Boolean Algebra — Definition, Formula & Examples

Boolean algebra is a branch of algebra where variables take only two values — true (1) or false (0) — and are combined using the operations AND, OR, and NOT.

A Boolean algebra is an algebraic structure $(B, \land, \lor, \lnot, 0, 1)$ consisting of a set $B$ with two binary operations (meet $\land$ and join $\lor$ ), a unary operation (complement $\lnot$ ), and two identity elements (0 and 1), satisfying the commutative, associative, distributive, identity, and complement laws.

Key Formula

A \land (B \lor C) = (A \land B) \lor (A \land C)

Where:

$A, B, C$ = Boolean variables, each equal to 0 or 1
$\land$ = AND operation
$\lor$ = OR operation

How It Works

You manipulate Boolean expressions using a small set of laws, much like simplifying algebraic expressions with real numbers. The AND operation (

\land

) returns 1 only when both inputs are 1. The OR operation (

\lor

) returns 1 when at least one input is 1. The NOT operation (

\lnot

) flips 0 to 1 and 1 to 0. Key laws include De Morgan's laws:

\lnot(A \land B) = \lnot A \lor \lnot B

and

\lnot(A \lor B) = \lnot A \land \lnot B

. You can verify any Boolean identity by testing all possible input combinations in a truth table.

Worked Example

Problem: Simplify the Boolean expression: A ∧ (A ∨ B).

Apply the distributive law: Distribute A over the OR inside the parentheses.

A \land (A \lor B) = (A \land A) \lor (A \land B)

Apply the idempotent law: Since A AND A always equals A, replace that term.

= A \lor (A \land B)

Apply the absorption law: The absorption law states that A ∨ (A ∧ B) = A, because if A is 1 the whole expression is 1 regardless of B, and if A is 0 both terms are 0.

= A

Answer: The simplified expression is

A

Why It Matters

Boolean algebra is the mathematical foundation of digital circuit design — every logic gate in a computer chip implements a Boolean operation. It is also central to database queries, programming conditionals, and any discrete mathematics or computer science course.

Common Mistakes

Mistake: Assuming AND distributes over OR the same way multiplication distributes over addition, but forgetting that OR also distributes over AND.

Correction: In Boolean algebra, both distributive laws hold:

A \land (B \lor C) = (A \land B) \lor (A \land C)

AND

A \lor (B \land C) = (A \lor B) \land (A \lor C)

. The second has no analogue in ordinary algebra.