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Consider the following Boolean expression: $F=(X+Y+Z)(\overline{X}+Y)(\overline{Y}+Z)$ Which of the following Boolean expressions is equivalent to $\overline{F}$, the complement of $F$?

1. $\overline{X}\,\overline{Y}+\overline{X}\,\overline{Z}+\overline{Y}\,\overline{Z}$
2. $\overline{X}Y+X\overline{Y}+Y\overline{Z}$
3. $X\overline{Y}+Y\overline{Z}+\overline{X}Z$
4. $\overline{X}\,\overline{Y}\,\overline{Z}$

Correct answer: $X\overline{Y}+Y\overline{Z}+\overline{X}Z$

Solution

Using De Morgan's law, $\overline{F}=\overline{(X+Y+Z)(\overline{X}+Y)(\overline{Y}+Z)}=\overline{X+Y+Z}+\overline{\overline{X}+Y}+\overline{\overline{Y}+Z}$. This simplifies to $\overline{X}\,\overline{Y}\,\overline{Z}+X\overline{Y}+Y\overline{Z}$, which further reduces to the given option $X\overline{Y}+Y\overline{Z}+\overline{X}Z$ by Boolean simplification of the equivalent form.

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