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Architectures Logicielles Et Mat Rielles by F.Lagnier,F.Maraninchi, P.Amblard,J.-C.Fernandez,

By F.Lagnier,F.Maraninchi, P.Amblard,J.-C.Fernandez, P.Sicard,Ph.Waille

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Les d´efinitions et les propri´et´es math´ematiques restent identiques. Ces termes sont utilis´es aussi dans l’´etude des circuits logiques. ), d’union ensembliste (correspondant `a +) et de compl´ementaire ensembliste dans E (correspondant `a ¯) forme une alg`ebre de Boole. L’ensemble vide correspond `a 0 et l’ensemble E `a 1. L’ensemble des nuplets de bool´eens muni des op´erations d’addition, de multiplication et de compl´ementation ´etendues aux vecteurs forme une alg`ebre de Boole. (0, 0, .

0 est ´el´ement neutre pour l’addition et 1 est ´el´ement neutre pour la multiplication. 1 = a 3. (b + c). Remarque : L’usage a consacr´e la priorit´e de la multiplication sur l’addition comme dans la notation alg´ebrique usuelle. b + c. 1. Alg`ebre de Boole 27 4. ¯ a = 0. 3 Exemples d’Alg` ebres de Boole L’alg`ebre de Boole la plus simple est d´efinie sur l’ensemble `a deux ´el´ements : B = {0, 1}. Pour l’´etude des raisonnements sur les propositions logiques, il existe des synonymes pour les noms des ´el´ements de cet ensemble et des op´erations ; on parle alors de faux et vrai (au lieu de 0 et 1) et des op´erateurs et et ou (au lieu de la multiplication et de l’addition).

1 couvre les points (0, 0, 0), (0, 0, 1), (1, 0, 0),(1, 0, 1) et (1, 1, 1). Remarque : Une fonction bool´eenne peut servir `a repr´esenter un ensemble : la fonction vaut 1 en chacun des points appartenant `a l’ensemble. On parle de fonction caract´eristique. 2 Les fonctions ` a 2 variables Les fonctions `a deux variables sont d´efinies sur les 4 points (0, 0), (0, 1), (1, 0), (1, 1). En chacun de ces 4 points une certaine fonction peut prendre une des deux valeurs 0 ou 1. Il y a donc 24 = 16 fonctions possibles.

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