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\noindent
\parbox{2in}{\bf Math 2510} 
\hfill {\Large \bf Test \#2} \hfill
\parbox{2in}{\bf \hfill April $2^{\mathrm{nd}}$, 2010}

\vspace{0.3in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}\\
Don't merely state answers, prove your statements. {\bf Be sure to show your work!}

\begin{description}
\item[\large 1. (\underline{\hskip 0.35 truein}/28 points)] Converging Questions
\begin{enumerate}[(a)]

   \item Prove that $\displaystyle{ \left\langle \frac{3n^2}{n^2+2n+3} \right\rangle }$ converges.

\vspace{4in}

   \item Show that $\displaystyle{ \left\langle \frac{\sin(n)}{n^4+1} \right\rangle }$ converges.\\ {\it Hint:} Ignore $\sin(n)$, then prove $\sin(n)$ is
   bounded and use a theorem.


\newpage

   \item Prove that $\displaystyle{ \left\langle \frac{n^2+1}{n} \right\rangle }$ diverges.

\vspace{4in}

   \item Let $a_n \rightarrow A$ and $b_n \rightarrow B$. Show that $a_n+b_n \rightarrow A+B$.
\end{enumerate}

\newpage
\item[\large 2. (\underline{\hskip 0.35 truein}/20 points)] Some set stuff.
\begin{enumerate}[(a)]

   \item Let $A,B,C,D$ be sets. Suppose that $A \cup B \subseteq C \cup D$, $A \cap B = \phi$, and $C \subseteq A$.
             Prove that $B \subseteq D$. 

\vspace{4.5in}

   \item Let $f:X \rightarrow Y$ and let $T \subseteq Y$.  Prove that if $f$ is onto, then $f(f^{-1}(T))=T$. Point out which ``half'' of your
             proof does {\bf not} need the ``onto'' hypothesis.

\end{enumerate}

\newpage
\item[\large 3. (\underline{\hskip 0.35 truein}/28 points)] For each of the following functions, decide if $f$ is 1-1, onto, both, or neither. {\bf Prove your answers!}
\begin{enumerate}[(a)]

   \item Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=5x+2$ 
\vspace{4in}

   \item Let $f:\mathbb{Z} \rightarrow \mathbb{Z}$ be defined by $f(x)=5x+2$ 

\vspace{4in}

\newpage

   \item Let $f:\mathbb{Z} \rightarrow \mathbb{Z}$ be defined by $\displaystyle{ f(x) = \left\{ \begin{array}{cl} 2x & x \mathrm{\;is\;even} \\
             x+1 & x \mathrm{\;is\;odd} \end{array} \right. }$ 

\vspace{4in}

   \item Let $g:X \rightarrow Y$ and $f: Y \rightarrow Z$. Assume that $f \circ g: X \rightarrow Z$ is a bijection. If $f$ {\bf must be} 1-1, prove it. 
   If $f$ might not be 1-1, give a counter-example. If $f$ {\bf must be} onto, prove it. If $f$ might not be onto, give a counter-example. [A counter-example should specify {\bf both} $f$ and $g$ and be accompanied by a proof that it is in fact a counter-example.]

\end{enumerate}

\newpage
\item[\large 4. (\underline{\hskip 0.35 truein}/28 points)] Equivalent Nonsense.\\
         Recall that for integers $a$ and $b$, $a \cong b \;(\mathrm{mod\;} 4)$ if and only if there exists some $k \in \mathbb{Z}$ 
         such that $a=b+4k$.
         
\begin{enumerate}[(a)]

   \item Prove that $a \cong b \;(\mathrm{mod\;}4)$ is an equivalence relation on $\mathbb{Z}$.

\vspace{4in}

   \item List the equivalence classes of this equivalence relation.
   
\vspace{4in}

   \item Prove that the function $f:\mathbb{Z}_4 \rightarrow \mathbb{Z}_{10}$ defined by $f([n])=[5n]$ is well-defined.
             [$\mathbb{Z}_m$ are the equivalence classes of integers mod $m$.]

\vspace{4in}

   \item Let $n \in \mathbb{Z}$ be a sum of squares. [That is there exist integers $a$ and $b$ such that $n=a^2+b^2$.] 
             Prove that $n \not\cong 3 \;(\mathrm{mod\;}4)$.

\end{enumerate}

\end{description}
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