\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{enumerate}
\usepackage{ifthen}
\usepackage{graphicx}

\setlength{\unitlength}{0.1in}

\setlength{\oddsidemargin}{-0.50in}
\setlength{\evensidemargin}{-0.50in}
\setlength{\topmargin}{-0.50in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\topskip}{0.0in}
\setlength{\textheight}{9.4in}
\setlength{\textwidth}{7.5in}

\newcommand{\comp}{ \,{\scriptstyle \stackrel{\circ}{}}\, } 
\newcommand{\nullset}{\mathrm{O}\!\!\!\!\big/\,}
\newcommand{\divides}{\,\Big|\,}
\newcommand{\myspace}{\mbox{\parbox[c][0.35in]{0.35in}{\hfill}}}
\newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}}

\newcommand{\prob}[2]{\noindent {\bf\large #1. (#2 points)}\   }

\begin{document}

\pagestyle{empty}

\noindent
\parbox{2in}{\bf Math 2510} 
\hfill {\Large \bf Test \#2} \hfill
\parbox{2in}{\bf \hfill April $15^{\mathrm{th}}$, 2015}

\vspace{0.2in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}\hfill \hfill {\bf Be sure to show your work!}

\vspace{0.2in}

\prob{1}{30} Converging Questions
\begin{enumerate}[(a)]

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

\vfill

   \item Prove that $\displaystyle{ \left\langle \frac{(-1)^n n}{n^2+1} \right\rangle }$ converges.

\vfill

\newpage

   \item Prove that $\displaystyle{ \left\langle \frac{n^4}{n-1} \right\rangle_{n=2}^\infty }$ diverges.

\vfill

   \item Let $a_n \rightarrow L$ (for some $L \in \mathbb{R}$) and $b_n \rightarrow 0$. Show that $a_nb_n \rightarrow 0$.

\vfill

\end{enumerate}

\newpage

\prob{2}{20} Some set stuff.
\begin{enumerate}[(a)]

   \item Let $A,B,C$ be sets. Show $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

\vfill

   \item Let $f: X \to Y$ be a function and $T_1,T_2 \subseteq Y$. Show that $f^{-1}(T_1 - T_2) = f^{-1}(T_1)-f^{-1}(T_2)$.

\vfill

\newpage

   \item Let $f:X \rightarrow Y$ and let $R_1,R_2 \subseteq X$.  Prove that if $f$ is one-to-one, then $f(R_1 \cap R_2)=f(R_1)\cap f(R_2)$. Point out which ``half'' of your proof does {\bf not} need the ``one-to-one'' hypothesis.

% containment always holds the reverse doesn't

\vfill


\end{enumerate}

\newpage

\prob{3}{25} 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 $f(x)=x^3+1$ 
   
\vfill

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

\vfill

\newpage

   \item Let $f:\mathbb{Z} \rightarrow \mathbb{Z}$ be defined by $\displaystyle{ f(x) = \left\{ \begin{array}{cl} \dfrac{x}{2} & x \mathrm{\;is\;even} \vspace{0.1in} \\ 
             \dfrac{x-1}{2} & x \mathrm{\;is\;odd} \end{array} \right. }$ 

\vfill

   \item Show that composing two one-to-one functions yields a one-to-one function and that composing two onto functions yields and onto function.  Specifically, let $f: X \to Y$ and $g: Y \to Z$. Show that $g \circ f$ is one-to-one if we assume both $f$ and $g$ are one-to-one. Then show $g \circ f$ is onto if we assume both $f$ and $g$ are onto.

\vfill

\end{enumerate}

\newpage

\prob{4}{25} Equivalent Nonsense.
         
\begin{enumerate}[(a)]

   \item Give the definition of an equivalence relation (with details). Then consider $x \sim y$ only if $x \geq y$. This is a relation on $\mathbb{R}$. Why isn't it an equivalence relation?

\vfill

   \item List the equivalence classes of the relation ``mod $5$'' (on $\mathbb{Z}$).
   
\vfill

\newpage

   \item Prove that the function $f:\mathbb{Z}_6 \rightarrow \mathbb{Z}_{22}$ defined by $f([n])=[11n]$ is well-defined. Then show that $g:\mathbb{Z}_6 \to \mathbb{Z}_{10}$ ``defined'' by $g([n])=[11n]$ is not well-defined.
             [$\mathbb{Z}_m$ are the equivalence classes of integers mod $m$.]

\vfill

   \item Let $x \in \mathbb{Z}_{>0}$ (a positive integer). We can write $x$ in decimal form: $x=d_\ell \cdots d_1d_0$ ($d_j$ is the $j^\mathrm{th}$ digit). Actually, $x = d_\ell \cdot 10^\ell + \cdots + d_2 \cdot 10^2 + d_1 \cdot 10 + d_0$ (where $d_j \in \{0,\dots,9\}$).
   
   Explain why $x$ is divisible by $3$ if and only if $d_\ell+\cdots+d_0$ (the sum of its digits) is divisible by $3$. [{\it Hint:} mod 3]
   
   \vfill

\end{enumerate}

\end{document}
