From 81667db1c0b6b3fce22d970904a0e472c4a02c8b Mon Sep 17 00:00:00 2001 From: Mark Date: Wed, 31 Jan 2024 18:21:28 -0800 Subject: [PATCH] Edits --- Advanced/Nonstandard Analysis/main.tex | 2 +- .../parts/1 extensions.tex | 47 ++++++++++++------- .../Nonstandard Analysis/parts/2 dual.tex | 10 ++-- .../Nonstandard Analysis/parts/x supremum.tex | 20 +++++--- 4 files changed, 50 insertions(+), 29 deletions(-) diff --git a/Advanced/Nonstandard Analysis/main.tex b/Advanced/Nonstandard Analysis/main.tex index 20da0d2..e9edcc7 100755 --- a/Advanced/Nonstandard Analysis/main.tex +++ b/Advanced/Nonstandard Analysis/main.tex @@ -17,7 +17,7 @@ % use [nosolutions] flag to hide solutions. % use [solutions] flag to show solutions. \documentclass[ - solutions, + nosolutions, singlenumbering, ]{../../resources/ormc_handout} \usepackage{../../resources/macros} diff --git a/Advanced/Nonstandard Analysis/parts/1 extensions.tex b/Advanced/Nonstandard Analysis/parts/1 extensions.tex index 84dfbac..b88508f 100644 --- a/Advanced/Nonstandard Analysis/parts/1 extensions.tex +++ b/Advanced/Nonstandard Analysis/parts/1 extensions.tex @@ -24,8 +24,8 @@ An ordered field must satisfy the following properties: \begin{itemize} \item Commutativity: $a + b = b + a$ \item Associativity: $a + (b + c) = (a + b) + c$ - \item Identity: there exists an element $0$ so that $a + 0 = a \forall a \in S$ - \item Inverse: for every $-a$, there exists a $-a$ so that $a + (-a) = 0$ + \item Identity: there exists an element $0$ so that $a + 0 = a$ for all $a \in S$ + \item Inverse: for every $a$, there exists a $-a$ so that $a + (-a) = 0$ \end{itemize} \item \textbf{Properties of $\times$:} @@ -33,7 +33,7 @@ An ordered field must satisfy the following properties: \item Commutativity \item Associativity \item Identity (which we label $1$) - \item Inverse (which we label $a^{-1}$, and which doesn't exist for $0$) + \item For every $a \neq 0$, there exists an inverse $a^{-1}$ so that $aa^{-1} = 1$ \item Distributivity: $a(b + c) = ab + ac$ \end{itemize} @@ -41,7 +41,7 @@ An ordered field must satisfy the following properties: \begin{itemize} \item Non-reflexive: $x < x$ is always false \item Transitive: $x < y$ and $y < z$ imply $x < z$ - \item Connected: for all $x, y \in S$, either $x < y$, $y > x$, or $x = y$. + \item Connected: for all $x, y \in S$, either $x < y$, $x > y$, or $x = y$. \item If $x < y$ then $x + z < y + z$ \item If $x < y$ and $z > 0$, then $xz < yz$ \item $0 < 1$ @@ -51,11 +51,30 @@ An ordered field must satisfy the following properties: \definition{} An ordered field that contains $\mathbb{R}$ is called a \textit{nonarchimedian extension} of $\mathbb{R}$. +\vfill +\pagebreak + + +\problem{} +Which of the following are ordered fields? +\begin{itemize}[itemsep=2mm] + \item $\mathbb{R}$ with the usual definitions of $+$, $\times$, $<$ + \item $\mathbb{R}$ with the usual definitions of $+$, $\times$, $\leq$ \par + \note{Note that our relation here is $\leq$, not $<$} + \item $\mathbb{Z}$ with the usual definitions of $+$, $\times$, $<$ + \item $\mathbb{Q}$ with the usual definitions of $+$, $\times$, $<$ + \item $\mathbb{C}$ with the usual definitions of $+$, $\times$, \par + and with $(a + bi) < (c + di)$ iff $a < c$. +\end{itemize} + +\vfill + + \problem{} Show that each of the following is true in any ordered field. \begin{enumerate} \item if $x \neq 0$ then $(x^{-1})^{-1} = x$ - \item $0 \times x = x$ + \item $0 \times x = 0$ \item $(-x)(-y) = xy$ \item if $0 < x < y$, then $x^{-1} > y^{-1}$ \end{enumerate} @@ -93,7 +112,7 @@ In an ordered field, the \textit{magnitude} of a number x is defined as follows: \definition{} We say an element $\delta$ of an ordered field is \textit{infinitesimal} if $|nd| < 1$ for all $n \in \mathbb{Z^+}$. \par -\note{Note that $\mathbb{Z}^+$ is a subset of any nonarchimedian extension.} \par +\note{Note that $\mathbb{Z}^+$ is a subset of any nonarchimedian extension of $\mathbb{R}$.} \par \vspace{2mm} @@ -107,7 +126,7 @@ We say $x$ is \textit{negative} if $x < 0$. \par \problem{} Show that a positive $\delta$ is infinitesimal if and only if $\delta < x$ for all $x \in \mathbb{R}^+$. \par -Then, show that a negative $\delta$ is infinitesimal if and only if it is bigger than every $x \in \mathbb{R}^-$. +Then, show that a negative $\delta$ is infinitesimal if and only if $\delta > x$ for every $x \in \mathbb{R}^-$. \vfill @@ -122,31 +141,27 @@ Prove the following statements: \par \end{itemize} \vfill +\pagebreak + \problem{} Let $\delta$ be a positive infinitesimal. Which is greater? \begin{itemize} \item $\delta$ or $\delta^2$ - \item $(1 - \delta)$ or $(1 + \delta^2)^{-1!