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%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem}     % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
\usepackage{wasysym}            % Symbols
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% Document Info
\author{Marius Van Nieuwenhuyse (mariusvnh)}
\pdfinfo{
  /Title (logic.pdf)
  /Creator (Cheatography)
  /Author (Marius Van Nieuwenhuyse (mariusvnh))
  /Subject (Logic Cheat Sheet)
}

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% Header and Footer
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\fancyhead{} % Set header to blank
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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Logic Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Marius Van Nieuwenhuyse (mariusvnh)} via \textcolor{DarkBackground}{\uline{cheatography.com/134859/cs/27858/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Marius Van Nieuwenhuyse (mariusvnh) \\
  \uline{cheatography.com/mariusvnh} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 6th May, 2021.\\
   Updated 6th May, 2021.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{1.50372 cm} x{1.37841 cm} x{0.87717 cm} p{0.4177 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{5.377cm}}{\bf\textcolor{white}{Logical Operators}}  \tn
% Row 0
\SetRowColor{LightBackground}
Not & Negation & NOT & ¬ \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
And & Conjunction & AND & ∧ \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
Or & Disjunction & OR & ∨ \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
Exclusive Or & N/A & XOR & ⊕ \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
N/A & Implication & IMPLIES & → \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
Equal & Equivalence & EQUAL & ↔ \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
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\par\addvspace{1.3em}

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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Precedence}}  \tn
% Row 0
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\mymulticolumn{1}{x{5.377cm}}{NOT} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{AND, NAND} \tn 
% Row Count 2 (+ 1)
% Row 2
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\mymulticolumn{1}{x{5.377cm}}{OR, NOR} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{IMPLIES} \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{EQUAL, XOR} \tn 
% Row Count 5 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Commutativity}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{`A op B == B op A` \{\{nl\}\} Works for AND, OR, EQUAL, XOR, NAND, and NOR% Row Count 2 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{The only operator that isn't commutative is IMPLIES}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Associativity}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{`A op (B op C) == (A op B) op C` \{\{nl\}\} Works for AND, OR,  EQUAL, and XOR.% Row Count 2 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.32733 cm} p{0.68655 cm} x{2.56312 cm} }
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\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Distributivity}}  \tn
% Row 0
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A ∨ (B ∧ C) & \{\{ac\}\}== & \{\{ar\}\}(A ∨ B) ∧ (A ∨ C) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
A ∧ (B ∨ C) & \{\{ac\}\}== & \{\{ar\}\}(A ∧ B) ∨ (A ∧ C) \tn 
% Row Count 4 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.32733 cm} x{1.32733 cm} x{1.92234 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Implies Truth Table}}  \tn
% Row 0
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\{\{ac\}\} {\bf{A}} & \{\{ac\}\} {\bf{B}} & \{\{ac\}\} {\bf{Result}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\{\{ac\}\}False & \{\{ac\}\}False & \{\{ac\}\}True \tn 
% Row Count 3 (+ 1)
% Row 2
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\{\{ac\}\}True & \{\{ac\}\}False & \{\{ac\}\}False \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
\{\{ac\}\}False & \{\{ac\}\}True & \{\{ac\}\}True \tn 
% Row Count 5 (+ 1)
% Row 4
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\{\{ac\}\}True & \{\{ac\}\}True & \{\{ac\}\}True \tn 
% Row Count 6 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{0.68655 cm} p{0.77809 cm} x{3.11236 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Equivalents Formulae}}  \tn
% Row 0
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A ↔ B & \{\{ac\}\}== & \{\{ar\}\}(A → B) ∧ (B → A) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
A ⊕ B & \{\{ac\}\}== & \{\{ar\}\}¬(A → B) ∨ ¬(B → A) \tn 
% Row Count 4 (+ 2)
% Row 2
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A → B & \{\{ac\}\}== & \{\{ar\}\}¬A ∨ B \tn 
% Row Count 6 (+ 2)
% Row 3
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A → B & \{\{ac\}\}== & \{\{ar\}\}¬(A ∧ ¬B) \tn 
% Row Count 8 (+ 2)
% Row 4
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A ∨ B & \{\{ac\}\}== & \{\{ar\}\}¬(¬A ∧ ¬B) \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
A ∨ B & \{\{ac\}\}== & \{\{ar\}\}¬A → B \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
A ∧ B & \{\{ac\}\}== & \{\{ar\}\}¬(¬A ∨ ¬B) \tn 
% Row Count 14 (+ 2)
% Row 7
\SetRowColor{white}
A ∧ B & \{\{ac\}\}== & \{\{ar\}\}¬(A → ¬B) \tn 
% Row Count 16 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Sementic Tableaux}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{If you are asked to {\bf{prove}} something with sementic tableaux, negate the formulae first.\{\{nl\}\} If you are asked to {\bf{refute}} something with the sementic tableaux, {\bf{do not}} negate the formulae.% Row Count 4 (+ 4)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}