Write answers to second exercise.

1 files changed, 117 insertions, 2 deletions

diff --git a/controle-20180206.tex b/controle-20180206.tex
index 4ea67f4..be3a599 100644
--- a/controle-20180206.tex
+++ b/controle-20180206.tex
@@ -350,18 +350,133 @@ démontrer).
 (c) $x\mathbin{@}y\mathbin{\#}z$, (d) $x\mathbin{@}y\mathbin{@}z$,
 (e) $(x\mathbin{\#}y)\mathbin{@}z$.
 
+\begin{corrige}
+On obtient les arbres d'analyse suivants :
+(a) % S<S<S<T<U<V<x.>>>>#.T<U<V<y.>>>>#.T<U<V<z.>>>>
+\begin{tikzpicture}[line join=bevel,baseline=(S0.base)]
+\node (S0) at (53.333bp,0.000bp) [draw=none] {$S$};
+\node (S1) at (20.000bp,-30.000bp) [draw=none] {$S$};  \draw (S0) -- (S1);
+\node (S2) at (0.000bp,-60.000bp) [draw=none] {$S$};  \draw (S1) -- (S2);
+\node (T0) at (0.000bp,-80.000bp) [draw=none] {$T$};  \draw (S2) -- (T0);
+\node (U0) at (0.000bp,-100.000bp) [draw=none] {$U$};  \draw (T0) -- (U0);
+\node (V0) at (0.000bp,-120.000bp) [draw=none] {$V$};  \draw (U0) -- (V0);
+\node (x0) at (0.000bp,-140.000bp) [draw=none] {$x$};  \draw (V0) -- (x0);
+\node (o0) at (20.000bp,-60.000bp) [draw=none] {$\#$};  \draw (S1) -- (o0);
+\node (T1) at (40.000bp,-60.000bp) [draw=none] {$T$};  \draw (S1) -- (T1);
+\node (U1) at (40.000bp,-80.000bp) [draw=none] {$U$};  \draw (T1) -- (U1);
+\node (V1) at (40.000bp,-100.000bp) [draw=none] {$V$};  \draw (U1) -- (V1);
+\node (y0) at (40.000bp,-120.000bp) [draw=none] {$y$};  \draw (V1) -- (y0);
+\node (o1) at (60.000bp,-30.000bp) [draw=none] {$\#$};  \draw (S0) -- (o1);
+\node (T2) at (80.000bp,-30.000bp) [draw=none] {$T$};  \draw (S0) -- (T2);
+\node (U2) at (80.000bp,-50.000bp) [draw=none] {$U$};  \draw (T2) -- (U2);
+\node (V2) at (80.000bp,-70.000bp) [draw=none] {$V$};  \draw (U2) -- (V2);
+\node (z0) at (80.000bp,-90.000bp) [draw=none] {$z$};  \draw (V2) -- (z0);
+\end{tikzpicture}
+\hskip1cmplus5cmminus.5cm
+(b) % S<S<T<U<V<x.>>>>#.T<U<V<y.>>@.T<U<V<z.>>>>>
+\begin{tikzpicture}[line join=bevel,baseline=(S0.base)]
+\node (S0) at (26.667bp,0.000bp) [draw=none] {$S$};
+\node (S1) at (0.000bp,-30.000bp) [draw=none] {$S$};  \draw (S0) -- (S1);
+\node (T0) at (0.000bp,-50.000bp) [draw=none] {$T$};  \draw (S1) -- (T0);
+\node (U0) at (0.000bp,-70.000bp) [draw=none] {$U$};  \draw (T0) -- (U0);
+\node (V0) at (0.000bp,-90.000bp) [draw=none] {$V$};  \draw (U0) -- (V0);
+\node (x0) at (0.000bp,-110.000bp) [draw=none] {$x$};  \draw (V0) -- (x0);
+\node (o0) at (20.000bp,-30.000bp) [draw=none] {$\#$};  \draw (S0) -- (o0);
+\node (T1) at (60.000bp,-30.000bp) [draw=none] {$T$};  \draw (S0) -- (T1);
+\node (U1) at (40.000bp,-60.000bp) [draw=none] {$U$};  \draw (T1) -- (U1);
+\node (V1) at (40.000bp,-80.000bp) [draw=none] {$V$};  \draw (U1) -- (V1);
+\node (y0) at (40.000bp,-100.000bp) [draw=none] {$y$};  \draw (V1) -- (y0);
+\node (o1) at (60.000bp,-60.000bp) [draw=none] {$@$};  \draw (T1) -- (o1);
+\node (T2) at (80.000bp,-60.000bp) [draw=none] {$T$};  \draw (T1) -- (T2);
