\input{config}
\input{preambles/preamble}
\input{preambles/definitions}
\input{preambles/environments}

\title{Problem Set 1}
\author[Daniel Gonzalez Cedre]{Discrete Mathematics}
\publisher{University of Notre Dame}
\date{Due on the \red{28\textsuperscript{th} of January, 2024}}

\begin{document}

\maketitle

\section*{Justify each of your answers with an argument.\footnotemark[1]}
% \marginnote{\emph{An answer provided with bad or no justification is as good as a wrong answer.}}
\makeatletter \@footnotetext[-2ex]{An answer provided with bad or no justification is as good as a wrong answer.
Think carefully, and think deeply.} \makeatother

\begin{enumerate}
  \item[(50 pts) \quad 1.]
    Determine the truth value of each sentence below.
    \begin{enumerate}
      \item
        ``Madrid is the capital of Spain.''
      \item
        ``Santa Claus lives on the north pole.''
      \item
        ``This sentence is \emph{false.}''
      \item
        ``The set of all sets that don't contain themselves contains itself.''%
        \sidenote{A \defn{set} is a collection of objects.
          When we talk about \emph{``the set of all $x$ with a property,''} we mean the collection of all those $x$ that have that property and \emph{only} those $x$.}
      \item
        ``Red is a beautiful color.''
      \item
        ``Every declarative sentence is either \emph{true} or \emph{false} but not both.''
      \item
        ``If this sentence is \emph{false,} then $7$ is a prime number.''%
        \sidenote{Note that $7$ really is a prime number.}
      \item
        ``The set of all sets contains itself.''
      \item
        ``This sentence is \emph{true.}''
      \item
        ``If this sentence is \emph{true,} then $2$ is an odd number.''%
        \sidenote{Note that $2$ is not really an odd number.}
    \end{enumerate}

  \item[(25 pts) \quad 2.]
    Suppose we have an infinite sequence of sentences
    \begin{equation*}
      S_0, S_1, S_2, \dots S_i, \dots
    \end{equation*}
    where each sentence asserts that every sentence following it is \emph{false.}
    \marginnote[3ex]{In this definition, $i$ ranges over all of the natural numbers $0, 1, 2, \dots$}
    \begin{equation*}
      S_i \defeq \text{``}S_j \text{ is \emph{false} for all } j > i\text{.''}
    \end{equation*}
    What are the truth values of the sentences in this sequence?

  \item[(25 pts) \quad 3.]
    In the sentence below, \emph{``you''} refers to \emph{you}\emph{,} the student reading these sentences and solving this problem set.
    Determine the truth value of the following sentence.
    \begin{equation*}
      \text{``You have finitely many beliefs.''}
    \end{equation*}
\end{enumerate}

\end{document}
