Merge branch 'master' of ssh://git.betalupi.com:33/Mark/ormc-handouts

Advanced/Intro to Proofs/main.tex

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 	\maketitle
 
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 	\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
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 	\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 $r = \frac{a}{b}$
 	
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 		a proof.
 	\end{itemize}
 
-
 	\vfill
 	\pagebreak
 
 
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 	\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.
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+	\problem{}
+	Show that there are infinitely may primes. \par
+	You may use the fact that every integer has a prime factorization.
+
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+	\vfill
+	\pagebreak
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 	\problem{}
 	For a set $X$, define its \textit{diagonal} as $\text{D}(X) = \{ (x, x) \in X \times X ~\bigl|~ x \in X \}$.
 
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 		Friendship is always mutual.
 	\end{itemize}
 
+	\vfill
+	\pagebreak
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+	\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
 
 
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 	\problem{}
 	Let $X = \{1, 2, ..., n\}$ for some $n \geq 2$. Let $k \in \mathbb{Z}$ so that $1 \leq k \leq n - 1$. \par
 	Let $E = \{Y \subset X ~\bigl|~ |Y| = k\}$, $E_1 = \{Y \in E ~\bigl|~ 1 \in Y\}$, and $E_2 = \{Y \in E ~\bigl|~ 1 \notin Y\}$
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 	\vfill
 	\pagebreak
 
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 	\problem{}
 	Let $x, y \in \mathbb{N}$ be natural numbers.
 	Consider the set $S = \{ax + by ~\bigl|~ a, b \in \mathbb{Z}, ax + by = 0\}$. \par
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 	\vfill
 	\pagebreak
 
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 	\problem{}
 	
 	\begin{itemize}[itemsep=4mm]
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 	\vfill
 	\pagebreak
 
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 	\problem{}
 	In this problem we prove the binomial theorem:
 	for $a, b \in \mathbb{R}$ and $n \in \mathbb{Z}^+$\hspace{-0.5ex}, we have
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 	\problem{}
 	A \textit{relation} on a set $X$ is an $R \subset X \times X$. \par
 	\begin{itemize}
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 	Let $R$ be an equivalence relation on a set $X$. \par
 	Show that the set of equivalence classes is a partition of $X$.
 
+	\vfill
+	\pagebreak
+
 \end{document}