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Expressions prĂ©fixĂ©es, infixĂ©es ou postfixĂ©es⚓

Expressions préfixées ou infixées

On peut remarquer qu'avec nos opérateurs booléens, nous devons mettre l'opérateur avant les deux valeurs :

  • \(a\) et \(b\) devient et(a, b)
  • \(a\) ou \(b\) devient ou(a, b)
  • \(a\) ou (\(b\) et \(c\)) devient ou(a, et(b,c))

On dit que la notation est préfixée. C'est-à-dire qu'on met d'abord l'opérateur et ensuite les paramÚtres. La notation "classique" est appelée infixée.

Expression postfixée

Il existe également des expressions postifixées. Par exemple, en Postscript, le langage développé par Adobe à partir de 1982 pour décrire les documents et utilisé par la plupart des imprimantes, les expressions mathématiques sont postfixée.

Ainsi b b mul 4 a mul c mul sub correspond Ă  \(b^2-4ac\). Cela peut se traduire par l'arbre ci-contre.

Expressions booléennes préfixées

De la mĂȘme façon, on peut reprĂ©senter les expressions boolĂ©eenes afin ensuite de les Ă©crire sous forme prĂ©fixĂ©e. Ainsi, l'expression (\(a\) ou \(b\)) et ((non \(a\)) ou \(b\)) correspond Ă  l'arbre ci-contre.

Cela se traduit par la notation préfixée et(ou(a, b), ou(non(a), b)).

Fonction anonyme

Pour obtenir la table de vérité de cette expression, on peut définir une nouvelle fonction :

def expression(a,b):
    return et(ou(a, b), ou(non(a), b))

Puisqu'il y a des chances pour qu'on n'utilise cette fonction qu'une seule fois, il est possible d'utiliser une fonction anonyme :

>>> table2(lambda a, b : et(ou(a, b), ou(non(a), b)))
a b f(a,b)
0 0 0
0 1 1
1 0 0
1 1 1

La notation est :

lambda PARAM : EXPRESSION

S'il y a plusieurs paramÚtres, il faut les séparer par des virgules.

C'est l'équivalent de la notation mathématique \(f\colon x \mapsto x^2\) pour \(f(x)=x^2\).

En Python, cela se noterait lambda x : x**2.

Exercice 9 : xor

En utilisant les fonctions nand et non, écrire la fonction ou qui prend en paramÚtres deux booléens a et b et qui renvoie le résultat de a ou b.

Indice

Il faut utiliser non(a) et non(b).

Exemples
>>> table2(xor)
a b f(a,b)
0 0 0
0 1 1
1 0 1
1 1 0

###(DĂ©s-)Active le code aprĂšs la ligne # Tests (insensible Ă  la casse)
(Ctrl+I)
Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)
Évaluations restantes : 5/5

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976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43990.43993.43990.43996.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43921.43908.43927.43979.43976.43993.43990.43910.43954.43908.43927.43995.43991.43996.43997.43910.43908.43927.43976.43978.43997.43910.43908.43927.43996.43985.43982.43910.43908.43927.43980.43996.43910.43908.43927.43980.43978.43910.43908.43927.43980.43993.43994.43988.43997.43910.43908.43927.43996.43985.43982.43910.43954.43908.43976.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43978.43997.43989.44007.43998.43993.43987.43997.44007.43984.43915.43930.43910.44010.43997.43989.43993.43978.43977.43981.43997.43979.43928.43906.43908.43927.43979.43976.43993.43990.43910.43908.43927.43976.43910.43954.43908.43976.43910.44020.43985.43999.43990.43997.43928.43982.43985.43996.43997.43908.43927.43976.43910.43954.43908.43976.43910.44023.43990.43928.43976.43991.43981.43982.43993.43985.43980.43928.43993.43981.43979.43979.43985.43928.43998.43993.43985.43978.43997.43928.43906.43908.43927.43976.43910.43954.43908.43996.43985.43982.43928.43995.43988.43993.43979.43979.43909.43930.43984.43985.43999.43984.43988.43985.43999.43984.43980.43930.43910.43908.43976.43978.43997.43910.43908.43979.43976.43993.43990.43910.43908.43927.43979.43976.43993.43990.43910.43908.43995.43991.43996.43997.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43996.43997.43998.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43998.43930.43910.43968.43991.43978.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43906.43908.43927.43979.43976.43993.43990.43910.43954.43928.43928.43928.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43978.43997.43980.43981.43978.43990.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43991.43981.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43997.43980.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43990.43991.43990.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43997.43980.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43990.43991.43990.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43921.43921.43908.43927.43979.43976.43993.43990.43910.43954.43908.43927.43995.43991.43996.43997.43910.43908.43927.43976.43978.43997.43910.43908.43927.43996.43985.43982.43910.43954.43908.43976.43910.44020.43985.43999.43990.43997.43928.43982.43985.43996.43997.43908.43927.43976.43910.43954.43908.43927.43996.43985.43982.43910.43954.43908.43927.43996.43997.43980.43993.43985.43988.43979.43910
Exercice 10

