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Advanced/Intro to Proofs/main.tex
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@ -20,7 +20,6 @@
\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
@ -30,11 +29,11 @@
\begin{itemize}[itemsep=4mm] \begin{itemize}[itemsep=4mm]
\item \item
Show that the product of two odd integers is odd. Show that the product of two odd integers is odd.
\item \item
Let $a, b \in \mathbb{Z}, a \neq 0$. 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 $ak = b$. We say $a$ \textit{divides} $b$ and write $a~|~b$ if there is a $k \in \mathbb{Z}$ so that $ak = b$.
Show that $a~|~b \implies a~|~2b$ Show that $a~|~b \implies a~|~2b$
\item \item
@ -63,7 +62,7 @@
\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 $r = \frac{a}{b}$
\vspace{2mm} \vspace{2mm}
\begin{itemize}[itemsep=4mm] \begin{itemize}[itemsep=4mm]
\item Show that $\sqrt{2}$ is irrational. \item Show that $\sqrt{2}$ is irrational.
@ -120,7 +119,7 @@
\begin{itemize}[itemsep=4mm] \begin{itemize}[itemsep=4mm]
\item Make sense of the conditions on $E$. \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 \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. 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 $X$. Recall that $|X|$ denotes the size of a set $X$.
@ -168,14 +167,14 @@
\item Compute $|E_1|$, $|E_2|$, and $|E|$. \par \item Compute $|E_1|$, $|E_2|$, and $|E|$. \par
Recall that a set of size $n$ has $\binom{n}{k}$ subsets of size $k$. 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, \item Conclude that for any $n$ and $k$ satisfying the conditions above,
$$ $$
\binom{n-1}{k} + \binom{n-1}{k-1} = \binom{n}{k} \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. \item For $t \in \mathbb{N}$, show that $\binom{2t}{t}$ is even.
\end{itemize} \end{itemize}
@ -209,7 +208,7 @@
\problem{} \problem{}
\begin{itemize}[itemsep=4mm] \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 $h: Z \to X$, \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 if $f \circ g = f \circ h$ from $Z$ to $Y$ then $g = h$ from $Z$ to $X$. \par
@ -261,7 +260,7 @@
\vfill \vfill
\pagebreak \pagebreak