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diff --git a/exercices-ordinaux.tex b/exercices-ordinaux.tex new file mode 100644 index 0000000..3f5849e --- /dev/null +++ b/exercices-ordinaux.tex @@ -0,0 +1,209 @@ +%% 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} |