\documentclass[12pt,HW250,ifthen]{article}
\usepackage{comment}
\newcommand{\into}{{\rightarrow}}
\newcommand{\lf}{\left\lfloor}
\newcommand{\rf}{\right\rfloor}
\newcommand{\lc}{\left\lceil}
\newcommand{\rc}{\right\rceil}
\newcommand{\Ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
\newcommand{\Z}{{\sf Z}}
\newcommand{\N}{{\sf N}}
\newcommand{\Q}{{\sf Q}}
\newcommand{\R}{{\sf R}}
\newcommand{\Rpos}{{\sf R}^+}
\usepackage{amsmath}

\begin{document}

\centerline{Homework 12, Morally due Tue May 14, 3:30PM}

\centerline{\bf THIS HW IS ONE PAGES!!!!!!!!!!}

WHEN IS THE FINAL?  Saturday May 18, 4-6 

WHERE IS THE FINAL?  PHYSICS 1201

\newif{\ifshowsoln}


\begin{enumerate}

\item
(30 points --- 10 each)
Show that the following sets are uncountable
\begin{enumerate}
\item
The set of functions from $\N$ to $\N$ that are strictly increasing.
(That means that, for all $x,y\in \N$, if $x<y$ then $f(x)<f(y)$.)
\item
The set of functions from $\N$ to PRIMES.
\item
The set of functions from $\N$ to PRIMES that are strictly increasing.
\end{enumerate}


\item
(40 points --- 20 points each)
Let $(A,\le_1)$ and $(B,\le_2)$ be ordered sets.
An {\it order preserving bijection $f$ from $A$ to $B$}
is a bijection from $A$ to $B$ such that, for all $x,y\in A$.

$$x\le_1 y \rightarrow f(x) \le_2 f(y).$$

\begin{enumerate}
\item
Show that there is NO order preserving bijections from $\N$ to $\Z$.
\item
Show that there is NO order preserving bijections from $\N$ to $\Q^{\ge 0}$.
(Thats the rationals $\ge 0$.)
\end{enumerate}

(30 points)
Prove or disprove: If $A_1,A_2,\ldots$ are countable and disjoint then
$A_1\times A_2 \times A_3 \times\cdots$ is countable.


\end{enumerate}

\end{document}