EDI18701 AULA07

Teoremas Booleanos

(1) $A.0=0\,$

(2) $A.1=A\,$

(3) $A.A=A\,$

(4) $A.{\bar {A}}=0\,$

(5) $A+0=A\,$

(6) $A+1=1\,$

(7) $A+A=A\,$

(8) $A+{\bar {A}}=1\,$

(9) $A+B=B+A\,$

(10) $A.B=B.A\,$

(11) $A+(B+C)=(A+B)+C=A+B+C\,$

(12) $A(B.C)=(A.B).C=A.B.C\,$

(13a) $A(B+C)=AB+AC\,$

(13b) $(A+B).(C+D)=AC+AD+BC+BD\,$

(14) $A+(AB)=A\,$

(15a) $A+({\bar {A}}B)=A+B\,$

(15b) ${\bar {A}}+(AB)={\bar {A}}+B\,$

Exemplos

(a) $A{\bar {B}}C+{\bar {A}}{\bar {B}}{\bar {C}}=...\,$

(b) $A{\bar {B}}D+A{\bar {B}}{\bar {D}}=...\,$

(c) $({\bar {A}}+B).(A+B)=...\,$

(d) $ACD+{\bar {A}}BCD=...\,$

(e) $S=A{\bar {C}}+AB{\bar {C}}=...\,$

(f) $S={\bar {A}}{\bar {B}}C{\bar {D}}+{\bar {A}}{\bar {B}}{\bar {C}}{\bar {D}}=...\,$

(g) $S={\bar {A}}D+ABD=...\,$

DeMORGAN

(16) $({\overline {A+B}})={\bar {A}}.{\bar {B}}\,$

(17) $({\overline {A.B}})={\bar {A}}+{\bar {B}}\,$

Exemplos

$S={\bar {A}}{\bar {B}}{\bar {C}}{\bar {D}}+{\bar {A}}{\bar {B}}{\bar {C}}D+{\bar {A}}B{\bar {C}}{\bar {D}}+{\bar {A}}B{\bar {C}}D+ABCD+A{\bar {B}}{\bar {C}}{\bar {D}}+A{\bar {B}}{\bar {C}}D+A{\bar {B}}C{\bar {D}}+A{\bar {B}}CD$

$S={\bar {A}}{\bar {B}}{\bar {C}}{\bar {D}}+{\bar {A}}{\bar {B}}C{\bar {D}}+{\bar {A}}B{\bar {C}}D+{\bar {A}}BCD+AB{\bar {C}}D+ABCD+A{\bar {B}}{\bar {C}}{\bar {D}}+A{\bar {B}}C{\bar {D}}$

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