Expressions préfixées, infixées ou postfixées
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
, Ă©crire 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
43908.43996.43997.43980.43993.43985.43988.43979.43928.43995.43988.43993.43979.43979.43909.43930.43980.43985.43976.43930.43910.43954.43908.43979.43981.43989.43989.43993.43978.43969.43910.44011.43991.43988.43981.43980.43985.43991.43990.43928.43934.43993.43989.43976.43907.43928.44010.43997.43989.43993.43978.43977.43981.43997.43979.43908.43927.43979.43981.43989.43989.43993.43978.43969.43910.43954.43908.43996.43985.43982.43928.43989.43993.43978.43987.43996.43991.43983.43990.43909.43930.43913.43930.43928.43979.43980.43969.43988.43997.43909.43930.43989.43993.43978.43999.43985.43990.43906.43913.43926.43919.43997.43989.43928.43913.43997.43989.43930.43928.43910.43954.43908.43996.43985.43982.43928.43995.43988.43993.43979.43979.43909.43930.43984.43985.43999.43984.43988.43985.43999.43984.43980.43930.43910.43908.43980.43993.43994.43988.43997.43928.43995.43988.43993.43979.43979.43909.43930.43984.43985.43999.43984.43988.43985.43999.43984.43980.43980.43993.43994.43988.43997.43930.43910.43908.43980.43978.43910.43908.43980.43984.43928.43995.43991.43988.43979.43976.43993.43990.43909.43930.43914.43930.43928.43995.43988.43993.43979.43979.43909.43930.43998.43985.43988.43997.43990.43993.43989.43997.43930.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43998.43985.43988.43997.43990.43993.43989.43997.43930.43910.90037.43928.44008.43978.43991.43976.43991.43979.43985.43980.43985.43991.43990.43928.43996.43997.43928.43995.43991.43978.43978.43997.43995.43980.43985.43991.43990.43908.43927.43979.43976.43993.43990.43910.43908.43927.43980.43984.43910.43908.43927.43980.43978.43910.43908.43980.43978.43910.43908.43980.43996.43928.43995.43988.43993.43979.43979.43909.43930.43988.43985.43990.43997.43990.43991.43979.43930.43910.43908.43996.43985.43982.43928.43995.43988.43993.43979.43979.43909.43930.43988.43985.43990.43997.43990.43991.43996.43985.43982.43930.43910.43908.43976.43978.43997.43910.43908.43979.43976.43993.43990.43910.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43913.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43914.43908.43927.43979.43976.43993.43990.43910.43908.43927.43976.43978.43997.43910.43908.43927.43996.43985.43982.43910.43908.43927.43980.43996.43910.43908.43980.43996.43928.43995.43988.43993.43979.43979.43909.43930.43995.43991.43996.43997.43930.43910.43908.43996.43985.43982.43910.43908.43976.43978.43997.43910.43908.43979.43976.43993.43990.43910.43908.43927.43979.43976.43993.43990.43910.43908.43995.43991.43996.43997.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43996.43997.43998.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43998.43930.43910.43968.43991.43978.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43906.43908.43927.43979.43976.43993.43990.43910.43954.43928.43928.43928.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43978.43997.43980.43981.43978.43990.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43997.43980.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43991.43981.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43990.43993.43990.43996.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43921.43908.43927.43979.43976.43993.43990.43910.43954.43908.43927.43995.43991.43996.43997.43910.43908.43927.43976.43978.43997.43910.43908.43927.43996.43985.43982.43910.43908.43927.43980.43996.43910.43908.43927.43980.43978.43910.43908.43927.43980.43993.43994.43988.43997.43910.43908.43927.43996.43985.43982.43910.43954.43908.43976.