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\title{\sc Math 2100 / 2350 -- Homework 5}
\author{
	Fall 2019\\[10pt]
	due Wednesday, \textbf{November 13}, at the beginning of class\\[10pt]
	Sections 2.5, 3.3, 4.1
}
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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.

\begin{enumerate}
	\item Prove that if any five points other than $(0,0)$ are placed on the coordinate plane, then there are two points, call them $A$ and $B$, such that the angle formed by the rays from $(0,0)$ to $A$ and from $(0,0)$ to $B$ is acute.
		
	\item Use a proof by contradiction to prove that an even perfect square cannot have the form $4k+2$.

	\item Prove that if $a + b + c \geq 35$, then either $a \geq 10$, $b \geq 12$, or $c \geq 13$.
	
	\item Prove or disprove: For any two sets $A$ and $B$: $\PP(A) \cup \PP(B) \subseteq \PP(A \cup B)$.

	\item Prove or disprove: For any three sets $A$, $B$, and $C$: $(A \cup B) \times C = (A \times C) \cup (B \times C)$.

	\item Prove the following set inequality:
	\[
		(\{n^2 - 1 : n \in \Z\} \cap \{2k : k \in \N\}) \subseteq \{4m:m\in\Z\}.
	\]
	
	\item Prove the following set inequality:
	\[
		(\{6k + 1 : k \in \Z\} \cup \{6m - 1: m \in \Z\}) \subseteq \{2n+1 : n \in \Z\}.
	\]
	
	\item Draw the one-sided and two-sided arrow diagrams for the function $f : \PP(\{4,5,6\}) \to \PP(\{1,2,3\})$ defined by 
	\[
		f(S) = \{x-3:x\in S\} \smallsetminus \{2\}.
	\]


		
\end{enumerate}
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