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\usepackage{seqsplit}           % Splits long words.
%\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
%\usepackage[english,german,french,spanish,italian]{babel}              % Languages

% Document Info
\author{TheEmu001}
\pdfinfo{
  /Title (discrete-exam-1.pdf)
  /Creator (Cheatography)
  /Author (TheEmu001)
  /Subject (Discrete Exam 1 Cheat Sheet)
}

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% Commands
\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
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% Header and Footer
\pagestyle{fancy}
\fancyhead{} % Set header to blank
\fancyfoot{} % Set footer to blank
\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}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Discrete Exam 1 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{TheEmu001} via \textcolor{DarkBackground}{\uline{cheatography.com/30732/cs/9211/}}}
\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}TheEmu001 \\
  \uline{cheatography.com/theemu001} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 21st September, 2016.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \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*}{4}

\begin{tabularx}{3.833cm}{p{0.65227 cm} x{2.78073 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Useful Symbols}}  \tn
% Row 0
\SetRowColor{LightBackground}
∀ & for all (universal operator) \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
∃ & exists (existential operator) \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
∈ & in the set \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
∧ & and \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
∨ & or \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
∼ & not \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
≡ & equivalent \tn 
% Row Count 7 (+ 1)
% Row 7
\SetRowColor{white}
⊂ & subset \tn 
% Row Count 8 (+ 1)
% Row 8
\SetRowColor{LightBackground}
⊃ & superset \tn 
% Row Count 9 (+ 1)
% Row 9
\SetRowColor{white}
\{\}, ∅ & empty set \tn 
% Row Count 10 (+ 1)
% Row 10
\SetRowColor{LightBackground}
↔ & biconditional (both are true) \tn 
% Row Count 11 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.68217 cm} x{1.75083 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Statements}}  \tn
% Row 0
\SetRowColor{LightBackground}
Universal & For all, for each \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Existential & At least, there exists \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Conditional & If → then \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
Universal Conditional & For all \& if-then \tn 
% Row Count 6 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Universal Existential & For all \& there exists \tn 
% Row Count 8 (+ 2)
% Row 5
\SetRowColor{white}
Existential Universal & There exists \& for all \tn 
% Row Count 10 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.15254 cm} x{0.94023 cm} x{0.94023 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{3.833cm}}{\bf\textcolor{white}{List of Equivalences}}  \tn
% Row 0
\SetRowColor{LightBackground}
Conditional Statemnts & p→q≡\textasciitilde{}(p∧\textasciitilde{}q) & p→q≡\textasciitilde{}p∨q \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
contrapositive & p→q≡\textasciitilde{}q→\textasciitilde{}p &  \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Converse & p→q (cond) & q→p (converse) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
inverse & p→q (cond) & \textasciitilde{}p→\textasciitilde{}p (inverse) \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{3.833cm}}{vacuously true = true by absence \newline converse and inverse are the {\bf{SAME}}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Set-Builder Notation}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/theemu001_1474497580_Screen Shot 2016-09-21 at 5.39.04 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Set-Roster Notation}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{A = \{1, 2, 3 ... 100\}} \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{use ellipses for larger sets}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.6866 cm} x{2.7464 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Subsets}}  \tn
% Row 0
\SetRowColor{LightBackground}
B⊆A & B=subset, A=superset \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Proper Subsets: elements that belong to superset but NOT subset}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.75526 cm} x{2.67774 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Relations}}  \tn
% Row 0
\SetRowColor{LightBackground}
\seqsplit{Relations=} & subsets of cartesian product \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
R ⊆ A x B & Relation ⊆ Domain x Codomain \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Domain & {\bf{SET}} that includes every element from source \tn 
% Row Count 6 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{don't always have to include ordered pairs}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Functions}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Requirements: \newline % Row Count 1 (+ 1)
- Arrow coming out of {\bf{every element}} in domain \newline % Row Count 2 (+ 1)
- Every element can only have {\bf{one}} element of {\emph{domain}} connected to one element of {\emph{codomain}}% Row Count 4 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{unsatisfied requirement = relation \newline y can be used repeatedly but x values only have one arrow coming out}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.2132 cm} x{0.94023 cm} x{0.87957 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{3.833cm}}{\bf\textcolor{white}{Predicates and Quantified Statements}}  \tn
