From d90605c043f66c027c71b8ffa34d69f6e70d7444 Mon Sep 17 00:00:00 2001 From: deilann Date: Wed, 3 Jul 2024 01:18:48 -0600 Subject: [PATCH 1/3] Create Intro and Examples.tex add typeset copy of Intro and Examples --- Intro and Examples.tex | 138 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 138 insertions(+) create mode 100644 Intro and Examples.tex diff --git a/Intro and Examples.tex b/Intro and Examples.tex new file mode 100644 index 0000000..f9bbe27 --- /dev/null +++ b/Intro and Examples.tex @@ -0,0 +1,138 @@ +\title{Groups 4 Subgroups} +\date{} +\author{} +\documentclass{article} +\usepackage{amsmath,amsfonts,amssymb} +\usepackage[usenames,dvipsnames]{color} +\usepackage{amsthm, thmtools} +\usepackage{fullpage, changepage, tabto} +\usepackage{enumerate, enumitem} +\usepackage{mathtools} +\usepackage[colorlinks=true, linkcolor=BlueGreen, citecolor=BlueGreen, urlcolor=BlueGreen]{hyperref} +\usepackage{asymptote} +\usepackage{wrapfig} +\usepackage{parskip} + +% set tab distances +\NumTabs{16} + +%%% theorem environments for styling +% notation +\declaretheoremstyle[ + notefont={\normalfont}, + spaceabove={0.5em}, + spacebelow={0.5em}, + notebraces={}{}, + headformat={\NOTE}, + headpunct={}, + postheadhook={\hspace{0em}\tabto{4em}}, + ]{notation} +\declaretheorem[style=notation]{notation} + +% definition +\declaretheoremstyle[ + headfont={\itshape}, + notefont={\normalfont\bfseries}, + spaceabove={1em}, + notebraces={}{}, + headformat={def \NOTE \newline}, + headpunct={ }, + postheadhook={\hspace{0em}\vspace{0.25em}\tab}, + ]{definition} +\declaretheorem[style=definition,name=Definition, +refname={definition,definitions}]{definition} + +% note +\declaretheoremstyle[ + headfont={\bfseries}, + shaded={margin={1.5em}, textwidth={20em}}, + headformat={note: \newline}, + headpunct={ }, + ]{note} +\declaretheorem[style=note]{note} + + +% prettier empty set +\newcommand{\nullset}{\varnothing} + +% logic spacing +\newcommand{\spaced}[1]{\, #1 \,} +\newcommand{\scomp}{\spaced{\backslash}} +\newcommand{\sland}{\spaced{\land}} +\newcommand{\slor}{\spaced{\lor}} +\newcommand{\simplies}{\spaced{\Rightarrow}} +\newcommand{\slexists}{\, \exists} + +% set macros +\newcommand{\buildset}[2]{\{\spaced{#1} \mid \spaced{#2} \}} +\newcommand{\finiteset}[3]{\{ \spaced{#1,} \spaced{#2,} \spaced{#3,} \spaced{\dots} \}} + +\begin{document} + \maketitle + \tableofcontents + \newpage + \section{intro} + $$\text{Consider }U = \buildset{z \in \mathbb{C}}{\lvert z \rvert = 1} \subseteq \mathbb{C}.$$ + \begin{wrapfigure}[2]{l}{0.35\textwidth} + \begin{asy} + settings.outformat="pdf"; + unitsize(1cm); + draw((0,0) -- (2,0), arrow=Arrow(TeXHead)); + draw((0,0) -- (-2,0), arrow=Arrow(TeXHead)); + draw((0,0) -- (0,-2), arrow=Arrow(TeXHead)); + draw((0,0) -- (0,2), arrow=Arrow(TeXHead)); + draw(unitcircle); + \end{asy} + \end{wrapfigure} + \vspace{24pt} + \begin{note} + each elem of $U$ is defined by $\theta \in [0, 2\pi) = \mathbb{R}_{2\pi}$ + \end{note} + \vspace{48pt} + \begin{center} + Every angle $\theta \in \mathbb{R}_{2\pi}$ given by $f: U \rightarrow R_{2\pi}, \, f(z) = \theta$ for $z = e^{i\theta}$\\ + \vspace{8pt} + +\\ + \vspace{8pt} + $f^{-1}: \mathbb{R}_{2\pi} \rightarrow U, \, f^{-1}(\theta) = e^{i\theta}$\\ + \vspace{24pt} + Consider $U$ with multiplication:\\ + \medskip + Let $z_1 = e^{i\theta_1},\, z_2 = e^{i\theta_2} \in U$\\ + \vspace{8pt} + then $z_1 \cdot z_2 = e^{i\theta_1}\cdot e^{i\theta_2} = e^{i\left(\theta_1 + \theta_2\right)}$ where $\theta_1, \, \theta_2 \in \mathbb{R}_{2\pi},\, \theta_1 + \theta_2 \in \mathbb{R}_{2\pi}$\\ + \vspace{8pt} + Set $U$ is \underline{closed} under multiplication. + \end{center} + +\section{definition} + \begin{definition}[algebraic structure]\label{algebraic structure} + \hspace{0em}\vspace{-1em}\\ + \tab\tab Let $S$ be a set.