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Tables

Boolean Functions and Propositional Calculus

Table A1. Propositional Forms on Two Variables

$latex
\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}
&fg=000000$

Table A2. Propositional Forms on Two Variables

$latex
\begin{tabular}{|*{7}{c|}}
\multicolumn{7}{c}{Table A2. 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&
\(0\)
\\
\hline
\(f_{1}\)&
\(f_{0001}\)&&
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 \(x\)&
\(\lnot x \land y\)
\\
\(f_{4}\)&
\(f_{0100}\)&&
0 1 0 0&
\(~x~(y)\)&
\(x\) without \(y\)&
\(x \land \lnot y\)
\\
\(f_{8}\)&
\(f_{1000}\)&&
1 0 0 0&
\(~x~~y~\)&
\(x\) and \(y\)&
\(x \land y\)
\\
\hline
\(f_{3}\)&
\(f_{0011}\)&&
0 0 1 1&
\((x)\)&
not \(x\)&
\(\lnot x\)
\\
\(f_{12}\)&
\(f_{1100}\)&&
1 1 0 0&
\(x\)&
\(x\)&
\(x\)
\\
\hline
\(f_{6}\)&
\(f_{0110}\)&&
0 1 1 0&
\((x,~y)\)&
\(x\) not equal to \(y\)&
\(x \ne y\)
\\
\(f_{9}\)&
\(f_{1001}\)&&
1 0 0 1&
\(((x,~y))\)&
\(x\) equal to \(y\)&
\(x = y\)
\\
\hline
\(f_{5}\)&
\(f_{0101}\)&&
0 1 0 1&
\((y)\)&
not \(y\)&
\(\lnot y\)
\\
\(f_{10}\)&
\(f_{1010}\)&&
1 0 1 0&
\(y\)&
\(y\)&
\(y\)
\\
\hline
\(f_{7}\)&
\(f_{0111}\)&&
0 1 1 1&
\((~x~~y~)\)&
not both \(x\) and \(y\)&
\(\lnot x \lor \lnot y\)
\\
\(f_{11}\)&
\(f_{1011}\)&&
1 0 1 1&
\((~x~(y))\)&
not \(x\) without \(y\)&
\(x \Rightarrow y\)
\\
\(f_{13}\)&
\(f_{1101}\)&&
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 \(y\)&
\(x \lor y\)
\\
\hline
\(f_{15}\)&
\(f_{1111}\)&&
1 1 1 1&
\(((~))\)&
true&
\(1\)
\\
\hline
\end{tabular}
&fg=000000$

