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\noindent
\parbox{2.25in}{\bf Math 3220}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Jan. $25^\mathrm{th}$, 2019}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
%
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\begin{description}
\item[\#1] (Zorn 1.6 \#4) Let $a,b,c,d \in \mathbb{R}$ where $a<b$ and $c<d$. Prove that the intervals $I=(a,b)$ and $J=(c,d)$ have the same cardinality. Do this by finding an appropriate linear function $f:I \to J$ (linear means $f(x)=Ax+B$ for some $A,B \in \mathbb{R}$). Prove that your $f$ is both one-to-one and onto.

\item[\#2] Let $A \subseteq B \subseteq \mathbb{R}$. Suppose that $|A|=|B|$ (i.e., $A$ and $B$ have the same cardinality). It does not follow that $A=B$. Give a counterexample to show why. Then, what assumption on $B$ would force this to be true?

\item[\#3] (Zorn 1.7 \#4) Find all values $x$ that satisfy the given inequality; express answers in interval notation.

\begin{enumerate}[(a)]
\item $|2x+7|<11$
\item $|2x+7|>11$
\item $|2x^2+7|<11$
\end{enumerate}

\item[\#4] Let $N$ be a positive integer and $x_1,\dots,x_N \in \mathbb{R}$. Show that $|x_1+\cdots+x_N| \leq |x_1|+\cdots+|x_N|$. 

{\it Note:} Prove this using induction on $N$.

\item[\#5] Let $a,b \in \mathbb{R}$ where $a<b$. Suppose that $x,y \in (a,b)$. Show that $|x-y|<b-a$.

\end{description}

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