Skip to main content\(\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}}
\newcommand{\lb}{\mathrm{lb}} \newcommand{\ub}{\mathrm{ub}}
\newcommand{\vx}{\mathbf{x}} \newcommand{\va}{\mathbf{a}}
\newcommand{\vb}{\mathbf{b}} \newcommand{\vc}{\mathbf{c}}
\newcommand{\vy}{\mathbf{y}} \newcommand{\vz}{\mathbf{z}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section A.2 Exercises
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Show that
\(\N^2\cup\{(0,0)\}\) is countable.
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Show that any subset of a countable set is countable.
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Show that
\((0,1)\) and
\(\R\) have the same cardinality by constructing an explicit bijection.
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Show that the intersection of two intervals is empty, a singleton, or an interval.
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Show that the union of two overlapping intervals is an interval.
