Algebradeboole
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Transcript of Algebradeboole
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Instituto Universitario Politécnico“Santiago Mariño”Extensión MaturínEsc. Ing. Sistemas
Álgebra de Boole
2
Maturín, 2012
Facilitadora : Ing. Mariángela Pollonais
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Variable booleana:
Solo puede tomar dos valores (0 ó 1)
Como recordarán las operaciones básicas son:
Adición booleana OR
Multiplicación booleana AND
Negación NOT(complemento)
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Adición booleana(OR):
0+ 0 = 0 0 + 1 = 1
1 + 1 = 1 1 + 0 = 1
VB L=A+B
A
B
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Multiplicación Booleana(AND):
0 · 0 = 00 · 1 = 01 · 1 = 11 · 0 = 0
VB L=AB
A B
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Negación NOT(complemento)
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Álgebra de BoolePostulado 2)
A + 0 = A ; A . 1 = APostulado 5)
A +A´= 1; A .A´=0Teorema 1)
A+A=A; A.A=A
Teorema 2)
A+1=1; A.0=0
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Teorema 3, involución:
(A) = A
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Postulado 3, Conmutativo:
A+B = B+A; A.B = B.A;
Teorema 4, asociativo
A+(B+C) = (A+B) + C; A.(B.C) = (A.B) . C
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Postulado 4,Distributivo:
A(B+C) = AB + AC;
A + BC=(A+B)(A+C)
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Teorema 5, de De Morgan:
A · B = A + B
A + B = A · B
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Ley de absorción:
A + AB = A
A(A+B) = A
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Demostrar que :A +AB = A.
Aplicando los teoremas y postulados de Boole
A+AB = A(1+B)
= A x 1
= A
Ley distributiva
Teorema 2: (1+B)=1
Postulado 2: (A.1)=A
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(A+B)(A+C)=AA+AC+AB+BC
= A +AC+AB+BC
= A +AC+BC
= A + BC
Ley distributiva
AA=A
A+AB=A
A+AC=A
Otro ejemplo : Compruebe que
(A +B)(A+C) = A+BC
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Simplifique la siguiente expresión aplicando el Álgebra de Boole:
ab + a(b+c) + b (b+c) =
=ab + ab + ac + bb + bc
= ab + ac + b (1+ c) = ab + ac + b 1 = ab + ac + b = b (a +1) + ac = b 1 + ac = b +ac
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Otra:[ab (c+bd) +ab]c = [abc+ abbd + ab]c
= [ abc+abd+ ab]c
= abcc+ abdc+abc = abc+ abdc+abc
= abc+ abdc
= (1 + d) abc
= 1 abc
= abc
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Un comportamiento de un sistema puede expresarse formalmente como
Estas funciones también suelen denominarse “funciones booleanas”, ya que responden al “álgebra de Boole”.
CBACBAf ,,