The GATE Grind

GATE 2021 CS – Question 52

Digital Logic · Boolean Algebra and Minimization · 2 marks · Multiple select

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. $(X+\overline{Z})(\overline{Y}+\overline{Z})$
  4. $X\overline{Y}+Y\overline{Z}+\overline{X}\,\overline{Y}\,\overline{Z}$

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Correct answer: (B) $X\overline{Y}+\overline{Z}$; (C) $(X+\overline{Z})(\overline{Y}+\overline{Z})$; (D) $X\overline{Y}+Y\overline{Z}+\overline{X}\,\overline{Y}\,\overline{Z}$

Explanation

F simplifies to XYZ + $\overline{X}\,\overline{Y}Z$ + ... i.e. F is 1 only for 001, 011 (check) and 111. Its complement is $X\overline{Y}+\overline{Z}$, which equals $(X+\overline{Z})(\overline{Y}+\overline{Z})$ and also the expanded form in D. Option A is the dual, not the complement.