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\begin{document}

\begin{center}
	\Large{Math 31 - Homework 1}
	
	\vspace{2mm}
	\large{Due Friday, June 28}
\end{center}

\subsection*{Easy}

\paragraph{1.} Find $\gcd(a, b)$ and express $\gcd(a, b)$ as $ma+nb$ for:
\begin{thmenum}
	\item $(116, -84)$
		
	\item $(85, 65)$
		
	\item $(72, 26)$
		
	\item $(72, 25)$
\end{thmenum}

\paragraph{2.} Verify that the following elements of $\la \Z_n, \cdot \ra$ are invertible, and find their multiplicative inverses.
\begin{thmenum}
	\item 4 in $\Z_{15}$
	\item 14 in $\Z_{19}$
\end{thmenum}

\paragraph{3.} In each case, determine whether $*$ defines a binary operation on the given set. If not, give reason(s) why $*$ fails to be a binary operation.
\begin{thmenum}
	\item $*$ defined on $\Z^+$ by $a*b=a-b$.
	\item $*$ defined on $\Z^+$ by $a*b=a^b$.
	\item $*$ defined on $\Z$ by $a*b=a/b$.
	\item $*$ defined on $\R$ by $a*b=c$, where $c$ is at least 5 more than $a+b$.
\end{thmenum}

\paragraph{4.} Determine whether the binary operation $*$ is associative, and state whether it is commutative or not.
\begin{thmenum}
	\item $*$ defined on $\Z$ by $a*b=a-b$.
	\item $*$ defined on $\Q$ by $a*b=ab+1$.
	\item $*$ defined on $\Z^+$ by $a*b=a^b$.
\end{thmenum}

\paragraph{5.} [Saracino, Section 1, \#1.9] If $S$ is a finite set, then we can define a binary operation on $S$ by writing down all the values of $s_1 * s_2$ in a 
table. For instance, if $S = \{a, b, c, d\}$, then the following gives a binary operation on $S$.
\begin{center}
\begin{tabular}{c||c|c|c|c}
	$*$ & a & b & c & d \\
	\hline
	\hline
	a & a & c & b & d \\
	\hline
	b & c & a & d & b \\
	\hline
	c & b & d & a & c \\
	\hline
	d & d & b & c & a \\
\end{tabular}
\end{center}
Here, for $s_1, s_2 \in S$, $s_1 * s_2$ is the element in row $s_1$ and column $s_2$. For example, $c*b=d$. Is the above binary operation commutative? Is it
associative? (\textbf{Note:} The sort of table described in this problem is sometimes called a \textbf{Cayley table}.)

\paragraph{6.} Compute the Cayley table for $\la \Z_6, +_6 \ra$.

\subsection*{Medium}
\paragraph{7.} Suppose that $*$ is an associative and commutative binary operation on a set $S$. Show that the subset
\[
	H = \left\{a \in S : a * a = a \right\}
\]
of $S$ is closed under $*$. (The elements of $H$ are called \textbf{idempotents} for $*$.)














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