% 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}


	\vfill
	\pagebreak


	\problem{}
	Let $f$ be a function from a set $X$ to a set $Y$. We say $f$ is \textit{injective} if $f(x) = f(y) \implies x = y$. \par
	We say $f$ is \textit{surjective} if for all $y \in Y$ there exists an $x \in X$ so that $f(x) = y$. \par
	Let $A, B, C$ be sets, and let $f: A \to B$, $g: B \to C$ be functions. Let $h = g \circ f$.

	\vspace{2mm}
	\begin{itemize}
		\item Show that if $h$ is injective, $f$ must be injective and $g$ may not be injective.
		\item Show that if $h$ is surjective, $g$ must be surjective and $f$ may not be surjective.
	\end{itemize}

	\vfill
	\pagebreak

	\problem{}
	


\end{document}