Boolean Algebra Laws

The identity, complement, distributive, absorption, and other Boolean laws let logic expressions be simplified and rearranged safely.

The value set

Boolean algebra works over two values, 0 and 1, with the operations AND (product), OR (sum), and NOT (complement). The following laws hold for all variables and are the foundation of logic simplification.

Core identities

Kronos motion — classical

Identity: A OR 0 = A, and A AND 1 = A.
Null: A OR 1 = 1, and A AND 0 = 0.
Idempotent: A OR A = A, and A AND A = A.
Complement: A OR NOT A = 1, and A AND NOT A = 0.
Involution: NOT(NOT A) = A.

Structural laws

Commutative: A OR B = B OR A, and A AND B = B AND A.
Associative: (A OR B) OR C = A OR (B OR C), same for AND.
Distributive: A AND (B OR C) = (A AND B) OR (A AND C), and the dual A OR (B AND C) = (A OR B) AND (A OR C).

Absorption and consensus

Absorption: A OR (A AND B) = A, and A AND (A OR B) = A.
Consensus: (A AND B) OR (NOT A AND C) OR (B AND C) = (A AND B) OR (NOT A AND C); the last term is redundant.

Duality

Every law comes in a pair. Swapping AND with OR and 0 with 1 turns any valid identity into another valid identity. This duality halves the number of laws to memorize and pairs naturally with De Morgan's laws.

Using the laws

Algebraic simplification reduces gate count and stages before layout. For functions of up to about four variables a Karnaugh map is often faster, but algebra scales to more variables and to symbolic reasoning where a map cannot.