Work 3

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

\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}

Working Example

\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}

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}

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