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\noindent
\parbox{2in}{\bf Math 2110} 
\hfill {\Large \bf Test \#2} \hfill
\parbox{2in}{\bf \hfill November $2^{\mathrm{nd}}$, 2020}

\vspace{0.3in}

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

\vspace{0.2in}

\noindent {\large 1. (20 points)} Divides and Induction

\begin{enumerate}[(a)]

\item Give a careful definition of the set of odd integers using {\it set builder} notation.

\vspace{0.2in}

$\mathbb{O}=$ 

\vspace{0.2in}

\item Suppose the $a,b,c \in \mathbb{Z}$ and that $a$ divides $b$ and $a$ divides $c$. Prove that $a$ divides $b+c$.

\vfill

\item Using induction, prove that for any positive integer $n$, we have $\displaystyle \sum\limits_{i=1}^n (2i-1)=n^2$. 

\vfill

\vfill

\end{enumerate}

\newpage

\noindent {\large 2. (20 points)} Contradiction and Strategy

\begin{enumerate}[(a)]

\item Suppose I want to prove the statement: ``Given any $m \in \mathbb{Z}$, there exists a unique $n \in \mathbb{Z}$ such that $\displaystyle m<\dfrac{n}{2}<m+1$.''

In addition, suppose we have already shown existence. Sketch the beginning and end of the next part of the proof.

\vfill

\item Prove there is no  smallest integer.

\vfill

\item Prove $\sqrt{2}$ is irrational.

\vfill

\end{enumerate}

\newpage

\noindent {\large 3. (20 points)} Sequences

\begin{enumerate}[(a)]

\item Give the definition of $\langle a_n \rangle_{n=0}^\infty \to L$. 

\vfill

\item Show $\langle n^2+1 \rangle_{n=0}^\infty$ is bounded below but not above.

\vfill

\item Prove that $\displaystyle \left\langle \dfrac{n+5}{3n^2+9n-1} \right\rangle_{n=1}^\infty \to 0$.

The following calculation might be helpful: \qquad $\displaystyle  \dfrac{n+5}{3n^2+9n-1} \leq \dfrac{n+5n}{3n^2+9n-1} < \dfrac{6n}{3n^2-1} \leq \dfrac{6n}{3n^2-n^2}=\dfrac{6n}{2n^2}=\dfrac{3}{n} $

\vfill

\end{enumerate}

\newpage

\noindent {\large 4. (20 points)} Sets!

\begin{enumerate}[(a)]

\item Let $\mathcal{A}$ be a set of subsets of $\mathbb{R}$ (i.e., $\mathcal{A} \subseteq \mathcal{P}(\mathbb{R})$). Define what ``$x \in \bigcup \mathcal{A}$'' means.

\vfill

\item Let $A = \{ 12n \;|\; n \in \mathbb{Z} \}$ and $B = \{ x \;|\; x\in \mathbb{Z} \text{ and $x$ is divisible by } 20\}$. 

Also, let $C =\{ n \;|\; \text{ there exists some } k\in\mathbb{Z} \text{ such that } n=2k \}$. Prove that $A \cup B \subseteq C$.

\vfill

\item Let $f:A \to B$ and $T_1,T_2 \subseteq B$. Prove that $f^{-1}(T_1 \cap T_2)=f^{-1}(T_1) \cap f^{-1}(T_2)$.

\vfill

\end{enumerate}

\newpage

\noindent {\large 5. (20 points)} Functions!

\begin{enumerate}[(a)]

\item Let $f:A \to B$. Define what it means for $f$ to be invertible.

\vfill

\item Suppose $f:A \to B$ and $g:B \to C$ are onto. Show that $g\circ f:A \to C$ is onto.

\vfill

\item Consider $f:\mathbb{Z} \to \mathbb{R}$ defined by $f(n)=5n-2$. Prove that $f$ is one-to-one but not onto.

\vfill

\end{enumerate}


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