From 7c2398bcf5c5933cf841c454f7e95d2ecc64de6d Mon Sep 17 00:00:00 2001
From: Mark <mark@betalupi.com>
Date: Fri, 6 Oct 2023 14:12:42 -0700
Subject: [PATCH 1/2] Added a problem

---
 Advanced/Intro to Proofs/main.tex | 22 ++++++++++++++++++++++
 1 file changed, 22 insertions(+)

diff --git a/Advanced/Intro to Proofs/main.tex b/Advanced/Intro to Proofs/main.tex
index ba22a3e..8e8779a 100755
--- a/Advanced/Intro to Proofs/main.tex	
+++ b/Advanced/Intro to Proofs/main.tex	
@@ -101,4 +101,26 @@
 	\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}
\ No newline at end of file

From f8d66bdc226fcb18e37eb02bad64283de1e1b9a3 Mon Sep 17 00:00:00 2001
From: Mark <mark@betalupi.com>
Date: Fri, 6 Oct 2023 14:17:34 -0700
Subject: [PATCH 2/2] Added a problem

---
 Advanced/Intro to Proofs/main.tex | 23 +++++++++++++++++++++++
 1 file changed, 23 insertions(+)

diff --git a/Advanced/Intro to Proofs/main.tex b/Advanced/Intro to Proofs/main.tex
index c6d05ea..126d59b 100755
--- a/Advanced/Intro to Proofs/main.tex	
+++ b/Advanced/Intro to Proofs/main.tex	
@@ -16,6 +16,10 @@
 
 	\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
@@ -53,6 +57,9 @@
 
 
 
+
+
+
 	\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 $r = \frac{a}{b}$
 	
@@ -82,6 +89,22 @@
 
 
 
+
+
+
+
+	\problem{}
+	Show that there are infinitely may primes. \par
+	You may use the fact that every integer has a prime factorization.
+
+
+	\vfill
+	\pagebreak
+
+
+
+
+
 	\problem{}
 	For a set $X$, define its \textit{diagonal} as $\text{D}(X) = \{ (x, x) \in X \times X ~\bigl|~ x \in X \}$.