## Two Dimensional rules for case statements structures. Locale: nemeth, Style: Verbose.

 0 $\left\{\begin{array}{l}x=y\\ 5x-y=4\end{array}$ ⠨⠠⠷⠭⠀⠨⠅⠀⠽⠀⠀⠀⠀⠀ ⠨⠠⠷⠼⠢⠭⠤⠽⠀⠨⠅⠀⠼⠲

## Two Dimensional rules for tabular layout equation. Locale: nemeth, Style: Verbose.

 0 $\begin{array}{cc}{\mathbf{e}}_{1}& =\left(100\cdots 00{\right)}^{\text{t}}\\ {\mathbf{e}}_{2}& =\left(010\cdots 00{\right)}^{\text{t}}\\ & ⋮\\ {\mathbf{e}}_{n}& =\left(000\cdots 01{\right)}^{\text{t}}\end{array}$ ⠸⠰⠑⠂⠀⠀⠀⠨⠅⠀⠷⠂⠴⠴⠀⠄⠄⠄⠀⠴⠴⠾⠘⠞⠐ ⠸⠰⠑⠆⠀⠀⠀⠨⠅⠀⠷⠴⠂⠴⠀⠄⠄⠄⠀⠴⠴⠾⠘⠞⠐ ⠄⠄⠄⠀⠀⠀ ⠸⠰⠑⠰⠝⠐⠀⠨⠅⠀⠷⠴⠴⠴⠀⠄⠄⠄⠀⠴⠂⠾⠘⠞ 1 $\begin{array}{cc}{x}_{2}+{x}_{3}+{x}_{4}& =0\\ {x}_{1}+{x}_{2}+{x}_{5}& =0\\ {x}_{1}+{x}_{3}+{x}_{6}& =0\text{.}\end{array}$ ⠭⠆⠬⠭⠒⠬⠭⠲⠀⠨⠅⠀⠼⠴⠀ ⠭⠂⠬⠭⠆⠬⠭⠢⠀⠨⠅⠀⠼⠴⠀ ⠭⠂⠬⠭⠒⠬⠭⠖⠀⠨⠅⠀⠼⠴⠲ 2 $\begin{array}{c}0+H=3+H=\left\{0,3\right\}\\ 1+H=4+H=\left\{1,4\right\}\\ 2+H=5+H=\left\{2,5\right\}\text{.}\end{array}$ ⠼⠴⠬⠠⠓⠀⠨⠅⠀⠼⠒⠬⠠⠓⠀⠨⠅⠀⠨⠷⠴⠠⠀⠒⠨⠾⠀ ⠼⠂⠬⠠⠓⠀⠨⠅⠀⠼⠲⠬⠠⠓⠀⠨⠅⠀⠨⠷⠂⠠⠀⠲⠨⠾⠀ ⠼⠆⠬⠠⠓⠀⠨⠅⠀⠼⠢⠬⠠⠓⠀⠨⠅⠀⠨⠷⠆⠠⠀⠢⠨⠾⠲ 3 $\begin{array}{c}{\varphi }_{\alpha }\left(f+g\right)=\left(f+g\right)\left(\alpha \right)=f\left(\alpha \right)+g\left(\alpha \right)={\varphi }_{\alpha }\left(f\right)+{\varphi }_{\alpha }\left(g\right)\\ {\varphi }_{\alpha }\left(fg\right)=\left(fg\right)\left(\alpha \right)=f\left(\alpha \right)g\left(\alpha \right)={\varphi }_{\alpha }\left(f\right){\varphi }_{\alpha }\left(g\right)\end{array}$ ⠋⠰⠨⠁⠐⠷⠋⠬⠛⠾⠀⠨⠅⠀⠷⠋⠬⠛⠾⠷⠨⠁⠾⠀⠨⠅⠀⠋⠷⠨⠁⠾⠬⠛⠷⠨⠁⠾⠀⠨⠅⠀⠋⠰⠨⠁⠐⠷⠋⠾⠬⠋⠰⠨⠁⠐⠷⠛⠾ ⠋⠰⠨⠁⠐⠷⠋⠛⠾⠀⠨⠅⠀⠷⠋⠛⠾⠷⠨⠁⠾⠀⠨⠅⠀⠋⠷⠨⠁⠾⠛⠷⠨⠁⠾⠀⠨⠅⠀⠋⠰⠨⠁⠐⠷⠋⠾⠋⠰⠨⠁⠐⠷⠛⠾⠀⠀⠀⠀ 