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\title{Contradiction and contrapositive proofs}
\date{October 8, 2021}

\begin{document} \ 

\vspace{-.25in}

\maketitle

\vspace{-.2in}

\prop{ \label{one}
Let $a, b \in \ZZ$.
If 7 does not divide $ab$, then 7 divides neither $a$ nor $b$.
}

\prob{
Write the \textbf{contrapositive} and the \textbf{negation} of proposition \ref{one}.
}

\vspace{2in}

Two proofs of proposition \ref{one} are given below.
Both proofs basically work, but neither is perfect.
Read the proofs carefully and note places where they could be improved.
Then write a better proof of the theorem (using whatever approach you deem best).

\vspace{.1in}

\begin{quote}

\begin{proof}
%We prove the contrapositive: If $7|a$ or $7|b$, then $7|(ab)$.
Suppose $a, b \in \ZZ$ and $7|a$ or $7|b$.
By definition %, there is an $n$ such that 
$7n = a$ or $7n = b$.
Then $7na$ or $7nb$ $=$ $ab$.
By definition $7|(ab)$.
This proves the contrapositive.
\end{proof}

\end{quote}

\vspace{.1in}

\begin{quote}

\begin{proof}[Proof by contradiction]
Suppose $7 \nmid (ab)$ and $7|a$ or $7|b$.
WLOG, suppose $7|a$.
Then $ab = 7nb$ for some $n \in \ZZ$.
%By definition $7|(ab)$, which is a contradiction.
This is a contradiction.
\end{proof}

\end{quote}


\prob{
Write a better proof of proposition \ref{one}.
}


\newpage

\prob{
Prove that for every $n \in \NN$, $ 4 \nmid (n^2 +2)$.
}

\vfill

\chal{
Prove that $\log_2 3$ is irrational.
}

\vspace{3in}

\end{document}