Table A3. Ef Expanded Over Differential Features {dx, dy}

$latex
\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{Table A3. \(\mathrm{E}f\) Expanded Over Differential Features \(\{\mathrm{d}x, \mathrm{d}y\}\)} \\
\hline
&
\(~~~~~~~~ f ~~~~~~~~\)&
\(~~~~\mathrm{T}_{11}f~~~~\)&
\(~~~~\mathrm{T}_{10}f~~~~\)&
\(~~~~\mathrm{T}_{01}f~~~~\)&
\(~~~~\mathrm{T}_{00}f~~~~\)
\\
&&
\(\mathrm{E}f|_{~\mathrm{d}x ~\mathrm{d}y~}~\)&
\(\mathrm{E}f|_{~\mathrm{d}x~(\mathrm{d}y)}~\)&
\(\mathrm{E}f|_{(\mathrm{d}x)~\mathrm{d}y~}~\)&
\(\mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)}~\)
\\
\hline\hline
\(f_{0}\)&
\(0\)&
\(0\)&
\(0\)&
\(0\)&
\(0\)
\\
\hline
\(f_{1}\)&
\((x)(y)\)&
\(~x~~y~\)&
\(~x~(y)\)&
\((x)~y~\)&
\((x)(y)\)
\\
\(f_{2}\)&
\((x)~y~\)&
\(~x~(y)\)&
\(~x~~y~\)&
\((x)(y)\)&
\((x)~y~\)
\\
\(f_{4}\)&
\(~x~(y)\)&
\((x)~y~\)&
\((x)(y)\)&
\(~x~~y~\)&
\(~x~(y)\)
\\
\(f_{8}\)&
\(~x~~y~\)&
\((x)(y)\)&
\((x)~y~\)&
\(~x~(y)\)&
\(~x~~y~\)
\\
\hline
\(f_{3}\)&
\((x)\)&
\( x \)&
\( x \)&
\((x)\)&
\((x)\)
\\
\(f_{12}\)&
\( x \)&
\((x)\)&
\((x)\)&
\( x \)&
\( x \)
\\
\hline
\(f_{6}\)&
\( (x,y) \)&
\( (x,y) \)&
\(((x,y))\)&
\(((x,y))\)&
\( (x,y) \)
\\
\(f_{9}\)&
\(((x,y))\)&
\(((x,y))\)&
\( (x,y) \)&
\( (x,y) \)&
\(((x,y))\)
\\
\hline
\(f_{5}\)&
\((y)\)&
\( y \)&
\((y)\)&
\( y \)&
\((y)\)
\\
\(f_{10}\)&
\( y \)&
\((y)\)&
\( y \)&
\((y)\)&
\( y \)
\\
\hline
\(f_{7}\)&
\((~x~~y~)\)&
\(((x)(y))\)&
\(((x)~y~)\)&
\((~x~(y))\)&
\((~x~~y~)\)
\\
\(f_{11}\)&
\((~x~(y))\)&
\(((x)~y~)\)&
\(((x)(y))\)&
\((~x~~y~)\)&
\((~x~(y))\)
\\
\(f_{13}\)&
\(((x)~y~)\)&
\((~x~(y))\)&
\((~x~~y~)\)&
\(((x)(y))\)&
\(((x)~y~)\)
\\
\(f_{14}\)&
\(((x)(y))\)&
\((~x~~y~)\)&
\((~x~(y))\)&
\(((x)~y~)\)&
\(((x)(y))\)
\\
\hline
\(f_{15}\)&
\(1\)&
\(1\)&
\(1\)&
\(1\)&
\(1\)
\\
\hline\hline
\multicolumn{2}{|c||}{Fixed Point Total}&
4&
4&
4&
16
\\
\hline
\end{tabular}
&fg=000000$

Table A4. Df Expanded Over Differential Features {dx, dy}

$latex
\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{Table A4. \(\mathrm{D}f\) Expanded Over Differential Features \(\{\mathrm{d}x, \mathrm{d}y\}\)} \\
\hline
&
\(~~~~~~~~ f ~~~~~~~~\)&
\(\mathrm{D}f|_{~\mathrm{d}x\;\mathrm{d}y~}~\)&
\(\mathrm{D}f|_{~\mathrm{d}x~(\mathrm{d}y)}~\)&
\(\mathrm{D}f|_{(\mathrm{d}x)~\mathrm{d}y~}~\)&
\(\mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)}~\)
\\
\hline\hline
\( f_{0} \)&
\( 0 \)&
\( 0 \)&
\( 0 \)&
\( 0 \)&
\( 0 \)
\\
\hline
\( f_{1} \)&
\( (x)(y) \)&
\( ((x,y)) \)&
\( (y) \)&
\( (x) \)&
\( 0 \)
\\
\( f_{2} \)&
\( (x)~y~ \)&
\( (x,y) \)&
\( y \)&
\( (x) \)&
\( 0 \)
\\
\( f_{4} \)&
\( ~x~(y) \)&
\( (x,y) \)&
\( (y) \)&
\( x \)&
\( 0 \)
\\
\( f_{8} \)&
\( ~x~~y~ \)&
\( ((x,y)) \)&
\( y \)&
\( x \)&
\( 0 \)
\\
\hline
\( f_{3} \)&
\( (x) \)&
\( 1 \)&
\( 1 \)&
\( 0 \)&
\( 0 \)
\\
\( f_{12} \)&
\( x \)&
\( 1 \)&
\( 1 \)&
\( 0 \)&
\( 0 \)
\\
\hline
\( f_{6} \)&
\( (x,y) \)&
\( 0 \)&
\( 1 \)&
\( 1 \)&
\( 0 \)
\\
\( f_{9} \)&
\( ((x,y)) \)&
\( 0 \)&
\( 1 \)&
\( 1 \)&
\( 0 \)
\\
\hline
\( f_{5} \)&
\( (y) \)&
\( 1 \)&
\( 0 \)&
\( 1 \)&
\( 0 \)
\\
\( f_{10} \)&
\( y  \)&
\( 1 \)&
\( 0 \)&
\( 1 \)&
\( 0 \)
\\
\hline
\( f_{7} \)&
\( (~x~~y~) \)&
\( ((x,y)) \)&
\( y \)&
\( x \)&
\( 0 \)
\\
\( f_{11}\) &
\( (~x~(y)) \)&
\( (x,y) \)&
\( (y) \)&
\( x \)&
\( 0 \)
\\
\( f_{13}\) &
\( ((x)~y~) \)&
\( (x,y) \)&
\( y \)&
\( (x) \)&
\( 0 \)
\\
\( f_{14} \)&
\( ((x)(y)) \)&
\( ((x,y)) \)&
\( (y) \)&
\( (x) \)&
\( 0 \)
\\
\hline
\(f_{15}\)&
\( 1 \)&
\( 0 \)&
\( 0 \)&
\( 0 \)&
\( 0 \)
\\
\hline
\end{tabular}
&fg=000000$

