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