De Morgan's laws

De_Morgan's laws (or De_Morgan's theorems) are used to simplify the Boolean expressions.

There are two theorems:

1. The complement of two or more AND variables is equal to the OR of the complements of each variable.
   $\overline{xy}=\overline{x}+\overline{y}$
2. The complement of two or more OR variables is equal to the AND of the complements of each variable.
   $\overline{x+y}=\overline{x}\overline{y}$

Note that this theorems are true in both direction

Examples

• $\overline{xyz}=\overline{x}+\overline{y}+\overline{z}$

• $\overline{x+y+z}=\overline{x}\overline{y}\overline{z}$

• Note that x can be a combination of other variables.

lets say $x=(A+B)$ and $y=(C+D)$ so:

$\overline{xy}=\overline{x}+\overline{y}$

will be:

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

• In case we have more than one livile bar (or complement or negate) start with the top bar first

$\overline{\stackrel{‾}{(A+B)}+\stackrel{‾}{(C+D)}}=\overline{\stackrel{‾}{(A+B)}}.\overline{\stackrel{‾}{(C+D)}}=(A+B)(C+D)$

Note that $\overline{\stackrel{‾}{A}}=A$ Boolean rule.

• $\overline{\stackrel{‾}{ABC}+D+E}=\overline{\stackrel{‾}{ABC}}\overline{D}\overline{E}=ABC\overline{D}\overline{E}$

• $\overline{(A+B)\overline{C}+D+\overline{E}}=(\overline{(A+B)\overline{C})}\overline{D}\overline{\stackrel{‾}{E}}=(\overline{(A+B)}+\overline{\stackrel{‾}{C}})\overline{D}E=(\overline{A}\overline{B}+C)\overline{D}E$