Table A5. Ef Expanded Over Ordinary Features {x, y}

$latex
\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{Table A5. \(\mathrm{E}f\) Expanded Over Ordinary Features \(\{x, y\}\)} \\
\hline
&
\(~~~~~~~~ f ~~~~~~~~\)&
\(~~\mathrm{E}f|_{ x\;y }~~~\)&
\(~~\mathrm{E}f|_{ x~(y)}\,~~\)&
\(~~\mathrm{E}f|_{(x)~y }\,~~\)&
\(~~\mathrm{E}f|_{(x)(y)}\;~\)
\\
\hline\hline
\(f_{0}\)&
0&
0&
0&
0&
0
\\
\hline
\(f_{1}\)&
\((x)(y)\)&
~d\(x\)~~d\(y~\)&
~d\(x\)~(d\(y\))&
(d\(x\))~d\(y~\)&
(d\(x\))(d\(y\))
\\
\(f_{2}\)&
\((x)~y~\)&
~d\(x\)~(d\(y\))&
~d\(x\)~~d\(y~\)&
(d\(x\))(d\(y\))&
(d\(x\))~d\(y~\)
\\
\(f_{4}\)&
\(~x~(y)\)&
(d\(x\))~d\(y~\)&
(d\(x\))(d\(y\))&
~d\(x\)~~d\(y~\)&
~d\(x\)~(d\(y\))
\\
\(f_{8}\)&
\(~x~~y~\)&
(d\(x\))(d\(y\))&
(d\(x\))~d\(y~\)&
~d\(x\)~(d\(y\))&
~d\(x\)~~d\(y~\)
\\
\hline
\(f_{3}\)&
\((x)\)&
 d\(x\) &
 d\(x\) &
(d\(x\))&
(d\(x\))
\\
\(f_{12}\)&
\( x \)&
(d\(x\))&
(d\(x\))&
 d\(x\) &
 d\(x\) 
\\
\hline
\(f_{6}\)&
\( (x,y) \)&
 (d\(x\), d\(y\)) &
((d\(x\), d\(y\)))&
((d\(x\), d\(y\)))&
 (d\(x\), d\(y\)) 
\\
\(f_{9}\)&
\(((x,y))\)&
((d\(x\), d\(y\)))&
 (d\(x\), d\(y\)) &
 (d\(x\), d\(y\)) &
((d\(x\), d\(y\)))
\\
\hline
\(f_{5}\)&
\((y)\)&
 d\(y\) &
(d\(y\))&
 d\(y\) &
(d\(y\))
\\
\(f_{10}\)&
\( y \)&
(d\(y\))&
 d\(y\) &
(d\(y\))&
 d\(y\) 
\\
\hline
\(f_{7}\)&
\((~x~~y~)\)&
((d\(x\))(d\(y\)))&
((d\(x\))~d\(y\)~)&
(~d\(x\)~(d\(y\)))&
(~d\(x\)~~d\(y\)~)
\\
\(f_{11}\)&
\((~x~(y))\)&
((d\(x\))~d\(y\)~)&
((d\(x\))(d\(y\)))&
(~d\(x\)~~d\(y\)~)&
(~d\(x\)~(d\(y\)))
\\
\(f_{13}\)&
\(((x)~y~)\)&
(~d\(x\)~(d\(y\)))&
(~d\(x\)~~d\(y\)~)&
((d\(x\))(d\(y\)))&
((d\(x\))~d\(y\)~)
\\
\(f_{14}\)&
\(((x)(y))\)&
(~d\(x\)~~d\(y\)~)&
(~d\(x\)~(d\(y\)))&
((d\(x\))~d\(y\)~)&
((d\(x\))(d\(y\)))
\\
\hline
\(f_{15}\)&
1&
1&
1&
1&
1
\\
\hline
\end{tabular}
&fg=000000$

