Added nonstandard analysis handout
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Advanced/Nonstandard Analysis/main.tex
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Advanced/Nonstandard Analysis/main.tex
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% Copyright (C) 2023 <Mark (mark@betalupi.com)>
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% You may have received a copy of the GNU General Public License
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% along with this program. If not, see <https://www.gnu.org/licenses/>.
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%
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%
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%
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% If you edit this, please give credit!
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% Quality handouts take time to make.
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% 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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solutions,
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singlenumbering,
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]{../../resources/ormc_handout}
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\usepackage{../../resources/macros}
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\usepackage{units}
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\uptitlel{Advanced 2}
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\uptitler{Winter 2024}
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\title{Nonstandard Analysis}
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\subtitle{
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Prepared by \githref{Mark} on \today{} \\
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Based on handouts by Nikita and Stepan
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}
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\begin{document}
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\maketitle
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\input{parts/1 extensions}
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\input{parts/2 dual}
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\input{parts/x supremum}
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\end{document}
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166
Advanced/Nonstandard Analysis/parts/1 extensions.tex
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166
Advanced/Nonstandard Analysis/parts/1 extensions.tex
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% Copyright (C) 2023 <Mark (mark@betalupi.com)>
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% You may have received a copy of the GNU General Public License
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% along with this program. If not, see <https://www.gnu.org/licenses/>.
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%
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%
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%
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% If you edit this, please give credit!
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% Quality handouts take time to make.
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\section{Nonarchimedian Extensions}
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\definition{}
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An \textit{ordered field} consists of a set $S$, the operations $+$ and $\times$, and the relation $<$. \par
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An ordered field must satisfy the following properties:
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\begin{itemize}
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\item \textbf{Properties of $+$:}
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\begin{itemize}
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\item Commutativity: $a + b = b + a$
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\item Associativity: $a + (b + c) = (a + b) + c$
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\item Identity: there exists an element $0$ so that $a + 0 = a \forall a \in S$
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\item Inverse: for every $-a$, there exists a $-a$ so that $a + (-a) = 0$
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\end{itemize}
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\item \textbf{Properties of $\times$:}
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\begin{itemize}
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\item Commutativity
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\item Associativity
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\item Identity (which we label $1$)
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\item Inverse (which we label $a^{-1}$, and which doesn't exist for $0$)
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\item Distributivity: $a(b + c) = ab + ac$
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\end{itemize}
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\item \textbf{Properties of $<$:}
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\begin{itemize}
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\item Non-reflexive: $x < x$ is always false
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\item Transitive: $x < y$ and $y < z$ imply $x < z$
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\item Connected: for all $x, y \in S$, either $x < y$, $y > x$, or $x = y$.
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\item If $x < y$ then $x + z < y + z$
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\item If $x < y$ and $z > 0$, then $xz < yz$
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\item $0 < 1$
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\end{itemize}
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\end{itemize}
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\definition{}
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An ordered field that contains $\mathbb{R}$ is called a \textit{nonarchimedian extension} of $\mathbb{R}$.
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\vfill
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\pagebreak
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\problem{}
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Show that each of the following is true in any ordered field.
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\begin{enumerate}
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\item if $x \neq 0$ then $(x^{-1})^{-1} = x$
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\item $0 \times x = x$
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\item $(-x)(-y) = xy$
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\item if $0 < x < y$, then $x^{-1} > y^{-1}$
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\end{enumerate}
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\begin{solution}
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\textbf{Part A:}
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We know that $x^{-1} \times (x^{-1})^{-1} = 1$ \par
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Thus $x \times (x^{-1} \times (x^{-1})^{-1}) = x \times 1 = x$ \par
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We can rewrite this as $(x \times x^{-1}) \times (x^{-1})^{-1} = x$ \par
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When then becomes $1 \times (x^{-1})^{-1} = x$ \par
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And thus $(x^{-1})^{-1} = x$
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\end{solution}
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\vfill
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\pagebreak
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\definition{}
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In an ordered field, the \textit{magnitude} of a number x is defined as follows: \par
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\begin{equation*}
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|x| =
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\begin{cases}
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x & \text{\tab} x \geq 0 \\
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-x & \text{\tab otherwise}
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\end{cases}
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\end{equation*}
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\definition{}
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We say an element $\delta$ of an ordered field is \textit{infinitesimal} if $|nd| < 1$ for all $n \in \mathbb{Z^+}$. \par
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\note{Note that $\mathbb{Z}^+$ is a subset of any nonarchimedian extension.} \par
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\vspace{2mm}
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Likewise, we say $x$ is \textit{limited} if $|x| < n$ for some $n \in \mathbb{Z}^+$. \par
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Elements that are not limited are \textit{unlimited}.
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\definition{}
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We say an element $x$ of a field is \textit{positive} if $x > 0$. \par
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We say $x$ is \textit{negative} if $x < 0$. \par
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\problem{}
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Show that a positive $\delta$ is infinitesimal if and only if $\delta < x$ for all $x \in \mathbb{R}^+$. \par
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Then, show that a negative $\delta$ is infinitesimal if and only if it is bigger than every $x \in \mathbb{R}^-$.
