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\begin{center}\textbf{Problem Set 1 \hspace{.7in} Foundations \hspace{.7in} Due 1/19/22} \end{center}

\noindent Four of these problems will be graded (my choice, not yours!), with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. Submit your work either on paper or as a pdf on the course Moodle page.

\begin{enumerate}

\item If $p, q \in \mathbb{Z}$ are both even, then the product of $p$ and $q$ is even.

\item If $p, q \in \mathbb{Z}$ with the product of $p$ and $q$ even, then $p$ and $q$ are even.

\item The cube of an odd integer is odd.

\item The sum of a throddodd and a throddodd integer is throdd.

\item If $n \in \mathbb{Z}$ is throdd, then $-n$ is throdd.

\item The product of a threven and a throdd integer is threven.

\item The sum of any three consecutive integers is threven

\item Critique the following proof of the proposition ``The sum of any two throdd integers must be even'':

%  \vspace{6pt}
 \begin{quote}Well, let $m$ and $n$ be throdd integers, so $m=3a+1$ and $n=3a+1$, where $a \in \mathbb{Z}$.  Then $m+n=(3a+1)+(3a+1)=6a+2=2(3a+1)$, and $3a+1$ is an integer by closure, so $m+n$ is 2 times an integer and thus even by definition. $\qed$
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\end{enumerate}

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