Table A6. Df Expanded Over Ordinary Features {x, y}

$latex
\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{Table A6. \(\mathrm{D}f\) Expanded Over Ordinary Features \(\{x, y\}\)} \\
\hline
&
\(~~~~~~~~ f ~~~~~~~~\)&
\(~~\mathrm{D}f|_{ x\;y }~~~\)&
\(~~\mathrm{D}f|_{ x~(y)}\,~~\)&
\(~~\mathrm{D}f|_{(x)~y }\,~~\)&
\(~~\mathrm{D}f|_{(x)(y)}\,~\)
\\
\hline\hline
\(f_{0}\)&
0&
0&
0&
0&
0
\\
\hline
\(f_{1}\)&
\((x)(y)\)&
~~d\(x\)~~d\(y~~\)&
\;d\(x\)~(d\(y\))~&
~(d\(x\))~d\(y~~\)&
((d\(x\))(d\(y\)))
\\
\(f_{2}\)&
\((x)~y~\)&
\;d\(x\)~(d\(y\))~&
~~d\(x\)~~d\(y~~\)&
((d\(x\))(d\(y\)))&
~(d\(x\))~d\(y~~\)
\\
\(f_{4}\)&
\(~x~(y)\)&
~(d\(x\))~d\(y~~\)&
((d\(x\))(d\(y\)))&
~~d\(x\)~~d\(y~~\)&
~~d\(x\)~(d\(y\))~
\\
\(f_{8}\)&
\(~x~~y~\)&
((d\(x\))(d\(y\)))&
~(d\(x\))~d\(y~~\)&
\;d\(x\)~(d\(y\))~&
~~d\(x\)~~d\(y~~\)
\\
\hline
\(f_{3}\)&
\((x)\)&
d\(x\)&
d\(x\)&
d\(x\)&
d\(x\)
\\
\(f_{12}\)&
\( x \)&
d\(x\)&
d\(x\)&
d\(x\)&
d\(x\)
\\
\hline
\(f_{6}\)&
\( (x,y) \)&
(d\(x\), d\(y\))&
(d\(x\), d\(y\))&
(d\(x\), d\(y\))&
(d\(x\), d\(y\))
\\
\(f_{9}\)&
\(((x,y))\)&
(d\(x\), d\(y\))&
(d\(x\), d\(y\))&
(d\(x\), d\(y\))&
(d\(x\), d\(y\))
\\
\hline
\(f_{5}\)&
\((y)\)&
d\(y\)&
d\(y\)&
d\(y\)&
d\(y\)
\\
\(f_{10}\)&
\( y \)&
d\(y\)&
d\(y\)&
d\(y\)&
d\(y\)
\\
\hline
\(f_{7}\)&
\((~x~~y~)\)&
((d\(x\))(d\(y\)))&
~(d\(x\))~d\(y~~\)&
\;d\(x\)~(d\(y\))~&
~~d\(x\)~~d\(y~~\)
\\
\(f_{11}\)&
\((~x~(y))\)&
~(d\(x\))~d\(y~~\)&
((d\(x\))(d\(y\)))&
~~d\(x\)~~d\(y~~\)&
~~d\(x\)~(d\(y\))~
\\
\(f_{13}\)&
\(((x)~y~)\)&
\;d\(x\)~(d\(y\))~&
~~d\(x\)~~d\(y~~\)&
((d\(x\))(d\(y\)))&
~(d\(x\))~d\(y~~\)
\\
\(f_{14}\)&
\(((x)(y))\)&
~~d\(x\)~~d\(y~~\)&
\;d\(x\)~(d\(y\))~&
~(d\(x\))~d\(y~~\)&
((d\(x\))(d\(y\)))
\\
\hline
\(f_{15}\)&
1&
0&
0&
0&
0
\\
\hline
\end{tabular}
&fg=000000$

