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Computer Logic Design (CDA3201)
FALL 2003

 

Review Solutions

The following problems should help you to develop proficiency in using the laws of logic.
DO NOT SOLVE BY TRUTH TABLES OR K-MAP.

1.     A D' + A' B' + C' D + A' C' + B' D = A D' + (B C)'

Solution:

(     AD'         )   + A'B' + B'D + C'D  + A'C'
= AD' + AD'B' + AD'C' + A'B' + B'D + A'C' + C'D
         |_______________|______|
                 |____________________|______|
 
= AD' + AD'B' + A'B' + B'D + AD'C' + A'C' + C'D
 
= AD' + B'(AD' + A' + D)   + C'(AD' + A' + D)
 
= AD' + B'(A' + D' + D)    + C'(A' + D' + D)
 
= AD' + B'(A' + 1)         + C'(A' + 1)
 
= AD' + B'                 + C'
 
= AD' + (BC)'
 

2.     X Y' + Z ( X' + Y + W ) = Z + X Y'

Solution:

 
= (    XY' ) + Z ( X' + Y + W )
 
= XY' + XY'Z + Z ( X' + Y + W )
 
= XY' + Z ( XY' + X' + Y + W )
 
= XY' + Z ( X'  + Y' + Y + W )
 
= XY' + Z ( X'  +    1   + W )
 
= XY' + Z
 

3.     X' Z' + Y Z + X Y' = Y' Z' + X' Y + X Z

Solution:

  X'Z' + Y Z + X Y'
 
= X'Z'( Y + Y')   + Y Z ( X + X')  + X Y' ( Z + Z')
 
= X'Z'Y + X'Z'Y'  + X Y Z + X'Y Z + X Y' Z + X Y' Z'
           |___________________________________|
 
   |__________________________|
 
                      |_______________|
  
= Y'Z'(X + X') + X' Y ( Z + Z')+ X Z ( Y + Y' )
 
= Y'Z' + X'Y + X Z
 

4.    X' Y' Z' + X Y Z + (W Z')' Z' + X' Y Z' + W' X Y + X Y' Z' = W' X Y + X Y Z + X' Z' + Y' Z'

Solution:

X'Y'Z' + X Y Z + (W Z')' Z' + X'Y Z' + W'X Y + X Y'Z' 
 
   |____________________________|
 
   |_____________________________________________|
 
= X'Z'     +  Y'Z' + X Y Z + (W Z')'Z'   + W'X Y
 
= X'Z'     +  Y'Z' + X Y Z + (W'+ Z) Z'  + W'X Y
 
= X'Z'     +  Y'Z' + X Y Z + W'Z'        + W'X Y
 
= W'X Y    +  X'Z' + Y'Z'  + W'Z'        + X Y Z
 
= X Y (W') + (X' + Y')(Z') + W'Z'        + X Y Z
 
= X Y (W') + (X Y)'(Z')    + W'Z'        + X Y Z
 
= X Y (W') + (X Y)'(Z')                  + X Y Z
 
= X Y (W') + (X' + Y')(Z')               + X Y Z
 
= W'X Y    + X'Z' + Y'Z'                 + X Y Z
 
= W'X Y    + X Y Z  + X'Z'  + Y'Z'