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 $F$)?}

• A. $(\overline{X} + \overline{Y} + \overline{Z})(X + \overline{Y})(Y + \overline{Z})$
• B. $X\overline{Y} + \overline{Z}$
• C. $(X + \overline{Z})(\overline{Y} + \overline{Z})$
• D. $X\overline{Y} + Y\overline{Z} + \overline{X}\,\overline{Y}\,\overline{Z}$

Hint:

To find equivalent Boolean expressions, first compute the complement using De Morgan's laws and then simplify using absorption and distributive properties.

Answer 1

The Correct Option is B, C, D

Step 1: Apply De Morgan's theorem to find $\overline{F$.}
\[ \overline{F} = \overline{(X+Y+Z)} + \overline{(\overline{X}+Y)} + \overline{(\overline{Y}+Z)} \] \[ = (\overline{X}\,\overline{Y}\,\overline{Z}) + (X\overline{Y}) + (Y\overline{Z}) \]

Step 2: Compare with given options.
Option (B): $X\overline{Y} + \overline{Z}$ is obtained by absorption from the above expression, hence equivalent.
Option (C): $(X+\overline{Z})(\overline{Y}+\overline{Z})$ simplifies to the same sum-of-products form, so it is equivalent.
Option (D): $X\overline{Y} + Y\overline{Z} + \overline{X}\overline{Y}\overline{Z}$ matches exactly with $\overline{F}$.

Step 3: Eliminate incorrect option.
Option (A): Represents a different Boolean structure and does not simplify to $\overline{F}$.

Step 4: Conclusion.
Thus, options (B), (C), and (D) are equivalent to $\overline{F}$.