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| -rw-r--r-- | controle-20180206.tex | 119 |
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} % |