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  % or whatever (possibly just delete it)
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\usepackage{elephantbird}



\begin{document}


\title[] % (optional, use only with long paper titles)
{Relations}

\subtitle{INFO1 - Semaines 41 \& 42}

\author[] % (optional, use only with lots of authors)
{Guillaume CONNAN }
% - Give the names in the same order as the appear in the paper.
% - Use the inst{?} command only if the authors have different
%   affiliation.

\institute{\textsc{IUT} de Nantes - Dpt d'informatique }% (optional, but mostly needed)

\logo{\includegraphics[scale=0.15]{logo_iut}}

%\logo{\includegraphics[scale=0.15]{big_connan}}

\date[] % (optional, should be abbreviation of conference name)
{Dernière mise à jour: \today}
% - Either use conference name or its abbreviation.
% - Not really informative to the audience, more for people (including
%   yourself) who are reading the slides online

\subject{ }


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%   \begin{frame}<beamer>
%     \frametitle{Sommaire}
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 {\scriptsize
\begin{multicols}{2}
    \tableofcontents[currentsection]
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\begin{frame}
  \titlepage
\end{frame}

\begin{frame}
 \frametitle{Sommaire}
{\scriptsize
\begin{multicols}{2} 
 \tableofcontents
\end{multicols}
}

 
 \end{frame}



\section{Préliminaires...}


\subsection{Un peu de calcul matriciel}


\begin{frame}
  $$c_{ij}=\displaystyle\sum_{k=1}^{n}a_{ik}b_{kj}$$
\end{frame}


\subsection{Calcul booléen}


\begin{frame}
Soit  $$A=
\begin{pmatrix}
 0&1&1\\
 1&1&0\\
 0&0&1 
\end{pmatrix}
\qquad
\text{ et }
\qquad
B=
\begin{pmatrix}
  1&1&0\\
  1&0&1\\
  0&1&1
\end{pmatrix}
$$


Calculer $A\oplus B$ (somme booléenne terme à terme), $A\oplus \text{Attila}(3,3)$ et  $A\barwedge B$ (produit matriciel en utilisant le produit booléen $\barwedge$  pour le produit des  termes et $\oplus$ pour leur somme).
\end{frame}





\section{Relations binaires}


\subsection{À l'IUT}

\begin{frame}
  \begin{definition}
Une \textit{relation binaire}  entre deux ensemble E et  F est la donnée
de E, F et d'un
sous-ensemble du produit cartésien $E\times F$.

Une    \textit{relation   $n$-aire}    entre   $n$    ensembles   $E_1$,
$E_2$,...,$E_n$ est la donnée de ces ensembles et d'un sous-ensemble du produit cartésien $E_1\times
E_2\times \cdots E_n$.
\end{definition}
\end{frame}


\subsection{Au CE1}



\begin{frame}
  
%\begin{figure}
\begin{center}
  \includegraphics[width=0.7\linewidth]{CE1-relations}
\end{center}
%\caption{Comment j'ai étudié les relations en CE1...}
\label{ce1}
%\end{figure}


\end{frame}





\subsection{Généalogie}

 \begin{frame}


\begin{center}
\begin{tikzpicture}
\tikzstyle{every node}=[]
%\footnotesize
\node {Marcel \& Gisèle} [edge from parent fork down]
  [sibling distance=2cm]
  child {node {Berthe}}
  child {node {Kurt}
  };
\node at (5,0) {Günter \& Gertrude} [edge from parent fork down] 
  [sibling distance=2cm] 
  child {node {Tatiana}}
  child {node {Georgina}}
  child {node {Barbarella}};
\node at (1.9,-1.5) {\& } [edge from parent fork down] 
  [sibling distance=2cm] 
  child {node {Clovis}}
  child {node {Dagobert}}
  child {node {Mathémator}};
\end{tikzpicture}
\end{center}

Formez les couples $(x,y)$ tels que $x$ soit le grand-père de $y$.

\pause


Formez les couples $(x,y)$ tels que $x$ soit le frère de $y$.




