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%% ------ Custom solution environment ---------
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  {\par\noindent\textbf{Solution.}\ } 
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{\end{exmp}}

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%% Put these after \begin{solution}, as necessary
\newcommand\AIpolish{\textbf{(p)}\; }
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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due January 16 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-3mm}

%%------------------------------------------------------------------------

\begin{exercise} % HW 1
    Suppose $a$, $b$, and $c$ are integers. Prove that if $a\mid b$ and $a\mid c$, then $a\mid(b+c)$.
\end{exercise}

%\begin{solution} 
    %% Reminder to cite AI usage, if applicable. 
    % Put it immediately after \begin{solution} (on the same line)
    %\AIpolish
    %\AImajor
    %\AIminor
    %\AIpolishmajor
    %\AIpolishminor
    %
    % To do.
%\end{solution}

\medskip

%% You can add \smallskip, \medskip, \bigskip, or e.g., \vspace{5mm} if you want vertical spacing.

%%------------------------------------------------------------------------

\begin{exercise} % HW 1
    Prove that the sum of two odd integers is even. 
\end{exercise}

%\begin{solution} %% Reminder to cite AI usage here, if applicable. 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise}
    Consider the statement:
    \begin{center}
        \emph{If $x,y\in\Z$ are odd, then $xy$ is odd.}
    \end{center}
    Prove this three different ways: via a direct proof, a proof by contradiction, and by proving the contrapositive.
\end{exercise}

%\begin{solution} %% Reminder to cite AI usage here, if applicable.
%\noindent\emph{Direct proof}. 
%
%\noindent\emph{Proof by contradiction}. 
%
%\noindent\emph{Proof of the contrapositive}. 
%
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise}
    Consider the following ``theorem,'' and subsequent ``proof.'' \\

    \noindent\emph{Theorem}. The money you have is equal to the money that you need. \\

    \noindent\emph{Proof}. Let $M$ be the money that you have and $N$ the money that you need. We will show that $M=N$. Consider the average of these two quantities, $A =\frac{N+M}{2}$. We have
    \[
    2A = M + N,
    \]
    and multiplying both sides of the equation above by $M-N$ yields 
    \[
    2AM-2AN = M^2-N^2. 
    \]
    Equivalently, we can write $N^2 -2AN = M^2 -2AM$. Adding $A^2$
    to both sides gives
    \[
    N^2-2AN +A^2 = M^2-2AM +A^2,
    \]
    or equivalently,
    \[
    (N-A)^2=(M-A)^2.
    \]
    Taking the square root and adding $A$ to both sides gives $N=M$. $\hfill\Box$ \\

    \noindent Pinpoint the flaw in this ``proof'' as precisely as you can. 
\end{exercise}

%\begin{solution} %% Reminder to cite AI usage here, if applicable. 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise}
    Consider four cards, each of which has a letter on one side and a number on the other side. They are placed on the table in front of you, and you can see the following:
    \[
    \begin{tikzpicture}
        \draw (0,0) rectangle (1,2); \node at (.5,1) {$C$};
        \draw (2,0) rectangle (3,2); \node at (2.5,1) {$U$};
        \draw (4,0) rectangle (5,2); \node at (4.5,1) {$6$};
        \draw (6,0) rectangle (7,2); \node at (6.5,1) {$7$};
    \end{tikzpicture}
    \]
    What is the fewest number of cards that you need to flip over to verify the statement:
    \begin{center}
    ``\emph{if a card has a vowel on one side, then it has an even number on the other side}''?
    \end{center}
    Fully justify your answer. 
\end{exercise}

%\begin{solution} %% Reminder to cite AI usage here, if applicable. 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\end{document}
