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+% use [nosolutions] flag to hide solutions.
+% use [solutions] flag to show solutions.
+\documentclass[
+	solutions,
+	singlenumbering
+]{../../resources/ormc_handout}
+
+
+\uptitlel{Advanced 2}
+\uptitler{Fall 2023}
+\title{Intro to Proofs}
+\subtitle{Prepared by Mark on \today{}}
+
+
+\begin{document}
+
+	\maketitle
+
+	\problem{}
+	We say an integer $x$ is \textit{even} if $x = 2k$ for some $k \in \mathbb{Z}$.
+	We say $x$ is \textit{odd} if $x = 2k + 1$ for some $k \in \mathbb{Z}$. \par
+	Assume that every integer is even or odd, and never both.
+
+	\vspace{2mm}
+	\begin{itemize}[itemsep=4mm]
+		\item
+		Show that the product of two odd integers is odd.
+		
+		\item
+		Let $a, b \in \mathbb{Z}, a \neq 0$.
+		We say $a$ \textit{divides} $b$ and write $a~|~b$ if there is a $k \in \mathbb{Z}$ so that $ak = b$.
+	
+		Show that $a~|~b \implies a~|~2b$
+
+		\item
+		Show that $5n^2 + 3n + 7$ is odd for any $n \in \mathbb{Z}$.
+
+		\item
+		Let $a, b, c$ be integers so that $a^2 + b^2 = c^2$. \par
+		Show that one of $a, b$ is even.
+
+		\item
+		Show that every odd integer is the difference of two squares.
+
+		\item
+		Prove the assumption in the statement of this problem.
+	\end{itemize}
+
+
+	\vfill
+	\pagebreak
+
+
+
+	\problem{}
+	Let $r \in \mathbb{R}$. We say $r$ is \textit{rational} if there exist $p, q \in \mathbb{Z}, q \neq 0$ so that $r = \frac{a}{b}$
+	
+	\vspace{2mm}
+	\begin{itemize}[itemsep=4mm]
+		\item Show that $\sqrt{2}$ is irrational.
+		\item Show that the product of two rational numbers must be rational, while the product of
+		irrational numbers may be rational or irrational. If you claim a number is irrational, provide
+		a proof.
+	\end{itemize}
+
+
+	\vfill
+	\pagebreak
+
+
+	\problem{}
+	Let $X = \{x \in \mathbb{Z} ~\bigl|~ n \geq 2 \}$. For $k \geq 2$, degine $X_k = \{kx ~\bigl|~ x \in X \}$. \par
+	What is $X - (X_2 \cup X_3 \cup X_4 \cup ...)$? Prove your claim.
+
+	\vfill
+	\pagebreak
+
+
+
+	\problem{}
+	For a set $X$, define its \textit{diagonal} as $\text{D}(X) = \{ (x, x) \in X \times X ~\bigl|~ x \in X \}$.
+
+	\vspace{2mm}
+
+	An \textit{undirected graph} $G$ is an ordered pair $(V, E)$, where $V$ is a set, and $E \subset V \times V$
+	satisfies $(a, b) \in E \iff (b, a) \in E$ and $E \cap \text{D}(X) = \varnothing$.
+
+	The elements of $V$ are called \textit{vertices}; the elements of $E$ are called \textit{edges}.
+
+	\vspace{2mm}
+	\begin{itemize}[itemsep=4mm]
+
+		\item Make sense of the conditions on $E$.
+		
+		\item The \textit{degree} of a vertex $a$ is the number of edges connected to that vertex. \par
+		We'll denote this as $d(a)$. Write a formal definition of this function using set-builder notation and the definitions above.
+		Recall that $|X|$ denotes the size of a set $X$.
+
+		\item There are 9 people at a party. Show that they cannot each have 3 friends. \par
+		Friendship is always mutual.
+	\end{itemize}
+
+
+\end{document}
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