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Mark 2023-10-06 14:17:34 -07:00
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Advanced/Intro to Proofs/main.tex
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@ -16,6 +16,10 @@
\maketitle \maketitle
\problem{} \problem{}
We say an integer $x$ is \textit{even} if $x = 2k$ for some $k \in \mathbb{Z}$. 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 We say $x$ is \textit{odd} if $x = 2k + 1$ for some $k \in \mathbb{Z}$. \par
@ -53,6 +57,9 @@
\problem{} \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 $r = \frac{a}{b}$ 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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\problem{}
Show that there are infinitely may primes. \par
You may use the fact that every integer has a prime factorization.
\vfill
\pagebreak
\problem{} \problem{}
For a set $X$, define its \textit{diagonal} as $\text{D}(X) = \{ (x, x) \in X \times X ~\bigl|~ x \in X \}$. For a set $X$, define its \textit{diagonal} as $\text{D}(X) = \{ (x, x) \in X \times X ~\bigl|~ x \in X \}$.