\\ + \tab\tab Let $\star$ be a binary operation on $S$\\ + \tab\tab\tab $\star: S \times S \rightarrow S$\\ + \tab\tab $\left(S,\, \star \right)$ is an \textbf{algebraic structure} + \end{definition} + \vspace{12pt} + So $\left(U,\, \cdot\right)$ and $\left(\mathbb{R}_{2\pi},\, +_{2\pi}\right)$ are algebraic structures such that there is a ``1-1" relation between $U$ and $\mathbb{R}_{2\pi}$ and operations are preserved. + \vspace{8pt} + + $$\left(U,\, \cdot\right) \simeq \left(\mathbb{R}_{2\pi},\, +_{2\pi}\right)$$ + \begin{center}or\end{center} + $$U \simeq \mathbb{R}_{2\pi}$$ + +\section{isomorphisms} +To show two algebraic structures are isomorphic, find a bijective function that preserves the operations.\\ +Isomorphic structures share the same algebraic properties (associativity, commutative, distributive...).\\ +To show two algebraic structures are \underline{not} isomorphic, find a property that they don't share (the easiest to check is cardinality). + +\subsection{Examples} +\begin{center} +$$\left(\mathbb{N},\, +\right) \not\simeq \left(\mathbb{Z},\, +\right)$$ +because $\mathbb{N}$ has no additive identity +$$\left(\mathbb{Z}^{*},\, \cdot\right) \not\simeq \left(\mathbb{Q}^{*},\, \cdot\right)$$ +because not all elements of $\mathbb{Z}^*$ have inverses $\left(\frac{n}{m}\right)^{-1} = \frac{m}{n}$ +$$\left(\mathbb{R},\, +\right) \not\simeq \left(\mathbb{R},\, \cdot\right)$$ +because $x \star x + 1 = 0$ is solveable in $\left(\mathbb{R},\, +\right): 2x + 1 = 0$ but unsolveable in $\left(\mathbb{R},\, \cdot\right): x^2 + 1 = 0$ +\end{center} +\end{document} From dc6cd25ec372d55af6d20f521c1face20b670064 Mon Sep 17 00:00:00 2001 From: deilann Date: Wed, 3 Jul 2024 01:24:07 -0600 Subject: [PATCH 2/3] Revert "Create Intro and Examples.tex" This reverts commit d90605c043f66c027c71b8ffa34d69f6e70d7444. --- Intro and Examples.tex | 138 ----------------------------------------- 1 file changed, 138 deletions(-) delete mode 100644 Intro and Examples.tex diff --git a/Intro and Examples.tex b/Intro and Examples.tex deleted file mode 100644 index f9bbe27..0000000 --- a/Intro and Examples.tex +++ /dev/null @@ -1,138 +0,0 @@ -\title{Groups 4 Subgroups} -\date{} -\author{} -\documentclass{article} -\usepackage{amsmath,amsfonts,amssymb} -\usepackage[usenames,dvipsnames]{color} -\usepackage{amsthm, thmtools} -\usepackage{fullpage, changepage, tabto} -\usepackage{enumerate, enumitem} -\usepackage{mathtools} -\usepackage[colorlinks=true, linkcolor=BlueGreen, citecolor=BlueGreen, urlcolor=BlueGreen]{hyperref} -\usepackage{asymptote} -\usepackage{wrapfig} -\usepackage{parskip} - -% set tab distances -\NumTabs{16} - -%%% theorem environments for styling -% notation -\declaretheoremstyle[ - notefont={\normalfont}, - spaceabove={0.5em}, - spacebelow={0.5em}, - notebraces={}{}, - headformat={\NOTE}, - headpunct={}, - postheadhook={\hspace{0em}\tabto{4em}}, - ]{notation} -\declaretheorem[style=notation]{notation} - -% definition -\declaretheoremstyle[ - headfont={\itshape}, - notefont={\normalfont\bfseries}, - spaceabove={1em}, - notebraces={}{}, - headformat={def \NOTE \newline}, - headpunct={ }, - postheadhook={\hspace{0em}\vspace{0.25em}\tab}, - ]{definition} -\declaretheorem[style=definition,name=Definition, -refname={definition,definitions}]{definition} - -% note -\declaretheoremstyle[ - headfont={\bfseries}, - shaded={margin={1.5em}, textwidth={20em}}, - headformat={note: \newline}, - headpunct={ }, - ]{note} -\declaretheorem[style=note]{note} - - -% prettier