35 lines
1.0 KiB
TeX
35 lines
1.0 KiB
TeX
\section{Dense Orderings}
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\note[Note]{This section is fairly difficult. If you get stuck here, try the bonus problems on the next page.}
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\definition{}
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An \textit{ordered set} is a set with an \say{order} attached to it. \par
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A few examples are below:
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\begin{itemize}
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\item $\mathbb{Z}$ is an ordered set under $<$.
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\item $\{\texttt{A}, \texttt{B}, ..., \texttt{Z}\}$ is an ordered set under $\diamond$,\par
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Where $\alpha \diamond \beta$ holds iff the letter $\alpha$ comes before letter $\beta$ in the alphabet.
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\end{itemize}
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\definition{}
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We say an ordered set $A$ is \textit{dense} if for any $a, b \in A$ there is a $c \in A$ so that $a < c < b$.\par
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Intuitively, this means that there is an element of $A$ between any two elements of $A$.
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\problem{}
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Show that the ordered set $(\mathbb{Q}, <)$ is dense.
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\vfill
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\problem{}
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Show that the ordered set $(\mathbb{R}, <)$ is dense.
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\vfill
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\problem{}
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Show that there is a real number between every two rationals.
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\vfill
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\problem{}
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Show that there is a rational number between every two reals.
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\vfill
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\pagebreak |