À l'aide de table2(f), afficher les tables de valeurs des expressions de l'exercice 4 de la feuille de cours sur l'algùbre de Boole.

Par exemple, la table de \(a\) ou (\(a\) et \(b\)) s'obtient avec table2(lambda a, b: ou(a, et(a, b))).

Liste des expressions
  1. \(a\) ou (\(a\) et \(b\))
  2. \(a\) et (\(a\) ou \(b\))
  3. \(a\) ou ((non \(a\)) et \(b\))
  4. (\(a\) ou \(b\)) et ((non \(a\)) ou \(b\))
  5. (\(a\) xor \(b\)) ou (\(a\) et \(b\))
  6. (\(a\) xor \(b\)) et (\(a\) ou (non \(b\)))
Astuce pour le terminal

On rappelle qu'on peut naviguer dans l'historique du terminal Ă  l'aide des flĂšches haut et bas.

Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)

Exercice 11

En vous inspirant de table2, définir une fonction table3 qui affiche la table de vérité d'une fonction à 3 paramÚtres.

Exemples
>>> table3(lambda a, b, c: ou(a, et(b, c)))
a b c f(a,b,c)
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
>>> table3(lambda a, b, c: et(non(a), et(b, non(c))))
a b c f(a,b,c)
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 0
Le code de table2 
def table2(f):
    print("a", "b", "f(a,b)")
    for a in range(2):
        for b in range(2):
            print(a, b, f(a,b))

###(DĂ©s-)Active le code aprĂšs la ligne # Tests (insensible Ă  la casse)
(Ctrl+I)
Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)
Évaluations restantes : ∞/∞

Solution 
def table3(f):
    print("a", "b", "c", "f(a,b,c)")
    for a in range(2):
        for b in range(2):
            for c in range(2):
                print(a, b, c, f(a,b,c))

Si vous n'avez pas réussi à l'écrire, copiez ce code dans l'IDE pour pouvoir l'utiliser dans les exercices suivants.

Exercice 12 : impl

En utilisant les fonctions précédentes, écrire la fonction impl qui prend en paramÚtres deux booléens a et b et qui renvoie le résultat de a \(\Rightarrow\) b.

Indice

On rappelle que \(a \Rightarrow b\) est Ă©quivalent Ă  (non \(a\)) ou \(b\).

Exemples
>>> table2(impl)
a b f(a,b)
0 0 1
0 1 1
1 0 0
1 1 1

###(DĂ©s-)Active le code aprĂšs la ligne # Tests (insensible Ă  la casse)
(Ctrl+I)
Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)
Évaluations restantes : 5/5

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Exercice 13

À l'aide des fonctions table2(f) et table3(f), afficher les tables de vĂ©ritĂ©s des axiomes de Hilbert et vĂ©rifier que toutes les expressions sont toujours vraies.

Les axiomes d'Hilbert
  • \(a\Rightarrow (b \Rightarrow a)\)
  • \(\left(a \Rightarrow (b \Rightarrow c)\right) \Rightarrow \left((a \Rightarrow b) \Rightarrow (a \Rightarrow c)\right)\)
  • \(\left((\text{non}\ b) \Rightarrow (\text{non}\ a)\right) \Rightarrow (a \Rightarrow b)\)
  • \(a\Rightarrow \left(b\Rightarrow (a \ \text{et}\ b)\right)\)
  • \((a\ \text{et}\ b) \Rightarrow a\)
  • \((a\ \text{et}\ b) \Rightarrow b\)
  • \(a \Rightarrow (a\ \text{ou}\ b)\)
  • \(b \Rightarrow (a\ \text{ou}\ b)\)
  • \((a\ \text{ou}\ b) \Rightarrow \left( (a\Rightarrow c) \Rightarrow \left( (b\Rightarrow c) \Rightarrow c \right) \right)\)

Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)