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 $F$)?

Given the Boolean expression in product-of-sums (POS) form:

First, simplify $F$: Using the distributive law on $(\overline{X} + Y)(\overline{Y} + Z)$:

Multiplying by $(X + Y + Z)$:

Now, compute the complement $\overline{F}$ using De Morgan's laws:

Now evaluate each option:

• Option (A): $(\overline{X} + \overline{Y} + \overline{Z})(X + \overline{Y})(Y + \overline{Z})$ is the dual of $F$ with complemented literals, which is not equivalent to $\overline{F}$.

• Option (B): $X\overline{Y} + \overline{Z}$ is directly identical to $\overline{F}$.

• Option (C): $(X + \overline{Z})(\overline{Y} + \overline{Z}) = X\overline{Y} + \overline{Z}$, which matches $\overline{F}$.

• Option (D): $X\overline{Y} + Y\overline{Z} + \overline{X}\,\overline{Y}\,\overline{Z} = X\overline{Y} + \overline{Z}(Y + \overline{X}\,\overline{Y}) = X\overline{Y} + \overline{Z}(Y + \overline{X}) = X\overline{Y} + Y\overline{Z} + \overline{X}\,\overline{Z}$. By consensus between $X\overline{Y}$ and $\overline{X}\,\overline{Z}$, this simplifies to $X\overline{Y} + \overline{Z}(Y + \overline{Y}) = X\overline{Y} + \overline{Z}$, which also matches $\overline{F}$.

Thus, options B, C, and D are correct.