# User:Jon Awbrey/WORDPRESS

WordPress versions of HTML and LaTeX markup

## 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

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>
<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>

</tr>

<tr>
<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>

</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>