Fourier Transforms of Boolean Functions

Re: Another Problem

The problem is concretely about Boolean functions \(f\!\) of \(k\!\) variables, and seems not to involve prime numbers at all. For any subset \(S\!\) of the coordinates, the corresponding Fourier coefficient is given by:

\(\hat{f}(S) = \frac{1}{2^k} \sum_{x \in \mathbb{Z}_2^k} f(x)\chi_S(x)\!\)

where \(\chi_S(x)\!\) is \(-1\!\) if \(\textstyle \sum_{i \in S} x_i\!\) is odd, and \(+1\!\) otherwise.

\(k = 1\!\)

\(k = 2\!\)

For ease of reading formulas, let \(x = (x_1, x_2) = (u, v).\!\)

Table 2.1. Values of χS(x)

$latex
\begin{tabular}{|c||*{4}{c}|}
\multicolumn{5}{c}{Table 2.1. Values of \( \boldsymbol{\chi}_\mathcal{S}(x) \) for \( f : \mathbb{B}^2 \to \mathbb{B} \)} \\[4pt]
\hline
\( \mathcal{S} \backslash (u, v) \) &
\( (1, 1) \) &
\( (1, 0) \) &
\( (0, 1) \) &
\( (0, 0) \)
\\
\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}
&fg=000000$

Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables

$latex
\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}
&fg=000000$

Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables

$latex
\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&
\((u~~v)\)&
\( 3/4\)&
\( 1/4\)&
\( 1/4\)&
\(-1/4\)
\\
\hline
\(f_{11}\)&
\(f_{1011}\)&&
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}
&fg=000000$

Work 2

Array Test

$latex
|x| = \left\{
\begin{array}{ll}
x  & \text{if \( x \geq 0 \)};
\\
-x & \text{if \( x < 0 \)}.
\end{array}
\right.
&fg=000000$
$latex
|x| = \left\{
\begin{array}{ll}
x  & \text{if}~ x \geq 0;
\\
-x & \text{if}~ x < 0.
\end{array}
\right.
&fg=000000$
$latex
\begin{array}{*{9}{l}}
Alpha & Bravo & Charlie & Delta & Echo & Foxtrot & Golf & Hotel & India
\\
Juliet & Kilo & Lima & Mike & November & Oscar & Papa & Quebec & Romeo
\\
Sierra & Tango & Uniform & Victor & Whiskey & X\text{-}ray & Yankee & Zulu & \varnothing
\end{array}&fg=000000$

Matrix Test

$latex
\begin{matrix}
Alpha & Bravo & Charlie & Delta & Echo & Foxtrot & Golf & Hotel & India
\\
Juliet & Kilo & Lima & Mike & November & Oscar & Papa & Quebec & Romeo
\\
Sierra & Tango & Uniform & Victor & Whiskey & X\text{-}ray & Yankee & Zulu & \varnothing
\end{matrix}&fg=000000$

Tabular Test 1

$latex
\begin{tabular}{lll}
Chicago & U.S.A. & 1893
\\
Z\"{u}rich & Switzerland & 1897
\\
Paris & France & 1900
\\
Heidelberg & Germany & 1904
\\
Rome & Italy & 1908
\end{tabular}&fg=000000$

Tabular Test 2

$latex
\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}&fg=000000$

Tabular Test 3

$latex
\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}&fg=000000$

Tabular Test 4

$latex
\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}&fg=000000$

Table Test 1

<table border="0" style="border-width:0;width:100%;">

<tr>
<td style="border-top:1px solid white;width:35%;"></td>

<td style="border-top:1px solid white;width:65%;">
Can we ever become what we weren’t in eternity?
Can we ever learn what we weren’t born knowing?
Can we ever share what we never had in common?</td>
</tr>

</table>

Table Test 2

<table align="left" border="0" style="border-width:0;">

<tr>
<td style="border-top:1px solid white;">
<p>Everything considered, a determined soul will always manage.</p></td>

<td style="border-top:1px solid white;">(41)</td>
</tr>

<tr>
<td style="border-top:1px solid white;">
<p>To a man devoid of blinders, there is no finer sight than that of the intelligence at grips with a reality that transcends it.</p></td>

