Booleana

Prévia do material em texto

Simplificação de expressões booleanas
Item S1
S1 = (A+B + C) (A+B + C)
=
(
(A+B) + C
) (
(A+B) + C
)
(X+C)(Y+C)=XY+C
= (A+B)(A+B) + C
= (AA+ AB +BA+BB) + C
= AB + AB + C
⇒ S1 = C + AB + AB
Item S2
S2 = (AC +B +D) + C (ACD)
De Morgan
= AC BD + C(A+ C +D)
= (A+ C)BD + CA+ CC + CD
= ABD + C BD + CA+ CD
= D
(
B C + C
)
+ CA
X+XY=X+Y
= D
(
C +B
)
+ CA
⇒ S2 = D (C +B) + CA
(forma soma-de-produtos equivalente: S2 = CD +DB + CA.)
Item S3
S3 = (A+B)C + D(C +B)
De Morgan
= (A+B) + C + D + (C +B)
= AB + C +D + C B
= C +D + AB
⇒ S3 = C + D + AB
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Item S4
S4 = (A+B + C) (A+B + C)
=
(
(A+B) + C
) (
(A+B) + C
)
(X+Z)(Y+Z)=XY+Z
= (A+B)(A+B) + C
= (AB + AB) + C
⇒ S4 = C + AB + AB
Item S5
S5 = AB C + AB C + AB C + AB C + AB C
= A
(
BC +BC +BC
)
+ AB(C + C)
= A
(
C(B +B) +BC
)
+ AB
= A (C +BC) + AB
= A (B + C) + AB
= AB + AC + AB
= B(A+ A) + AC
⇒ S5 = B + AC
2. Simplifique as expressões booleanas
(a)
Fa = ABC + ABC + AB
= B
(
AC + AC + A
)
= B
(
A+ A
)
( AC + AC = A )
= B
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(b) (: (A+B) (A+B))
Fb = (A+B) (A+B) = (AB) (A+B) = AB ( XY (X + Y ) = XY )
= AB
(c)
Fc = ABC + AC = C
(
AB + A
)
= C
(
A+B
)
( A+ AB = A+B )
= C(A+B)
(d)
Fd = BC +B(AD + AD) = BC +BA(D +D) = BC + AB = B(A+ C)
= B(A+ C)
(e)
Fe = (A+B + AB) (AB + AC +BC)
= (A+B) (AB + AC)
(
A+ AB = A, XY +XZ + Y Z = XY +XZ
)
= A(AB + AC) +B(AB + AC)
= AB + ABC
= AB + ABC
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