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43978.43997.43989.44007.43998.43993.43987.43997.44007.43984.43915.43930.43910.44010.43997.43989.43993.43978.43977.43981.43997.43979.43928.43906.43908.43927.43979.43976.43993.43990.43910.43908.43927.43976.43910.43954.43908.43976.43910.44020.43985.43999.43990.43997.43928.43982.43985.43996.43997.43908.43927.43976.43910.43954.43908.43976.43910.44023.43990.43928.43976.43991.43981.43982.43993.43985.43980.43928.43993.43981.43979.43979.43985.43928.43998.43993.43985.43978.43997.43928.43906.43908.43927.43976.43910.43954.43908.43996.43985.43982.43928.43995.43988.43993.43979.43979.43909.43930.43984.43985.43999.43984.43988.43985.43999.43984.43980.43930.43910.43908.43976.43978.43997.43910.43908.43979.43976.43993.43990.43910.43908.43927.43979.43976.43993.43990.43910.43908.43995.43991.43996.43997.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43996.43997.43998.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43998.43930.43910.43968.43991.43978.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43906.43908.43927.43979.43976.43993.43990.43910.43954.43928.43928.43928.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43978.43997.43980.43981.43978.43990.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43991.43981.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43997.43980.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43990.43991.43990.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43997.43980.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43993.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43924.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43990.43991.43990.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43994.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43921.43921.43908.43927.43979.43976.43993.43990.43910.43954.43908.43927.43995.43991.43996.43997.43910.43908.43927.43976.43978.43997.43910.43908.43927.43996.43985.43982.43910.43954.43908.43976.43910.44020.43985.43999.43990.43997.43928.43982.43985.43996.43997.43908.43927.43976.43910.43954.43908.43927.43996.43985.43982.43910.43954.43908.43927.43996.43997.43980.43993.43985.43988.43979.43910
Exercice 10
Ă l'aide de table2(f)
, afficher les tables de valeurs des expressions de l'exercice 4 de la feuille de cours sur l'algĂšbre 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
- \(a\) ou (\(a\) et \(b\))
- \(a\) et (\(a\) ou \(b\))
- \(a\) ou ((non \(a\)) et \(b\))
- (\(a\) ou \(b\)) et ((non \(a\)) ou \(b\))
- (\(a\) xor \(b\)) ou (\(a\) et \(b\))
- (\(a\) 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.
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))
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
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979.43979.43909.43930.43976.43930.43910.43921.43908.43927.43979.43976.43993.43990.43910.43954.43908.43927.43995.43991.43996.43997.43910.43908.43927.43976.43978.43997.43910.43908.43927.43996.43985.43982.43910.43908.43927.43980.43996.43910.43908.43927.43980.43978.43910.43908.43927.43980.43993.43994.43988.43997.43910.43908.43927.43996.43985.43982.43910.43954.43908.43927.43996.43985.43982.43910.43954.43908.43927.43996.43997.43980.43993.43985.43988.43979.43910
Exercice 13
Ă l'aide des fonctions table2(f)
et table3(f)
, afficher les tables de vérités des axiomes de Hilbert et vérifier que toutes les expressions sont toujours vraies.
Les axiomes d'Hilbert
- \(a\Rightarrow (b \Rightarrow a)\)
- \(\left(a \Rightarrow (b \Rightarrow c)\right) \Rightarrow \left((a \Rightarrow b) \Rightarrow (a \Rightarrow c)\right)\)
- \(\left((\text{non}\ b) \Rightarrow (\text{non}\ a)\right) \Rightarrow (a \Rightarrow b)\)
- \(a\Rightarrow \left(b\Rightarrow (a \ \text{et}\ b)\right)\)
- \((a\ \text{et}\ b) \Rightarrow a\)
- \((a\ \text{et}\ b) \Rightarrow b\)
- \(a \Rightarrow (a\ \text{ou}\ b)\)
- \(b \Rightarrow (a\ \text{ou}\ b)\)
- \((a\ \text{ou}\ b) \Rightarrow \left( (a\Rightarrow c) \Rightarrow \left( (b\Rightarrow c) \Rightarrow c \right) \right)\)
# Tests
(insensible Ă la casse)(Ctrl+I)