<td style="border-top:1px solid white;">(55)</td>
</tr>

</table>

Table Test 3

<table align="center" border="0">

<tr>
<td>
<br>
<p>Everything considered, a determined soul will always manage.</p></td>

<td><p>(41)</p></td>
</tr>

<tr>
<td>
<br>
<p>To a man devoid of blinders, there is no finer sight than that
of the intelligence at grips with a reality that transcends it.</p></td>

<td><p>(55)</p></td>
</tr>

</table>

Table Test 4

<table align="center" border="0" style="border-width:0;text-align:center;">

<tr>
<td style="border-top:1px solid white;">
<a href="http://inquiryintoinquiry.files.wordpress.com/2008/09/logicalgraphfigure1visibleframe1.jpg" title="Logical Graph Figure 1">
<img src="http://inquiryintoinquiry.files.wordpress.com/2008/09/logicalgraphfigure1visibleframe1.jpg" alt="()()=()" width="500" height="168" border="0"></a></td>

<td style="border-top:1px solid white;">(1)</td>
</tr>

<tr> 
<td style="border-top:1px solid white;">
<a href="http://inquiryintoinquiry.files.wordpress.com/2008/09/logicalgraphfigure2visibleframe1.jpg" title="Logical Graph Figure 2">
<img src="http://inquiryintoinquiry.files.wordpress.com/2008/09/logicalgraphfigure2visibleframe1.jpg" alt="(())= " width="500" height="168" border="0"></a></td>

<td style="border-top:1px solid white;">(2)</td>
</tr>

</table>

Table Test 5

<table align="center" border="0" style="text-align:center;">

<tr>
<td style="padding:10px;">
<a href="http://inquiryintoinquiry.files.wordpress.com/2008/09/logicalgraphfigure1visibleframe1.jpg" title="Logical Graph Figure 1">
<img src="http://inquiryintoinquiry.files.wordpress.com/2008/09/logicalgraphfigure1visibleframe1.jpg" alt="()()=()" align="center" width="500" height="168" /></a></td>

<td style="padding:80px 10px;">(1)</td>
</tr>

<tr>
<td style="padding:10px;">
<a href="http://inquiryintoinquiry.files.wordpress.com/2008/09/logicalgraphfigure2visibleframe1.jpg" title="Logical Graph Figure 2">
<img src="http://inquiryintoinquiry.files.wordpress.com/2008/09/logicalgraphfigure2visibleframe1.jpg" alt="(())= " align="center" width="500" height="168" /></a></td>

<td style="padding:80px 10px;">(2)</td>
</tr>

</table>

Table Test 6

<table align="center" border="0" style="text-align:center;">

<caption><font size="+2">$latex \text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1) $</font></caption>

<tr>
<td style="border-bottom:2px solid black;">$latex m_{0} $</td>
<td style="border-bottom:2px solid black;">$latex m_{1} $</td>
<td style="border-bottom:2px solid black;">$latex m_{2} $</td>
<td style="border-bottom:2px solid black;">$latex m_{3} $</td>
<td style="border-bottom:2px solid black;">$latex m_{4} $</td>
<td style="border-bottom:2px solid black;">$latex m_{5} $</td>
<td style="border-bottom:2px solid black;">$latex m_{6} $</td>
<td style="border-bottom:2px solid black;">$latex m_{7} $</td>
</tr>

<tr>
<td style="background:white;color:black;">0</td>
<td style="background:black;color:white;">1</td>
<td style="background:white;color:black;">0</td>
<td style="background:black;color:white;">1</td>
<td style="background:white;color:black;">0</td>
<td style="background:black;color:white;">1</td>
<td style="background:white;color:black;">0</td>
<td style="background:black;color:white;">1</td>
</tr>

<tr>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:black;color:white;">1</td>
<td style="background:black;color:white;">1</td>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:black;color:white;">1</td>
<td style="background:black;color:white;">1</td>
</tr>

<tr>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:black;color:white;">1</td>
<td style="background:black;color:white;">1</td>
<td style="background:black;color:white;">1</td>
<td style="background:black;color:white;">1</td>
</tr>

<tr>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
<td style="background:white;color:black;">0</td>
</tr>

</table>