+\node (U2) at (80.000bp,-80.000bp) [draw=none] {$U$};  \draw (T2) -- (U2);
+\node (V2) at (80.000bp,-100.000bp) [draw=none] {$V$};  \draw (U2) -- (V2);
+\node (z0) at (80.000bp,-120.000bp) [draw=none] {$z$};  \draw (V2) -- (z0);
+\end{tikzpicture}
+\hskip1cmplus5cmminus.5cm
+(c) % S<S<T<U<V<x.>>@.T<U<V<y.>>>>>#.T<U<V<z.>>>>
+\begin{tikzpicture}[line join=bevel,baseline=(S0.base)]
+\node (S0) at (53.333bp,0.000bp) [draw=none] {$S$};
+\node (S1) at (20.000bp,-30.000bp) [draw=none] {$S$};  \draw (S0) -- (S1);
+\node (T0) at (20.000bp,-50.000bp) [draw=none] {$T$};  \draw (S1) -- (T0);
+\node (U0) at (0.000bp,-80.000bp) [draw=none] {$U$};  \draw (T0) -- (U0);
+\node (V0) at (0.000bp,-100.000bp) [draw=none] {$V$};  \draw (U0) -- (V0);
+\node (x0) at (0.000bp,-120.000bp) [draw=none] {$x$};  \draw (V0) -- (x0);
+\node (o0) at (20.000bp,-80.000bp) [draw=none] {$@$};  \draw (T0) -- (o0);
+\node (T1) at (40.000bp,-80.000bp) [draw=none] {$T$};  \draw (T0) -- (T1);
+\node (U1) at (40.000bp,-100.000bp) [draw=none] {$U$};  \draw (T1) -- (U1);
+\node (V1) at (40.000bp,-120.000bp) [draw=none] {$V$};  \draw (U1) -- (V1);
+\node (y0) at (40.000bp,-140.000bp) [draw=none] {$y$};  \draw (V1) -- (y0);
+\node (o1) at (60.000bp,-30.000bp) [draw=none] {$\#$};  \draw (S0) -- (o1);
+\node (T2) at (80.000bp,-30.000bp) [draw=none] {$T$};  \draw (S0) -- (T2);
+\node (U2) at (80.000bp,-50.000bp) [draw=none] {$U$};  \draw (T2) -- (U2);
+\node (V2) at (80.000bp,-70.000bp) [draw=none] {$V$};  \draw (U2) -- (V2);
+\node (z0) at (80.000bp,-90.000bp) [draw=none] {$z$};  \draw (V2) -- (z0);
+\end{tikzpicture}
+\hskip1cmplus5cmminus.5cm
+(d) % S<T<U<V<x.>>@.T<U<V<y.>>@.T<U<V<z.>>>>>>
+\begin{tikzpicture}[line join=bevel,baseline=(S0.base)]
+\node (S0) at (26.667bp,0.000bp) [draw=none] {$S$};
+\node (T0) at (26.667bp,-20.000bp) [draw=none] {$T$};  \draw (S0) -- (T0);
+\node (U0) at (0.000bp,-50.000bp) [draw=none] {$U$};  \draw (T0) -- (U0);
+\node (V0) at (0.000bp,-70.000bp) [draw=none] {$V$};  \draw (U0) -- (V0);
+\node (x0) at (0.000bp,-90.000bp) [draw=none] {$x$};  \draw (V0) -- (x0);
+\node (o0) at (20.000bp,-50.000bp) [draw=none] {$@$};  \draw (T0) -- (o0);
+\node (T1) at (60.000bp,-50.000bp) [draw=none] {$T$};  \draw (T0) -- (T1);
+\node (U1) at (40.000bp,-80.000bp) [draw=none] {$U$};  \draw (T1) -- (U1);
+\node (V1) at (40.000bp,-100.000bp) [draw=none] {$V$};  \draw (U1) -- (V1);
+\node (y0) at (40.000bp,-120.000bp) [draw=none] {$y$};  \draw (V1) -- (y0);
+\node (o1) at (60.000bp,-80.000bp) [draw=none] {$@$};  \draw (T1) -- (o1);
+\node (T2) at (80.000bp,-80.000bp) [draw=none] {$T$};  \draw (T1) -- (T2);
+\node (U2) at (80.000bp,-100.000bp) [draw=none] {$U$};  \draw (T2) -- (U2);
+\node (V2) at (80.000bp,-120.000bp) [draw=none] {$V$};  \draw (U2) -- (V2);
+\node (z0) at (80.000bp,-140.000bp) [draw=none] {$z$};  \draw (V2) -- (z0);
+\end{tikzpicture}
+\hskip1cmplus5cmminus.5cm
+(e) % S<T<U<(.S<S<T<U<V<x.>>>>#.T<U<V<y.>>>>).>@.T<U<V<z.>>>>>
