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 B C=X$ and $D E F=Y \\ $.

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

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

Next, apply DeMorgan's theorem to each of the terms, $ \overline{A B C}$ and $\overline{D E F} \\ $.

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

(c)

Let, $ A \bar{B}=X, \bar{C} D=Y$, and $E F=Z \\ $.

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

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

Next, apply DeMorgan's theorem to each of the terms, $ \overline{A \bar{B}}, \overline{\bar{C} D}$, and $\overline{E F} \\ $.

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