339 lines
9.3 KiB
TeX
Executable File
339 lines
9.3 KiB
TeX
Executable File
% use [nosolutions] flag to hide solutions.
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% use [solutions] flag to show solutions.
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\documentclass[
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nosolutions,
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singlenumbering
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]{../../resources/ormc_handout}
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\usepackage{../../resources/macros}
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\uptitlel{Advanced 2}
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\uptitler{\smallurl{}}
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\title{Intro to Proofs}
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\subtitle{Prepared by Mark on \today{}}
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\begin{document}
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\maketitle
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\section{}
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\problem{}
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We say an integer $x$ is \textit{even} if $x = 2k$ for some $k \in \mathbb{Z}$.
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We say $x$ is \textit{odd} if $x = 2k + 1$ for some $k \in \mathbb{Z}$. \par
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Assume that every integer is even or odd, and never both.
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\vspace{2mm}
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\begin{itemize}[itemsep=4mm]
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\item
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Show that the product of two odd integers is odd.
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\item
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Let $a, b \in \mathbb{Z}, a \neq 0$.
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We say $a$ \textit{divides} $b$ and write $a~|~b$ if there is a $k \in \mathbb{Z}$ so that $ak = b$.
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Show that $a~|~b \implies a~|~2b$
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\item
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Show that $5n^2 + 3n + 7$ is odd for any $n \in \mathbb{Z}$.
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\item
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Let $a, b, c$ be integers so that $a^2 + b^2 = c^2$. \par
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Show that one of $a, b$ is even.
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\item
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Show that every odd integer is the difference of two squares.
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\item
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Prove the assumption in the statement of this problem.
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}
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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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\vspace{2mm}
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\begin{itemize}[itemsep=4mm]
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\item Show that $\sqrt{2}$ is irrational.
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\item Show that the product of two rational numbers must be rational, while the product of
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irrational numbers may be rational or irrational. If you claim a number is irrational, provide
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a proof.
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}
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Let $X = \{x \in \mathbb{Z} ~\bigl|~ x \geq 2 \}$. For $k \geq 2$, define $X_k = \{kx ~\bigl|~ x \in X \}$. \par
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What is $X - (X_2 \cup X_3 \cup X_4 \cup ...)$? Prove your claim.
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\vfill
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\pagebreak
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\problem{}
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Show that there are infinitely may primes. \par
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You may use the fact that every integer has a prime factorization.
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\vfill
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\pagebreak
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\problem{}
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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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\vspace{2mm}
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An \textit{undirected graph} $G$ is an ordered pair $(V, E)$, where $V$ is a set, and $E \subset V \times V$
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satisfies $(a, b) \in E \iff (b, a) \in E$ and $E \cap \text{D}(X) = \varnothing$.
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The elements of $V$ are called \textit{vertices}; the elements of $E$ are called \textit{edges}.
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\vspace{2mm}
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\begin{itemize}[itemsep=4mm]
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\item Make sense of the conditions on $E$.
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\item The \textit{degree} of a vertex $a$ is the number of edges connected to that vertex. \par
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We'll denote this as $d(a)$. Write a formal definition of this function using set-builder notation and the definitions above.
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Recall that $|X|$ denotes the size of a set $X$.
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\item There are 9 people at a party. Show that they cannot each have 3 friends. \par
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Friendship is always mutual.
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}
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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
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We say $f$ is \textit{surjective} if for all $y \in Y$ there exists an $x \in X$ so that $f(x) = y$. \par
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Let $A, B, C$ be sets, and let $f: A \to B$, $g: B \to C$ be functions. Let $h = g \circ f$.
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\vspace{2mm}
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\begin{itemize}
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\item Show that if $h$ is injective, $f$ must be injective and $g$ may not be injective.
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\item Show that if $h$ is surjective, $g$ must be surjective and $f$ may not be surjective.
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}
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Let $X = \{1, 2, ..., n\}$ for some $n \geq 2$. Let $k \in \mathbb{Z}$ so that $1 \leq k \leq n - 1$. \par
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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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\vspace{2mm}
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\begin{itemize}[itemsep=4mm]
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\item Show that $\{E_1, E_2\}$ is a partition of $E$. \par
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In other words, show that $\varnothing \neq E_1$, $\varnothing \neq E_2$, $E_1 \cup E_2 = E$, and $E_1 \cap E_2 = \varnothing$. \par
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\hint{What does this mean in English?}
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\item Compute $|E_1|$, $|E_2|$, and $|E|$. \par
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Recall that a set of size $n$ has $\binom{n}{k}$ subsets of size $k$.
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\item Conclude that for any $n$ and $k$ satisfying the conditions above,
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$$
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\binom{n-1}{k} + \binom{n-1}{k-1} = \binom{n}{k}
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$$
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\item For $t \in \mathbb{N}$, show that $\binom{2t}{t}$ is even.
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\end{itemize}
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\vfill
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\pagebreak
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\theorem{The Division Algorithm}<divalgo>
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Given two integers $a, b$, we can find two integers $q, r$, where $0 \leq r < b$ and $a = qb + r$. \par
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In other words, we can divide $a$ by $b$ to get $q$ remainder $r$.
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\problem{}
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Let $x, y \in \mathbb{N}$ be natural numbers.
