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{\large MATH 108 Fall 2019 \ -  \ {\bf Problem Set 9}}

% Put your name and section here.
{\large due December 2}
\end{center}


\begin{enumerate}
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	\item Prove that $\mathfrak{c} + \mathfrak{c} = \mathfrak{c}$ (where $\mathfrak{c} = |\RR|$, the cardinality of the continuum).

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

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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 $(\ZZ_{>0}, +)$.
      \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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	\item Write the Cayley table for the following finite algebraic structures.
	 \begin{enumerate}
      \item $(\ZZ/4\ZZ, +)$.
	  \item $(\ZZ/4\ZZ, \cdot)$.
	  \item (The set of bijective functions $\{1,2,3\} \to \{1,2,3\}$, composition).
	 \end{enumerate}

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\end{enumerate}
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