4 $\begin{array}{c}{\varphi }_{\alpha }\left(f+g\right)=\left(f+g\right)\left(\alpha \right)=f\left(\alpha \right)+g\left(\alpha \right)={\varphi }_{\alpha }\left(f\right)+{\varphi }_{\alpha }\left(g\right)\\ {\varphi }_{\alpha }\left(fg\right)=\left(fg\right)\left(\alpha \right)=f\left(\alpha \right)g\left(\alpha \right)={\varphi }_{\alpha }\left(f\right){\varphi }_{\alpha }\left(g\right)\\ 2{\varphi }_{\alpha }\left(f\right)=2\left(f\right)\left(\alpha \right)=f\left(2\alpha \right)={\varphi }_{2\alpha }\left(f\right)\end{array}$ ⠋⠰⠨⠁⠐⠷⠋⠬⠛⠾⠀⠨⠅⠀⠷⠋⠬⠛⠾⠷⠨⠁⠾⠀⠨⠅⠀⠋⠷⠨⠁⠾⠬⠛⠷⠨⠁⠾⠀⠨⠅⠀⠋⠰⠨⠁⠐⠷⠋⠾⠬⠋⠰⠨⠁⠐⠷⠛⠾ ⠋⠰⠨⠁⠐⠷⠋⠛⠾⠀⠨⠅⠀⠷⠋⠛⠾⠷⠨⠁⠾⠀⠨⠅⠀⠋⠷⠨⠁⠾⠛⠷⠨⠁⠾⠀⠨⠅⠀⠋⠰⠨⠁⠐⠷⠋⠾⠋⠰⠨⠁⠐⠷⠛⠾⠀⠀⠀⠀ ⠼⠆⠋⠰⠨⠁⠐⠷⠋⠾⠀⠨⠅⠀⠼⠆⠷⠋⠾⠷⠨⠁⠾⠀⠨⠅⠀⠋⠷⠆⠨⠁⠾⠀⠨⠅⠀⠋⠰⠆⠨⠁⠐⠷⠋⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

## Two Dimensional rules for matrix structures. Locale: nemeth, Style: Verbose.

 0 $\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1,n-m}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2,n-m}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{m1}& {a}_{m2}& \cdots & {a}_{m,n-m}\end{array}\right)$ ⠠⠷⠁⠂⠂⠀⠀⠀⠀⠁⠂⠆⠀⠀⠀⠀⠄⠄⠄⠀⠁⠰⠼⠂⠠⠀⠝⠤⠍⠐⠠⠾ ⠠⠷⠁⠆⠂⠀⠀⠀⠀⠁⠆⠆⠀⠀⠀⠀⠄⠄⠄⠀⠁⠰⠼⠆⠠⠀⠝⠤⠍⠐⠠⠾ ⠠⠷⠄⠄⠄⠀⠀⠀⠀⠄⠄⠄⠀⠀⠀⠀⠄⠄⠄⠀⠄⠄⠄⠀⠀⠀⠀⠀⠀⠀⠠⠾ ⠠⠷⠁⠰⠍⠐⠂⠐⠀⠁⠰⠍⠐⠆⠐⠀⠄⠄⠄⠀⠁⠰⠍⠠⠀⠝⠤⠍⠀⠀⠠⠾ 1 $H\mathbf{x}=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}H\mathbf{y}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)\text{.