+\begin{tikzpicture}[line join=bevel,baseline=(S0.base)]
+\node (S0) at (86.667bp,0.000bp) [draw=none] {$S$};
+\node (T0) at (86.667bp,-20.000bp) [draw=none] {$T$};  \draw (S0) -- (T0);
+\node (U0) at (40.000bp,-50.000bp) [draw=none] {$U$};  \draw (T0) -- (U0);
+\node (p0) at (0.000bp,-80.000bp) [draw=none] {$($};  \draw (U0) -- (p0);
+\node (S1) at (40.000bp,-80.000bp) [draw=none] {$S$};  \draw (U0) -- (S1);
+\node (S2) at (20.000bp,-110.000bp) [draw=none] {$S$};  \draw (S1) -- (S2);
+\node (T1) at (20.000bp,-130.000bp) [draw=none] {$T$};  \draw (S2) -- (T1);
+\node (U1) at (20.000bp,-150.000bp) [draw=none] {$U$};  \draw (T1) -- (U1);
+\node (V0) at (20.000bp,-170.000bp) [draw=none] {$V$};  \draw (U1) -- (V0);
+\node (x0) at (20.000bp,-190.000bp) [draw=none] {$x$};  \draw (V0) -- (x0);
+\node (o0) at (40.000bp,-110.000bp) [draw=none] {$\#$};  \draw (S1) -- (o0);
+\node (T2) at (60.000bp,-110.000bp) [draw=none] {$T$};  \draw (S1) -- (T2);
+\node (U2) at (60.000bp,-130.000bp) [draw=none] {$U$};  \draw (T2) -- (U2);
+\node (V1) at (60.000bp,-150.000bp) [draw=none] {$V$};  \draw (U2) -- (V1);
+\node (y0) at (60.000bp,-170.000bp) [draw=none] {$y$};  \draw (V1) -- (y0);
+\node (p1) at (80.000bp,-80.000bp) [draw=none] {$)$};  \draw (U0) -- (p1);
+\node (o1) at (100.000bp,-50.000bp) [draw=none] {$@$};  \draw (T0) -- (o1);
+\node (T3) at (120.000bp,-50.000bp) [draw=none] {$T$};  \draw (T0) -- (T3);
+\node (U3) at (120.000bp,-70.000bp) [draw=none] {$U$};  \draw (T3) -- (U3);
+\node (V2) at (120.000bp,-90.000bp) [draw=none] {$V$};  \draw (U3) -- (V2);
+\node (z0) at (120.000bp,-110.000bp) [draw=none] {$z$};  \draw (V2) -- (z0);
+\end{tikzpicture}
+\end{corrige}
+
 (2) En considérant que $(w)$ a le même effet / la même valeur
 que $w$ : (a) Quelle opération parmi $\#$ et $@$ est
 prioritaire\footnote{On dit qu'une opération binaire $\boxtimes$ est
   \emph{prioritaire} sur $\boxplus$ lorsque « $u\boxtimes v\boxplus
   w$ » se comprend comme « $(u\boxtimes v)\boxplus w$ » et
   « $u\boxplus v\boxtimes w$ » comme « $u\boxplus (v\boxtimes w)$ ».}
-sur l'autre ?  (b) L'opération $\#$ s'associe-t-elle\footnote{On dit
+sur l'autre ?\quad (b) L'opération $\#$ s'associe-t-elle\footnote{On dit
   qu'une opération binaire $\boxdot$ \emph{s'associe à gauche} lorsque
   « $u\boxdot v\boxdot w$ » se comprend comme « $(u\boxdot v)\boxdot
   w$ », et \emph{s'associe à droite} lorsque « $u\boxdot v\boxdot w$ »
   se comprend comme « $u\boxdot (v\boxdot w)$ ».} à gauche ou à
-droite ?  (c) L'opération $@$ s'associe-t-elle à gauche ou à droite ?
+droite ?\quad (c) L'opération $@$ s'associe-t-elle à gauche ou à droite ?
+
+\begin{corrige}
+(a) Les arbres d'analyse (b) et (c) de la question (1) montrent que
+  $@$ est prioritaire sur $\#$.\quad (b) L'arbre d'analyse (a) de la
+  question (1) montre que $\#$ s'associe à gauche.\quad (c) L'arbre
+  d'analyse (d) de la question (1) montre que $@$ s'associe à droite.
+\end{corrige}
 
 
 %