\documentclass[12pt]{article}

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\usepackage{amsmath}                           % Symbols
\usepackage{amssymb}                           % Symbols
\usepackage[normalem]{ulem}                    % Underlining
\usepackage{enumitem}                          % Customizing Enumeration
\usepackage{multicol}                          % Multiple Columns
\usepackage{qtree}                             % Trees
\usepackage{hyperref}                          % Hyperlink
\hypersetup{colorlinks	=	true}

\let\oldamp\&
\renewcommand{\&}{\ \oldamp \ }                % Added more space around '&'

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\title{\TeX\ Sample, Phil 12A}
\author{My Cool Name}
\date{}                                        % Comment out to set date automatically


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\begin{document}

\maketitle

\begin{enumerate}
  \item Typical font styles:
  \begin{center}
  \begin{tabular}{|c|c|l|}
  \hline
  \textbf{Style}    &    \textbf{Example}    &    \textbf{Command} \\
  \hline
  Bold              &    \textbf{validity}   &    \verb+\textbf{validity}+ \\
  Italics           &    \emph{validity}     &    \verb+\emph{validity}+ \\
  Underline         &    \uline{validity}    &    \verb+\uline{validity}+ \\
  Sans Serif        &    \textsf{validity}   &    \verb+\textsf{validity}+ \\
  Quotations        &    ``validity''        &    \verb+``validity''+ \\
  \hline
  \end{tabular}
  \end{center}
  
  \item To type math symbols, surround the commands with \verb+$...$+, e.g. to type ``$A \vee B$'', use \verb+$A \vee B$+.
  
  \item A bunch of useful symbols for Phil 12A.
  
  \begin{center}
  \begin{tabular}{|c|l|} % l = left-align, c = center, r = right-align, | = line separating columns
  \hline
  \textbf{Symbol}    &    \textbf{Meaning} \\    % \\ needed after every row, & separates columns
  \hline
  $\alpha,\beta,\dots$ &  Greek Letters \\
  $\sim A$           &    Negation (``not'') \\ % quotation marks are ``...''.  "..." yields backwards-looking quotation marks
  $A \& B$           &    Conjunction (``and'') \\
  $A \vee B$         &    Disjunction (``or'') \\
  $A \supset B$      &    Conditional (``if\dots then'') \\
  $A \equiv B$       &    Biconditional (``if and only if'') \\
  $\therefore$       &    Therefore \\
  $\exists x Fx$     &    Existential Quantifier (``there exists an $x$\dots'') \\
  $\forall x Fx$     &    Universal Quantifier (``for all $x$\dots'') \\
  $F_a, G_{ab}$      &    Subscripts \\
  $F^1, G^{12}$      &    Superscripts \\
  $\sqrt{1}$         &    Square root \\
  $\times$           &    Closed branch \\
  \hline
  \end{tabular}
  \end{center}
  
  Be careful with subscripts and superscripts!! For instance, \verb+$F^12$+ yields ``$F^12$''. To get `$12$' as a superscript, type \verb+$F^{12}$+.
  
  
  % Problem 2
  \item Typing trees takes a bit more practice. There are a number of ways to do it. One method involves the command \verb+\Tree+ (from \verb+qtree+). See the \href{http://www.ling.upenn.edu/advice/latex/qtree/qtreenotes.pdf}{qtree documentation} for details. Some examples are provided below (\textbf{note}: the spaces and curly brackets are important!):
  
  \Tree [ B C D ].A
  
  \begin{center}
  \verb+\Tree [ B C D ].A+
  \end{center}
  
  \Tree [ [.B b c ] [.C d ] [.D ] ].A
  
  \begin{center}
  \verb+\Tree [ [.B b c ] [.C d ] [.D ] ].A+
  \end{center}
  
  \Tree [ [.{$\sim A$} {$A$} ] [.{$B$} ] ].{$\sim A \vee B$}
  
  \begin{center}
  \verb+\Tree [ [.{$\sim A$} {$A$} ] [.{$B$} ] ].{$\sim A \vee B$}+
  \end{center}
      
  \Tree [ [ [.{$\sim A$} {$A$} ] [.{$\sim B$} {$B$} ] ].{$\sim A \& \sim B$} [.{$\sim C$} {$C$} ] ].{$(\sim A \& \sim B) \vee (\sim C)$}

  \begin{center}
  \verb+\Tree [ [ [.{$\sim A$} {$A$} ] [.{$\sim B$} {$B$} ] ].+
  \verb+{$\sim A \& \sim B$}[.{$\sim C$} {$C$} ] ].+
  \verb+{$(\sim A \& \sim B) \vee (\sim C)$}+
  \end{center}

\end{enumerate}

\end{document}