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\vfill
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\problem{}
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Prove the following statements: \par
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\begin{itemize}
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\item If $\delta$ and $\varepsilon$ are infinitesimal, then $\delta + \varepsilon$ is infinitesimal.
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\item If $\delta$ is infinitesimal and $x$ is limited, then $a\delta$ is infinitesimal.
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\item If $x$ and $y$ are limited, $xy$ and $x+y$ are too.
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\item A nonzero $\delta$ is infinitesimal iff $\delta^{-1}$ is unlimited.
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\end{itemize}
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\vfill
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\problem{}
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Let $\delta$ be a positive infinitesimal. Which is greater?
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\begin{itemize}
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\item $\delta$ or $\delta^2$?
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\item $(1 - \delta)$ or $(1 + \delta^2)^{-1!}$?
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\item $\frac{1 + \delta}{1 + \delta^2}$ or $\frac{2 + \delta^2}{2 + \delta^3}$? \par
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\note[Note]{we define $\frac{1}{x}$ as $x^{-1}$, and thus $\frac{a}{b} = a \times b^{-1}$}
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\end{itemize}
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\vfill
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\pagebreak
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\definition{}
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We say two elements of an ordered field are \textit{infinitely close} if $x - y$ is infinitesimal. \par
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We say that $x_0 \in \mathbb{R}$ is a \textit{standard part} of $x$ if it is infinitely close to $x$. \par
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\problem{}
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We will denote the standard part of $x$ as $\text{st}(x)$. \par
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Show that $\text{st}(x)$ is well-defined for limited $x$. \par
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(In other words, Show that $x_0$ exists and is unique for limited $x$). \par
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\hint{To prove existance, consider $\text{sup}(\{a \in \mathbb{R} ~|~ a < x\}$)}
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\vfill
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%\problem{}
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%Let $H$ be positive unlimited. Determine which of the following are limited. \par
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\problem{}
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Show that $\text{st}(x + y) = \text{st}(x) + \text{st}(y)$ and $\text{st}(xy) = \text{st}(x) \text{st}(y)$. \par
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\vfill
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\pagebreak
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108
Advanced/Nonstandard Analysis/parts/2 dual.tex
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Advanced/Nonstandard Analysis/parts/2 dual.tex
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% Copyright (C) 2023 <Mark (mark@betalupi.com)>
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% You may have received a copy of the GNU General Public License
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% along with this program. If not, see <https://www.gnu.org/licenses/>.
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%
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%
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%
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% If you edit this, please give credit!
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% Quality handouts take time to make.
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\section{Dual Numbers}
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\definition{}
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In the problems below, $\varepsilon$ an infinitesimal so that $\varepsilon^2 = 0$. \par
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Note that $\varepsilon \neq 0$.
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\definition{}
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The set of \textit{dual numbers} is a nonarchimedian extension of $\mathbb{R}$ \par
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that consists of elements that look like $a + b\varepsilon$, where $a, b \in \mathbb{R}$.
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\problem{}
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Compute $(a + b\varepsilon) \times (c + d\varepsilon)$.
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\vfill
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\definition{}
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Let $f(x)$ be an algebraic function $\mathbb{R} \to \mathbb{R}$. \par
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(that is, a function we can write using the operators $+-\times\div$, powers, and roots) \par
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\note[Note]{
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Why this condition? These are the only operations we have in an ordered field! \\
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Powers, roots, and division aren't directly available, but are fairly easy to define.
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}
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\vspace{2mm}
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the \textit{derivative} of such an $f$ is a function $f'$ that satisfies the following:
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\begin{equation*}
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f(x + \varepsilon) = f(x) + f'(x)\varepsilon
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\end{equation*}
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If $f(x + \varepsilon)$ is not defined, we will say that $f$ is not \textit{differentiable} at $x$.
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\problem{}
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What is the derivative of $f(x) = x^2$?
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\vfill
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\problem{}
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What is the derivative of $f(x) = x^n$?
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\vfill
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\problem{}
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Say the derivatives of $f$ and $g$ are known. \par
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Find the derivatives of $h(x) = f(x) + g(x)$ and $k(x) = f(x) \times g(x)$.
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\vfill
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\pagebreak
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\problem{}
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When can you divide dual numbers? \par
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That is, for what numbers $(a + b\varepsilon)$ is there a $(x + y\varepsilon)$ such that $(a +b\varepsilon)(x+y\varepsilon) = 1$?
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\vfill
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\problem{}
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Find an explicit formula for the inverse of a dual number, \par
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and use it to find the derivative of $f(x) = \frac{1}{x}$.
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\vfill
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\problem{}
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Which dual numbers have a square root? \par
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That is, for which dual numbers $(a + b\varepsilon)$ is there a dual number
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$(x + y\varepsilon)$ such that $(x + y\varepsilon)^2 = a + b\varepsilon$?