% Row 0
\SetRowColor{LightBackground}
Statement type\{\{bb\}\}\{\{ac\}\} & original\{\{bb\}\}\{\{ac\}\} & negated\{\{bb\}\}\{\{ac\}\} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Universal & ∀x∈D, P(x) & ∃x∈D, \textasciitilde{}P(x) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{3.833cm}}{Existential} \tn 
% Row Count 5 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{3}{x{3.833cm}}{Universal Conditional} \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{DeMorgan's Law}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{• Tells us how to handle conjunction and disjunction negations \newline % Row Count 2 (+ 2)
\textasciitilde{}(p∧q) ≡ \textasciitilde{}p∨\textasciitilde{}q         \newline % Row Count 3 (+ 1)
\textasciitilde{}(p∨q) ≡ \textasciitilde{}p∧\textasciitilde{}q \newline % Row Count 4 (+ 1)
"The connector is loose(l) or the machine is unplugged(u)" \newline % Row Count 6 (+ 2)
l ∨ u -{}- negation -{}-\textgreater{} \textasciitilde{}(l ∨ u) ≡ \textasciitilde{}l ∧\textasciitilde{}u \newline % Row Count 7 (+ 1)
"The connector is {\bf{not}} loose {\emph{and}} the machine is {\bf{not}} unplugged" \newline % Row Count 9 (+ 2)
\textasciitilde{}p∨q is the opposite of p∧\textasciitilde{}q% Row Count 10 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{When using DeMorgan's law, no need for truth table}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.06155 cm} x{1.42551 cm} p{0.54594 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{3.833cm}}{\bf\textcolor{white}{Tautologies and Contradictions}}  \tn
% Row 0
\SetRowColor{LightBackground}
Tautologies & Always true statements & t \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\seqsplit{Contradictions} & Always false statements & c \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
p∧\textasciitilde{}p≡{\bf{c}} & T∧F≡{\bf{c}} & \seqsplit{F∧T≡F} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
p∨{\bf{t}}≡{\bf{t}} & p∧{\bf{c}}≡{\bf{c}} &  \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{3.833cm}}{{\bf{Absorption law}}: variable absorbing operator \newline ⇒use truth table to prove law \newline ⇒other variables don't play a role in statement validity \newline p∨(p∧q)≡ p; p∧(p∨q)≡ p}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{p→q truth table}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/theemu001_1474495020_Screen Shot 2016-09-21 at 4.56.14 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Argument Truth Table}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/theemu001_1474495842_Screen Shot 2016-09-21 at 5.10.16 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Critical row = row where both premises are true \newline premises and conclusion = TRUE is a {\bf{valid}} argument}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.65227 cm} x{2.78073 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Arguments}}  \tn
% Row 0
\SetRowColor{LightBackground}
p→q & major premise \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
p & minor premise \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
∴q & therefore, conclusion \tn 
% Row Count 3 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{premises aka assumptions or hypotheses \newline verified using truth table}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{2.57475 cm} x{0.85825 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Argument Forms (VALID)}}  \tn
% Row 0
\SetRowColor{LightBackground}
Modus Pones & p→q \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
 & p \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
 & ∴q \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
Modus Tollens & p→q \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
 & \textasciitilde{}q \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
 & ∴\textasciitilde{}p \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
Gneralization & p \tn 
% Row Count 7 (+ 1)
% Row 7
\SetRowColor{white}
 & ∴p∨q \tn 
% Row Count 8 (+ 1)
% Row 8
\SetRowColor{LightBackground}
Specialization & p∧q \tn 
% Row Count 9 (+ 1)
% Row 9
\SetRowColor{white}
 & ∴q \tn 
% Row Count 10 (+ 1)
% Row 10
\SetRowColor{LightBackground}
Elimination & p∨q \tn 
% Row Count 11 (+ 1)
% Row 11
\SetRowColor{white}
 & \textasciitilde{}q \tn 
% Row Count 12 (+ 1)
% Row 12
\SetRowColor{LightBackground}
 & ∴p \tn 
% Row Count 13 (+ 1)
% Row 13
\SetRowColor{white}
Transitivity & p→q \tn 
% Row Count 14 (+ 1)
% Row 14
\SetRowColor{LightBackground}
 & q→r \tn 
% Row Count 15 (+ 1)
% Row 15
\SetRowColor{white}
 & ∴p→r \tn 
% Row Count 16 (+ 1)
% Row 16
\SetRowColor{LightBackground}
Proof by div. into cases & p∨q \tn 
% Row Count 17 (+ 1)
% Row 17
\SetRowColor{white}
 & p→r \tn 
% Row Count 18 (+ 1)
% Row 18
\SetRowColor{LightBackground}
 & q→r \tn 
% Row Count 19 (+ 1)
% Row 19
\SetRowColor{white}
 & ∴r \tn 
% Row Count 20 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.99114 cm} x{1.44186 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Fallacy (INVALID ARGUMENTS)}}  \tn
% Row 0
\SetRowColor{LightBackground}
Converse Error & p→q \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
 & q \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
⇒\{\{bb\}\} & \{\{bb\}\}∴p \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
Inverse Error & q→p \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
 & \textasciitilde{}p \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
 & ∴\textasciitilde{}q \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


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

\end{document}