4-1. Simplify The Following Expressions Using Boolean Algebra

4-1. Simplify The Following Expressions Using Boolean Algebra

A)

X

 = ABC +  ) = C (AB +     ) = C (    = C 

B)

Y

  ) = (Q + R)(            = Q       = 0 + Q      = Q 

C)

W

 C +   = ABC + A  )+   = AC (B+  = AC +   = C + 

D)

Q

̅ ̅    ) =  ( +  (    ) =  +    =   +     =

E)

X

=

     +   +A    BC+ABC+A  C  

F)

G)

Z

Y

=

  + A   +   BC+ABC+ A C  

=

 (    +A)+BC(   +A)+A C 

=

 +BC+A  C 

=

 +C(B+A  ) 

=

 +CB+CA  

=

  +C) ̅  ( B + ) ( 

=

+A+  ) + (BC) + (   )+(    ( 

=

 +0+A+  +C 0+(BC)+(  

=

 + A+  +C (BC)+  

=

 +A (BC+  

=

1+A

=

A

 C  + A    = (̅  ) +   +  CD+AC   +   C  + AC  + A   =  +  CD   + C  (    ) + A   =  +  CD  + C) + A   (   =  +  CD  + A   =  +  CD   +  (A  ) =

H)

X

 ) +   +    = AB(̅  ) +   +    = AB (C+   +AB   ̅    = ABC +  ̅ )  ) +   (AB+ = B(AC+ 

PROBLEMS

Answer: A = 5 O’Clock B = All Machines are Shut Down C = Friday D = The Production is Complete

̅D +  BCD CD + A CD + A  ** X = A ̅D ) +   BCD  (CD +  = A BCD D +  = 

(A)

  ̅  +   ̅  +    +      +  ̅  + ̅ + ̅  +    +     

̅ +    (̅  + ̅ + CD + C  )+ B (̅  + ̅D) + AB ̅ (CD + C ) =  (̅ + C) +   B̅ + ABC  + A C =   +  B̅ + ABC  + A C =  (  + B̅) + ABC  + A C =  (  + ̅) + ABC  + A C =  +  ̅ + ABC  + A C =  (  + AC ) +  ̅ + ABC  =  ( +C)+  ̅ + ABC  =  + C +   ̅ + ABC  = (B)

  ̅  +    +      +  ̅  +    +   ̅  +    +     

  (̅  + CD + C   )+  B (̅  + CD) + A  (̅  + CD + C ) =  (1 + C )   ( 1 + C  ) + AB  + A =  +   C  + AB  + A  (1+C ) =  +   C  + AB  + A  +     =  ( +  C  ) + A + AC   ) + A   =  (1+CD) + AB  = +  CD + AB  =  +AB  =

(C)

 C +    C + AB̅ + A  ̅ + A C    + AB + A   ) + C (   + A ) = ̅ (  ) + C  =  (1 + A ̅A  + CB = ̅ +   (̅A + C) = ̅ +   (C+ A) = ̅ +   ̅ +  A = ̅ +  + A = ̅ +  = ̅ + B + A