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\centerline{\bf Math 365 -- Monday 2/11/19}
\centerline{\bf Section 2.5: Cardinality of Sets}
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\begin{try}
Show that each of the following sets are countably infinite by giving a bijective function between that set and the positive integers.
\begin{enumerate}[(a)]
\item the integers greater than $10$.
\item the odd negative integers
\item the set $A\times \ZZ^+$, where $A=\{2,3\}$
\item the integers that are multiples of $10$
\end{enumerate}
\end{try}
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\begin{try}~
\begin{enumerate}[(a)]
\item Determine whether each of these sets is finite, countable, or uncountable. For those that are countably infinite, exhibit a bijective correspondence between the set of positive integers and that set.
\begin{enumerate}[(i)]
\item The integers that are multiples of $10$.
\item Integers not divisible by $3$.
\item The real numbers with decimal representations consisting of all $1$s.
\item The real numbers with decimal representations of all $1$s or $9$s.
\item The integers with absolute value less than $1,000$.
\item The real numbers between $0$ and $2$
\end{enumerate}
\item Give an example of two uncountable sets $A$ and $B$ such that $A-B$ is
\begin{enumerate}[(i)]
\item finite;
\item countably infinite;
\item uncountable.
\end{enumerate}
\item Explain why the power set of $\ZZ_{\ge 1}$ is not countable as follows:
\begin{enumerate}[(i)]
\item First, for each subset $A \subset \ZZ_{\ge 1}$, represent $A$ as an infinite bit string (a sequence of 1's and 0's with no end to the right) with $i$th bit 1 if $i$ belongs to the subset and $0$ otherwise. For example, we represent
\begin{align*}
\{3\}& \quad \text{as } 001000000000\dots,\\
\{1,3,4\}& \quad \text{as } 101100000000\dots, \text{ and}\\
\{2x ~|~ x \in \ZZ_{\ge 1}\} & \quad \text{as } 010101010101\dots.
\end{align*}
Give the bit-string expansions for the sets $\{2,4,6,7\}$ and $\{3x ~|~ x \in \ZZ_{\ge 1}\}$ (i.e.\ the positive multiples of 3); and give the set corresponding to the bitstring expansions 0000000000000\dots and 111111111\dots.
Finally, explain why this coding of sets as bit strings is actually a bijection between $\{ \text{infinite bit strings} \}$ and $\cP(\ZZ_{\ge 1})$.
\item Suppose that you can list these infinite strings in a list labeled by the positive integers (as we saw, this is the same as saying that there is some bijective map $f: \{ \text{infinite bit strings}\} \to \ZZ_{\ge 1}$). Construct a new bit string one bit at a time, so that it doesn't match the $i$th string in the $i$th bit. Conclude that your new string can't be in the list, so that the list wasn't actually complete.
\item Finally, explain how to use (i) and (ii) together to show that the sets in $\cP(\ZZ_{\ge 1})$ aren't listable (and therefore aren't countable).
\end{enumerate}
\item Show that if $A$ and $B$ are sets and $A \subset B$, then $|A| \leq |B|$. \\
{[Hint: Start with thinking about the definition of what it means for $|A| \leq |B|$.]}
\item Show that a subset of a countable set is also countable.\\
{[Hint: Start with ``Suppose $A$ is a countable set, and that $B \subseteq A$. Since $A$ is countable, there is a bijective function\dots''.]}
\item Use the Schr\"{o}der-Bernstein theorem to show that $(0,1)$ and $[0,1]$ have the same cardinality.
\end{enumerate}
\end{try}
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