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\begin{document}
\title{Math 101 Homework}
\author{Mary Radcliffe}
\date{due 25 Jan 2015}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item State the Axiom of Choice (in your own words). Explain at least one circumstance where we can select elements from a family of sets that does NOT require Choice.

\item Let $X=\{(a, b)\ | \ a, b\geq 0\}$, the first quadrant. Define $\leq$ by $(a, b)\leq (c, d)$ whenever $ac> bd$. Is this a partial order? (Hint: look at transitivity)

\item Let $X=\{(a, b)\ | \ a, b\geq 0\}$, the first quadrant. Define $\leq$ as follows:
\begin{enumerate}
\item If $a<c$, then $(a, b)<(c, d)$.
\item If $a=c$ and $b<d$, then $(a, b)<(c, d)$.
\end{enumerate}

Show that this is a partial order on $X$. (This is a common ordering, called lexicographical ordering, or dictionary ordering.)

\item State the De Bruijn-Erd\H{o}s Theorem, and explain its relevance to the Hadwiger-Nelson problem.

\end{enumerate}









\end{document}