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\title{\sc Math 2100 / 2105 / 2350 -- Homework 12 (Last one!)}
\author{due Thursday, \textbf{December 6}, at the beginning of class}
\date{}
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\begin{center} 
	\emph{This homework assignment was written in \LaTeX{}. You can find the source code on the course website.}
\end{center}

\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. \textbf{Staple your work} together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive credit. Explain all reasoning.

\ \\[20pt]
\textbf{For each relation below, answer the following questions. You must, as always, prove your answer.}
\begin{enumerate}[(a)]
	\item Is it reflexive?
	\item Is it irreflexive?
	\item Is it symmetric?
	\item Is it antisymmetric?
	\item Is it transitive?
	\item Is it a partial order? If so, draw the Hasse diagram.
	\item Is it an equivalence relation?
\end{enumerate}

\begin{enumerate}
	\item $S = \PP(\{1,2,3\})$ and $R = \{(A,B) \in S \times S : A \cap B \neq \emptyset\}$
	\item $S = \PP(\{1,2,3\})$ and $R = \{(A,B) \in S \times S : A \cup B \neq \emptyset\}$
	\item For a finite set $S$ of numbers, define $\Sigma(S)$ to be the sum of the numbers in $S$.\\
			$S = \PP(\{1,2,3,4\})$ and $R = \{(A,B) \in S \times S : \Sigma(A) \leq \Sigma(B)\}$
	\item $S = \{0,1,2,3,4,5,6,7,8,9,10\}$ and $R = \{(a,b) \in S \times S : |a-b| = 4\}$
	\item $S = \{\text{Marquette Students}\}$ and $R = \{(a,b) \in S \times S : \text{the first letters of the last name of students $a$ and $b$ are the same}\}$
	\item $S = [0, 2\pi)$ and $R = \{(a,b) \in S \times S : \cos(a) = \cos(b)\}$
	
	\end{enumerate}
\end{document}



