 0& $~u~~v~$& $0$& $1$& $1$& $1$ \\ $f_{9}$& $f_{1001}$&& 1 0 0 1& $((u,~v))$& $1$& $0$& $0$& $1$ \\ $f_{10}$& $f_{1010}$&& 1 0 1 0& $v$& $1$& $0$& $1$& $0$ \\ $f_{11}$& $f_{1011}$&& 1 0 1 1& $(~u~(v))$& $0$& $1$& $0$& $0$ \\ $f_{12}$& $f_{1100}$&& 1 1 0 0& $u$& $1$& $1$& $0$& $0$ \\ $f_{13}$& $f_{1101}$&& 1 1 0 1& $((u)~v~)$& $0$& $0$& $1$& $0$ \\ $f_{14}$& $f_{1110}$&& 1 1 1 0& $((u)(v))$& $0$& $0$& $0$& $1$ \\ $f_{15}$& $f_{1111}$&& 1 1 1 1& $((~))$& $1$& $1$& $1$& $1$ \\ \hline \end{tabular}$

$\begin{tabular}{|*{9}{c|}} \multicolumn{9}{c}{Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $L_1$& $L_2$&& $L_3$& $L_4$& $\hat{f}(f_{7} )$& $\hat{f}(f_{11})$& $\hat{f}(f_{13})$& $\hat{f}(f_{14})$ \\ ~&~&~&~&~&~&~&~&~\\ \hline && $u =$& 1 1 0 0&&&&& \\ && $v =$& 1 0 1 0&&&&& \\ \hline $f_{0}$& $f_{0000}$&& 0 0 0 0& $(~)$& $0$& $0$& $0$& $0$ \\ \hline $f_{1}$& $f_{0001}$&& 0 0 0 1& $(u)(v)$& $1$& $1$& $1$& $0$ \\ $f_{2}$& $f_{0010}$&& 0 0 1 0& $(u)~v~$& $1$& $1$& $0$& $1$ \\ $f_{4}$& $f_{0100}$&& 0 1 0 0& $~u~(v)$& $1$& $0$& $1$& $1$ \\ $f_{8}$& $f_{1000}$&& 1 0 0 0& $~u~~v~$& $0$& $1$& $1$& $1$ \\ \hline $f_{3}$& $f_{0011}$&& 0 0 1 1& $(u)$& $0$& $0$& $1$& $1$ \\ $f_{12}$& $f_{1100}$&& 1 1 0 0& $u$& $1$& $1$& $0$& $0$ \\ \hline $f_{6}$& $f_{0110}$&& 0 1 1 0& $(u,~v)$& $0$& $1$& $1$& $0$ \\ $f_{9}$& $f_{1001}$&& 1 0 0 1& $((u,~v))$& $1$& $0$& $0$& $1$ \\ \hline $f_{5}$& $f_{0101}$&& 0 1 0 1& $(v)$& $0$& $1$& $0$& $1$ \\ $f_{10}$& $f_{1010}$&& 1 0 1 0& $v$& $1$& $0$& $1$& $0$ \\ \hline $f_{7}$& $f_{0111}$&& 0 1 1 1& $(u~~v)$& $1$& $0$& $0$& $0$ \\ $f_{11}$& $f_{1011}$&& 1 0 1 1& $(~u~(v))$& $0$& $1$& $0$& $0$ \\ $f_{13}$& $f_{1101}$&& 1 1 0 1& $((u)~v~)$& $0$& $0$& $1$& $0$ \\ $f_{14}$& $f_{1110}$&& 1 1 1 0& $((u)(v))$& $0$& $0$& $0$& $1$ \\ \hline $f_{15}$& $f_{1111}$&& 1 1 1 1& $((~))$& $1$& $1$& $1$& $1$ \\ \hline \end{tabular}$

Array Format

$\begin{array}{c|c} t & x \\ \hline 0 & 0 \\ 1 & 1 \\ 2 & 0 \\ 3 & 1 \\ 4 & 0 \\ 5 & 1 \\ 6 & 0 \\ 7 & 1 \\ 8 & 0 \\ 9 & 1 \\ \ldots & \ldots \end{array}$

$\begin{array}{c|cc} t & x & \mathrm{d}x \\ \hline 0 & 0 & 1 \\ 1 & 1 & 1 \\ 2 & 0 & 1 \\ 3 & 1 & 1 \\ 4 & 0 & 1 \\ 5 & 1 & 1 \\ 6 & 0 & 1 \\ 7 & 1 & 1 \\ 8 & 0 & 1 \\ 9 & 1 & 1 \\ \ldots & \ldots & \ldots \end{array}$

$\begin{array}{c|cccc} t & x & \mathrm{d}x & \mathrm{d}^2 x & \ldots \\ \hline 0 & 0 & 1 & 0 & \ldots \\ 1 & 1 & 1 & 0 & \ldots \\ 2 & 0 & 1 & 0 & \ldots \\ 3 & 1 & 1 & 0 & \ldots \\ 4 & 0 & 1 & 0 & \ldots \\ 5 & 1 & 1 & 0 & \ldots \\ 6 & 0 & 1 & 0 & \ldots \\ 7 & 1 & 1 & 0 & \ldots \\ 8 & 0 & 1 & 0 & \ldots \\ 9 & 1 & 1 & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{array}$