\end{frame}


\subsection{Repr\'{e}sentation d'une relation}




\begin{frame}
 $$E=\left\{ a,b,c,d,e\right\} \qquad F=\left\{ \alpha ,\beta
,\gamma ,\delta \right\} $$ 

\pause

$$
G_{\mathcal{R}}=\left\{ \left( a,\gamma \right) ,\left( b,\beta \right)
,\left( b,\delta \right) ,\left( d,\gamma \right) ,\left( e,\delta \right)
\right\}
$$



\pause

$$G_{\mathcal{R}}=\left\{ a\mapsto \gamma ,b\mapsto \beta ,b\mapsto \delta
,d\mapsto \gamma ,e\mapsto \delta \right\}
$$

\end{frame}


\begin{frame}[fragile]
  
\pgfdeclarelayer{background}
\pgfsetlayers{background,main}

\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm} 
\SetGraphUnit{0.75cm}
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             ball color= gray!10,
                             color = black,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{be,ga,de} }
\foreach \v/\n in {be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(ga)
\Edge(b)(be)
\Edge(b)(de)
\Edge(d)(ga)
\Edge(e)(de) 


\SetGraphUnit{1.5cm} 
\SO(e){c}
\SO[L=$\alpha$](de){al}


\begin{pgfonlayer}{background} 
\node[draw=blue,fill=blue!20,inner sep=1.5pt,fit=(a) (b) (d) (e) ] {};
\node[color=red,fill=red!20,inner  sep=2pt,fit=(be) (ga) (de)]{};
\node[inner sep=6pt,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
\node[inner sep=6pt,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}

\end{tikzpicture}
\end{center}
\caption{Relation non totale}
\end{figure}

\end{frame}


\begin{frame}

  \begin{multicols}{2}
\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm} 
\SetGraphUnit{0.75cm}
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             ball color= gray!10,
                             color = black,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{be,ga,de} }
\foreach \v/\n in {be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(ga)
\Edge(b)(be)
\Edge(b)(de)
\Edge(d)(ga)
\Edge(e)(de) 


\SetGraphUnit{1.5cm} 
\SO(e){c}
\SO[L=$\alpha$](de){al}


\begin{pgfonlayer}{background} 
\node[draw=blue,fill=blue!20,inner sep=1.5pt,fit=(a) (b) (d) (e) ] {};
\node[color=red,fill=red!20,inner  sep=2pt,fit=(be) (ga) (de)]{};
\node[inner sep=6pt,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
\node[inner sep=6pt,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}

\end{tikzpicture}
\end{center}
%\caption{Relation non totale}
\end{figure}
    




  
\begin{center}
             \begin{tabularx}{0.25\textwidth}{Y{0}}
	     \whline
             De & Vers\\
\whline
 a & $\gamma$\\
\hline
b & $\beta$,$\delta$\\
\hline
d & $\gamma$\\
\hline
e& $\delta$\\

             \whline
             \end{tabularx}
             \end{center}

  \end{multicols}




\end{frame}



\begin{frame}
  \begin{multicols}{2}
    \begin{tabular}{|c|c|c|c|c|}
\hline
$\curvearrowright $ & $\alpha $ & $\beta $ & $\gamma $ & $\delta $ \\ \hline
$a$ & \textbf{0} & \textbf{0} & \textbf{1} & \textbf{0} \\ \hline
$b$ & \textbf{0} & \textbf{1} & \textbf{0} & \textbf{1} \\ \hline
$c$ & \textbf{0} & \textbf{0} & \textbf{0} & \textbf{0} \\ \hline
$d$ & \textbf{0} & \textbf{0} & \textbf{1} & \textbf{0} \\ \hline
$e$ & \textbf{0} & \textbf{0} & \textbf{0} & \textbf{1} \\ \hline
\end{tabular}


$
R=
\begin{pmatrix}
0&0&1&0\\
0&1&0&1\\
0&0&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix}
$

  \end{multicols}
\end{frame}



\caml

\begin{frame}[fragile]
  
  \begin{lstlisting}[caption={}]
type ('a,'b) relation =
  Rel of ('a * 'b ens)  ens ;;
\end{lstlisting}


