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{\bf Supplementary problems (assigned 8/29/03)}
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\begin{enumerate}
\item Use the ordered pair definition of functions to give a definition
of 'injective' and to show that a composition of injective functions is
injective.
\item Show that '+' is a well-defined operation on $\Q$.
\item A very important feature of the natural numbers $\mathbf{N}$ is that
every element $n = \mathbf{N}$ can be uniquely factored into primes.  That
is, there are unique prime numbers $p_1<p_2<\dots<p_k$
and natural numbers $j_1,j_2,\dots j_k$ such that
$$
n = p_1^{j_1} p_2^{j_2} \dots p_k^{j_k}
$$
Use this fact to show that if $r \in \mathbf{Q}$ is a  
rational number satisfying $r^j = n$, then in fact $r\in\mathbf{Z}$.
\end{enumerate}

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