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\begin{center}
{\huge  \sc   Homework 8} \\
\smallskip
{ Due Date: 05/24/2022}\\
\smallskip
{Name:  }\\
{\large  }
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\begin{enumerate}

\item[21.4] Let $S$ be the set consisting of all infinite sequences of 0s and 1s. Use Cantor's diagonal argument to prove that $S$ is uncountable. 
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\item[21.5]
\begin{enumerate}
    \item Let $S$ be the set consisting of all the finite subsets of $N$. Prove that $S$ is countable. 
    
     \item Let $T$ be the set consisting of all the infinite subsets of $N$. Prove that $T$ is uncountable. 
\end{enumerate}
\newpage 
\item[21.6]

Every Tuesday, critic Ivor Smallbrain drinks a little too much, staggers out of the pub, and performs a kind of random walk towards his home. At each step of this walk, he stumbles either forwards or backwards, and the walk ends either when he collapses in a heap or when he reaches his front door (one of these always happens after a finite [possibly very large] number of steps). Ivor’s Irish friend Gerry O’Laughing always accompanies him and records each random walk as a sequence of 0s and1s: backwards: at each step he writes 1 if the step is forwards and 0 if it is backwards.
Prove that the set of all possible random walks is countable.

\newpage 
\item[(1)] Suppose that $A$ and $B$ are sets, and that there exists an injection $f:A\rightarrow B$ and surjection $g:B\rightarrow A$. Either prove that there exists a bijection $h:A\rightarrow B$, or give an example to show that such a bijection need to exist. 
\newpage 

\item[(2)] Suppose that $A$ and $B$ are sets, and that there exists surjections $f:A\rightarrow B$ and $g:B\rightarrow A$. Use the Schorder-Bernstein Theorem to show that there exists a bijection between $A$ and $B$.

\end{enumerate}




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