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\vfill
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\problem{}
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Find an explicit formula for the square root and use it to find the derivative of $f(x) = \sqrt{x}$
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\vfill
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\problem{}
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Find the derivative of the following functions:
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\begin{itemize}
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\item $f(x) = \frac{x}{1 + x^2}$
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\item $g(x) = \sqrt{1 - x^2}$
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\end{itemize}
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\vfill
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\problem{}
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Say the derivatives of $f$ and $g$ are known. \par
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What is the derivative of $f(g(x))$?
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\vfill
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\pagebreak
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146
Advanced/Nonstandard Analysis/parts/x supremum.tex
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146
Advanced/Nonstandard Analysis/parts/x supremum.tex
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% Copyright (C) 2023 <Mark (mark@betalupi.com)>
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% You may have received a copy of the GNU General Public License
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% along with this program. If not, see <https://www.gnu.org/licenses/>.
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%
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%
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%
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% If you edit this, please give credit!
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% Quality handouts take time to make.
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\section*{Bonus: The supremum \& infimum}
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\definition{}
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In this section, we'll define a \say{real number} as a decimal, infinite or finite.
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\problem{}
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Write $2.317171717...$ as a simple fraction.
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\vfill
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\problem{}
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Write $\nicefrac{2}{11}$ as an infinite decimal and prove that your answer is correct.
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\vfill
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\problem{}
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Show that $0.999... = 1$
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\note[Note]{
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There is no real number $0.0...1$ with a digit $1$ \say{at infinity.} \\
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Some numbers have two decimal representations, some have only one.
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}
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\vfill
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\problem{}
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Concatenate all the natural numbers in order to form $0.12345678910111213...$. \par
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Show that the resulting decimal is irrational.
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\vfill
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\problem{}
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Show that a rational number exists between any two real numbers.
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\vfill
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\pagebreak
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\definition{}
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Let $M$ be a subset of $\mathbb{R}$.\par
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We say $c \in \mathbb{R}$ is an \textit{upper bound} of $M$ if $c \geq m$ for all $m \in M$. \par
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The smallest such $c$ is called the \textit{supremum} of $M$, and is denoted $\text{sup}(M)$. \par
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\vspace{2mm}
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Similarly, $x \in \mathbb{R}$ is a \textit{lower bound} of $M$ if $x \leq m \forall m \in M$. \par
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The largest upper bound of $M$ is called the \textit{infimum} of $M$, denoted $\text{inf}(M)$.
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\problem{}
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Show that $x$ is the supremum of $M$ if and only if...
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\begin{itemize}
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\item For all $m \in M$, $m \leq x$
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\item For any $x_0 < x$, there exists an $m \in M$ so that $m > x_0$
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\end{itemize}
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\vfill
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\problem{}
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Show that any subset of $\mathbb{R}$ has at most one supremum and at most one infimum.
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\vfill
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\problem{}
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Find the supremum and infimum of the following sets:
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\begin{itemize}
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\item $\bigl\{ a^2 + 2a \bigl| -5 < a < 5\bigr\}$
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\item $\bigl\{\pm \frac{n}{2n + 1} \bigl| n \in \mathbb{N}\bigr\}$
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\end{itemize}
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\vfill
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\problem{}
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Let $A$ and $B$ be subsets of $\mathbb{R}$, and let $\text{sup}(A)$ and $\text{sup}(B)$ be known.
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\begin{itemize}
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\item $\text{sup}(A \cup B)$
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\item $\text{sup}(A + B)$, where $A + B = {a + b \forall (a, b) \in A \times B}$,
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\item $\text{inf}(A \cdot B)$, where $A \cdot B = {ab \forall (a, b) \in A \times B}$
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\end{itemize}
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\vfill
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\pagebreak
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\theorem{Completeness Axiom}<completeness>
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Every non-empty subset of $\mathbb{R}$ that is bounded above has a least upper bound.
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\problem{}
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Show that $a < \text{sup}(A)$ if and only if there is a $c$ in $A$ where $a < c$
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\vfill
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\problem{}
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Use the definitions in this handout to prove \ref{completeness}. \par
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\hint{Build the supremum one digit at a time.}
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\vfill
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\problem{}
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Let $[a_1, b_1] \subseteq [a_2, b_3] \subseteq [a_3, b_3] \subseteq ...$ be an infinite sequence of
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closed line intervals. Show that there exists a $c \in \mathbb{R}$ that lies in all of them. \par
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Is this true for open intervals?
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\vfill
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\pagebreak
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\problem{Bonus}
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Show that every real number in $[0, 1]$ can be written as a sum of 9 numbers \par
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Whose decimal representations only contain 0 and 8. \par
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\vfill
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\problem{Bonus}
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Two genies take an infinite amount of turns and write the digits of an infinite
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decimal. The first genie, on every turn, writes any finite amount of digits to the tail of the decimal.
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The second genie writes one digit to the end. If the resulting decimal after an infinite amount
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of turns is periodic, the first genie wins; otherwise, the second genie wins. Who has a winning
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strategy? \par
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\vfill
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\pagebreak
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