Boolean Algebra

Simplify with complement, absorption, idempotence, and De Morgan.

The idea

Boolean algebra is the algebra of the two values $0$ and $1$ — the two states of a wire in a digital circuit — and we use it to simplify logical expressions. Its operations come from logic: AND, written as a product $x \cdot y$, is $1$ only when both inputs are $1$; OR, written as a sum $x + y$, is $1$ when at least one input is; NOT, written with a prime $x'$, exchanges $0$ and $1$.

Much of ordinary algebra carries over: both operations are commutative and associative, products distribute over sums, and $x \cdot 1 = x$, $x + 0 = x$. So we can factor and expand Boolean expressions by the familiar rules.

Some laws are new, because the only values are $0$ and $1$:

$x + x = x, \qquad x + 1 = 1, \qquad x \cdot x' = 0, \qquad x + x' = 1.$

Each of these replaces an expression with a shorter one, and that is the purpose of the algebra: a shorter expression describes a smaller circuit that computes the same thing.

Ways to work on it

• Walkthrough. The core Boolean laws and what they simplify.
• Practice. Simplify with a single Boolean law.
• Hardest. Simplify an expression that takes several laws in sequence.

Not sure where to start? Take the ten-question placement test.