\pause


\begin{lstlisting}[caption={}]
let r = Rel( 
  Ens((1, Ens ('a', Ens ('b', Vide))),
     Ens((2, Ens ('c', Vide)),
        Ens((3, Ens ('b', Vide)),
           Ens((4, Vide),
              Ens((5, Ens ('b', Vide)),
                 Vide)))))
);;
  \end{lstlisting}

\pause

\begin{lstlisting}[caption={}]
# r;;

- : (int, char) relation
\end{lstlisting}


\end{frame}




\subsection{Relation transposée}


\begin{frame}
  \begin{definition}
La \textbf{relation transposée} de la relation $\mathcal{R}=(E,F,G_{%
\mathcal{R}})$ est la relation not\'{e}e $\mathcal{R}^{\sim}$ d\'{e}finie par $%
\mathcal{R}^{\sim}=(F,E,G_{\mathcal{R}^{\sim}})$ avec 
$
G_{\mathcal{R}^{\sim}}=\left\{ (y,x)\in F\times E\mid (x,y)\in G_{\mathcal{R}%
}\right\}
$,
c'est-\`{a}-dire $y\mathcal{R}^{\sim}x\Leftrightarrow x\mathcal{R}y$.
\end{definition}
\end{frame}


\begin{frame}
  \begin{exercice}
    Considérons les ensembles  et la relation
suivants:
\begin{itemize}
\item Étudiants$=\bigl\{$Roger, Berthe, Jean-Pierre, Gudrun$\bigr\}$;
\item Chaînes$=\bigl\{$ TF1, Gulli, Zen TV$\bigr\}$;
\item  TV$=\bigl\{$(Roger,Gulli),(Berthe,Gulli),
  (Jean-Pierre,Zen TV),(Gudrun,Gulli)$\bigr\}$.
\end{itemize}

\pause 

Que pensez-vous de la relation transposée $TV^\sim$?
  \end{exercice}
\end{frame}




\begin{frame}
$$
R=
\begin{pmatrix}
0&0&1&0\\
0&1&0&1\\
0&0&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{pmatrix}
\qquad
$$


\pause


$$
\tr R=
\begin{pmatrix}
0&0&0&0&0\\
0&1&0&0&0\\
1&0&0&1&0\\
0&1&0&0&1
\end{pmatrix}
$$
\end{frame}




\begin{frame} \frametitle{Image Contre-image}
  \begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{be,ga,de} }
\foreach \v/\n in {be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(ga)
\Edge(b)(be)
\Edge(b)(de)
\Edge(d)(ga)
\Edge(e)(de) 


\SetGraphUnit{1.5cm} 
\SO(e){c}
\SO[L=$\alpha$](de){al}


\begin{pgfonlayer}{background} 
\node[draw=blue,fill=blue!20,inner sep=1.5pt,fit=(a) (b) (d) (e) ] {};
\node[color=red,fill=red!20,inner  sep=2pt,fit=(be) (ga) (de)]{};
\node[inner sep=6pt,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
\node[inner sep=6pt,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}

\end{tikzpicture}
\end{center}
\caption{Image et contre-image}
\end{figure}
\end{frame}


\begin{frame} \frametitle{Égalité}
  \begin{center}
    $E_{1}=E_{2}$ \textbf{et} $F_{1}=F_{2}$ \textbf{et} $G_{%
\mathcal{R}_{1}}=G_{\mathcal{R}_{2}}.$
  \end{center}
\end{frame}


\subsubsection{Inclusion}


\begin{frame}\frametitle{Inclusion}
  {\small

\begin{minipage}[t]{0.5\linewidth}
\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color= black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(be)
\Edge(a)(ga)
\Edge(c)(ga)
\Edge(d)(ga)
\Edge(d)(de)
\Edge(e)(de)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!20,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!20,inner sep=2pt,thick,ellipse,fit=(al) (be) (ga) (de)]{};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\caption{$\RR_1$}
\end{figure}
\end{minipage}



\pause



\begin{flushright}

\begin{minipage}[t]{0.5\linewidth}\vspace{-7cm}
\hspace{5.5cm} 
\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             ball color= gray!10,
                             color = black,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(be)
%\Edge(a)(ga)
\Edge(c)(ga)
%\Edge(d)(ga)
%\Edge(d)(de)
\Edge(e)(de)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!20,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!20,inner  sep=2pt,thick,ellipse,fit=(al)(be) (ga) (de)]{};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\caption{$\RR_2$ est une sous-relation de $\RR_1$}
\end{figure}

