Apply DeMorgan's theorems to each of the following expressions:

(a) $\overline{(A+B+C) D}$

(b) $\overline{A B C+D E F}$

(c) $\overline{A \bar{B}+\bar{C} D+E F}$

Solution:

(a)

Let, $ A+B+C=X$ and $D=Y \\ $.

The expression $\overline{(A+B+C) D}$ is of the form $\overline{X Y}=\bar{X}+\bar{Y}$ and can be rewritten as,

$$ \overline{(A+B+C) D}=\overline{A+B+C}+\bar{D} \\ $$

Next, apply DeMorgan's theorem to the term $ \overline{A+B+C} \\ $.

$$ \overline{A+B+C}+\bar{D}=\bar{A} \bar{B} \bar{C}+\bar{D} \\ $$

(b)

Let, $ A …