empty set -\newcommand{\nullset}{\varnothing} - -% logic spacing -\newcommand{\spaced}[1]{\, #1 \,} -\newcommand{\scomp}{\spaced{\backslash}} -\newcommand{\sland}{\spaced{\land}} -\newcommand{\slor}{\spaced{\lor}} -\newcommand{\simplies}{\spaced{\Rightarrow}} -\newcommand{\slexists}{\, \exists} - -% set macros -\newcommand{\buildset}[2]{\{\spaced{#1} \mid \spaced{#2} \}} -\newcommand{\finiteset}[3]{\{ \spaced{#1,} \spaced{#2,} \spaced{#3,} \spaced{\dots} \}} - -\begin{document} - \maketitle - \tableofcontents - \newpage - \section{intro} - $$\text{Consider }U = \buildset{z \in \mathbb{C}}{\lvert z \rvert = 1} \subseteq \mathbb{C}.$$ - \begin{wrapfigure}[2]{l}{0.35\textwidth} - \begin{asy} - settings.outformat="pdf"; - unitsize(1cm); - draw((0,0) -- (2,0), arrow=Arrow(TeXHead)); - draw((0,0) -- (-2,0), arrow=Arrow(TeXHead)); - draw((0,0) -- (0,-2), arrow=Arrow(TeXHead)); - draw((0,0) -- (0,2), arrow=Arrow(TeXHead)); - draw(unitcircle); - \end{asy} - \end{wrapfigure} - \vspace{24pt} - \begin{note} - each elem of $U$ is defined by $\theta \in [0, 2\pi) = \mathbb{R}_{2\pi}$ - \end{note} - \vspace{48pt} - \begin{center} - Every angle $\theta \in \mathbb{R}_{2\pi}$ given by $f: U \rightarrow R_{2\pi}, \, f(z) = \theta$ for $z = e^{i\theta}$\\ - \vspace{8pt} - +\\ - \vspace{8pt} - $f^{-1}: \mathbb{R}_{2\pi} \rightarrow U, \, f^{-1}(\theta) = e^{i\theta}$\\ - \vspace{24pt} - Consider $U$ with multiplication:\\ - \medskip - Let $z_1 = e^{i\theta_1},\, z_2 = e^{i\theta_2} \in U$\\ - \vspace{8pt} - then $z_1 \cdot z_2 = e^{i\theta_1}\cdot e^{i\theta_2} = e^{i\left(\theta_1 + \theta_2\right)}$ where $\theta_1, \, \theta_2 \in \mathbb{R}_{2\pi},\, \theta_1 + \theta_2 \in \mathbb{R}_{2\pi}$\\ - \vspace{8pt} - Set $U$ is \underline{closed} under multiplication. - \end{center} - -\section{definition} - \begin{definition}[algebraic structure]\label{algebraic structure} - \hspace{0em}\vspace{-1em}\\ - \tab\tab Let $S$ be a set.\\ - \tab\tab Let $\star$ be a binary operation on $S$\\ - \tab\tab\tab $\star: S \times S \rightarrow S$\\ - \tab\tab $\left(S,\, \star \right)$ is an \textbf{algebraic structure} - \end{definition} - \vspace{12pt} - So $\left(U,\, \cdot\right)$ and $\left(\mathbb{R}_{2\pi},\, +_{2\pi}\right)$ are algebraic structures such that there is a ``1-1" relation between $U$ and $\mathbb{R}_{2\pi}$ and operations are preserved. - \vspace{8pt} - - $$\left(U,\, \cdot\right) \simeq \left(\mathbb{R}_{2\pi},\, +_{2\pi}\right)$$ - \begin{center}or\end{center} - $$U \simeq \mathbb{R}_{2\pi}$$ - -\section{isomorphisms} -To show two algebraic structures are isomorphic, find a bijective function that preserves the operations.\\ -Isomorphic structures share the same algebraic properties (associativity, commutative, distributive...).\\ -To show two algebraic structures are \underline{not} isomorphic, find a property that they don't share (the easiest to check is cardinality). - -\subsection{Examples} -\begin{center} -$$\left(\mathbb{N},\, +\right) \not\simeq \left(\mathbb{Z},\, +\right)$$ -because $\mathbb{N}$ has no additive identity -$$\left(\mathbb{Z}^{*},\, \cdot\right) \not\simeq \left(\mathbb{Q}^{*},\, \cdot\right)$$ -because not all elements of $\mathbb{Z}^*$ have inverses $\left(\frac{n}{m}\right)^{-1} = \frac{m}{n}$ -$$\left(\mathbb{R},\, +\right) \not\simeq \left(\mathbb{R},\, \cdot\right)$$ -because $x \star x + 1 = 0$ is solveable in $\left(\mathbb{R},\, +\right): 2x + 1 = 0$ but unsolveable in $\left(\mathbb{R},\, \cdot\right): x^2 + 1 = 0$ -\end{center} -\end{document} From 6869e457af1ccbcf3b7bab98a5e8f1022a01efc0 Mon Sep 17 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