Question

Consider the following Boolean expression.

\(F = (X + Y + Z)(\overline X + Y)(\overline Y + Z)\)

Which of the following Boolean expressions is/are equivalent to \(\overline F\) (complement of F)?

1. \((\overline X + \overline Y + \overline Z )(X + \overline Y )(Y + \overline Z )\)
2. \(X\overline Y + \overline Z\)
3. \(X\overline Y + Y\overline Z + \overline X \overline Y \overline Z \)
4. \((X + \overline Z ) . (\overline Y + \overline Z )\)

Answer 1

Correct Answer - Option :

Formula:

(A + B)(A + C) = A + B.C

Calculation:

\(F = (X + Y + Z)(\overline X + Y)(\overline Y + Z)\)

F = (X + Y + Z).(Y̅ + Z).(X̅ + Y)

F = (Z + Y̅.(X + Y)).(X̅ + Y)

F = (Z + X.Y̅ + Y.Y̅).(X̅ + Y)

F = (Z + X.Y̅).(X̅ + Y)   

F = (Z + X.Y̅).(X̅ + Y)

F = Z.(X̅ + Y) + (X.Y̅).(X̅ + Y)

F = Z.(X̅ + Y) + X.Y̅.X̅ + X.Y̅.Y

F = Z.(X̅ + Y)   

Taking complement on both sides

F̅ = \(\overline{Z.(\overline X + Y)}\)    

By Demorgan's law

F̅ = Z̅ + X.Y̅  =  X.Y̅ + Z ................. (B)

\(\overline F= (X + \overline Z ) . (\overline Y + \overline Z ) ...........(D)\)

Option C:

\(F = (X + Y + Z)(\overline X + Y)(\overline Y + Z)\)

Taking complement on both sides

\(\overline F= \overline X \overline Y \overline Z +X\overline Y + Y\overline Z \)

Hence option B, C and D are correct