\newcommand{\ang}{{\bf {\mathcal A }}}
\newcommand{\bb}{{\bf {\mathcal B }}}
\newcommand{\pp}{{\bf {\mathcal P }}}
\newcommand{\cc}{{\bf C}}
\newcommand{\et}{\mathop{\,\rm et \,}}
\newcommand{\etc}{\mathop{\rm etc }}
\newcommand{\ou}{\mathop{\,\rm ou \,}}
\newcommand{\Ima}{\mathop{\rm Im }}
\newcommand{\Ker}{\mathop{\rm Ker }}
\newcommand{\si}{\mathop{\rm si }}
\newcommand{\re}{\mathop{\rm r\'e\'ecrire }}
\newcommand{\reu}{\mathop{\rm r\'eu }}
\newcommand{\alors}{\mathop{\rm alors }}
\newcommand{\Arctg}{\mathop{\rm Arctg }}
\newcommand{\Rea}{\mathop{\rm Re}}
\newcommand{\Arg}{\mathop{\rm Arg }}
\newcommand{\sinon}{\mathop{\rm sinon }}
\newcommand{\non}{\mathop{\rm non }}
\newcommand{\rr}{{\bf R}}
\newcommand{\rdeu}{{\bf R^2}}
\newcommand{\rrn}{{\bf R^n}}
\newcommand{\nn}{{\bf N}}
\newcommand{\kk}{{\bf K}}
\newcommand{\kx}{{\bf K}[X]}
\newcommand{\rX}{{\bf R}[X]}
\newcommand{\qq}{{\bf Q}}
\newcommand{\zz}{{\bf Z}}
\newcommand{\rar}{\rightarrow}
\newcommand{\imp}{\Longrightarrow}
\newcommand{\limp}{\longleftarrow}
\newcommand{\lrimp}{\longleftrightarrow}
\newcommand{\ssi}{\Longleftrightarrow}
\newcommand{\ove}{\overline}
\documentclass[12pt]{article} \usepackage{vmargin}
\usepackage{amsmath}
\usepackage{amssymb}
\pagestyle{empty} \setmarginsrb{0.5cm}{0.5cm}{1.5cm}{1cm}{1cm}{1cm}{1cm}{1cm}

%\documentstyle[12pt]{article} \pagestyle{empty} \setlength{\textwidth}{13cm}
%\setlength{\parindent}{0pt} \setlength{\parskip}{.2em} 
\begin{document}
 M\'ethodo Exos2~~~~~~~~~~~~~~~~~~~~~{\Huge {\bf 
                      ~Enonc\'es }}  \hfill 8/10/01 ~Deug MIASSM TC\\ %
		     
		     \vspace{0.3cm}
%Pour mettre le prof au parfum

                      \vspace{0.3cm}

                                                
\begin{enumerate}

\item {\Large {\bf Valider}}

$$0=1  ~~~~~\frac {i}{1+e^{i\pi}} \neq \frac{0}{1} 
~~~~~2^4 \neq 2 ~[4] ~~~~~\Arg~i= \pi /2  
~~~~~\Arg (e^{i\pi} +e^{-i\pi})=\Arg (e^{-i\pi} -e^{i\pi}) \,[2\pi]$$
                      
		      
		      \vspace{-0.6cm}


$$\sqrt 
2 > 1 ~~~~\Rea i < \Ima i ~~~~i \leq 2|i| ~~~~ \Arg~i \ge 0 ~~~~
 0=1 \et \Arg~0= 0 \, [2\pi]
~~~~1+1=2 \ou \sqrt {-1} =0$$
                      
		      
		      \vspace{-0.6cm}


 $$(\sqrt {-1})^2=-1 \imp 1+1=2
~~~~\si~ 1 > 2 ~\alors~1/0 > 2/0 ~~~~\Arg~i = \pi/2 \ssi \Arg~(i/2) = \pi/4$$
                      
		      
		      \vspace{-0.3cm}


$$\forall x \in \rr, \, \frac {1}{x^2}= (\frac {1}{x})^2 ~~~~~~~
\exists z \in \cc, \,z^5+ez^3+z^2+\sqrt 2 z=i\pi \et \Rea z \neq 0$$
		      
		      \vspace{-0.5cm}


$$\forall z \in \cc, \, \Arg z = \pi /2 \,[2 \pi]  \imp \Ima z =0
~~~~~\forall x \in \cc, \,\exists y \in \cc,  \sqrt y=x
~~~~~\forall x,y \in {\bf R},\,  x/y =y/x.$$

Quel est le domaine de d\'efinition de
l'\'enonc\'e:
\hfill  $\exists y \in \rr, \, y \neq x \imp 1/y \neq 1/x$.


		      
		      \vspace{0.8cm}



\item {\Large {\bf Calculer}}

		      \vspace{0.5cm}


{\bf a) Evaluer }
		      
		      \vspace{-0.5cm}


$$ 0=1  ~~~~~~4^5 = 4 ~[5] 
~~~~~~\frac {i}{e^{i\pi} + e^{-i\pi}}=\frac{0}{1}  
~~~~~~2^4 \neq 2 \,[4] ~~~~~~\Arg~i \neq \pi /2 \, [2 \pi] 
~~~~~~\sqrt 2<1 ~~~|i| \leq 0$$
		      
		      \vspace{-0.3cm}


 $$0=1 \et \Arg~1= \Arg~2 \, [2\pi]
~~~~~~~~~~~1+1=2 \ou \sqrt {1} =0 $$
		      
		      \vspace{-0.5cm}


$$(\sqrt {1})^2=-1 \imp 1+1=2
~~~~~~~~~~\si~ 1 > 2 ~\alors~0/1 > 0/2 ~~~~~~~~~~ 1=2 \ssi 2=4$$
		      
		      \vspace{-0.7cm}


$$\forall x,y \in {\bf R},\,  xy=yx ~~~~~~~~~~~~\exists x \in \rr, \,x^2=x
~~~~~~~~~~~~\forall z \in \cc, \,\Arg z = \pi /2 \,[2 \pi]  \imp \Ima z =0$$
		      
		      \vspace{-0.5cm}



$$\forall x \in \rr, \,\exists y \in \rr, \, y^2=x 
~~~~~~~~~~~~\forall x \in \cc, \,\exists y \in \cc, \, y^2=x.$$

{\bf b) Nier} \hfill ($x$ et $y$ sont des r\'eels)
		      
		      \vspace{-0.3cm}


$$x=x+1 ~~~~~x \neq y ~~~~~ |x| > \sqrt 2 ~~~~~x \leq e^2 ~~~~~
x < 0 \ou e^{x}=-2 ~~~~~x \neq 0 \imp \frac{1}{x}=1$$
		      
		      \vspace{-0.7cm}


$$
x>0 \ssi y<1  ~~~~\forall x \in \rr, \,x \leq x^2 ~~~~
\exists x \in \rr, \, x=x+1 
~~~~ \forall x \in \rr, \, \exists n \in \nn, \, x \ge n \et x < n+1. $$

{\bf c) Formuler} la contrapos\'ee puis la r\'eciproque

		      
		      \vspace{-0.2cm}


$$\forall x,y \in \rr, \, \sin x = \sin y \imp x = y 
~~~~~~~~~~~~\forall a,b \in \nn, \,a^3 \neq b^3 \imp a \neq b$$
		      
		      \vspace{-0.5cm}


$$\forall u,v \in \rr,\,  e^u < e^v \imp u < v
~~~~~~~~~~~~ \forall x,y \in \nn,  \,x \leq y \imp \cos x \leq \cos y \,.$$



\end {enumerate}
\end{document}
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%
\end{document}