\end{minipage}

\end{flushright}

}
\end{frame}











\begin{frame} \frametitle{Réunion}
  

\begin{minipage}[t]{0.5\linewidth}
\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(be)
\Edge(a)(ga)
%\Edge(c)(ga)
\Edge(d)(ga)
%\Edge(d)(de)
\Edge(e)(de)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!20,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!20,inner sep=2pt,thick,ellipse,fit=(al)(be) (ga) (de)]{};
%\node[draw=blue,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=red,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\caption{$\RR_1$}
\end{figure}
\end{minipage}\hspace{0.5cm}

\vspace{-7cm}
\begin{flushright}
\begin{minipage}[t]{0.5\linewidth}%\vspace{-0.5cm}
\hspace{5.5cm} 
\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge[color=orange!50!yellow](a)(be)
\Edge[color=orange!50!yellow](a)(de)
\Edge[color=orange!50!yellow](c)(de)
%\Edge(d)(ga)
%\Edge(d)(de)
\Edge[color=orange!50!yellow](d)(ga)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!20,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!20,inner sep=2pt,thick,ellipse,fit=(al) (be) (ga) (de)]{};
%\node[draw=blue,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=red,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\caption{$\RR_2$}
\end{figure}
\end{minipage}
\end{flushright}

\end{frame}






\begin{frame} \frametitle{Réunion}
  
\vspace{-0.33cm}

\begin{minipage}[t]{0.5\linewidth}
\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(be)
\Edge(a)(ga)
%\Edge(c)(ga)
\Edge(d)(ga)
%\Edge(d)(de)
\Edge(e)(de)
\Edge[color=gray!80!yellow](a)(be)
\Edge[color=orange!50!yellow](a)(de)
\Edge[color=orange!50!yellow](c)(de)
%\Edge(d)(ga)
%\Edge(d)(de)
\Edge[color=gray!80!yellow](d)(ga)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!20,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!20,inner sep=2pt,thick,ellipse,fit=(al)(be) (ga) (de)]{};
%\node[draw=blue,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=red,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\caption{$\RR_1\cup\RR_2$}
\end{figure}
\end{minipage}\hspace{0.5cm}
  
\end{frame}





\begin{frame} \frametitle{Intersection}
  

\begin{minipage}[t]{0.5\linewidth}
\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(be)
\Edge(a)(ga)
%\Edge(c)(ga)
\Edge(d)(ga)
%\Edge(d)(de)
\Edge(e)(de)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!20,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!20,inner sep=2pt,thick,ellipse,fit=(al)(be) (ga) (de)]{};
%\node[draw=blue,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=red,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\caption{$\RR_1$}
\end{figure}
\end{minipage}\hspace{0.5cm}

\vspace{-7cm}
\begin{flushright}
\begin{minipage}[t]{0.5\linewidth}%\vspace{-0.5cm}
\hspace{5.5cm} 
\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge[color=orange!50!yellow](a)(be)
\Edge[color=orange!50!yellow](a)(de)
\Edge[color=orange!50!yellow](c)(de)
%\Edge(d)(ga)
%\Edge(d)(de)
\Edge[color=orange!50!yellow](d)(ga)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!20,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!20,inner sep=2pt,thick,ellipse,fit=(al) (be) (ga) (de)]{};
%\node[draw=blue,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=red,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\caption{$\RR_2$}
\end{figure}
\end{minipage}
\end{flushright}

\end{frame}









\begin{frame} \frametitle{Intersection}
  
\vspace{-0.33cm}

\begin{minipage}[t]{0.5\linewidth}
\begin{figure}
\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
%\Edge(a)(be)
%\Edge(a)(ga)
%\Edge(c)(ga)
%\Edge(d)(ga)
%\Edge(d)(de)
%\Edge(e)(de)
\Edge[color=gray!80!yellow](a)(be)
%\Edge[color=orange!50!yellow](a)(de)
%\Edge[color=orange!50!yellow](c)(de)
%\Edge(d)(ga)
%\Edge(d)(de)
\Edge[color=gray!80!yellow](d)(ga)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!20,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!20,inner sep=2pt,thick,ellipse,fit=(al)(be) (ga) (de)]{};
%\node[draw=blue,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=red,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\caption{$\RR_1\cap\RR_2$}
\end{figure}
\end{minipage}\hspace{0.5cm}
  
