% -----------------------acknowledge for overleaf and designer of this template-----------------------
\documentclass[12pt]{exam}
\usepackage{amsthm}
\usepackage{libertine}
\usepackage[utf8]{inputenc}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb}
\usepackage{multicol}
\usepackage[shortlabels]{enumitem}
\usepackage{siunitx}
\usepackage{cancel}
\usepackage{graphicx}
\usepackage{pgfplots}
\usepackage{listings}


% automata
\usepackage{tikz}
\usetikzlibrary{automata, positioning, arrows}


\pgfplotsset{width=10cm,compat=1.9}
\usepgfplotslibrary{external}
\tikzexternalize

\newcommand{\class}{\Large Logic and Computation I, Autumn 2024 } % This is the name of the course 
\newcommand{\examnum}{Exercise 02-01} % This is the index of the homework
\newcommand{\examdate}{Due Date: } % This is the due date
\newcommand{\timelimit}{}





\begin{document}
\pagestyle{plain}
\thispagestyle{empty}

\noindent
\begin{tabular*}{\textwidth}{l @{\extracolsep{\fill}} r @{\extracolsep{8pt}} l}
\textbf{\class} & & \textit{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\\
&&\\
\textbf{\examnum} & &\textit{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\\ 
\textbf{\examdate} &&\\
\end{tabular*}\\
\rule[2ex]{\textwidth}{2pt}
% ---


\noindent \textbf{Exercise 2.1.1}

\noindent Show that P2 and P3 are  tautologies.
\begin{itemize}
%\item[P1.]  $\varphi \to (\psi \to \varphi)$
\item[P2.]  $\Big(\varphi \to (\psi \to \theta)\Big) \to \Big((\varphi \to \psi) \to (\varphi \to \theta)\Big)$
\item[P3.]  $(\neg \psi \to  \neg \varphi) \to (\varphi \to \psi)$
\end{itemize}
\bigskip

\noindent\underline{\textsf{Solution:}}

\vspace{8em}


\noindent \textbf{Exercise 2.1.2}

\noindent Construct proofs of the following propositions.

\begin{itemize}
\item[$(1)$] $\neg \varphi \to (\varphi \to \psi)$.
\item[$(2)$] $\neg \neg \varphi \to \varphi$.
\end{itemize}
\bigskip

\noindent\underline{\textsf{Solution:}}

\vspace{8em}
\end{document}



