...

Commits (1)
 ... ... @@ -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 ... ...
This diff is collapsed.