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Consider the set $S = \{ax + by ~\bigl|~ a, b \in \mathbb{Z}, ax + by > 0\}$. \par
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The well-ordering principle states that every nonempty subset of the natural numbers has a least element.
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\vspace{4mm}
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\begin{itemize}[itemsep=4mm]
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\item Show that $S$ has a least element. Call it $d$.
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\item Let $z = \text{gcd}(x, y)$. Show that $z$ divides $d$.
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\item Show that $d$ divides $x$ and $d$ divides $y$.
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\item Prove or disprove $\text{gcd}(x, y) \in S$.
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}
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\begin{itemize}[itemsep=4mm]
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\item Let $f: X \to Y$ be an injective function. Show that for any two functions $g: Z \to X$ and $h: Z \to X$,
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if $f \circ g = f \circ h$ from $Z$ to $Y$ then $g = h$ from $Z$ to $X$. \par
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By definition, functions are equal if they agree on every input in their domain. \par
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\hint{This is a one-line proof.}
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\item Let $f: X \to Y$ be a surjective function.
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Show that for any two functions $g: Y \to W$ and $h: Y \to W$, if
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$g \circ f = h \circ f \implies g = h$.
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\item[$\star$] Let $f: X \to Y$ be a function where for any set $Z$ and functions $g: Z \to X$ and $h: Z \to X$,
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$f \circ g = f \circ h \implies g = h$. Show that $f$ is injective.
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\item[$\star$] Let $f: X \to Y$ be a function where for any set $W$ and functions $g: Y \to W$ and $h: Y \to W$,
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$g \circ f = h \circ f \implies g = h$. Show f is surjective.
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}
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In this problem we prove the binomial theorem:
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for $a, b \in \mathbb{R}$ and $n \in \mathbb{Z}^+$\hspace{-0.5ex}, we have
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$$
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(a + b)^n = \sum_{k=0}^n \binom{n}{k}a^kb^{N-k}
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$$
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In the proof below, we let $a$ and $b$ be arbitrary numbers.
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\vspace{4mm}
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\begin{itemize}
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\item Check that this formula works for $n = 0$. Also, check a few small $n$
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to get a sense of what's going on.
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\item Let $N \in \mathbb{N}$. Suppose we know that for a specific value of $N$,
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$$
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(a + b)^N = \sum_{k=0}^N \binom{N}{k}a^kb^{N-k}
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$$
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Now, show that this formula also works for $N + 1$.
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\item Conclude that this formula works for all $a, b \in \mathbb{R}$ and $n \in \mathbb{Z}^+$\hspace{-0.5ex}.
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}
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A \textit{relation} on a set $X$ is an $R \subset X \times X$. \par
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\begin{itemize}
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\item We say $R$ is \textit{reflexive} if $(x,x) \in R$ for all $x \in X$.
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\item We say $R$ is \textit{symmetric} if $(x, y) \in R \implies (y, x) \in R$.
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\item We say $R$ is \textit{transitive} if $(x, y) \in R$ and $(y, z) \in R$ imply $(x, z) \in R$.
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\item We say $R$ is an \textit{equivalence relation} if it is reflexive, symmetric, and transitive.
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\end{itemize}
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Say we have a set $X$ and an equivalence relation $R$. \par
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The \textit{equivalence class} of an element $x \in X$ is the set $\{y \in X ~\bigl|~ (x, y) \in R\}$.
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\vspace{4mm}
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Let $R$ be an equivalence relation on a set $X$. \par
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Show that the set of equivalence classes is a partition of $X$.
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\vfill
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\pagebreak
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\problem{}
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Show that there exist two positive irrational numbers $a$ and $b$ so that $a^b$ is rational.
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% Solution: a = b = root 2, check all cases
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% if irrational,a=rt, s = rt^rt, which is rational
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\vfill
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\problem{}
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Show that the following holds for any planar graph:
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$$
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\text{vertices} - \text{edges} + \text{faces} = 2
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$$
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\hint{If you don't know what these words mean, ask an instructor.}
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\vfill
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\pagebreak
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\problem{}
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Consider a rectangular chocolate bar of arbitrary size. \par
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What is the minimum number of breaks you need to make to
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seperate all its pieces?
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\begin{solution}
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number of squares minus one, simple proof by induction.
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\end{solution}
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\vfill
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\problem{}
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Four travellers are on a plane, each moving along a straight line at an arbitrary constant speed. \par
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No two of their paths are parallel, and no three intersect at the same point. \par
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We know that traveller A has met traveler B, C, and D, and that B has met C and D (and A). \par
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Show that C and D must also have met.
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\begin{solution}
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When a body travels at a constant speed, its graph with respect to time is a straight line. \par
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So, we add time axis in the third dimension, perpendicular to our plane. \par
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Naturally, the projection of each of these onto the plane corresponds to a road.
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Now, note that two intersecting lines define a plane and use the conditions in the problem to show that no two lines are parallel.
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\end{solution}
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\vfill
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\problem{}
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Say we have an $n$-gon with non-intersecting edges. \par
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What is the size of the smallet set of vertices that can \say{see} every point inside the polygon?
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\vfill
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\pagebreak
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\end{document} |