<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\section*{Uncountable Sets}

\problem{}&lt;binarystrings&gt;
Let $B$ be the set of infinite binary strings. Show that $B$ is not countable. \par
Here's how you should start:

\vspace{2mm}

Assume we have some enumeration $n(b)$ that assigns a natural number to every $b \in B$.\par
Now, arrange the elements of $B$ in a table, in order of increasing index: \par

\begin{center}
\begin{tikzpicture}[scale=0.5]
	\node at (0, 0) {$n(b)$};
	\node at (4.5, 0) {digits of $b$};

	% Vertical lines
	\draw (1, 0.5) -- (1, -8);
	\draw (-1, 0.5) -- (-1, -8);

	% Horizontal title
	\draw (-1, -0.5) -- (8, -0.5);

	\foreach \i/\j in {
		0/1010100110011110,
		1/0101101011010010,
		2/1101011001010101,
		3/0001100101010110,
		4/1101011101000110,
		5/1101100010100111,
		6/1011001101001010%
	} {
		\node at (0, -\i-1) {$\i$};
		\draw (-1, -1.5 - \i) -- (8, -1.5 - \i);
		\node[anchor=west] at (1, -\i-1) {\texttt{\j}...};
	}

	\node at (0, -7-1) {...};
	\node at (4.5, -7-1) {.....};
\end{tikzpicture}
\end{center}


First, convince yourself that if $B$ is countable, this table will contain every element of $B$, \par
then construct a new element of $B$ that is guaranteed to \textit{not} be in this table.\par
\hint{What should the first digit of this new string be? What should its second digit be? \\
Or, even better, what \textit{shouldn't} they be?}
\vfill


\problem{}
Using \ref{binarystrings}, show that $\mathcal{P}(\mathbb{N})$ is uncountable.
\vfill
\pagebreak

\problem{}
Show that $\mathbb{R}$ is not countable. \par
\hint{Earlier in this handout, we defined a real number as \say{a decimal, finite or infinite.}}
\vfill

\problem{}
Find a bijection from $(0, 1)$ to $\mathbb{R}$.\par
\hint{$(0, 1)$ is the set of all real numbers between 0 and 1, not including either endpoint.}
\vspace{2mm}
This problem brings us to the surprising conclusion that there are \say{just as many} numbers between 0 and 1 as there are in the entire real line.

\vfill

\problem{}
Find a bijection between $(0, 1)$ and $[0, 1]$. \par
\hint{$[0, 1]$ is the set of all real numbers between 0 and 1, including both endpoints.}
\vfill

\pagebreak</pre></body></html>