\end{frame}





\begin{frame} \frametitle{Composition}

{\footnotesize
\begin{tabular}[t]{ccc}

\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick}] 
\SetGraphUnit{.65cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,color=black,
                             ball color= gray!10,
                             minimum size= 10pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
{\SetVertexNoLabel
\Vertices[empty,x=0,y=0,dir=\SO]{line}{a,b,c,d,e} }
\foreach \v in {a,b,c,d,e}{%
\Vertex[Node]{\v}}   
{\SetVertexNoLabel
\Vertices[empty,x=2,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(be)
\Edge(a)(ga)
\Edge(c)(ga)
\Edge(d)(ga)
\Edge(d)(de)
\Edge(e)(de)
\begin{pgfonlayer}{background}
\node[draw=blue,fill=blue!20,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
\node[draw=black,fill=green!10,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F=U}] {};
\end{pgfonlayer}
\end{tikzpicture}

&

\pause

\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick}] 
\SetGraphUnit{.65cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,color=black,
                             ball color= gray!10,
                             minimum size= 10pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
{\SetVertexNoLabel
\Vertices[empty,x=2,y=0,dir=\SO]{line}{1,2,3,4,5} }
\foreach \v in {1,2,3,4,5}{%
\Vertex[Node]{\v}}   
{\SetVertexNoLabel
\Vertices[empty,x=0,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(be)(2)
\Edge(be)(4)
\Edge(ga)(5)
\begin{pgfonlayer}{background}
\node[draw=red,fill=red!20,,inner sep=4pt,thick,ellipse,fit=(1) (2) (3) (4) (5),pin={[pin edge={thick}]V}] {};
\node[draw=black,fill=green!10,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]U=F}] {};
\end{pgfonlayer}
\end{tikzpicture}


&

\pause

\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick}] 
\SetGraphUnit{.65cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color=black,
                             ball color= gray!10,
                             minimum size= 10pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
{\SetVertexNoLabel
\Vertices[empty,x=0,y=0,dir=\SO]{line}{a,b,c,d,e} }
\foreach \v in {a,b,c,d,e}{%
\Vertex[Node]{\v}}   
{\SetVertexNoLabel
\Vertices[empty,x=2,y=0,dir=\SO]{line}{al,be,ga,de,ep} }
\foreach \v/\n in {al/1,be/2,ga/3,de/4,ep/5}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(be)
\Edge(a)(de)
\Edge(a)(ep)
\Edge(c)(ep)
\Edge(d)(ep)
\begin{pgfonlayer}{background}
\node[draw=blue,fill=blue!20,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
\node[draw=red,fill=red!20,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de) (ep),pin={[pin edge={thick}]V}] {};
\end{pgfonlayer}
\end{tikzpicture}


\end{tabular}
}
\end{frame}






\begin{frame} \frametitle{Composition}

{\footnotesize
\begin{tabular}[t]{ccc}

\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick}] 
\SetGraphUnit{.65cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,color=black,
                             ball color= gray!10,
                             minimum size= 10pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
{\SetVertexNoLabel
\Vertices[empty,x=0,y=0,dir=\SO]{line}{a,b,c,d,e} }
\foreach \v in {a,b,c,d,e}{%
\Vertex[Node]{\v}}   
{\SetVertexNoLabel
\Vertices[empty,x=2,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge[color=yellow](a)(be)
\Edge[color=yellow](a)(ga)
\Edge(c)(ga)
\Edge(d)(ga)
\Edge(d)(de)
\Edge(e)(de)
\begin{pgfonlayer}{background}
\node[draw=blue,fill=blue!40,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
\node[draw=black,fill=green!30,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F=U}] {};
\end{pgfonlayer}
\end{tikzpicture}