Tabular Format

Test A

$\begin{tabular}{|*{7}{c|}} \multicolumn{7}{c}{Table A1. Propositional Forms on Two Variables} \\ \hline $L_1$&$L_2$&&$L_3$&$L_4$&$L_5$&$L_6$ \\ \hline &&$x=$&1 1 0 0&&& \\ &&$y=$&1 0 1 0&&& \\ \hline $f_{0}$& $f_{0000}$&& 0 0 0 0& $(~)$& false& $0$ \\ $f_{1}$& $f_{0001}$&& 0 0 0 1& $(x)(y)$& neither $x$ nor $y$& $\lnot x \land \lnot y$ \\ $f_{2}$& $f_{0010}$&& 0 0 1 0& $(x)~y~$& $y$ without $x$& $\lnot x \land y$ \\ $f_{3}$& $f_{0011}$&& 0 0 1 1& $(x)$& not $x$& $\lnot x$ \\ $f_{4}$& $f_{0100}$&& 0 1 0 0& $~x~(y)$& $x$ without $y$& $x \land \lnot y$ \\ $f_{5}$& $f_{0101}$&& 0 1 0 1& $(y)$& not $y$& $\lnot y$ \\ $f_{6}$& $f_{0110}$&& 0 1 1 0& $(x,~y)$& $x$ not equal to $y$& $x \ne y$ \\ $f_{7}$& $f_{0111}$&& 0 1 1 1& $(x~~y)$& not both $x$ and $y$& $\lnot x \lor \lnot y$ \\ \hline $f_{8}$& $f_{1000}$&& 1 0 0 0& $~x~~y~$& $x$ and $y$& $x \land y$ \\ $f_{9}$& $f_{1001}$&& 1 0 0 1& $((x,~y))$& $x$ equal to $y$& $x = y$ \\ $f_{10}$& $f_{1010}$&& 1 0 1 0& $y$& $y$& $y$ \\ $f_{11}$& $f_{1011}$&& 1 0 1 1& $(~x~(y))$& not $x$ without $y$& $x \Rightarrow y$ \\ $f_{12}$& $f_{1100}$&& 1 1 0 0& $x$& $x$& $x$ \\ $f_{13}$& $f_{1101}$&& 1 1 0 1& $((x)~y~)$& not $y$ without $x$& $x \Leftarrow y$ \\ $f_{14}$& $f_{1110}$&& 1 1 1 0& $((x)(y))$& $x$ or $y$& $x \lor y$ \\ $f_{15}$& $f_{1111}$&& 1 1 1 1& $((~))$& true& $1$ \\ \hline \end{tabular}$

Test B

Working Example

Tabular Test 1

$\begin{tabular}{lll} Chicago & U.S.A. & 1893 \\ Z\"{u}rich & Switzerland & 1897 \\ Paris & France & 1900 \\ Heidelberg & Germany & 1904 \\ Rome & Italy & 1908 \end{tabular}$

Tabular Test 2

$\begin{tabular}{|r|r|} \hline $ n $ & $ n! $ \\ \hline 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 24 \\ 5 & 120 \\ 6 & 720 \\ 7 & 5040 \\ 8 & 40320 \\ 9 & 362880 \\ 10 & 3628800 \\ \hline \end{tabular}$

Tabular Test 3

$\begin{tabular}{|c|c|*{16}{c}|} \multicolumn{18}{c}{Table 1. Higher Order Propositions $ (n = 1) $} \\[4pt] \hline $ f $ & $ f $ & $ m_{0} $ & $ m_{1} $ & $ m_{2} $ & $ m_{3} $ & $ m_{4} $ & $ m_{5} $ & $ m_{6} $ & $ m_{7} $ & $ m_{8} $ & $ m_{9} $ & $ m_{10} $ & $ m_{11} $ & $ m_{12} $ & $ m_{13} $ & $ m_{14} $ & $ m_{15} $ \\[4pt] \hline $ f_0 $ & \texttt{()} & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\[4pt] $ f_1 $ & \texttt{(}$ x $\texttt{)} & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\[4pt] $ f_2 $ & $ x $ & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\[4pt] $ f_3 $ & \texttt{(())} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\[4pt] \hline \end{tabular}$

