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\begin{center}
{\large MATH 108 Winter 2019 \ -  \ {\bf Problem Set 7}}

% Put your name and section here.
{\large due March 8}
\end{center}


\begin{enumerate}
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	\item
	 \begin{enumerate}
	  \item Prove (with the Axiom of Choice) that every infinite set has a countably infinite subset.
	  \item Prove that every infinite set has a proper subset with the same cardinality.
	 \end{enumerate}

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	\item Prove that the set of irrational numbers, $\RR \setminus \QQ$, is uncountable.
	
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	\item Let $\NN_0^\infty$ denote the set of all infinite sequences of nonnegative integers,
	 \[ \NN_0^\infty = \{(a_1,a_2,a_3,\ldots)\mid a_1,a_2,a_3,\ldots \in \NN_0\}. \]
	Use Cantor's diagonalization argument to prove that $\NN_0^\infty$ is uncountable.

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	\item Prove that the following sets have cardinality $\mathfrak{c}$.
	 \begin{enumerate}
	  \item The set of all functions from $\NN_1$ to $\{0,1\}$.
	  \item The closed interval $[0,1]$.
	  \item $\mathcal{P}(\NN_1) \times \mathcal{P}(\NN_1)$.
	  %\item $\RR^2$.
	 \end{enumerate}

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	\item Order the following cardinal numbers: $|(0,1)|$, $|[0,1]|$, $|\{0,1\}|$, $|\{0\}|$, $|\mathcal{P}(\RR)|$, $|\QQ|$, $|\emptyset|$, $|\RR^2|$, $|\mathcal{P}(\mathcal{P}(\RR))|$, $|\RR|$, $|\mathcal{P}(\QQ)|$.

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	\item Determine whether each algebraic structure is a group.  If no, which properties does it fail?  If yes, is it abelian?  Find an identity element if one exists.
	 \begin{enumerate}
	  \item $(\NN_1, +)$.
	  \item $(\QQ, \cdot)$.
	  \item $(\QQ\setminus\{0\}, \cdot)$.
      \item $(\ZZ/4\ZZ, +)$.
	  \item $(\ZZ/4\ZZ\setminus \{\overline{0}\}, \cdot)$.
	  \item (The set of functions $\{1,2,3\} \to \{1,2,3\}$, composition).
	  \item (The set of bijective functions $\{1,2,3\} \to \{1,2,3\}$, composition).
	  \item (The set of $2\times 2$ real matrices with determininant 1, matrix multiplication).
	 \end{enumerate}
    
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\end{enumerate}
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