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\begin{center}
{\large MATH 108 Fall 2019 \ -  \ {\bf Problem Set 5}}

% Put your name and section here.
{\large due November 4}
\end{center}


\begin{enumerate}
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	\item Let $\sim$ be the relation on $\RR$ defined by $x \sim y$ if and only if $x - y \in \ZZ$.
	 \begin{enumerate}
	  \item Prove that $\sim$ is an equivalence relation.
	  \item Prove for all real numbers $x,y,z,w$ that if $\overline{x} = \overline{z}$ and $\overline{y} = \overline{w}$ then $\overline{x+y} = \overline{z+w}$.
	 \end{enumerate}

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	\item Using modular arithmetic, prove that for all postive integers $n$,
	 \begin{enumerate}
	  \item $10^n - 1$ is divisible by 3.
	  \item $n^4 + 2n^3 - n^2 - 2n$ is divisible by 4.
	  \item $1^n + 2^n + 3^n + 4^n$ is a multiple of 5 or one less than a multiple of 5.
	 \end{enumerate}

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	\item The ``Cancellation Law'' for $\ZZ/m\ZZ$ is the statement: For all $x,y,z \in \ZZ$, if $xy \equiv xz \pmod{m}$ and $x \not\equiv 0 \pmod{m}$ then $y \equiv z \pmod{m}$.
	 \begin{enumerate}
	  \item Prove that if $m$ is prime then the Cancellation Law for $\ZZ/m\ZZ$ is true.
	  \item Prove that if $m$ is composite then the Cancellation Law for $\ZZ/m\ZZ$ is false.
	 \end{enumerate}

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	\item Let $\preceq$ be the relation on $\ZZ^2$ defined by $(a,b) \preceq (c,d)$ if and only if $a \leq c$ and $b \leq d$.
   \begin{enumerate}
    \item Prove that $\preceq$ is a partial order.
    \item Find the greatest lower bound of $\{(1,5),(3,3)\}$.
    \item Is $\preceq$ a total order?  Justify your answer.
   \end{enumerate}

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	\item Let $A$ be the set of divisors of 36,
	 $A = \{1,2,3,4,6,9,12,18,36\}$. 
	 Draw the Hasse diagram for the poset $(A, |)$.

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\end{enumerate}
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