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%\begin{comment}
\noindent
\textbf{Name:} \smallskip  \\
\textbf{Collaborators:} \smallskip \\ %%% List anyone with whom you discussed the problems
\textbf{Outside resources:} \smallskip \\ %%% List all resources used OTHER than the textbook, lecture notes, quizzes, worksheets, and previous homework assignments.
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\begin{center}
{
Math 2106, Foundations of Mathematical Proof \\
HW 1 --- Due January 20, 2017 (Fri)
}
\end{center}


\noindent Turn in the following problems from Hammack's book.  
\begin{center}
\begin{tabular}{ll}
\begin{minipage}{0.3\textwidth}
\begin{itemize}
\setlength\itemsep{-0.2em}
\item[1.1:] 16, 28, 52
\item[1.2:] 20
\item[1.3:] 2, 10, 14
\item[1.4:] 6, 14, 16, 18, 20
\item[1.8:] 4, 8, 14
\end{itemize}
\end{minipage}
&
\begin{minipage}{0.3\textwidth}
\begin{itemize}
\setlength\itemsep{-0.2em}
\item[2.3:] 2, 6, 10, 12
\item[2.5:] 10
\item[2.6:] 12, 14
\item[2.7:] 4, 6, 8, 10
\item[2.9:] 6, 10
\item[2.10:] 4, 6, 8
\end{itemize}
\end{minipage}
\end{tabular}
\end{center}


\noindent{\bf Additional exercises} (to be turned in)

\begin{enumerate}[label=A\arabic*]

\item Describe the following sets using the set builder notation.
\begin{enumerate}
\item the set of odd integers
\item the set of rational numbers that may be written with denominator greater than $100$
\item the set of rational numbers that may be written with positive denominator less than $4$
\end{enumerate}


\item Using truth tables, prove that each of the following compound propositions is a tautology.  


\begin{enumerate}
%%%  Modus ponens
	\item $[p \And (p \Implies q)] \Implies q$ 
%%% Modus tollens
	\item $[\Not q \And (p \Implies q)] \Implies\ \Not p$ 
%%% Hypothetical syllogism
	\item $[(p \Implies q) \And (q \Implies r)] \Implies (p \Implies r)$
%%% Disjunctive syllogism
	\item $[(p \Or q)\ \And \Not p] \Implies q$
\end{enumerate}

\textit{These implications are four of the most important ``rules 
of inference'' in propositional logic.
Each rule gives a conclusion which follows logically from a set of hypotheses.
As such, these rules are the building blocks of a correct proof.}


\item Prove that each of the following propositions is \emph{not} a tautology, with or without using truth tables.

\begin{enumerate}
%%% Affirming the conclusion
	\item $[(p \Implies q) \And q] \Implies p$
%%% Denying the hypothesis
	\item $[(p \Implies q) \And \Not p] \Implies\ \Not q$
\end{enumerate}

\textit{
These implications are common logical fallacies 
(errors in reasoning) since the conclusion does \emph{not} follow logically
from the set of hypotheses.  
}
\end{enumerate}
\end{document}





 





