74 lines
2.1 KiB
TeX
74 lines
2.1 KiB
TeX
|
\section*{Uncountable Sets}
|
||
|
|
|
||
|
|
\problem{}<binarystrings>
|
||
|
|
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
|