}$ ⠠⠓⠸⠰⠭⠀⠨⠅⠀⠠⠷⠼⠴⠠⠾⠁⠝⠙⠠⠓⠸⠰⠽⠀⠨⠅⠀⠠⠷⠼⠂⠠⠾⠲ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠼⠴⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠼⠴⠠⠾⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠼⠴⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠼⠂⠠⠾⠀ 2 $H\mathbf{x}=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$ ⠠⠓⠸⠰⠭⠀⠨⠅⠀⠠⠷⠼⠴⠠⠾ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠼⠴⠠⠾ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠼⠴⠠⠾ 3 $H\mathbf{y}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$ ⠠⠓⠸⠰⠽⠀⠨⠅⠀⠠⠷⠼⠂⠠⠾ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠼⠴⠠⠾ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠼⠂⠠⠾ 4 $H=\left(\begin{array}{ccccc}1& 1& 1& 0& 0\\ 0& 1& 0& 1& 0\\ 1& 0& 0& 0& 1\end{array}\right)$ ⠠⠓⠀⠨⠅⠀⠠⠷⠼⠂⠀⠼⠂⠀⠼⠂⠀⠼⠴⠀⠼⠴⠠⠾ ⠀⠀⠀⠀⠀⠀⠠⠷⠼⠴⠀⠼⠂⠀⠼⠴⠀⠼⠂⠀⠼⠴⠠⠾ ⠀⠀⠀⠀⠀⠀⠠⠷⠼⠂⠀⠼⠴⠀⠼⠴⠀⠼⠴⠀⠼⠂⠠⠾ 5 $⟨\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8\\ 9\\ 10\end{array}⟩$ ⠨⠨⠠⠷⠼⠂⠀⠨⠨⠠⠾ ⠨⠨⠠⠷⠼⠆⠀⠨⠨⠠⠾ ⠨⠨⠠⠷⠼⠒⠀⠨⠨⠠⠾ ⠨⠨⠠⠷⠼⠲⠀⠨⠨⠠⠾ ⠨⠨⠠⠷⠼⠢⠀⠨⠨⠠⠾ ⠨⠨⠠⠷⠼⠖⠀⠨⠨⠠⠾ ⠨⠨⠠⠷⠼⠶⠀⠨⠨⠠⠾ ⠨⠨⠠⠷⠼⠦⠀⠨⠨⠠⠾ ⠨⠨⠠⠷⠼⠔⠀⠨⠨⠠⠾ ⠨⠨⠠⠷⠼⠂⠴⠨⠨⠠⠾ 6 $\left[\begin{array}{ccc}1& 9& -13\\ 20& 5& -6\end{array}\right]$ ⠈⠠⠷⠼⠂⠀⠀⠼⠔⠀⠤⠼⠂⠒⠈⠠⠾ ⠈⠠⠷⠼⠆⠴⠀⠼⠢⠀⠤⠼⠖⠀⠈⠠⠾ 7 $|\begin{array}{cc}a& b\\ c& d\end{array}|$ ⠠⠳⠁⠀⠃⠠⠳ ⠠⠳⠉⠀⠙⠠⠳ 8 $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ ⠠⠁⠀⠨⠅⠀⠠⠷⠁⠀⠃⠠⠾ ⠀⠀⠀⠀⠀⠀⠠⠷⠉⠀⠙⠠⠾ 9 $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ ⠠⠷⠁⠀⠃⠠⠾ ⠠⠷⠉⠀⠙⠠⠾ 10 $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ ⠈⠠⠷⠁⠀⠃⠈⠠⠾ ⠈⠠⠷⠉⠀⠙⠈⠠⠾ 11 $|\begin{array}{cc}a& b\\ c& d\end{array}|$ ⠠⠳⠁⠀⠃⠠⠳ ⠠⠳⠉⠀⠙⠠⠳ 12 $\left(\begin{array}{ccc}\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)& r& s\\ 1& 2& 3\end{array}\right)$ ⠠⠷⠠⠷⠁⠀⠃⠠⠾⠀⠗⠀⠀⠎⠀⠠⠾ ⠠⠷⠠⠷⠉⠀⠙⠠⠾⠀⠀⠀⠀⠀⠀⠠⠾ ⠠⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾ ⠠⠷⠼⠂⠀⠀⠀⠀⠀⠀⠼⠆⠀⠼⠒⠠⠾ 13 $\begin{array}{r}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\phantom{\rule{1em}{0ex}}\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)\phantom{\rule{1em}{0ex}}\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)\\ \left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)\phantom{\rule{1em}{0ex}}\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\phantom{\rule{1em}{0ex}}\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)\end{array}$ ⠠⠷⠼⠂⠀⠼⠴⠀⠼⠴⠠⠾⠠⠷⠼⠂⠀⠼⠴⠀⠼⠴⠠⠾⠠⠷⠼⠴⠀⠼⠂⠀⠼⠴⠠⠾ ⠠⠷⠼⠴⠀⠼⠂⠀⠼⠴⠠⠾⠠⠷⠼⠴⠀⠼⠴⠀⠼⠂⠠⠾⠠⠷⠼⠂⠀⠼⠴⠀⠼⠴⠠⠾ ⠠⠷⠼⠴⠀⠼⠴⠀⠼⠂⠠⠾⠠⠷⠼⠴⠀⠼⠂⠀⠼⠴⠠⠾⠠⠷⠼⠴⠀⠼⠴⠀⠼⠂⠠⠾ ⠠⠷⠼⠴⠀⠼⠴⠀⠼⠂⠠⠾⠠⠷⠼⠴⠀⠼⠴⠀⠼⠂⠠⠾⠠⠷⠼⠴⠀⠼⠂⠀⠼⠴⠠⠾ ⠠⠷⠼⠂⠀⠼⠴⠀⠼⠴⠠⠾⠠⠷⠼⠴⠀⠼⠂⠀⠼⠴⠠⠾⠠⠷⠼⠴⠀⠼⠴⠀⠼⠂⠠⠾ ⠠⠷⠼⠴⠀⠼⠂⠀⠼⠴⠠⠾⠠⠷⠼⠂⠀⠼⠴⠀⠼⠴⠠⠾⠠⠷⠼⠂⠀⠼⠴⠀⠼⠴⠠⠾ 14 $\left(\begin{array}{ccc}1& x& y\\ 0& 1& z\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& {x}^{\prime }& {y}^{\prime }\\ 0& 1& {z}^{\prime }\\ 0& 0& 1\end{array}\right)=\left(\begin{array}{ccc}1& x+{x}^{\prime }& y+{y}^{\prime }+x{z}^{\prime }\\ 0& 1& z+{z}^{\prime }\\ 0& 0& 1\end{array}\right)\text{.}$ ⠠⠷⠼⠂⠀⠭⠀⠀⠽⠀⠠⠾⠠⠷⠼⠂⠀⠭⠘⠄⠐⠀⠽⠘⠄⠐⠠⠾⠀⠨⠅⠀⠠⠷⠼⠂⠀⠭⠬⠭⠘⠄⠐⠀⠽⠬⠽⠘⠄⠐⠬⠭⠵⠘⠄⠐⠠⠾⠲ ⠠⠷⠼⠴⠀⠼⠂⠀⠵⠀⠠⠾⠠⠷⠼⠴⠀⠼⠂⠀⠀⠀⠵⠘⠄⠐⠠⠾⠀⠀⠀⠀⠠⠷⠼⠴⠀⠼⠂⠀⠀⠀⠀⠀⠵⠬⠵⠘⠄⠐⠀⠀⠀⠀⠀⠀⠠⠾⠀ ⠠⠷⠼⠴⠀⠼⠴⠀⠼⠂⠠⠾⠠⠷⠼⠴⠀⠼⠴⠀⠀⠀⠼⠂⠀⠀⠠⠾⠀⠀⠀⠀⠠⠷⠼⠴⠀⠼⠴⠀⠀⠀⠀⠀⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀

## Two Dimensional rules for table structures. Locale: nemeth, Style: Verbose.

 0 $\begin{array}{lll}\cdot & 0& 1\\ 0& 0& 0\\ 1& 0& 1\end{array}$ ⠡⠀⠀⠀⠼⠴⠀⠀⠼⠂ ⠐⠒⠀⠀⠐⠒⠀⠀⠐⠒ ⠼⠴⠀⠀⠼⠴⠀⠀⠼⠴ ⠼⠂⠀⠀⠼⠴⠀⠀⠼⠂ 1 $\begin{array}{cccc}+& 0& 1& 2\\ 0& 0& 1& 2\\ 1& 1& 2& 0\\ 2& 2& 0& 1\end{array}$ ⠬⠀⠀⠀⠼⠴⠀⠀⠼⠂⠀⠀⠼⠆ ⠐⠒⠀⠀⠐⠒⠀⠀⠐⠒⠀⠀⠐⠒ ⠼⠴⠀⠀⠼⠴⠀⠀⠼⠂⠀⠀⠼⠆ ⠼⠂⠀⠀⠼⠂⠀⠀⠼⠆⠀⠀⠼⠴ ⠼⠆⠀⠀⠼⠆⠀⠀⠼⠴⠀⠀⠼⠂ 2 $\begin{array}{ccc}& N& \left(1\phantom{\rule{0.167em}{0ex}}2\right)N\\ N& N& \left(1\phantom{\rule{0.167em}{0ex}}2\right)N\\ \left(1\phantom{\rule{0.167em}{0ex}}2\right)N& \left(1\phantom{\rule{0.167em}{0ex}}2\right)N& N\end{array}$ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠼⠂⠀⠼⠆⠾⠠⠝ ⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒ ⠠⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠼⠂⠀⠼⠆⠾⠠⠝ ⠷⠼⠂⠀⠼⠆⠾⠠⠝⠀⠀⠷⠼⠂⠀⠼⠆⠾⠠⠝⠀⠀⠠⠝⠀⠀⠀⠀⠀⠀⠀ 3 $\begin{array}{cccc}+& 0+3\mathbb{Z}& 1+3\mathbb{Z}& 2+3\mathbb{Z}\\ 0+3\mathbb{Z}& 0+3\mathbb{Z}& 1+3\mathbb{Z}& 2+3\mathbb{Z}\\ 1+3\mathbb{Z}& 1+3\mathbb{Z}& 2+3\mathbb{Z}& 0+3\mathbb{Z}\\ 2+3\mathbb{Z}& 2+3\mathbb{Z}& 0+3\mathbb{Z}& 1+3\mathbb{Z}\end{array}$ ⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠴⠬⠒⠈⠰⠠⠵⠀⠀⠼⠂⠬⠒⠈⠰⠠⠵⠀⠀⠼⠆⠬⠒⠈⠰⠠⠵ ⠐⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒ ⠼⠴⠬⠒⠈⠰⠠⠵⠀⠀⠼⠴⠬⠒⠈⠰⠠⠵⠀⠀⠼⠂⠬⠒⠈⠰⠠⠵⠀⠀⠼⠆⠬⠒⠈⠰⠠⠵ ⠼⠂⠬⠒⠈⠰⠠⠵⠀⠀⠼⠂⠬⠒⠈⠰⠠⠵⠀⠀⠼⠆⠬⠒⠈⠰⠠⠵⠀⠀⠼⠴⠬⠒⠈⠰⠠⠵ ⠼⠆⠬⠒⠈⠰⠠⠵⠀⠀⠼⠆⠬⠒⠈⠰⠠⠵⠀⠀⠼⠴⠬⠒⠈⠰⠠⠵⠀⠀⠼⠂⠬⠒⠈⠰⠠⠵