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Intro to Mathematical Reasoning (Math 300) -- 
Homework 9\footnote{Version 3/29/04}
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\begin{enumerate}
\item For each of the following functions, determine if the function
is one-to-one, onto (the given target set), or both. Prove your
answers.
\begin{enumerate}
\item Let $f:\mathbb{Z} \longrightarrow \mathbb{Z}$ be given
by the rule $f(n)=n+12$ for all $n \in \mathbb{Z}$.
\item Let $g:\mathbb{N} \longrightarrow \mathbb{N}$ be given
by the rule $f(n)=n+12$ for all $n \in \mathbb{N}$.
\item Let $h:\mathbb{Z} \longrightarrow \mathbb{Z}$ be given
by the  rule $h(x)=x/2$ for $x$ even and $h(x)=(x-1)/2$ for $x$
odd.
\end{enumerate}
\item Let $r: \mathbb{N} \times \mathbb{N} \longrightarrow \mathbb{N}$
be given by the rule $r(a,b)=2^{a-1}(2b-1)$.  Prove that $r$ is
one-to-one and onto.  (Hint: To prove that $r$ is one-to-one
let $(a_1,b_1)$ and $(a_2,b_2)$ be elements of $\mathbb{N} \times \mathbb{N}$
and assume $f(a_1,b_1)=f(a_2,b_2)$.  Use the fact that a number can't be
both odd and even to prove that $a_1=a_2$ and then prove that $b_1=b_2$.
To prove that $r$ is onto, use complete induction to prove
that for all natural numbers $n$, there are natural numbers $a$ and $b$
such that $n=2^{a-1}(2b-1)$.  The basis case for the induction should
be the case that $n$ is odd.)
\item End of chapter 5, problem 5 (a) and (b).
\item End of chapter 5, problem 9.
\item End of chapter 5, problem 24.
\item Prove Theorem 8.2.4.
\item Problem 8.3.8.
\end{enumerate}


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