Set edits

Advanced/Size of Sets/parts/6 uncountable.tex

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+\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