Tabular Test 4

$\begin{tabular}{|*{7}{c|}} \multicolumn{7}{c}{\textbf{Table A1. Propositional Forms on Two Variables}} \\ \hline $ L_1 $ & $ L_2 $ && $ L_3 $ & $ L_4 $ & $ L_5 $ & $ L_6 $ \\ \hline & & $ x = $ & 1 1 0 0 & & & \\ & & $ y = $ & 1 0 1 0 & & & \\ \hline $ f_{0} $ & $ f_{0000} $ && 0 0 0 0 & $ (~) $ & false & $ 0 $ \\ $ f_{1} $ & $ f_{0001} $ && 0 0 0 1 & $ (x)(y) $ & neither $ x $ nor $ y $ & $ \lnot x \land \lnot y $ \\ $ f_{2} $ & $ f_{0010} $ && 0 0 1 0 & $ (x)\ y $ & $ y $ without $ x $ & $ \lnot x \land y $ \\ $ f_{3} $ & $ f_{0011} $ && 0 0 1 1 & $ (x) $ & not $ x $ & $ \lnot x $ \\ $ f_{4} $ & $ f_{0100} $ && 0 1 0 0 & $ x\ (y) $ & $ x $ without $ y $ & $ x \land \lnot y $ \\ $ f_{5} $ & $ f_{0101} $ && 0 1 0 1 & $ (y) $ & not $ y $ & $ \lnot y $ \\ $ f_{6} $ & $ f_{0110} $ && 0 1 1 0 & $ (x,\ y) $ & $ x $ not equal to $ y $ & $ x \ne y $ \\ $ f_{7} $ & $ f_{0111} $ && 0 1 1 1 & $ (x\ y) $ & not both $ x $ and $ y $ & $ \lnot x \lor \lnot y $ \\ \hline $ f_{8} $ & $ f_{1000} $ && 1 0 0 0 & $ x\ y $ & $ x $ and $ y $ & $ x \land y $ \\ $ f_{9} $ & $ f_{1001} $ && 1 0 0 1 & $ ((x,\ y)) $ & $ x $ equal to $ y $ & $ x = y $ \\ $ f_{10} $ & $ f_{1010} $ && 1 0 1 0 & $ y $ & $ y $ & $ y $ \\ $ f_{11} $ & $ f_{1011} $ && 1 0 1 1 & $ (x\ (y)) $ & not $ x $ without $ y $ & $ x \Rightarrow y $ \\ $ f_{12} $ & $ f_{1100} $ && 1 1 0 0 & $ x $ & $ x $ & $ x $ \\ $ f_{13} $ & $ f_{1101} $ && 1 1 0 1 & $ ((x)\ y) $ & not $ y $ without $ x $ & $ x \Leftarrow y $ \\ $ f_{14} $ & $ f_{1110} $ && 1 1 1 0 & $ ((x)(y)) $ & $ x $ or $ y $ & $ x \lor y $ \\ $ f_{15} $ & $ f_{1111} $ && 1 1 1 1 & $ ((~)) $ & true & $ 1 $ \\ \hline \end{tabular}$

Tabular Test 5

$\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{Table 2.1 Values of $ \chi_S(x) $ for $ f : \mathbb{B}^2 \to \mathbb{B} $} \\[4pt] \hline $ (x_1, x_2) \backslash S $ & $ \varnothing $ & $ \{ x_1 \} $ & $ \{ x_2 \} $ & $ \{ x_1, x_2 \} $ \\ \hline\hline $ (0, 0) $ & $ +1 $ & $ +1 $ & $ +1 $ & $ +1 $ \\ $ (1, 0) $ & $ +1 $ & $ -1 $ & $ +1 $ & $ -1 $ \\ $ (0, 1) $ & $ +1 $ & $ +1 $ & $ -1 $ & $ -1 $ \\ $ (1, 1) $ & $ +1 $ & $ -1 $ & $ -1 $ & $ +1 $ \\ \hline \end{tabular}$

$\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{Table 2.1 Values of $ \chi_S(x) $ for $ f : \mathbb{B}^2 \to \mathbb{B} $} \\[4pt] \hline $ S \backslash (x_1, x_2) $ & $ (0, 0) $ & $ (1, 0) $ & $ (0, 1) $ & $ (1, 1) $ \\ \hline\hline $ \varnothing $ & $ +1 $ & $ +1 $ & $ +1 $ & $ +1 $ \\ $ \{ x_1 \} $ & $ +1 $ & $ -1 $ & $ +1 $ & $ -1 $ \\ $ \{ x_2 \} $ & $ +1 $ & $ +1 $ & $ -1 $ & $ -1 $ \\ $ \{ x_1, x_2 \} $ & $ +1 $ & $ -1 $ & $ -1 $ & $ +1 $ \\ \hline \end{tabular}$

$\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{Table 2.1 Values of $ \chi_S(x) $ for $ f : \mathbb{B}^2 \to \mathbb{B} $} \\[4pt] \hline $ S \backslash (x_1, x_2) $ & $ (1, 1) $ & $ (1, 0) $ & $ (0, 1) $ & $ (0, 0) $ \\ \hline\hline $ \varnothing $ & $ +1 $ & $ +1 $ & $ +1 $ & $ +1 $ \\ $ \{ x_1 \} $ & $ -1 $ & $ -1 $ & $ +1 $ & $ +1 $ \\ $ \{ x_2 \} $ & $ -1 $ & $ +1 $ & $ -1 $ & $ +1 $ \\ $ \{ x_1, x_2 \} $ & $ +1 $ & $ -1 $ & $ -1 $ & $ +1 $ \\ \hline \end{tabular}$

Work 3

Table Work

New Versions

Old Versions

Array Format