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authorDavid A. Madore <david+git@madore.org>2016-04-04 13:09:40 (GMT)
committerDavid A. Madore <david+git@madore.org>2016-04-04 13:09:40 (GMT)
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Start writing an exercise sheet on ordinals.
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+%% This is a LaTeX document. Hey, Emacs, -*- latex -*- , get it?
+\documentclass[12pt,a4paper]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+%\usepackage{ucs}
+\usepackage{times}
+% A tribute to the worthy AMS:
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{amsthm}
+%
+\usepackage{mathrsfs}
+\usepackage{wasysym}
+\usepackage{url}
+%
+\usepackage{makeidx}
+%% Self-note: compile index with:
+%% xindy -M texindy -C utf8 -L french notes-mitro206.idx
+%
+\usepackage{graphics}
+\usepackage[usenames,dvipsnames]{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{matrix,calc}
+\usepackage{hyperref}
+%
+\theoremstyle{definition}
+\newtheorem{comcnt}{Tout}
+\newcommand\thingy{%
+\refstepcounter{comcnt}\smallbreak\noindent\textbf{\thecomcnt.} }
+\newcommand\exercice{%
+\refstepcounter{comcnt}\bigbreak\noindent\textbf{Exercice~\thecomcnt.}}
+\renewcommand{\qedsymbol}{\smiley}
+%
+\newcommand{\outnb}{\operatorname{outnb}}
+\newcommand{\downstr}{\operatorname{downstr}}
+\newcommand{\precs}{\operatorname{precs}}
+\newcommand{\mex}{\operatorname{mex}}
+\newcommand{\id}{\operatorname{id}}
+\newcommand{\limp}{\Longrightarrow}
+\newcommand{\gr}{\operatorname{gr}}
+\newcommand{\rk}{\operatorname{rk}}
+%
+\DeclareUnicodeCharacter{00A0}{~}
+%
+\DeclareMathSymbol{\tiret}{\mathord}{operators}{"7C}
+\DeclareMathSymbol{\traitdunion}{\mathord}{operators}{"2D}
+%
+\DeclareFontFamily{U}{manual}{}
+\DeclareFontShape{U}{manual}{m}{n}{ <-> manfnt }{}
+\newcommand{\manfntsymbol}[1]{%
+ {\fontencoding{U}\fontfamily{manual}\selectfont\symbol{#1}}}
+\newcommand{\dbend}{\manfntsymbol{127}}% Z-shaped
+\newcommand{\danger}{\noindent\hangindent\parindent\hangafter=-2%
+ \hbox to0pt{\hskip-\hangindent\dbend\hfill}}
+%
+\newif\ifcorrige
+\corrigetrue
+\newenvironment{corrige}%
+{\ifcorrige\relax\else\setbox0=\vbox\bgroup\fi%
+\smallbreak\noindent{\underbar{\textit{Corrigé.}}\quad}}
+{{\hbox{}\nobreak\hfill\checkmark}%
+\ifcorrige\relax\else\egroup\fi\par}
+%
+%
+%
+\begin{document}
+\title{Exercices sur les ordinaux}
+\author{David A. Madore}
+\maketitle
+
+\centerline{\textbf{MITRO206}}
+
+{\footnotesize
+\immediate\write18{sh ./vc > vcline.tex}
+\begin{center}
+Git: \input{vcline.tex}
+\end{center}
+\immediate\write18{echo ' (stale)' >> vcline.tex}
+\par}
+
+\pretolerance=8000
+\tolerance=50000
+
+
+
+%
+%
+%
+
+\exercice
+
+\newcommand{\spaceout}{\hskip1emplus2emminus.5em}
+
+Ranger les ordinaux suivants par ordre croissant :
+\spaceout $\omega^{\omega+1} + \omega^\omega\cdot 33$ ;
+\spaceout $\omega\cdot 3 + 42$ ;
+\spaceout $\omega^{\omega+1} + \omega + 33$ ;
+\spaceout $\omega^{\omega+2} + \omega^\omega$ ;
+\spaceout $\omega^2\cdot 42 + 1000$ ;
+\spaceout $\omega^2 + \omega$ ;
+\spaceout $\omega^2\cdot 42 + \omega$ ;
+\spaceout $\omega^{\omega^2 + 1}$ ;
+\spaceout $\omega^{\omega^{(\omega\cdot 2)}}$ ;
+\spaceout $\omega^{\omega^\omega} + 1$ ;
+\spaceout $\omega^{\omega+1} + \omega\cdot 33$ ;
+\spaceout $\omega^{\omega^2}$ ;
+\spaceout $\omega^{\omega^2 + 1} + \omega^{\omega\cdot 2}\cdot 1000$ ;
+\spaceout $\omega^{\omega^2 + \omega}$ ;
+\spaceout $\omega\cdot 3$ ;
+\spaceout $\omega^{(\omega^\omega\cdot 2)}$ ;
+\spaceout $\omega^{\omega^3}$ ;
+\spaceout $\omega^{\omega+1} + 1000$ ;
+\spaceout $\omega^{\omega+2}$ ;
+\spaceout $\omega^{\omega+1}\cdot 2$ ;
+\spaceout $\omega\cdot 2 + 1729$ ;
+\spaceout $\omega^2 + 1000$ ;
+\spaceout $42$ ;
+\spaceout $\omega^{\omega^2 + \omega} + \omega^{\omega^2 + 1}$ ;
+\spaceout $\omega^{\omega\cdot 2}\cdot 1000$ ;
+\spaceout $\omega^2\cdot 42$ ;
+\spaceout $\omega^{\omega+1} + \omega^2\cdot 33$ ;
+\spaceout $\omega^2$ ;
+\spaceout $\omega$ ;
+\spaceout $\omega^{\omega+1}$ ;
+\spaceout $\omega^{\omega^2 + 1} + \omega^{\omega^2}\cdot 42$ ;
+\spaceout $\omega^{\omega\cdot 2} + \omega^{\omega+2}$ ;
+\spaceout $\omega^{\omega^2 + \omega} + \omega^{\omega^2} + \omega^{\omega+1}$ ;
+\spaceout $\omega^{\omega^{(\omega^2)}}$ ;
+\spaceout $\omega^{\omega^2 + 1}\cdot 2$ ;
+\spaceout $\omega^2 + \omega\cdot 42$ ;
+\spaceout $\omega + 42$ ;
+\spaceout $\omega^{\omega^2\cdot 2}$ ;
+\spaceout $\omega^{\omega\cdot 2 + 42}$ ;
+\spaceout $\omega^{\omega\cdot 2}$ ;
+\spaceout $\omega\cdot 2$ ;
+\spaceout $\omega^{\omega+1} + \omega^2 + 33$ ;
+\spaceout $\omega^{\omega^{(\omega+1)}}$ ;
+\spaceout $\omega^\omega$ ;
+\spaceout $\omega^{\omega^2 + \omega + 1}$ ;
+\spaceout $\omega^{\omega^\omega}\cdot 2$ ;
+\spaceout $\omega^{\omega^\omega}$ ;
+\spaceout $0$ ;
+\spaceout $\omega^{\omega^2} + \omega^{\omega+1}$ ;
+\spaceout $\omega^{(\omega^\omega + 1)}$.