&

\pause

\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick}] 
\SetGraphUnit{.65cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,color=black,
                             ball color= gray!10,
                             minimum size= 10pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
{\SetVertexNoLabel
\Vertices[empty,x=2,y=0,dir=\SO]{line}{1,2,3,4,5} }
\foreach \v in {1,2,3,4,5}{%
\Vertex[Node]{\v}}   
{\SetVertexNoLabel
\Vertices[empty,x=0,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge[color=yellow](be)(2)
\Edge[color=yellow](be)(4)
\Edge[color=yellow](ga)(5)
\begin{pgfonlayer}{background}
\node[draw=red,fill=red!40,,inner sep=4pt,thick,ellipse,fit=(1) (2) (3) (4) (5),pin={[pin edge={thick}]V}] {};
\node[draw=black,fill=green!30,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]U=F}] {};
\end{pgfonlayer}
\end{tikzpicture}


&

\pause

\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick}] 
\SetGraphUnit{.65cm}  
\SetUpEdge[lw= 1.5pt,color= black!70]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,
                             color=black,
                             ball color= gray!10,
                             minimum size= 10pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
{\SetVertexNoLabel
\Vertices[empty,x=0,y=0,dir=\SO]{line}{a,b,c,d,e} }
\foreach \v in {a,b,c,d,e}{%
\Vertex[Node]{\v}}   
{\SetVertexNoLabel
\Vertices[empty,x=2,y=0,dir=\SO]{line}{al,be,ga,de,ep} }
\foreach \v/\n in {al/1,be/2,ga/3,de/4,ep/5}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge[color=yellow](a)(be)
\Edge[color=yellow](a)(de)
\Edge[color=yellow](a)(ep)
\Edge(c)(ep)
\Edge(d)(ep)
\begin{pgfonlayer}{background}
\node[draw=blue,fill=blue!40,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
\node[draw=red,fill=red!30,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de) (ep),pin={[pin edge={thick}]V}] {};
\end{pgfonlayer}
\end{tikzpicture}


\end{tabular}
}
\end{frame}

\begin{frame}
  \begin{exercice}
     Si  $R$  est  la  matrice  de  $\mathcal{R}$  et  $S$  la  matrice  de
  $\mathcal{S}$ alors quelle sera la matrice de $R\ ;\ S$?
  \end{exercice}

\pause

\begin{exercice}
    Que pensez de  $\RR^{\sim}\circ \RR$ ou de $\RR  \circ \RR^{\sim}$? Regardez
ce que cela donne avec la relation $\RR$ introduite précédemment.

Quelle est la relation transposée de $\mathcal{S\circ R}$
\end{exercice}

\end{frame}



\section{Fonctions}




\begin{frame} \frametitle{fonction?}

\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!50]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(a)(be)
\Edge(a)(ga)
\Edge(c)(ga)
\Edge(d)(ga)
\Edge(d)(de)
\Edge(e)(de)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!40,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!40,inner sep=2pt,thick,ellipse,fit=(al) (be) (ga) (de)]{};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
  
\end{frame}







\begin{frame} \frametitle{fonction?}

\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!50]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
%\Edge(a)(be)
\Edge(a)(ga)
\Edge(c)(ga)
%\Edge(d)(ga)
\Edge(d)(de)
\Edge(e)(de)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!40,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!40,inner sep=2pt,thick,ellipse,fit=(al) (be) (ga) (de)]{};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
  
\end{frame}








\begin{frame} \frametitle{fonction?}

\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!50]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
                             shading= ball,color=black,
                             ball color= gray!10,
                             minimum size= 15pt,
                             draw}}
\tikzstyle{EdgeStyle}=[->]
\Vertices[dir=\SO]{line}{a,b,c,d,e}
{\SetVertexNoLabel
\Vertices[empty,x=3,y=0,dir=\SO]{line}{al,be,ga,de} }
\foreach \v/\n in {al/\alpha,be/\beta,ga/\gamma,de/\delta}{%
\Vertex[Node,L=$\n$]{\v}}   
\Edge(b)(al)
\Edge(a)(ga)
\Edge(c)(ga)
%\Edge(d)(ga)
\Edge(d)(de)
\Edge(e)(de)
\begin{pgfonlayer}{background}
 \node[draw=blue,fill=blue!40,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d) (e)] {};
 \node[draw=red,fill=red!40,inner sep=2pt,thick,ellipse,fit=(al) (be) (ga) (de)]{};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(a) (b) (c) (d) (e),pin={[pin edge={thick}]E}] {};
%\node[draw=black,inner sep=4pt,thick,ellipse,fit=(al) (be) (ga) (de),pin={[pin edge={thick}]F}] {};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
  