}$ + \item $(1 - \delta)$ or $(1 + \delta^2)^{-1}$ \item $\frac{1 + \delta}{1 + \delta^2}$ or $\frac{2 + \delta^2}{2 + \delta^3}$ \par \note[Note]{we define $\frac{1}{x}$ as $x^{-1}$, and thus $\frac{a}{b} = a \times b^{-1}$} \end{itemize} \vfill -\pagebreak -\definition{} +\definition{} We say two elements of an ordered field are \textit{infinitely close} if $x - y$ is infinitesimal. \par We say that $x_0 \in \mathbb{R}$ is the \textit{standard part} of $x$ if it is infinitely close to $x$. \par -\problem{} We will denote the standard part of $x$ as $\text{st}(x)$. \par -Show that $\text{st}(x)$ is well-defined for limited $x$. \par -(In other words, show that $x_0$ exists and is unique for limited $x$). \par -\hint{To prove existence, consider $\text{sup}(\{a \in \mathbb{R} ~|~ a < x\}$)} - -\vfill +You may assume that $\text{st}(x)$ exists and is unique for limited $x$. \par %\problem{} %Let $H$ be positive unlimited. Determine which of the following are limited. \par diff --git a/Advanced/Nonstandard Analysis/parts/2 dual.tex b/Advanced/Nonstandard Analysis/parts/2 dual.tex index bfce456..020b21c 100644 --- a/Advanced/Nonstandard Analysis/parts/2 dual.tex +++ b/Advanced/Nonstandard Analysis/parts/2 dual.tex @@ -17,12 +17,12 @@ \section{Dual Numbers} \definition{} -In the problems below, $\varepsilon$ an infinitesimal so that $\varepsilon^2 = 0$. \par -Note that $\varepsilon \neq 0$. +In the problems below, let $\varepsilon$ a positive infinitesimal so that $\varepsilon^2 = 0$. \par +\note{Note that $\varepsilon \neq 0$.} \definition{} The set of \textit{dual numbers} is a nonarchimedian extension of $\mathbb{R}$ \par -that consists of elements that look like $a + b\varepsilon$, where $a, b \in \mathbb{R}$. +that consists of elements of the form $a + b\varepsilon$, where $a, b \in \mathbb{R}$. \problem{} Compute $(a + b\varepsilon) \times (c + d\varepsilon)$. @@ -74,8 +74,8 @@ That is, for what numbers $(a + b\varepsilon)$ is there a $(x + y\varepsilon)$ s \vfill \problem{} -Find an explicit formula for the inverse of a dual number, \par -and use it to find the derivative of $f(x) = \frac{1}{x}$. +Find an explicit formula for the inverse of a dual number $(a + b\varepsilon)$, assuming one exists. \par +Then, use this find the derivative of $f(x) = \frac{1}{x}$. \vfill diff --git a/Advanced/Nonstandard Analysis/parts/x supremum.tex b/Advanced/Nonstandard Analysis/parts/x supremum.tex index afe27ee..8078f18 100644 --- a/Advanced/Nonstandard Analysis/parts/x supremum.tex +++ b/Advanced/Nonstandard Analysis/parts/x supremum.tex @@ -60,14 +60,14 @@ The smallest such $c$ is called the \textit{supremum} of $M$, and is denoted $\t \vspace{2mm} -Similarly, $x \in \mathbb{R}$ is a \textit{lower bound} of $M$ if $x \leq m \forall m \in M$. \par +Similarly, $x \in \mathbb{R}$ is a \textit{lower bound} of $M$ if $x \leq m$ for all $m \in M$. \par The largest upper bound of $M$ is called the \textit{infimum} of $M$, denoted $\text{inf}(M)$. \problem{} Show that $x$ is the supremum of $M$ if and only if... \begin{itemize} - \item For all $m \in M$, $m \leq x$ - \item For any $x_0 < x$, there exists an $m \in M$ so that $m > x_0$ + \item for all $m \in M$, $m \leq x$, and + \item for any $x_0 < x$, there exists an $m \in M$ so that $m > x_0$ \end{itemize} \vfill @@ -92,7 +92,8 @@ Find the supremum and infimum of the following sets: \problem{} -Let $A$ and $B$ be subsets of $\mathbb{R}$, and let $\text{sup}(A)$ and $\text{sup}(B)$ be known. +Let $A$ and $B$ be subsets of $\mathbb{R}$, and let $\text{sup}(A)$ and $\text{sup}(B)$ be known. \par +Compute the following in terms of $\text{sup}(A)$ and $\text{sup}(B)$. \begin{itemize} \item $\text{sup}(A \cup B)$ \item $\text{sup}(A + B)$, where $A + B = {a + b \forall (a, b) \in A \times B}$, @@ -100,6 +101,12 @@ Let $A$ and $B$ be subsets of $\mathbb{R}$, and let $\text{sup}(A)$ and $\text{s \end{itemize} +\vfill + +\problem{} +Prove the assumptions in \ref{stpart}: \par +Show that $\text{st}(x)$ is exists and is unique for limited $x$. + \vfill \pagebreak @@ -119,9 +126,8 @@ Use the definitions in this handout to prove \ref{completeness}. \par \vfill \problem{} -Let $[a_1, b_1] \subseteq [a_2, b_3] \subseteq [a_3, b_3] \subseteq ...$ be an infinite sequence of -closed line intervals. Show that there exists a $c \in \mathbb{R}$ that lies in all of them. \par -Is this true for open intervals? +Let $[a_1, b_1] \supseteq [a_2, b_3] \supseteq [a_3, b_3] \supseteq ...$ be an infinite sequence of closed line intervals. +\par Show that there exists a $c \in \mathbb{R}$ that lies in all of them. Is this true of open intervals? \vfill