Edits

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}

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 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 $a + (-a) = 0$
+		\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{}<stpart>
 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
+You may assume that $\text{st}(x)$ exists and is unique 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
 
 %\problem{}
 %Let $H$ be positive unlimited. Determine which of the following are limited. \par

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
+In the problems below, let $\varepsilon$ a positive infinitesimal so that $\varepsilon^2 = 0$. \par
-Note that $\varepsilon \neq 0$.
+\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
+Find an explicit formula for the inverse of a dual number $(a + b\varepsilon)$, assuming one exists. \par
-and use it to find the derivative of $f(x) = \frac{1}{x}$.
+Then, use this find the derivative of $f(x) = \frac{1}{x}$.
 
 \vfill
 

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 all $m \in M$, $m \leq x$, and
-	\item For any $x_0 < x$, there exists an $m \in M$ so that $m > x_0$
+	\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
+Let $[a_1, b_1] \supseteq [a_2, b_3] \supseteq [a_3, b_3] \supseteq ...$ be an infinite sequence of closed line intervals.
-closed line intervals. Show that there exists a $c \in \mathbb{R}$ that lies in all of them. \par
+\par Show that there exists a $c \in \mathbb{R}$ that lies in all of them. Is this true of open intervals?
-Is this true for open intervals?
 
 
 \vfill