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\begin{document}
 
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\title{Devoir 9}
\author{Louis-Olivier Brassard\\
PHI1005 -- Logique 1}
\date{Automne 2019}

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Soit le dictionnaire suivant :

\begin{itemize}
    \item $Qx$ : $x$ est un Québécois
    \item $Cx$ : $x$ est un Canadien
    \item $Fx$ : $x$ est francophone
\end{itemize}


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    \node at (0,5) {\LARGE\textbf{Diagramme 1}};
    \node at (-4,3.5) {\LARGE\textit{Q}};
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\section*{Question 1.}

Traduisez les énoncés suivants en calcul des prédicats et dites s’ils sont vrais ou faux selon l’interprétation donnée par le diagramme 1.

\begin{enumerate}[label=\alph*)]
  % a)
  \item Ce n’est pas le cas qu’il y a un Canadien francophone qui est un Québécois.
  \[\]
  
  % b)
  \item Quelques francophones ne sont ni des Québécois, ni des Canadiens.
  \[\]
 
  % c)
  \item Tout le monde est un Canadien, un francophone ou un Québécois.
  \[\]
 
  % d)
  \item S’il y a un Québécois, alors il y a un Canadien.
  \[\]

  % e)
  \item Il y a des francophones si et seulement s’il y a des Québécois.
  \[\]

  % f)
  \item Tous les Canadiens francophones sont des Québécois.
  \[\]
 
\end{enumerate}

\section*{Question 2.}

Traduisez les énoncés suivants en langue naturelle et dites s’ils sont vrais ou faux selon l’interprétation donnée par le diagramme 1.

\begin{enumerate}[label=\alph*)]
  % a)
  \item \[(\forall)((Cx \land \lnot Qx) \supset \lnot Fx)\]
   \textbf{}
  
  % b)
  \item \[(\forall x)((Fx \land Qx) \supset Cx)\]
   \textbf{}
 
  % c)
  \item \[(\exists x)(Cx \land \lnot (Qx \land Fx))\]
   \textbf{}
 
  % d)
  \item \[(\forall x)(Fx \supset (Qx \lor Cx)))\]
   \textbf{}
 
\end{enumerate}

\section*{Question 3.}

À l’aide de la méthode des arbres, déterminez si les propositions suivantes sont logiquement valides.
\begin{enumerate}[label=\alph*)]
  % a)
  \item \[\vdash (\forall) Ax \supset \lnot (\exists x) \lnot Ax\]
  
  
  % b)
  \item \[\vdash (\exists x)Ax \supset \lnot (\forall x) \lnot Ax\]
  
 
 \end{enumerate}

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