From 5cfba49bf0825f46d1eecd16cfbc2fd7716504f7 Mon Sep 17 00:00:00 2001
From: Mark <mark@betalupi.com>
Date: Wed, 10 Jan 2024 21:00:01 -0800
Subject: [PATCH] Whitespace

---
 Advanced/Intro to Proofs/main.tex | 17 ++++++++---------
 1 file changed, 8 insertions(+), 9 deletions(-)

diff --git a/Advanced/Intro to Proofs/main.tex b/Advanced/Intro to Proofs/main.tex
index aa75495..4ce45b1 100755
--- a/Advanced/Intro to Proofs/main.tex	
+++ b/Advanced/Intro to Proofs/main.tex	
@@ -20,7 +20,6 @@
 
 
 
-
 	\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
@@ -30,11 +29,11 @@
 	\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
@@ -63,7 +62,7 @@
 
 	\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.
@@ -120,7 +119,7 @@
 	\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 $X$.
@@ -168,14 +167,14 @@
 
 		\item Compute $|E_1|$, $|E_2|$, and $|E|$. \par
 		Recall that a set of size $n$ has $\binom{n}{k}$ subsets of size $k$.
-	
+
 		\item Conclude that for any $n$ and $k$ satisfying the conditions above,
 		$$
 			\binom{n-1}{k} + \binom{n-1}{k-1} = \binom{n}{k}
 		$$
 
 		\item For $t \in \mathbb{N}$, show that $\binom{2t}{t}$ is even.
-	
+
 	\end{itemize}
 
 
@@ -209,7 +208,7 @@
 
 
 	\problem{}
-	
+
 	\begin{itemize}[itemsep=4mm]
 		\item Let $f: X \to Y$ be an injective function. Show that for any two functions $g: Z \to X$ and $h: Z \to X$,
 		if $f \circ g = f \circ h$ from $Z$ to $Y$ then $g = h$ from $Z$ to $X$. \par
@@ -261,7 +260,7 @@
 	\vfill
 	\pagebreak
 
-	
+