+
+\begin{corrige}
+On vérifie que tous ces ordinaux sont écrits en forme normale de
+Cantor (et les exposants de $\omega$ aussi, etc.). On les compare
+donc en comparant à chaque fois la plus grande puissance de $\omega$.
+
+Dans l'ordre croissant : \spaceout $0$ ;
+\spaceout $42$ ;
+\spaceout $\omega$ ;
+\spaceout $\omega + 42$ ;
+\spaceout $\omega\cdot 2$ ;
+\spaceout $\omega\cdot 2 + 1729$ ;
+\spaceout $\omega\cdot 3$ ;
+\spaceout $\omega\cdot 3 + 42$ ;
+\spaceout $\omega^2$ ;
+\spaceout $\omega^2 + 1000$ ;
+\spaceout $\omega^2 + \omega$ ;
+\spaceout $\omega^2 + \omega\cdot 42$ ;
+\spaceout $\omega^2\cdot 42$ ;
+\spaceout $\omega^2\cdot 42 + 1000$ ;
+\spaceout $\omega^2\cdot 42 + \omega$ ;
+\spaceout $\omega^\omega$ ;
+\spaceout $\omega^{\omega+1}$ ;
+\spaceout $\omega^{\omega+1} + 1000$ ;
+\spaceout $\omega^{\omega+1} + \omega + 33$ ;
+\spaceout $\omega^{\omega+1} + \omega\cdot 33$ ;
+\spaceout $\omega^{\omega+1} + \omega^2 + 33$ ;
+\spaceout $\omega^{\omega+1} + \omega^2\cdot 33$ ;
+\spaceout $\omega^{\omega+1} + \omega^\omega\cdot 33$ ;
+\spaceout $\omega^{\omega+1}\cdot 2$ ;
+\spaceout $\omega^{\omega+2}$ ;
+\spaceout $\omega^{\omega+2} + \omega^\omega$ ;
+\spaceout $\omega^{\omega\cdot 2}$ ;
+\spaceout $\omega^{\omega\cdot 2} + \omega^{\omega+2}$ ;
+\spaceout $\omega^{\omega\cdot 2}\cdot 1000$ ;
+\spaceout $\omega^{\omega\cdot 2 + 42}$ ;
+\spaceout $\omega^{\omega^2}$ ;
+\spaceout $\omega^{\omega^2} + \omega^{\omega+1}$ ;
+\spaceout $\omega^{\omega^2 + 1}$ ;
+\spaceout $\omega^{\omega^2 + 1} + \omega^{\omega\cdot 2}\cdot 1000$ ;
+\spaceout $\omega^{\omega^2 + 1} + \omega^{\omega^2}\cdot 42$ ;
+\spaceout $\omega^{\omega^2 + 1}\cdot 2$ ;
+\spaceout $\omega^{\omega^2 + \omega}$ ;
+\spaceout $\omega^{\omega^2 + \omega} + \omega^{\omega^2} + \omega^{\omega+1}$ ;
+\spaceout $\omega^{\omega^2 + \omega} + \omega^{\omega^2 + 1}$ ;
+\spaceout $\omega^{\omega^2 + \omega + 1}$ ;
+\spaceout $\omega^{\omega^2\cdot 2}$ ;
+\spaceout $\omega^{\omega^3}$ ;
+\spaceout $\omega^{\omega^\omega}$ ;
+\spaceout $\omega^{\omega^\omega} + 1$ ;
+\spaceout $\omega^{\omega^\omega}\cdot 2$ ;
+\spaceout $\omega^{(\omega^\omega + 1)}$ ;
+\spaceout $\omega^{(\omega^\omega\cdot 2)}$ ;
+\spaceout $\omega^{\omega^{(\omega+1)}}$ ;
+\spaceout $\omega^{\omega^{(\omega\cdot 2)}}$ ; et enfin
+\spaceout $\omega^{\omega^{(\omega^2)}}$.
+\end{corrige}
+
+
+%
+%
+%
+\end{document}