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\centerline{MATH 243 -- Mathematical Structures}
\centerline{Problem Set 4}
\centerline{{\bf Due:} October 6, 2017}
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\noindent
I.  Let $m,b$ be integers and consider the mapping $f : \mathbb{Z} \to \mathbb{Z}$ defined
by $f(x) = mx + b$.
\begin{enumerate}
\item[(A)] Prove that $f$ is injective if and only if $m \ne 0$.
\item[(B)] Find conditions on $m,b$ equivalent to saying $f$ is surjective and prove your assertion.
\end{enumerate}
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\noindent
II.  Let $b,c$ be integers and define $f : \mathbb{Z} \to \mathbb{Z}$ by $f(x) = x^2 + bx + c$.  
\begin{enumerate}
\item[(A)]  Show that $f$ is \emph{not injective}.
\item[(B)]  Show that $f$ is \emph{not surjective}.  
\end{enumerate}
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\noindent
III.  For each of the following pairs of integers $N,n$, find the integer
quotient $q$ and remainder $0\le r < n-1$ satisfying $N = qn + r$ as in 
Theorem 4.8.
\begin{enumerate}
\item[(A)] $N = 796$, $n = 26$
\item[(B)] $N = 1205$, $n = 37$
\item[(C)] $N = -27$, $n = 7$.  
\end{enumerate}
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\noindent
{\it From the Text:}  Exercises 4.4, 4.5, 4.6, 4.8.
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