\end{frame}





\begin{frame} \frametitle{Projection canonique}
  \begin{definition}[Projection canonique]
  
La fonction totale 
$$
\pi_{i}\colon\begin{array}{rll} E_{1}\times E_{2}\times \ldots \times E_{n} &\to &  E_{i} \\ (x_{1},x_{2},\ldots ,x_{n})  & \mapsto &  x_i\end{array}$$

 est appelée
projection canonique de 
$E_{1}\times E_{2}\times \ldots \times E_{n}$ sur $E_{i}$.

Si tous les $E_{i}$ sont égaux au même ensemble $E$, $\pi_{i}$ est
appelée la $i$\eme projection.

\end{definition}

\end{frame}
















\begin{frame} \frametitle{Loi de composition}
  

\begin{definition}[Loi de composition]
  On appelle loi de composition (ou opération) dans $E$ toute fonction de $%
A\times E$ dans $E$.

 Si $A=E$, on dit que la loi est une loi de composition
interne, sinon on parle de loi de composition externe ou loi mixte. 
\end{definition}





 Si $f$ est
une  telle loi  de  composition  et $(a,x)\in  A\times  E,$ on  remplace
souvent la \textbf{notation préfixée} $f\left(
(a,x)\right) $ par une \textbf{notation infixée} du style $a\circledcirc x$ ou $a\ast x$ ou $a\boxplus x$ ou $%
a\boxdot x$ ou $a+x$  ou...








\end{frame}






\begin{frame} \frametitle{fonction injective}

\begin{center}
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\GraphInit[vstyle=Normal] 

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\Edge(b)(al)
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%\Edge(d)(ga)
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 \node[draw=blue,fill=blue!40,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (c)] {};
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\end{frame}





\begin{frame} \frametitle{fonction surjective}

\begin{center}
\begin{tikzpicture}[ every fit/.style={ellipse,draw,thick},] 
\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!50]
\GraphInit[vstyle=Normal] 

\tikzset{VertexStyle/.style= {shape= circle,
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                             ball color= gray!10,
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{\SetVertexNoLabel
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\Vertex[Node,L=$\n$]{\v}}   
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\Edge(a)(be)
\Edge(c)(ga)
\Edge(d)(ga)
%\Edge(d)(de)
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 \node[draw=blue,fill=blue!40,inner sep=1.5pt,thick,ellipse,fit=(a) (b) (d)] {};
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\end{frame}





\begin{frame} \frametitle{fonction bijective}

\begin{center}
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\SetGraphUnit{0.75cm}  
\SetUpEdge[lw= 1.5pt,color= black!50]
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                             ball color= gray!10,
                             minimum size= 15pt,
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{\SetVertexNoLabel
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\begin{frame}\frametitle{Méthode B}
  En méthode B on note:
\begin{itemize}
\item ${D\foncpart\ \ C}$: fonctions partielles de D vers C;
\item ${D\fonctot C}$: fonctions totales de D vers C;
\item ${D\injpart\ \ C}$: injections partielles de D vers C;
\item ${D\injtot C}$: injections totales de D vers C;
\item ${D\surjpart\ \ C}$: surjections partielles de D vers C;
\item ${D\surjtot C}$: surjections totales de D vers C;
\item ${D\bijpart C}$: bijections partielles de D sur C;
\item ${D\bijtot C}$: bijections totales de D sur C.
\end{itemize}
\end{frame}





\section{Relations binaires sur un ensemble}

\subsection{Au CE1}





%\begin{figure}
\begin{center}
  \includegraphics[width=\linewidth]{CE1-rel-ens}
\end{center}
%\end{figure}












\end{document}
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