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......@@ -13,8 +13,8 @@ We write $A\sim B$.
\begin{problem} \label{ex:EqCardOfSomePairs}
For each, prove the two have the same cardinality.
\begin{items}
\item $\N$ and $\mathcal{E}=\set{2k\suchthat k\in\N}$,
\item $\open{-\pi/2}{\pi/2}$ and $\R$,
\item $\N$ and $\mathcal{E}=\set{2k\suchthat k\in\N}$
\item $\open{-\pi/2}{\pi/2}$ and $\R$
\item $\open{0}{1}$ and $\R$
\end{items}
\begin{answer}
......@@ -62,7 +62,7 @@ A set is \definend{countable} if it is either finite or denumerable.
\end{df}
\begin{problem} \label{ex:IntegersCountable}
For each, prove the two sets have the same cardinality.
Prove each.
\begin{exes}
\begin{exercise}
The set of integers is countable.
......@@ -98,7 +98,7 @@ For each, prove the two sets have the same cardinality.
\end{center}
This is clearly a correspondence.
\remark sketch helps.
\remark a sketch helps.
The values associated with successive arguments trace out the
diagonals of the array~$\N\times\N$.
\begin{center}
......@@ -291,7 +291,7 @@ possible if $j_0=j_1$).
\begin{exercise}
$\powerset(\N)$
\ \hint let $\map{f}{\powerset(\N)}{\N}$
and consider $\setbuilder{n\in\N}{n\notin f(n)}$
and consider $\setbuilder{n\in\N}{n\notin f(n)}$.
\end{exercise}
\begin{answer}
We will prove that no function $\map{f}{\powerset(\N)}{\N}$ is onto.
......@@ -311,7 +311,7 @@ possible if $j_0=j_1$).
\end{answer}
\begin{exercise}
$\R$
\ \hint find a one-to-one map from $\powerset(\N)$ to $\R$
\ \hint find a one-to-one map from $\powerset(\N)$ to $\R$.
\end{exercise}
\begin{answer}
We will show that there is a one-to-one
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