Boolean Algebra

Often, we find extended complementation “bars” in Boolean expressions. A simple example is shown here, where a long bar extends over the Boolean expression \(A + B\):

$$\overline{A+B}$$

In this particular case, the expression represents the functionality of a NOR gate. Many times in the manipulation of Boolean expressions, it is good to be able to know how to eliminate such long bars. We can’t just get rid of the bar, though. There are specific rules to follow for “breaking” long bars into smaller bars in Boolean expressions.

What other type of logic gate has the same functionality (the same truth table) as a NOR gate, and what is its equivalent Boolean expression? The answer to this question will demonstrate what rule(s) we need to follow when we “break” a long complementation bar in a Boolean expression.

Another example we could use for learning how to “break bars” in Boolean algebra is that of the NAND gate:

$$\overline{AB}$$

What other type of logic gate has the same functionality (the same truth table) as a NAND gate, and what is its equivalent Boolean expression? The answer to this question will likewise demonstrate what rule(s) we need to follow when we “break” a long complementation bar in a Boolean expression.

What is DeMorgan’s Theorem?

Use DeMorgan’s Theorem, as well as any other applicable rules of Boolean algebra, to simplify the following expression so there are no more complementation bars extending over multiple variables:

$$\overline{\overline{AB}+\overline{AC}}$$