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\noindent{\Large{\textsc{Number Theory \hfill Assignment $1.1$  \\ Fall 2022 \hfill Due February 7}}}

\large

\vspace{1cm}

\begin{exercise}
Let $a, b \in \Z$.  Recall the ``useful fact'' that if $|b|>1$ and $a \ne 0$, then $|a| < |ab|$.  Prove that the converse implication is also true.  
\end{exercise}

\begin{exercise}
Give a careful proof that the only elements of $\Z$ with multiplicative inverses are the members of the set
\[
\Z^{\times} := \{ \pm 1 \}.
\]
\end{exercise}

\begin{exercise}
Given pairs of natural numbers $(a,b), (c,d) \in \N \times \N$, define a relation $\sim$ on $\N \times \N$ by
\[
(a,b) \sim (c,d)  \,\,\, \Leftrightarrow \,\,\, a+d = b+c.
\]

\begin{itemize}

\item[\bf a.]  Show that $\sim$ is an equivalence relation using only the operation $+$.

\item[\bf b.]  Let $\overline{(a,b)}$ denote the equivalence class of $(a,b)$ under $\sim$, and let $(\N \times \N) /\!\! \sim$ be the set of all equivalence classes.  Show that the rule $\overline{(a,b)} \mapsto a-b$ gives a well-defined bijection between $(\N \times \N) / \!\! \sim$ and $\Z$.\footnote{If you aren't sure what ``well-defined'' means, please ask.  It's a very important technical term.}

\item[\bf c.] What happens if we replace $+$ in the definition of $\sim$ by $\times$?

\end{itemize}

\end{exercise}

In Zermelo-Fraenkel set theory, which is one way to axiomatize modern mathematics, every object is a set, and the axioms give rules for how these sets behave.  Starting with these axioms alone, one defines $\N$ to be the set of cardinalities of nonempty finite sets.  So where does $\Z$ comes from?  The set $(\N \times \N) / \!\! \sim$ in Exercise 3 is actually the {\em definition} of $\Z$ as a set, so that part {\bf b} is completely circular.  Nonetheless, I think it demonstrates the intuition behind the decision to declare that $(\N \times \N) / \!\! \sim$ is the set of integers.  It isn't difficult to also define addition and multiplication in $(\N \times \N) / \!\! \sim$ using only the operation $+$ in $\N$, but checking that everything is well-defined is a bit tedious.  The end result is the familiar ring $(\Z,+,  \times)$, built formally from first principles.

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